diff --git a/lib/hcrypto/libtommath/LICENSE b/lib/hcrypto/libtommath/LICENSE index 5baa792a6..b23b3c867 100644 --- a/lib/hcrypto/libtommath/LICENSE +++ b/lib/hcrypto/libtommath/LICENSE @@ -1,4 +1,26 @@ -LibTomMath is hereby released into the Public Domain. + The LibTom license --- Tom St Denis +This is free and unencumbered software released into the public domain. +Anyone is free to copy, modify, publish, use, compile, sell, or +distribute this software, either in source code form or as a compiled +binary, for any purpose, commercial or non-commercial, and by any +means. + +In jurisdictions that recognize copyright laws, the author or authors +of this software dedicate any and all copyright interest in the +software to the public domain. We make this dedication for the benefit +of the public at large and to the detriment of our heirs and +successors. We intend this dedication to be an overt act of +relinquishment in perpetuity of all present and future rights to this +software under copyright law. + +THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, +EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF +MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. +IN NO EVENT SHALL THE AUTHORS BE LIABLE FOR ANY CLAIM, DAMAGES OR +OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, +ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR +OTHER DEALINGS IN THE SOFTWARE. + +For more information, please refer to diff --git a/lib/hcrypto/libtommath/README.md b/lib/hcrypto/libtommath/README.md new file mode 100644 index 000000000..be5b20783 --- /dev/null +++ b/lib/hcrypto/libtommath/README.md @@ -0,0 +1,44 @@ +# libtommath + +This is the git repository for [LibTomMath](http://www.libtom.net/LibTomMath/), a free open source portable number theoretic multiple-precision integer (MPI) library written entirely in C. + +## Build Status + +### Travis CI + +master: [![Build Status](https://api.travis-ci.org/libtom/libtommath.png?branch=master)](https://travis-ci.org/libtom/libtommath) + +develop: [![Build Status](https://api.travis-ci.org/libtom/libtommath.png?branch=develop)](https://travis-ci.org/libtom/libtommath) + +### AppVeyor + +master: [![Build status](https://ci.appveyor.com/api/projects/status/b80lpolw3i8m6hsh/branch/master?svg=true)](https://ci.appveyor.com/project/libtom/libtommath/branch/master) + +develop: [![Build status](https://ci.appveyor.com/api/projects/status/b80lpolw3i8m6hsh/branch/develop?svg=true)](https://ci.appveyor.com/project/libtom/libtommath/branch/develop) + +### ABI Laboratory + +API/ABI changes: [check here](https://abi-laboratory.pro/tracker/timeline/libtommath/) + +## Summary + +The `develop` branch contains the in-development version. Stable releases are tagged. + +Documentation is built from the LaTeX file `bn.tex`. There is also limited documentation in `tommath.h`. +There is also a document, `tommath.pdf`, which describes the goals of the project and many of the algorithms used. + +The project can be build by using `make`. Along with the usual `make`, `make clean` and `make install`, +there are several other build targets, see the makefile for details. +There are also makefiles for certain specific platforms. + +## Testing + +Tests are located in `demo/` and can be built in two flavors. +* `make test` creates a stand-alone test binary that executes several test routines. +* `make mtest_opponent` creates a test binary that is intended to be run against `mtest`. + `mtest` can be built with `make mtest` and test execution is done like `./mtest/mtest | ./mtest_opponent`. + `mtest` is creating test vectors using an alternative MPI library and `test` is consuming these vectors to verify correct behavior of ltm + +## Building and Installing + +Building is straightforward for GNU Linux only, the section "Building LibTomMath" in the documentation in `doc/bn.pdf` has the details. diff --git a/lib/hcrypto/libtommath/VERSION b/lib/hcrypto/libtommath/VERSION deleted file mode 100644 index a7d6788f4..000000000 --- a/lib/hcrypto/libtommath/VERSION +++ /dev/null @@ -1 +0,0 @@ -0.41 diff --git a/lib/hcrypto/libtommath/appveyor.yml b/lib/hcrypto/libtommath/appveyor.yml new file mode 100644 index 000000000..efe45688c --- /dev/null +++ b/lib/hcrypto/libtommath/appveyor.yml @@ -0,0 +1,20 @@ +version: 1.2.0-{build} +branches: + only: + - master + - develop + - /^release/ + - /^travis/ +image: +- Visual Studio 2019 +- Visual Studio 2017 +- Visual Studio 2015 +build_script: +- cmd: >- + if "Visual Studio 2019"=="%APPVEYOR_BUILD_WORKER_IMAGE%" call "C:\Program Files (x86)\Microsoft Visual Studio\2019\Community\VC\Auxiliary\Build\vcvars64.bat" + if "Visual Studio 2017"=="%APPVEYOR_BUILD_WORKER_IMAGE%" call "C:\Program Files (x86)\Microsoft Visual Studio\2017\Community\VC\Auxiliary\Build\vcvars64.bat" + if "Visual Studio 2015"=="%APPVEYOR_BUILD_WORKER_IMAGE%" call "C:\Program Files\Microsoft SDKs\Windows\v7.1\Bin\SetEnv.cmd" /x64 + if "Visual Studio 2015"=="%APPVEYOR_BUILD_WORKER_IMAGE%" call "C:\Program Files (x86)\Microsoft Visual Studio 14.0\VC\vcvarsall.bat" x86_amd64 + nmake -f makefile.msvc all +test_script: +- cmd: test.exe diff --git a/lib/hcrypto/libtommath/astylerc b/lib/hcrypto/libtommath/astylerc new file mode 100644 index 000000000..c5ff77940 --- /dev/null +++ b/lib/hcrypto/libtommath/astylerc @@ -0,0 +1,30 @@ +# Artistic Style, see http://astyle.sourceforge.net/ +# full documentation, see: http://astyle.sourceforge.net/astyle.html +# +# usage: +# astyle --options=astylerc *.[ch] + +# Do not create backup, annonying in the times of git +suffix=none + +## Bracket Style Options +style=kr + +## Tab Options +indent=spaces=3 + +## Bracket Modify Options + +## Indentation Options +min-conditional-indent=0 + +## Padding Options +pad-header +unpad-paren +align-pointer=name + +## Formatting Options +break-after-logical +max-code-length=120 +convert-tabs +mode=c diff --git a/lib/hcrypto/libtommath/bn.ilg b/lib/hcrypto/libtommath/bn.ilg deleted file mode 100644 index 3c859f034..000000000 --- a/lib/hcrypto/libtommath/bn.ilg +++ /dev/null @@ -1,6 +0,0 @@ -This is makeindex, version 2.14 [02-Oct-2002] (kpathsea + Thai support). -Scanning input file bn.idx....done (79 entries accepted, 0 rejected). -Sorting entries....done (511 comparisons). -Generating output file bn.ind....done (82 lines written, 0 warnings). -Output written in bn.ind. -Transcript written in bn.ilg. diff --git a/lib/hcrypto/libtommath/bn.ind b/lib/hcrypto/libtommath/bn.ind deleted file mode 100644 index c099b521d..000000000 --- a/lib/hcrypto/libtommath/bn.ind +++ /dev/null @@ -1,82 +0,0 @@ -\begin{theindex} - - \item mp\_add, \hyperpage{31} - \item mp\_add\_d, \hyperpage{56} - \item mp\_and, \hyperpage{31} - \item mp\_clear, \hyperpage{12} - \item mp\_clear\_multi, \hyperpage{13} - \item mp\_cmp, \hyperpage{25} - \item mp\_cmp\_d, \hyperpage{26} - \item mp\_cmp\_mag, \hyperpage{23} - \item mp\_div, \hyperpage{32} - \item mp\_div\_2, \hyperpage{28} - \item mp\_div\_2d, \hyperpage{30} - \item mp\_div\_d, \hyperpage{56} - \item mp\_dr\_reduce, \hyperpage{45} - \item mp\_dr\_setup, \hyperpage{45} - \item MP\_EQ, \hyperpage{23} - \item mp\_error\_to\_string, \hyperpage{9} - \item mp\_expt\_d, \hyperpage{47} - \item mp\_exptmod, \hyperpage{47} - \item mp\_exteuclid, \hyperpage{55} - \item mp\_gcd, \hyperpage{55} - \item mp\_get\_int, \hyperpage{20} - \item mp\_grow, \hyperpage{17} - \item MP\_GT, \hyperpage{23} - \item mp\_init, \hyperpage{11} - \item mp\_init\_copy, \hyperpage{14} - \item mp\_init\_multi, \hyperpage{13} - \item mp\_init\_set, \hyperpage{21} - \item mp\_init\_set\_int, \hyperpage{21} - \item mp\_init\_size, \hyperpage{15} - \item mp\_int, \hyperpage{10} - \item mp\_invmod, \hyperpage{56} - \item mp\_jacobi, \hyperpage{56} - \item mp\_lcm, \hyperpage{56} - \item mp\_lshd, \hyperpage{30} - \item MP\_LT, \hyperpage{23} - \item MP\_MEM, \hyperpage{9} - \item mp\_mod, \hyperpage{39} - \item mp\_mod\_d, \hyperpage{56} - \item mp\_montgomery\_calc\_normalization, \hyperpage{42} - \item mp\_montgomery\_reduce, \hyperpage{42} - \item mp\_montgomery\_setup, \hyperpage{42} - \item mp\_mul, \hyperpage{33} - \item mp\_mul\_2, \hyperpage{28} - \item mp\_mul\_2d, \hyperpage{29} - \item mp\_mul\_d, \hyperpage{56} - \item mp\_n\_root, \hyperpage{48} - \item mp\_neg, \hyperpage{31, 32} - \item MP\_NO, \hyperpage{9} - \item MP\_OKAY, \hyperpage{9} - \item mp\_or, \hyperpage{31} - \item mp\_prime\_fermat, \hyperpage{49} - \item mp\_prime\_is\_divisible, \hyperpage{49} - \item mp\_prime\_is\_prime, \hyperpage{51} - \item mp\_prime\_miller\_rabin, \hyperpage{50} - \item mp\_prime\_next\_prime, \hyperpage{51} - \item mp\_prime\_rabin\_miller\_trials, \hyperpage{50} - \item mp\_prime\_random, \hyperpage{51} - \item mp\_prime\_random\_ex, \hyperpage{52} - \item mp\_radix\_size, \hyperpage{53} - \item mp\_read\_radix, \hyperpage{53} - \item mp\_read\_unsigned\_bin, \hyperpage{54} - \item mp\_reduce, \hyperpage{40} - \item mp\_reduce\_2k, \hyperpage{46} - \item mp\_reduce\_2k\_setup, \hyperpage{46} - \item mp\_reduce\_setup, \hyperpage{40} - \item mp\_rshd, \hyperpage{30} - \item mp\_set, \hyperpage{19} - \item mp\_set\_int, \hyperpage{20} - \item mp\_shrink, \hyperpage{16} - \item mp\_sqr, \hyperpage{35} - \item mp\_sub, \hyperpage{31} - \item mp\_sub\_d, \hyperpage{56} - \item mp\_to\_unsigned\_bin, \hyperpage{54} - \item mp\_toradix, \hyperpage{53} - \item mp\_unsigned\_bin\_size, \hyperpage{54} - \item MP\_VAL, \hyperpage{9} - \item mp\_xor, \hyperpage{31} - \item MP\_YES, \hyperpage{9} - -\end{theindex} diff --git a/lib/hcrypto/libtommath/bn.pdf b/lib/hcrypto/libtommath/bn.pdf deleted file mode 100644 index 5be712382..000000000 Binary files a/lib/hcrypto/libtommath/bn.pdf and /dev/null differ diff --git a/lib/hcrypto/libtommath/bn_cutoffs.c b/lib/hcrypto/libtommath/bn_cutoffs.c new file mode 100644 index 000000000..b02ab7161 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_cutoffs.c @@ -0,0 +1,14 @@ +#include "tommath_private.h" +#ifdef BN_CUTOFFS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifndef MP_FIXED_CUTOFFS +#include "tommath_cutoffs.h" +int KARATSUBA_MUL_CUTOFF = MP_DEFAULT_KARATSUBA_MUL_CUTOFF, + KARATSUBA_SQR_CUTOFF = MP_DEFAULT_KARATSUBA_SQR_CUTOFF, + TOOM_MUL_CUTOFF = MP_DEFAULT_TOOM_MUL_CUTOFF, + TOOM_SQR_CUTOFF = MP_DEFAULT_TOOM_SQR_CUTOFF; +#endif + +#endif diff --git a/lib/hcrypto/libtommath/bn_deprecated.c b/lib/hcrypto/libtommath/bn_deprecated.c new file mode 100644 index 000000000..2056b20e6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_deprecated.c @@ -0,0 +1,321 @@ +#include "tommath_private.h" +#ifdef BN_DEPRECATED_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifdef BN_MP_GET_BIT_C +int mp_get_bit(const mp_int *a, int b) +{ + if (b < 0) { + return MP_VAL; + } + return (s_mp_get_bit(a, (unsigned int)b) == MP_YES) ? MP_YES : MP_NO; +} +#endif +#ifdef BN_MP_JACOBI_C +mp_err mp_jacobi(const mp_int *a, const mp_int *n, int *c) +{ + if (a->sign == MP_NEG) { + return MP_VAL; + } + if (mp_cmp_d(n, 0uL) != MP_GT) { + return MP_VAL; + } + return mp_kronecker(a, n, c); +} +#endif +#ifdef BN_MP_PRIME_RANDOM_EX_C +mp_err mp_prime_random_ex(mp_int *a, int t, int size, int flags, private_mp_prime_callback cb, void *dat) +{ + return s_mp_prime_random_ex(a, t, size, flags, cb, dat); +} +#endif +#ifdef BN_MP_RAND_DIGIT_C +mp_err mp_rand_digit(mp_digit *r) +{ + mp_err err = s_mp_rand_source(r, sizeof(mp_digit)); + *r &= MP_MASK; + return err; +} +#endif +#ifdef BN_FAST_MP_INVMOD_C +mp_err fast_mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_invmod_fast(a, b, c); +} +#endif +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C +mp_err fast_mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) +{ + return s_mp_montgomery_reduce_fast(x, n, rho); +} +#endif +#ifdef BN_FAST_S_MP_MUL_DIGS_C +mp_err fast_s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + return s_mp_mul_digs_fast(a, b, c, digs); +} +#endif +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C +mp_err fast_s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + return s_mp_mul_high_digs_fast(a, b, c, digs); +} +#endif +#ifdef BN_FAST_S_MP_SQR_C +mp_err fast_s_mp_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_sqr_fast(a, b); +} +#endif +#ifdef BN_MP_BALANCE_MUL_C +mp_err mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_balance_mul(a, b, c); +} +#endif +#ifdef BN_MP_EXPTMOD_FAST_C +mp_err mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) +{ + return s_mp_exptmod_fast(G, X, P, Y, redmode); +} +#endif +#ifdef BN_MP_INVMOD_SLOW_C +mp_err mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_invmod_slow(a, b, c); +} +#endif +#ifdef BN_MP_KARATSUBA_MUL_C +mp_err mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_karatsuba_mul(a, b, c); +} +#endif +#ifdef BN_MP_KARATSUBA_SQR_C +mp_err mp_karatsuba_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_karatsuba_sqr(a, b); +} +#endif +#ifdef BN_MP_TOOM_MUL_C +mp_err mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_toom_mul(a, b, c); +} +#endif +#ifdef BN_MP_TOOM_SQR_C +mp_err mp_toom_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_toom_sqr(a, b); +} +#endif +#ifdef S_MP_REVERSE_C +void bn_reverse(unsigned char *s, int len) +{ + if (len > 0) { + s_mp_reverse(s, (size_t)len); + } +} +#endif +#ifdef BN_MP_TC_AND_C +mp_err mp_tc_and(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_and(a, b, c); +} +#endif +#ifdef BN_MP_TC_OR_C +mp_err mp_tc_or(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_or(a, b, c); +} +#endif +#ifdef BN_MP_TC_XOR_C +mp_err mp_tc_xor(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_xor(a, b, c); +} +#endif +#ifdef BN_MP_TC_DIV_2D_C +mp_err mp_tc_div_2d(const mp_int *a, int b, mp_int *c) +{ + return mp_signed_rsh(a, b, c); +} +#endif +#ifdef BN_MP_INIT_SET_INT_C +mp_err mp_init_set_int(mp_int *a, unsigned long b) +{ + return mp_init_u32(a, (uint32_t)b); +} +#endif +#ifdef BN_MP_SET_INT_C +mp_err mp_set_int(mp_int *a, unsigned long b) +{ + mp_set_u32(a, (uint32_t)b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_SET_LONG_C +mp_err mp_set_long(mp_int *a, unsigned long b) +{ + mp_set_u64(a, b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_SET_LONG_LONG_C +mp_err mp_set_long_long(mp_int *a, unsigned long long b) +{ + mp_set_u64(a, b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_GET_INT_C +unsigned long mp_get_int(const mp_int *a) +{ + return (unsigned long)mp_get_mag_u32(a); +} +#endif +#ifdef BN_MP_GET_LONG_C +unsigned long mp_get_long(const mp_int *a) +{ + return (unsigned long)mp_get_mag_ul(a); +} +#endif +#ifdef BN_MP_GET_LONG_LONG_C +unsigned long long mp_get_long_long(const mp_int *a) +{ + return mp_get_mag_ull(a); +} +#endif +#ifdef BN_MP_PRIME_IS_DIVISIBLE_C +mp_err mp_prime_is_divisible(const mp_int *a, mp_bool *result) +{ + return s_mp_prime_is_divisible(a, result); +} +#endif +#ifdef BN_MP_EXPT_D_EX_C +mp_err mp_expt_d_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) +{ + (void)fast; + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_expt_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_EXPT_D_C +mp_err mp_expt_d(const mp_int *a, mp_digit b, mp_int *c) +{ + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_expt_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_N_ROOT_EX_C +mp_err mp_n_root_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) +{ + (void)fast; + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_root_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_N_ROOT_C +mp_err mp_n_root(const mp_int *a, mp_digit b, mp_int *c) +{ + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_root_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_UNSIGNED_BIN_SIZE_C +int mp_unsigned_bin_size(const mp_int *a) +{ + return (int)mp_ubin_size(a); +} +#endif +#ifdef BN_MP_READ_UNSIGNED_BIN_C +mp_err mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c) +{ + return mp_from_ubin(a, b, (size_t) c); +} +#endif +#ifdef BN_MP_TO_UNSIGNED_BIN_C +mp_err mp_to_unsigned_bin(const mp_int *a, unsigned char *b) +{ + return mp_to_ubin(a, b, SIZE_MAX, NULL); +} +#endif +#ifdef BN_MP_TO_UNSIGNED_BIN_N_C +mp_err mp_to_unsigned_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) +{ + size_t n = mp_ubin_size(a); + if (*outlen < (unsigned long)n) { + return MP_VAL; + } + *outlen = (unsigned long)n; + return mp_to_ubin(a, b, n, NULL); +} +#endif +#ifdef BN_MP_SIGNED_BIN_SIZE_C +int mp_signed_bin_size(const mp_int *a) +{ + return (int)mp_sbin_size(a); +} +#endif +#ifdef BN_MP_READ_SIGNED_BIN_C +mp_err mp_read_signed_bin(mp_int *a, const unsigned char *b, int c) +{ + return mp_from_sbin(a, b, (size_t) c); +} +#endif +#ifdef BN_MP_TO_SIGNED_BIN_C +mp_err mp_to_signed_bin(const mp_int *a, unsigned char *b) +{ + return mp_to_sbin(a, b, SIZE_MAX, NULL); +} +#endif +#ifdef BN_MP_TO_SIGNED_BIN_N_C +mp_err mp_to_signed_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) +{ + size_t n = mp_sbin_size(a); + if (*outlen < (unsigned long)n) { + return MP_VAL; + } + *outlen = (unsigned long)n; + return mp_to_sbin(a, b, n, NULL); +} +#endif +#ifdef BN_MP_TORADIX_N_C +mp_err mp_toradix_n(const mp_int *a, char *str, int radix, int maxlen) +{ + if (maxlen < 0) { + return MP_VAL; + } + return mp_to_radix(a, str, (size_t)maxlen, NULL, radix); +} +#endif +#ifdef BN_MP_TORADIX_C +mp_err mp_toradix(const mp_int *a, char *str, int radix) +{ + return mp_to_radix(a, str, SIZE_MAX, NULL, radix); +} +#endif +#ifdef BN_MP_IMPORT_C +mp_err mp_import(mp_int *rop, size_t count, int order, size_t size, int endian, size_t nails, + const void *op) +{ + return mp_unpack(rop, count, order, size, endian, nails, op); +} +#endif +#ifdef BN_MP_EXPORT_C +mp_err mp_export(void *rop, size_t *countp, int order, size_t size, + int endian, size_t nails, const mp_int *op) +{ + return mp_pack(rop, SIZE_MAX, countp, order, size, endian, nails, op); +} +#endif +#endif diff --git a/lib/hcrypto/libtommath/bn_error.c b/lib/hcrypto/libtommath/bn_error.c deleted file mode 100644 index fbba7aa1f..000000000 --- a/lib/hcrypto/libtommath/bn_error.c +++ /dev/null @@ -1,47 +0,0 @@ -#include -#ifdef BN_ERROR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -static const struct { - int code; - const char *msg; -} msgs[] = { - { MP_OKAY, "Successful" }, - { MP_MEM, "Out of heap" }, - { MP_VAL, "Value out of range" } -}; - -/* return a char * string for a given code */ -const char *mp_error_to_string(int code) -{ - int x; - - /* scan the lookup table for the given message */ - for (x = 0; x < (int)(sizeof(msgs) / sizeof(msgs[0])); x++) { - if (msgs[x].code == code) { - return msgs[x].msg; - } - } - - /* generic reply for invalid code */ - return "Invalid error code"; -} - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_error.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_fast_mp_invmod.c b/lib/hcrypto/libtommath/bn_fast_mp_invmod.c deleted file mode 100644 index f4780d8e8..000000000 --- a/lib/hcrypto/libtommath/bn_fast_mp_invmod.c +++ /dev/null @@ -1,148 +0,0 @@ -#include -#ifdef BN_FAST_MP_INVMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* computes the modular inverse via binary extended euclidean algorithm, - * that is c = 1/a mod b - * - * Based on slow invmod except this is optimized for the case where b is - * odd as per HAC Note 14.64 on pp. 610 - */ -int fast_mp_invmod (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x, y, u, v, B, D; - int res, neg; - - /* 2. [modified] b must be odd */ - if (mp_iseven (b) == 1) { - return MP_VAL; - } - - /* init all our temps */ - if ((res = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { - return res; - } - - /* x == modulus, y == value to invert */ - if ((res = mp_copy (b, &x)) != MP_OKAY) { - goto LBL_ERR; - } - - /* we need y = |a| */ - if ((res = mp_mod (a, b, &y)) != MP_OKAY) { - goto LBL_ERR; - } - - /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ - if ((res = mp_copy (&x, &u)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (&y, &v)) != MP_OKAY) { - goto LBL_ERR; - } - mp_set (&D, 1); - -top: - /* 4. while u is even do */ - while (mp_iseven (&u) == 1) { - /* 4.1 u = u/2 */ - if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { - goto LBL_ERR; - } - /* 4.2 if B is odd then */ - if (mp_isodd (&B) == 1) { - if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* B = B/2 */ - if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 5. while v is even do */ - while (mp_iseven (&v) == 1) { - /* 5.1 v = v/2 */ - if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { - goto LBL_ERR; - } - /* 5.2 if D is odd then */ - if (mp_isodd (&D) == 1) { - /* D = (D-x)/2 */ - if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* D = D/2 */ - if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 6. if u >= v then */ - if (mp_cmp (&u, &v) != MP_LT) { - /* u = u - v, B = B - D */ - if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } else { - /* v - v - u, D = D - B */ - if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* if not zero goto step 4 */ - if (mp_iszero (&u) == 0) { - goto top; - } - - /* now a = C, b = D, gcd == g*v */ - - /* if v != 1 then there is no inverse */ - if (mp_cmp_d (&v, 1) != MP_EQ) { - res = MP_VAL; - goto LBL_ERR; - } - - /* b is now the inverse */ - neg = a->sign; - while (D.sign == MP_NEG) { - if ((res = mp_add (&D, b, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - mp_exch (&D, c); - c->sign = neg; - res = MP_OKAY; - -LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &B, &D, NULL); - return res; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_fast_mp_invmod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_fast_mp_montgomery_reduce.c b/lib/hcrypto/libtommath/bn_fast_mp_montgomery_reduce.c deleted file mode 100644 index b6c0694bd..000000000 --- a/lib/hcrypto/libtommath/bn_fast_mp_montgomery_reduce.c +++ /dev/null @@ -1,172 +0,0 @@ -#include -#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* computes xR**-1 == x (mod N) via Montgomery Reduction - * - * This is an optimized implementation of montgomery_reduce - * which uses the comba method to quickly calculate the columns of the - * reduction. - * - * Based on Algorithm 14.32 on pp.601 of HAC. -*/ -int fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) -{ - int ix, res, olduse; - mp_word W[MP_WARRAY]; - - /* get old used count */ - olduse = x->used; - - /* grow a as required */ - if (x->alloc < n->used + 1) { - if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) { - return res; - } - } - - /* first we have to get the digits of the input into - * an array of double precision words W[...] - */ - { - register mp_word *_W; - register mp_digit *tmpx; - - /* alias for the W[] array */ - _W = W; - - /* alias for the digits of x*/ - tmpx = x->dp; - - /* copy the digits of a into W[0..a->used-1] */ - for (ix = 0; ix < x->used; ix++) { - *_W++ = *tmpx++; - } - - /* zero the high words of W[a->used..m->used*2] */ - for (; ix < n->used * 2 + 1; ix++) { - *_W++ = 0; - } - } - - /* now we proceed to zero successive digits - * from the least significant upwards - */ - for (ix = 0; ix < n->used; ix++) { - /* mu = ai * m' mod b - * - * We avoid a double precision multiplication (which isn't required) - * by casting the value down to a mp_digit. Note this requires - * that W[ix-1] have the carry cleared (see after the inner loop) - */ - register mp_digit mu; - mu = (mp_digit) (((W[ix] & MP_MASK) * rho) & MP_MASK); - - /* a = a + mu * m * b**i - * - * This is computed in place and on the fly. The multiplication - * by b**i is handled by offseting which columns the results - * are added to. - * - * Note the comba method normally doesn't handle carries in the - * inner loop In this case we fix the carry from the previous - * column since the Montgomery reduction requires digits of the - * result (so far) [see above] to work. This is - * handled by fixing up one carry after the inner loop. The - * carry fixups are done in order so after these loops the - * first m->used words of W[] have the carries fixed - */ - { - register int iy; - register mp_digit *tmpn; - register mp_word *_W; - - /* alias for the digits of the modulus */ - tmpn = n->dp; - - /* Alias for the columns set by an offset of ix */ - _W = W + ix; - - /* inner loop */ - for (iy = 0; iy < n->used; iy++) { - *_W++ += ((mp_word)mu) * ((mp_word)*tmpn++); - } - } - - /* now fix carry for next digit, W[ix+1] */ - W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT); - } - - /* now we have to propagate the carries and - * shift the words downward [all those least - * significant digits we zeroed]. - */ - { - register mp_digit *tmpx; - register mp_word *_W, *_W1; - - /* nox fix rest of carries */ - - /* alias for current word */ - _W1 = W + ix; - - /* alias for next word, where the carry goes */ - _W = W + ++ix; - - for (; ix <= n->used * 2 + 1; ix++) { - *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT); - } - - /* copy out, A = A/b**n - * - * The result is A/b**n but instead of converting from an - * array of mp_word to mp_digit than calling mp_rshd - * we just copy them in the right order - */ - - /* alias for destination word */ - tmpx = x->dp; - - /* alias for shifted double precision result */ - _W = W + n->used; - - for (ix = 0; ix < n->used + 1; ix++) { - *tmpx++ = (mp_digit)(*_W++ & ((mp_word) MP_MASK)); - } - - /* zero oldused digits, if the input a was larger than - * m->used+1 we'll have to clear the digits - */ - for (; ix < olduse; ix++) { - *tmpx++ = 0; - } - } - - /* set the max used and clamp */ - x->used = n->used + 1; - mp_clamp (x); - - /* if A >= m then A = A - m */ - if (mp_cmp_mag (x, n) != MP_LT) { - return s_mp_sub (x, n, x); - } - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_fast_mp_montgomery_reduce.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_fast_s_mp_mul_digs.c b/lib/hcrypto/libtommath/bn_fast_s_mp_mul_digs.c deleted file mode 100644 index 90f161b10..000000000 --- a/lib/hcrypto/libtommath/bn_fast_s_mp_mul_digs.c +++ /dev/null @@ -1,107 +0,0 @@ -#include -#ifdef BN_FAST_S_MP_MUL_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* Fast (comba) multiplier - * - * This is the fast column-array [comba] multiplier. It is - * designed to compute the columns of the product first - * then handle the carries afterwards. This has the effect - * of making the nested loops that compute the columns very - * simple and schedulable on super-scalar processors. - * - * This has been modified to produce a variable number of - * digits of output so if say only a half-product is required - * you don't have to compute the upper half (a feature - * required for fast Barrett reduction). - * - * Based on Algorithm 14.12 on pp.595 of HAC. - * - */ -int fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY]; - register mp_word _W; - - /* grow the destination as required */ - if (c->alloc < digs) { - if ((res = mp_grow (c, digs)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - pa = MIN(digs, a->used + b->used); - - /* clear the carry */ - _W = 0; - for (ix = 0; ix < pa; ix++) { - int tx, ty; - int iy; - mp_digit *tmpx, *tmpy; - - /* get offsets into the two bignums */ - ty = MIN(b->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = b->dp + ty; - - /* this is the number of times the loop will iterrate, essentially - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* execute loop */ - for (iz = 0; iz < iy; ++iz) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - - } - - /* store term */ - W[ix] = ((mp_digit)_W) & MP_MASK; - - /* make next carry */ - _W = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = c->used; - c->used = pa; - - { - register mp_digit *tmpc; - tmpc = c->dp; - for (ix = 0; ix < pa+1; ix++) { - /* now extract the previous digit [below the carry] */ - *tmpc++ = W[ix]; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpc++ = 0; - } - } - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_fast_s_mp_mul_digs.c,v $ */ -/* $Revision: 1.8 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_fast_s_mp_mul_high_digs.c b/lib/hcrypto/libtommath/bn_fast_s_mp_mul_high_digs.c deleted file mode 100644 index a03b9f132..000000000 --- a/lib/hcrypto/libtommath/bn_fast_s_mp_mul_high_digs.c +++ /dev/null @@ -1,98 +0,0 @@ -#include -#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* this is a modified version of fast_s_mul_digs that only produces - * output digits *above* digs. See the comments for fast_s_mul_digs - * to see how it works. - * - * This is used in the Barrett reduction since for one of the multiplications - * only the higher digits were needed. This essentially halves the work. - * - * Based on Algorithm 14.12 on pp.595 of HAC. - */ -int fast_s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY]; - mp_word _W; - - /* grow the destination as required */ - pa = a->used + b->used; - if (c->alloc < pa) { - if ((res = mp_grow (c, pa)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - pa = a->used + b->used; - _W = 0; - for (ix = digs; ix < pa; ix++) { - int tx, ty, iy; - mp_digit *tmpx, *tmpy; - - /* get offsets into the two bignums */ - ty = MIN(b->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = b->dp + ty; - - /* this is the number of times the loop will iterrate, essentially its - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* execute loop */ - for (iz = 0; iz < iy; iz++) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - } - - /* store term */ - W[ix] = ((mp_digit)_W) & MP_MASK; - - /* make next carry */ - _W = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = c->used; - c->used = pa; - - { - register mp_digit *tmpc; - - tmpc = c->dp + digs; - for (ix = digs; ix < pa; ix++) { - /* now extract the previous digit [below the carry] */ - *tmpc++ = W[ix]; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpc++ = 0; - } - } - mp_clamp (c); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_fast_s_mp_mul_high_digs.c,v $ */ -/* $Revision: 1.6 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_fast_s_mp_sqr.c b/lib/hcrypto/libtommath/bn_fast_s_mp_sqr.c deleted file mode 100644 index 848eaf046..000000000 --- a/lib/hcrypto/libtommath/bn_fast_s_mp_sqr.c +++ /dev/null @@ -1,114 +0,0 @@ -#include -#ifdef BN_FAST_S_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* the jist of squaring... - * you do like mult except the offset of the tmpx [one that - * starts closer to zero] can't equal the offset of tmpy. - * So basically you set up iy like before then you min it with - * (ty-tx) so that it never happens. You double all those - * you add in the inner loop - -After that loop you do the squares and add them in. -*/ - -int fast_s_mp_sqr (mp_int * a, mp_int * b) -{ - int olduse, res, pa, ix, iz; - mp_digit W[MP_WARRAY], *tmpx; - mp_word W1; - - /* grow the destination as required */ - pa = a->used + a->used; - if (b->alloc < pa) { - if ((res = mp_grow (b, pa)) != MP_OKAY) { - return res; - } - } - - /* number of output digits to produce */ - W1 = 0; - for (ix = 0; ix < pa; ix++) { - int tx, ty, iy; - mp_word _W; - mp_digit *tmpy; - - /* clear counter */ - _W = 0; - - /* get offsets into the two bignums */ - ty = MIN(a->used-1, ix); - tx = ix - ty; - - /* setup temp aliases */ - tmpx = a->dp + tx; - tmpy = a->dp + ty; - - /* this is the number of times the loop will iterrate, essentially - while (tx++ < a->used && ty-- >= 0) { ... } - */ - iy = MIN(a->used-tx, ty+1); - - /* now for squaring tx can never equal ty - * we halve the distance since they approach at a rate of 2x - * and we have to round because odd cases need to be executed - */ - iy = MIN(iy, (ty-tx+1)>>1); - - /* execute loop */ - for (iz = 0; iz < iy; iz++) { - _W += ((mp_word)*tmpx++)*((mp_word)*tmpy--); - } - - /* double the inner product and add carry */ - _W = _W + _W + W1; - - /* even columns have the square term in them */ - if ((ix&1) == 0) { - _W += ((mp_word)a->dp[ix>>1])*((mp_word)a->dp[ix>>1]); - } - - /* store it */ - W[ix] = (mp_digit)(_W & MP_MASK); - - /* make next carry */ - W1 = _W >> ((mp_word)DIGIT_BIT); - } - - /* setup dest */ - olduse = b->used; - b->used = a->used+a->used; - - { - mp_digit *tmpb; - tmpb = b->dp; - for (ix = 0; ix < pa; ix++) { - *tmpb++ = W[ix] & MP_MASK; - } - - /* clear unused digits [that existed in the old copy of c] */ - for (; ix < olduse; ix++) { - *tmpb++ = 0; - } - } - mp_clamp (b); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_fast_s_mp_sqr.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_2expt.c b/lib/hcrypto/libtommath/bn_mp_2expt.c index 11a508c7f..0ae3df1bf 100644 --- a/lib/hcrypto/libtommath/bn_mp_2expt.c +++ b/lib/hcrypto/libtommath/bn_mp_2expt.c @@ -1,48 +1,31 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_2EXPT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* computes a = 2**b * * Simple algorithm which zeroes the int, grows it then just sets one bit * as required. */ -int -mp_2expt (mp_int * a, int b) +mp_err mp_2expt(mp_int *a, int b) { - int res; + mp_err err; - /* zero a as per default */ - mp_zero (a); + /* zero a as per default */ + mp_zero(a); - /* grow a to accomodate the single bit */ - if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) { - return res; - } + /* grow a to accomodate the single bit */ + if ((err = mp_grow(a, (b / MP_DIGIT_BIT) + 1)) != MP_OKAY) { + return err; + } - /* set the used count of where the bit will go */ - a->used = b / DIGIT_BIT + 1; + /* set the used count of where the bit will go */ + a->used = (b / MP_DIGIT_BIT) + 1; - /* put the single bit in its place */ - a->dp[b / DIGIT_BIT] = ((mp_digit)1) << (b % DIGIT_BIT); + /* put the single bit in its place */ + a->dp[b / MP_DIGIT_BIT] = (mp_digit)1 << (mp_digit)(b % MP_DIGIT_BIT); - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_2expt.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_abs.c b/lib/hcrypto/libtommath/bn_mp_abs.c index d97e8db05..00900bbdd 100644 --- a/lib/hcrypto/libtommath/bn_mp_abs.c +++ b/lib/hcrypto/libtommath/bn_mp_abs.c @@ -1,43 +1,26 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_ABS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* b = |a| * * Simple function copies the input and fixes the sign to positive */ -int -mp_abs (mp_int * a, mp_int * b) +mp_err mp_abs(const mp_int *a, mp_int *b) { - int res; + mp_err err; - /* copy a to b */ - if (a != b) { - if ((res = mp_copy (a, b)) != MP_OKAY) { - return res; - } - } + /* copy a to b */ + if (a != b) { + if ((err = mp_copy(a, b)) != MP_OKAY) { + return err; + } + } - /* force the sign of b to positive */ - b->sign = MP_ZPOS; + /* force the sign of b to positive */ + b->sign = MP_ZPOS; - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_abs.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_add.c b/lib/hcrypto/libtommath/bn_mp_add.c index be2064477..dfa78de50 100644 --- a/lib/hcrypto/libtommath/bn_mp_add.c +++ b/lib/hcrypto/libtommath/bn_mp_add.c @@ -1,53 +1,38 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_ADD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* high level addition (handles signs) */ -int mp_add (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_add(const mp_int *a, const mp_int *b, mp_int *c) { - int sa, sb, res; + mp_sign sa, sb; + mp_err err; - /* get sign of both inputs */ - sa = a->sign; - sb = b->sign; + /* get sign of both inputs */ + sa = a->sign; + sb = b->sign; - /* handle two cases, not four */ - if (sa == sb) { - /* both positive or both negative */ - /* add their magnitudes, copy the sign */ - c->sign = sa; - res = s_mp_add (a, b, c); - } else { - /* one positive, the other negative */ - /* subtract the one with the greater magnitude from */ - /* the one of the lesser magnitude. The result gets */ - /* the sign of the one with the greater magnitude. */ - if (mp_cmp_mag (a, b) == MP_LT) { - c->sign = sb; - res = s_mp_sub (b, a, c); - } else { + /* handle two cases, not four */ + if (sa == sb) { + /* both positive or both negative */ + /* add their magnitudes, copy the sign */ c->sign = sa; - res = s_mp_sub (a, b, c); - } - } - return res; + err = s_mp_add(a, b, c); + } else { + /* one positive, the other negative */ + /* subtract the one with the greater magnitude from */ + /* the one of the lesser magnitude. The result gets */ + /* the sign of the one with the greater magnitude. */ + if (mp_cmp_mag(a, b) == MP_LT) { + c->sign = sb; + err = s_mp_sub(b, a, c); + } else { + c->sign = sa; + err = s_mp_sub(a, b, c); + } + } + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_add.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_add_d.c b/lib/hcrypto/libtommath/bn_mp_add_d.c index 8ca36c112..f30157561 100644 --- a/lib/hcrypto/libtommath/bn_mp_add_d.c +++ b/lib/hcrypto/libtommath/bn_mp_add_d.c @@ -1,112 +1,89 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_ADD_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* single digit addition */ -int -mp_add_d (mp_int * a, mp_digit b, mp_int * c) +mp_err mp_add_d(const mp_int *a, mp_digit b, mp_int *c) { - int res, ix, oldused; - mp_digit *tmpa, *tmpc, mu; + mp_err err; + int ix, oldused; + mp_digit *tmpa, *tmpc; - /* grow c as required */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { - return res; - } - } + /* grow c as required */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } - /* if a is negative and |a| >= b, call c = |a| - b */ - if (a->sign == MP_NEG && (a->used > 1 || a->dp[0] >= b)) { - /* temporarily fix sign of a */ - a->sign = MP_ZPOS; + /* if a is negative and |a| >= b, call c = |a| - b */ + if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) { + mp_int a_ = *a; + /* temporarily fix sign of a */ + a_.sign = MP_ZPOS; - /* c = |a| - b */ - res = mp_sub_d(a, b, c); + /* c = |a| - b */ + err = mp_sub_d(&a_, b, c); - /* fix sign */ - a->sign = c->sign = MP_NEG; + /* fix sign */ + c->sign = MP_NEG; - /* clamp */ - mp_clamp(c); + /* clamp */ + mp_clamp(c); - return res; - } + return err; + } - /* old number of used digits in c */ - oldused = c->used; + /* old number of used digits in c */ + oldused = c->used; - /* sign always positive */ - c->sign = MP_ZPOS; + /* source alias */ + tmpa = a->dp; - /* source alias */ - tmpa = a->dp; + /* destination alias */ + tmpc = c->dp; - /* destination alias */ - tmpc = c->dp; + /* if a is positive */ + if (a->sign == MP_ZPOS) { + /* add digits, mu is carry */ + mp_digit mu = b; + for (ix = 0; ix < a->used; ix++) { + *tmpc = *tmpa++ + mu; + mu = *tmpc >> MP_DIGIT_BIT; + *tmpc++ &= MP_MASK; + } + /* set final carry */ + ix++; + *tmpc++ = mu; - /* if a is positive */ - if (a->sign == MP_ZPOS) { - /* add digit, after this we're propagating - * the carry. - */ - *tmpc = *tmpa++ + b; - mu = *tmpc >> DIGIT_BIT; - *tmpc++ &= MP_MASK; + /* setup size */ + c->used = a->used + 1; + } else { + /* a was negative and |a| < b */ + c->used = 1; - /* now handle rest of the digits */ - for (ix = 1; ix < a->used; ix++) { - *tmpc = *tmpa++ + mu; - mu = *tmpc >> DIGIT_BIT; - *tmpc++ &= MP_MASK; - } - /* set final carry */ - ix++; - *tmpc++ = mu; + /* the result is a single digit */ + if (a->used == 1) { + *tmpc++ = b - a->dp[0]; + } else { + *tmpc++ = b; + } - /* setup size */ - c->used = a->used + 1; - } else { - /* a was negative and |a| < b */ - c->used = 1; + /* setup count so the clearing of oldused + * can fall through correctly + */ + ix = 1; + } - /* the result is a single digit */ - if (a->used == 1) { - *tmpc++ = b - a->dp[0]; - } else { - *tmpc++ = b; - } + /* sign always positive */ + c->sign = MP_ZPOS; - /* setup count so the clearing of oldused - * can fall through correctly - */ - ix = 1; - } + /* now zero to oldused */ + MP_ZERO_DIGITS(tmpc, oldused - ix); + mp_clamp(c); - /* now zero to oldused */ - while (ix++ < oldused) { - *tmpc++ = 0; - } - mp_clamp(c); - - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_add_d.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_addmod.c b/lib/hcrypto/libtommath/bn_mp_addmod.c index 6d8afe18c..1dcfb678c 100644 --- a/lib/hcrypto/libtommath/bn_mp_addmod.c +++ b/lib/hcrypto/libtommath/bn_mp_addmod.c @@ -1,41 +1,25 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_ADDMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* d = a + b (mod c) */ -int -mp_addmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +mp_err mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) { - int res; - mp_int t; + mp_err err; + mp_int t; - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } - if ((res = mp_add (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; + if ((err = mp_add(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); + +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_addmod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_and.c b/lib/hcrypto/libtommath/bn_mp_and.c index 8ea22878f..c259f8dec 100644 --- a/lib/hcrypto/libtommath/bn_mp_and.c +++ b/lib/hcrypto/libtommath/bn_mp_and.c @@ -1,57 +1,56 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_AND_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* AND two ints together */ -int -mp_and (mp_int * a, mp_int * b, mp_int * c) +/* two complement and */ +mp_err mp_and(const mp_int *a, const mp_int *b, mp_int *c) { - int res, ix, px; - mp_int t, *x; + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = ((a->sign == MP_NEG) && (b->sign == MP_NEG)) ? MP_NEG : MP_ZPOS; - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } - for (ix = 0; ix < px; ix++) { - t.dp[ix] &= x->dp[ix]; - } + for (i = 0; i < used; i++) { + mp_digit x, y; - /* zero digits above the last from the smallest mp_int */ - for (; ix < t.used; ix++) { - t.dp[ix] = 0; - } + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x & y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_and.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_clamp.c b/lib/hcrypto/libtommath/bn_mp_clamp.c index 2a565e8db..ac23bfd3f 100644 --- a/lib/hcrypto/libtommath/bn_mp_clamp.c +++ b/lib/hcrypto/libtommath/bn_mp_clamp.c @@ -1,19 +1,7 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CLAMP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* trim unused digits * @@ -22,23 +10,18 @@ * Typically very fast. Also fixes the sign if there * are no more leading digits */ -void -mp_clamp (mp_int * a) +void mp_clamp(mp_int *a) { - /* decrease used while the most significant digit is - * zero. - */ - while (a->used > 0 && a->dp[a->used - 1] == 0) { - --(a->used); - } + /* decrease used while the most significant digit is + * zero. + */ + while ((a->used > 0) && (a->dp[a->used - 1] == 0u)) { + --(a->used); + } - /* reset the sign flag if used == 0 */ - if (a->used == 0) { - a->sign = MP_ZPOS; - } + /* reset the sign flag if used == 0 */ + if (a->used == 0) { + a->sign = MP_ZPOS; + } } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_clamp.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_clear.c b/lib/hcrypto/libtommath/bn_mp_clear.c index a65f0a36c..ff78324d9 100644 --- a/lib/hcrypto/libtommath/bn_mp_clear.c +++ b/lib/hcrypto/libtommath/bn_mp_clear.c @@ -1,44 +1,20 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CLEAR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* clear one (frees) */ -void -mp_clear (mp_int * a) +void mp_clear(mp_int *a) { - int i; + /* only do anything if a hasn't been freed previously */ + if (a->dp != NULL) { + /* free ram */ + MP_FREE_DIGITS(a->dp, a->alloc); - /* only do anything if a hasn't been freed previously */ - if (a->dp != NULL) { - /* first zero the digits */ - for (i = 0; i < a->used; i++) { - a->dp[i] = 0; - } - - /* free ram */ - XFREE(a->dp); - - /* reset members to make debugging easier */ - a->dp = NULL; - a->alloc = a->used = 0; - a->sign = MP_ZPOS; - } + /* reset members to make debugging easier */ + a->dp = NULL; + a->alloc = a->used = 0; + a->sign = MP_ZPOS; + } } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_clear.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_clear_multi.c b/lib/hcrypto/libtommath/bn_mp_clear_multi.c index e5e3da340..794e45fa5 100644 --- a/lib/hcrypto/libtommath/bn_mp_clear_multi.c +++ b/lib/hcrypto/libtommath/bn_mp_clear_multi.c @@ -1,34 +1,19 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CLEAR_MULTI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + #include void mp_clear_multi(mp_int *mp, ...) { - mp_int* next_mp = mp; - va_list args; - va_start(args, mp); - while (next_mp != NULL) { - mp_clear(next_mp); - next_mp = va_arg(args, mp_int*); - } - va_end(args); + mp_int *next_mp = mp; + va_list args; + va_start(args, mp); + while (next_mp != NULL) { + mp_clear(next_mp); + next_mp = va_arg(args, mp_int *); + } + va_end(args); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_clear_multi.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_cmp.c b/lib/hcrypto/libtommath/bn_mp_cmp.c index ccd2c8eb9..ced484096 100644 --- a/lib/hcrypto/libtommath/bn_mp_cmp.c +++ b/lib/hcrypto/libtommath/bn_mp_cmp.c @@ -1,43 +1,26 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CMP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* compare two ints (signed)*/ -int -mp_cmp (mp_int * a, mp_int * b) +mp_ord mp_cmp(const mp_int *a, const mp_int *b) { - /* compare based on sign */ - if (a->sign != b->sign) { - if (a->sign == MP_NEG) { - return MP_LT; - } else { - return MP_GT; - } - } + /* compare based on sign */ + if (a->sign != b->sign) { + if (a->sign == MP_NEG) { + return MP_LT; + } else { + return MP_GT; + } + } - /* compare digits */ - if (a->sign == MP_NEG) { - /* if negative compare opposite direction */ - return mp_cmp_mag(b, a); - } else { - return mp_cmp_mag(a, b); - } + /* compare digits */ + if (a->sign == MP_NEG) { + /* if negative compare opposite direction */ + return mp_cmp_mag(b, a); + } else { + return mp_cmp_mag(a, b); + } } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_cmp.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_cmp_d.c b/lib/hcrypto/libtommath/bn_mp_cmp_d.c index 724c1c363..5a8337b5b 100644 --- a/lib/hcrypto/libtommath/bn_mp_cmp_d.c +++ b/lib/hcrypto/libtommath/bn_mp_cmp_d.c @@ -1,44 +1,28 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CMP_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* compare a digit */ -int mp_cmp_d(mp_int * a, mp_digit b) +mp_ord mp_cmp_d(const mp_int *a, mp_digit b) { - /* compare based on sign */ - if (a->sign == MP_NEG) { - return MP_LT; - } + /* compare based on sign */ + if (a->sign == MP_NEG) { + return MP_LT; + } - /* compare based on magnitude */ - if (a->used > 1) { - return MP_GT; - } + /* compare based on magnitude */ + if (a->used > 1) { + return MP_GT; + } - /* compare the only digit of a to b */ - if (a->dp[0] > b) { - return MP_GT; - } else if (a->dp[0] < b) { - return MP_LT; - } else { - return MP_EQ; - } + /* compare the only digit of a to b */ + if (a->dp[0] > b) { + return MP_GT; + } else if (a->dp[0] < b) { + return MP_LT; + } else { + return MP_EQ; + } } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_cmp_d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_cmp_mag.c b/lib/hcrypto/libtommath/bn_mp_cmp_mag.c index 4a505238a..f144ea9e6 100644 --- a/lib/hcrypto/libtommath/bn_mp_cmp_mag.c +++ b/lib/hcrypto/libtommath/bn_mp_cmp_mag.c @@ -1,55 +1,39 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CMP_MAG_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* compare maginitude of two ints (unsigned) */ -int mp_cmp_mag (mp_int * a, mp_int * b) +mp_ord mp_cmp_mag(const mp_int *a, const mp_int *b) { - int n; - mp_digit *tmpa, *tmpb; + int n; + const mp_digit *tmpa, *tmpb; - /* compare based on # of non-zero digits */ - if (a->used > b->used) { - return MP_GT; - } - - if (a->used < b->used) { - return MP_LT; - } - - /* alias for a */ - tmpa = a->dp + (a->used - 1); - - /* alias for b */ - tmpb = b->dp + (a->used - 1); - - /* compare based on digits */ - for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { - if (*tmpa > *tmpb) { + /* compare based on # of non-zero digits */ + if (a->used > b->used) { return MP_GT; - } + } - if (*tmpa < *tmpb) { + if (a->used < b->used) { return MP_LT; - } - } - return MP_EQ; + } + + /* alias for a */ + tmpa = a->dp + (a->used - 1); + + /* alias for b */ + tmpb = b->dp + (a->used - 1); + + /* compare based on digits */ + for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { + if (*tmpa > *tmpb) { + return MP_GT; + } + + if (*tmpa < *tmpb) { + return MP_LT; + } + } + return MP_EQ; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_cmp_mag.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_cnt_lsb.c b/lib/hcrypto/libtommath/bn_mp_cnt_lsb.c index 2d4a8d4f0..4b2d206eb 100644 --- a/lib/hcrypto/libtommath/bn_mp_cnt_lsb.c +++ b/lib/hcrypto/libtommath/bn_mp_cnt_lsb.c @@ -1,53 +1,37 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_CNT_LSB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ static const int lnz[16] = { 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 }; /* Counts the number of lsbs which are zero before the first zero bit */ -int mp_cnt_lsb(mp_int *a) +int mp_cnt_lsb(const mp_int *a) { int x; mp_digit q, qq; /* easy out */ - if (mp_iszero(a) == 1) { + if (MP_IS_ZERO(a)) { return 0; } /* scan lower digits until non-zero */ - for (x = 0; x < a->used && a->dp[x] == 0; x++); + for (x = 0; (x < a->used) && (a->dp[x] == 0u); x++) {} q = a->dp[x]; - x *= DIGIT_BIT; + x *= MP_DIGIT_BIT; /* now scan this digit until a 1 is found */ - if ((q & 1) == 0) { + if ((q & 1u) == 0u) { do { - qq = q & 15; + qq = q & 15u; x += lnz[qq]; q >>= 4; - } while (qq == 0); + } while (qq == 0u); } return x; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_cnt_lsb.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_complement.c b/lib/hcrypto/libtommath/bn_mp_complement.c new file mode 100644 index 000000000..fef1423c5 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_complement.c @@ -0,0 +1,12 @@ +#include "tommath_private.h" +#ifdef BN_MP_COMPLEMENT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = ~a */ +mp_err mp_complement(const mp_int *a, mp_int *b) +{ + mp_err err = mp_neg(a, b); + return (err == MP_OKAY) ? mp_sub_d(b, 1uL, b) : err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_copy.c b/lib/hcrypto/libtommath/bn_mp_copy.c index b0de16d8a..e72fcf6d5 100644 --- a/lib/hcrypto/libtommath/bn_mp_copy.c +++ b/lib/hcrypto/libtommath/bn_mp_copy.c @@ -1,68 +1,47 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_COPY_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* copy, b = a */ -int -mp_copy (mp_int * a, mp_int * b) +mp_err mp_copy(const mp_int *a, mp_int *b) { - int res, n; + int n; + mp_digit *tmpa, *tmpb; + mp_err err; - /* if dst == src do nothing */ - if (a == b) { - return MP_OKAY; - } + /* if dst == src do nothing */ + if (a == b) { + return MP_OKAY; + } - /* grow dest */ - if (b->alloc < a->used) { - if ((res = mp_grow (b, a->used)) != MP_OKAY) { - return res; - } - } + /* grow dest */ + if (b->alloc < a->used) { + if ((err = mp_grow(b, a->used)) != MP_OKAY) { + return err; + } + } - /* zero b and copy the parameters over */ - { - register mp_digit *tmpa, *tmpb; + /* zero b and copy the parameters over */ + /* pointer aliases */ - /* pointer aliases */ + /* source */ + tmpa = a->dp; - /* source */ - tmpa = a->dp; + /* destination */ + tmpb = b->dp; - /* destination */ - tmpb = b->dp; - - /* copy all the digits */ - for (n = 0; n < a->used; n++) { + /* copy all the digits */ + for (n = 0; n < a->used; n++) { *tmpb++ = *tmpa++; - } + } - /* clear high digits */ - for (; n < b->used; n++) { - *tmpb++ = 0; - } - } + /* clear high digits */ + MP_ZERO_DIGITS(tmpb, b->used - n); - /* copy used count and sign */ - b->used = a->used; - b->sign = a->sign; - return MP_OKAY; + /* copy used count and sign */ + b->used = a->used; + b->sign = a->sign; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_copy.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_count_bits.c b/lib/hcrypto/libtommath/bn_mp_count_bits.c index 5dfd5f375..b7c2cad10 100644 --- a/lib/hcrypto/libtommath/bn_mp_count_bits.c +++ b/lib/hcrypto/libtommath/bn_mp_count_bits.c @@ -1,45 +1,28 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_COUNT_BITS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* returns the number of bits in an int */ -int -mp_count_bits (mp_int * a) +int mp_count_bits(const mp_int *a) { - int r; - mp_digit q; + int r; + mp_digit q; - /* shortcut */ - if (a->used == 0) { - return 0; - } + /* shortcut */ + if (MP_IS_ZERO(a)) { + return 0; + } - /* get number of digits and add that */ - r = (a->used - 1) * DIGIT_BIT; + /* get number of digits and add that */ + r = (a->used - 1) * MP_DIGIT_BIT; - /* take the last digit and count the bits in it */ - q = a->dp[a->used - 1]; - while (q > ((mp_digit) 0)) { - ++r; - q >>= ((mp_digit) 1); - } - return r; + /* take the last digit and count the bits in it */ + q = a->dp[a->used - 1]; + while (q > 0u) { + ++r; + q >>= 1u; + } + return r; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_count_bits.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_decr.c b/lib/hcrypto/libtommath/bn_mp_decr.c new file mode 100644 index 000000000..c6a1572c6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_decr.c @@ -0,0 +1,34 @@ +#include "tommath_private.h" +#ifdef BN_MP_DECR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Decrement "a" by one like "a--". Changes input! */ +mp_err mp_decr(mp_int *a) +{ + if (MP_IS_ZERO(a)) { + mp_set(a,1uL); + a->sign = MP_NEG; + return MP_OKAY; + } else if (a->sign == MP_NEG) { + mp_err err; + a->sign = MP_ZPOS; + if ((err = mp_incr(a)) != MP_OKAY) { + return err; + } + /* There is no -0 in LTM */ + if (!MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + return MP_OKAY; + } else if (a->dp[0] > 1uL) { + a->dp[0]--; + if (a->dp[0] == 0u) { + mp_zero(a); + } + return MP_OKAY; + } else { + return mp_sub_d(a, 1uL,a); + } +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_div.c b/lib/hcrypto/libtommath/bn_mp_div.c index 2c364b396..71de55bea 100644 --- a/lib/hcrypto/libtommath/bn_mp_div.c +++ b/lib/hcrypto/libtommath/bn_mp_div.c @@ -1,88 +1,71 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DIV_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ #ifdef BN_MP_DIV_SMALL /* slower bit-bang division... also smaller */ -int mp_div(mp_int * a, mp_int * b, mp_int * c, mp_int * d) +mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) { mp_int ta, tb, tq, q; - int res, n, n2; + int n, n2; + mp_err err; - /* is divisor zero ? */ - if (mp_iszero (b) == 1) { - return MP_VAL; - } + /* is divisor zero ? */ + if (MP_IS_ZERO(b)) { + return MP_VAL; + } - /* if a < b then q=0, r = a */ - if (mp_cmp_mag (a, b) == MP_LT) { - if (d != NULL) { - res = mp_copy (a, d); - } else { - res = MP_OKAY; - } - if (c != NULL) { - mp_zero (c); - } - return res; - } + /* if a < b then q=0, r = a */ + if (mp_cmp_mag(a, b) == MP_LT) { + if (d != NULL) { + err = mp_copy(a, d); + } else { + err = MP_OKAY; + } + if (c != NULL) { + mp_zero(c); + } + return err; + } - /* init our temps */ - if ((res = mp_init_multi(&ta, &tb, &tq, &q, NULL) != MP_OKAY)) { - return res; - } + /* init our temps */ + if ((err = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) { + return err; + } - mp_set(&tq, 1); - n = mp_count_bits(a) - mp_count_bits(b); - if (((res = mp_abs(a, &ta)) != MP_OKAY) || - ((res = mp_abs(b, &tb)) != MP_OKAY) || - ((res = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) || - ((res = mp_mul_2d(&tq, n, &tq)) != MP_OKAY)) { - goto LBL_ERR; - } + mp_set(&tq, 1uL); + n = mp_count_bits(a) - mp_count_bits(b); + if ((err = mp_abs(a, &ta)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_abs(b, &tb)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul_2d(&tq, n, &tq)) != MP_OKAY) goto LBL_ERR; - while (n-- >= 0) { - if (mp_cmp(&tb, &ta) != MP_GT) { - if (((res = mp_sub(&ta, &tb, &ta)) != MP_OKAY) || - ((res = mp_add(&q, &tq, &q)) != MP_OKAY)) { - goto LBL_ERR; - } - } - if (((res = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) || - ((res = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY)) { - goto LBL_ERR; - } - } + while (n-- >= 0) { + if (mp_cmp(&tb, &ta) != MP_GT) { + if ((err = mp_sub(&ta, &tb, &ta)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&q, &tq, &q)) != MP_OKAY) goto LBL_ERR; + } + if ((err = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY) goto LBL_ERR; + } - /* now q == quotient and ta == remainder */ - n = a->sign; - n2 = (a->sign == b->sign ? MP_ZPOS : MP_NEG); - if (c != NULL) { - mp_exch(c, &q); - c->sign = (mp_iszero(c) == MP_YES) ? MP_ZPOS : n2; - } - if (d != NULL) { - mp_exch(d, &ta); - d->sign = (mp_iszero(d) == MP_YES) ? MP_ZPOS : n; - } + /* now q == quotient and ta == remainder */ + n = a->sign; + n2 = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + if (c != NULL) { + mp_exch(c, &q); + c->sign = MP_IS_ZERO(c) ? MP_ZPOS : n2; + } + if (d != NULL) { + mp_exch(d, &ta); + d->sign = MP_IS_ZERO(d) ? MP_ZPOS : n; + } LBL_ERR: mp_clear_multi(&ta, &tb, &tq, &q, NULL); - return res; + return err; } #else @@ -100,193 +83,168 @@ LBL_ERR: * The overall algorithm is as described as * 14.20 from HAC but fixed to treat these cases. */ -int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) { - mp_int q, x, y, t1, t2; - int res, n, t, i, norm, neg; + mp_int q, x, y, t1, t2; + int n, t, i, norm; + mp_sign neg; + mp_err err; - /* is divisor zero ? */ - if (mp_iszero (b) == 1) { - return MP_VAL; - } + /* is divisor zero ? */ + if (MP_IS_ZERO(b)) { + return MP_VAL; + } - /* if a < b then q=0, r = a */ - if (mp_cmp_mag (a, b) == MP_LT) { - if (d != NULL) { - res = mp_copy (a, d); - } else { - res = MP_OKAY; - } - if (c != NULL) { - mp_zero (c); - } - return res; - } + /* if a < b then q=0, r = a */ + if (mp_cmp_mag(a, b) == MP_LT) { + if (d != NULL) { + err = mp_copy(a, d); + } else { + err = MP_OKAY; + } + if (c != NULL) { + mp_zero(c); + } + return err; + } - if ((res = mp_init_size (&q, a->used + 2)) != MP_OKAY) { - return res; - } - q.used = a->used + 2; + if ((err = mp_init_size(&q, a->used + 2)) != MP_OKAY) { + return err; + } + q.used = a->used + 2; - if ((res = mp_init (&t1)) != MP_OKAY) { - goto LBL_Q; - } + if ((err = mp_init(&t1)) != MP_OKAY) goto LBL_Q; - if ((res = mp_init (&t2)) != MP_OKAY) { - goto LBL_T1; - } + if ((err = mp_init(&t2)) != MP_OKAY) goto LBL_T1; - if ((res = mp_init_copy (&x, a)) != MP_OKAY) { - goto LBL_T2; - } + if ((err = mp_init_copy(&x, a)) != MP_OKAY) goto LBL_T2; - if ((res = mp_init_copy (&y, b)) != MP_OKAY) { - goto LBL_X; - } + if ((err = mp_init_copy(&y, b)) != MP_OKAY) goto LBL_X; - /* fix the sign */ - neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; - x.sign = y.sign = MP_ZPOS; + /* fix the sign */ + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + x.sign = y.sign = MP_ZPOS; - /* normalize both x and y, ensure that y >= b/2, [b == 2**DIGIT_BIT] */ - norm = mp_count_bits(&y) % DIGIT_BIT; - if (norm < (int)(DIGIT_BIT-1)) { - norm = (DIGIT_BIT-1) - norm; - if ((res = mp_mul_2d (&x, norm, &x)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_mul_2d (&y, norm, &y)) != MP_OKAY) { - goto LBL_Y; - } - } else { - norm = 0; - } + /* normalize both x and y, ensure that y >= b/2, [b == 2**MP_DIGIT_BIT] */ + norm = mp_count_bits(&y) % MP_DIGIT_BIT; + if (norm < (MP_DIGIT_BIT - 1)) { + norm = (MP_DIGIT_BIT - 1) - norm; + if ((err = mp_mul_2d(&x, norm, &x)) != MP_OKAY) goto LBL_Y; + if ((err = mp_mul_2d(&y, norm, &y)) != MP_OKAY) goto LBL_Y; + } else { + norm = 0; + } - /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ - n = x.used - 1; - t = y.used - 1; + /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ + n = x.used - 1; + t = y.used - 1; - /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ - if ((res = mp_lshd (&y, n - t)) != MP_OKAY) { /* y = y*b**{n-t} */ - goto LBL_Y; - } + /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ + /* y = y*b**{n-t} */ + if ((err = mp_lshd(&y, n - t)) != MP_OKAY) goto LBL_Y; - while (mp_cmp (&x, &y) != MP_LT) { - ++(q.dp[n - t]); - if ((res = mp_sub (&x, &y, &x)) != MP_OKAY) { - goto LBL_Y; - } - } + while (mp_cmp(&x, &y) != MP_LT) { + ++(q.dp[n - t]); + if ((err = mp_sub(&x, &y, &x)) != MP_OKAY) goto LBL_Y; + } - /* reset y by shifting it back down */ - mp_rshd (&y, n - t); + /* reset y by shifting it back down */ + mp_rshd(&y, n - t); - /* step 3. for i from n down to (t + 1) */ - for (i = n; i >= (t + 1); i--) { - if (i > x.used) { - continue; - } + /* step 3. for i from n down to (t + 1) */ + for (i = n; i >= (t + 1); i--) { + if (i > x.used) { + continue; + } - /* step 3.1 if xi == yt then set q{i-t-1} to b-1, - * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ - if (x.dp[i] == y.dp[t]) { - q.dp[i - t - 1] = ((((mp_digit)1) << DIGIT_BIT) - 1); - } else { - mp_word tmp; - tmp = ((mp_word) x.dp[i]) << ((mp_word) DIGIT_BIT); - tmp |= ((mp_word) x.dp[i - 1]); - tmp /= ((mp_word) y.dp[t]); - if (tmp > (mp_word) MP_MASK) - tmp = MP_MASK; - q.dp[i - t - 1] = (mp_digit) (tmp & (mp_word) (MP_MASK)); - } + /* step 3.1 if xi == yt then set q{i-t-1} to b-1, + * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ + if (x.dp[i] == y.dp[t]) { + q.dp[(i - t) - 1] = ((mp_digit)1 << (mp_digit)MP_DIGIT_BIT) - (mp_digit)1; + } else { + mp_word tmp; + tmp = (mp_word)x.dp[i] << (mp_word)MP_DIGIT_BIT; + tmp |= (mp_word)x.dp[i - 1]; + tmp /= (mp_word)y.dp[t]; + if (tmp > (mp_word)MP_MASK) { + tmp = MP_MASK; + } + q.dp[(i - t) - 1] = (mp_digit)(tmp & (mp_word)MP_MASK); + } - /* while (q{i-t-1} * (yt * b + y{t-1})) > - xi * b**2 + xi-1 * b + xi-2 + /* while (q{i-t-1} * (yt * b + y{t-1})) > + xi * b**2 + xi-1 * b + xi-2 - do q{i-t-1} -= 1; + do q{i-t-1} -= 1; + */ + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1uL) & (mp_digit)MP_MASK; + do { + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & (mp_digit)MP_MASK; + + /* find left hand */ + mp_zero(&t1); + t1.dp[0] = ((t - 1) < 0) ? 0u : y.dp[t - 1]; + t1.dp[1] = y.dp[t]; + t1.used = 2; + if ((err = mp_mul_d(&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) goto LBL_Y; + + /* find right hand */ + t2.dp[0] = ((i - 2) < 0) ? 0u : x.dp[i - 2]; + t2.dp[1] = x.dp[i - 1]; /* i >= 1 always holds */ + t2.dp[2] = x.dp[i]; + t2.used = 3; + } while (mp_cmp_mag(&t1, &t2) == MP_GT); + + /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ + if ((err = mp_mul_d(&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) goto LBL_Y; + + if ((err = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) goto LBL_Y; + + if ((err = mp_sub(&x, &t1, &x)) != MP_OKAY) goto LBL_Y; + + /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ + if (x.sign == MP_NEG) { + if ((err = mp_copy(&y, &t1)) != MP_OKAY) goto LBL_Y; + if ((err = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) goto LBL_Y; + if ((err = mp_add(&x, &t1, &x)) != MP_OKAY) goto LBL_Y; + + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & MP_MASK; + } + } + + /* now q is the quotient and x is the remainder + * [which we have to normalize] */ - q.dp[i - t - 1] = (q.dp[i - t - 1] + 1) & MP_MASK; - do { - q.dp[i - t - 1] = (q.dp[i - t - 1] - 1) & MP_MASK; - /* find left hand */ - mp_zero (&t1); - t1.dp[0] = (t - 1 < 0) ? 0 : y.dp[t - 1]; - t1.dp[1] = y.dp[t]; - t1.used = 2; - if ((res = mp_mul_d (&t1, q.dp[i - t - 1], &t1)) != MP_OKAY) { - goto LBL_Y; - } + /* get sign before writing to c */ + x.sign = (x.used == 0) ? MP_ZPOS : a->sign; - /* find right hand */ - t2.dp[0] = (i - 2 < 0) ? 0 : x.dp[i - 2]; - t2.dp[1] = (i - 1 < 0) ? 0 : x.dp[i - 1]; - t2.dp[2] = x.dp[i]; - t2.used = 3; - } while (mp_cmp_mag(&t1, &t2) == MP_GT); + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + c->sign = neg; + } - /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ - if ((res = mp_mul_d (&y, q.dp[i - t - 1], &t1)) != MP_OKAY) { - goto LBL_Y; - } + if (d != NULL) { + if ((err = mp_div_2d(&x, norm, &x, NULL)) != MP_OKAY) goto LBL_Y; + mp_exch(&x, d); + } - if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { - goto LBL_Y; - } + err = MP_OKAY; - if ((res = mp_sub (&x, &t1, &x)) != MP_OKAY) { - goto LBL_Y; - } - - /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ - if (x.sign == MP_NEG) { - if ((res = mp_copy (&y, &t1)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_lshd (&t1, i - t - 1)) != MP_OKAY) { - goto LBL_Y; - } - if ((res = mp_add (&x, &t1, &x)) != MP_OKAY) { - goto LBL_Y; - } - - q.dp[i - t - 1] = (q.dp[i - t - 1] - 1UL) & MP_MASK; - } - } - - /* now q is the quotient and x is the remainder - * [which we have to normalize] - */ - - /* get sign before writing to c */ - x.sign = x.used == 0 ? MP_ZPOS : a->sign; - - if (c != NULL) { - mp_clamp (&q); - mp_exch (&q, c); - c->sign = neg; - } - - if (d != NULL) { - mp_div_2d (&x, norm, &x, NULL); - mp_exch (&x, d); - } - - res = MP_OKAY; - -LBL_Y:mp_clear (&y); -LBL_X:mp_clear (&x); -LBL_T2:mp_clear (&t2); -LBL_T1:mp_clear (&t1); -LBL_Q:mp_clear (&q); - return res; +LBL_Y: + mp_clear(&y); +LBL_X: + mp_clear(&x); +LBL_T2: + mp_clear(&t2); +LBL_T1: + mp_clear(&t1); +LBL_Q: + mp_clear(&q); + return err; } #endif #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_div.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_div_2.c b/lib/hcrypto/libtommath/bn_mp_div_2.c index 7ee3e5b70..f56ea8199 100644 --- a/lib/hcrypto/libtommath/bn_mp_div_2.c +++ b/lib/hcrypto/libtommath/bn_mp_div_2.c @@ -1,68 +1,49 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DIV_2_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* b = a/2 */ -int mp_div_2(mp_int * a, mp_int * b) +mp_err mp_div_2(const mp_int *a, mp_int *b) { - int x, res, oldused; + int x, oldused; + mp_digit r, rr, *tmpa, *tmpb; + mp_err err; - /* copy */ - if (b->alloc < a->used) { - if ((res = mp_grow (b, a->used)) != MP_OKAY) { - return res; - } - } + /* copy */ + if (b->alloc < a->used) { + if ((err = mp_grow(b, a->used)) != MP_OKAY) { + return err; + } + } - oldused = b->used; - b->used = a->used; - { - register mp_digit r, rr, *tmpa, *tmpb; + oldused = b->used; + b->used = a->used; - /* source alias */ - tmpa = a->dp + b->used - 1; + /* source alias */ + tmpa = a->dp + b->used - 1; - /* dest alias */ - tmpb = b->dp + b->used - 1; + /* dest alias */ + tmpb = b->dp + b->used - 1; - /* carry */ - r = 0; - for (x = b->used - 1; x >= 0; x--) { + /* carry */ + r = 0; + for (x = b->used - 1; x >= 0; x--) { /* get the carry for the next iteration */ - rr = *tmpa & 1; + rr = *tmpa & 1u; /* shift the current digit, add in carry and store */ - *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1)); + *tmpb-- = (*tmpa-- >> 1) | (r << (MP_DIGIT_BIT - 1)); /* forward carry to next iteration */ r = rr; - } + } - /* zero excess digits */ - tmpb = b->dp + b->used; - for (x = b->used; x < oldused; x++) { - *tmpb++ = 0; - } - } - b->sign = a->sign; - mp_clamp (b); - return MP_OKAY; + /* zero excess digits */ + MP_ZERO_DIGITS(b->dp + b->used, oldused - b->used); + + b->sign = a->sign; + mp_clamp(b); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_div_2.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_div_2d.c b/lib/hcrypto/libtommath/bn_mp_div_2d.c index 4f7fa59e3..c47d5ce35 100644 --- a/lib/hcrypto/libtommath/bn_mp_div_2d.c +++ b/lib/hcrypto/libtommath/bn_mp_div_2d.c @@ -1,97 +1,71 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DIV_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* shift right by a certain bit count (store quotient in c, optional remainder in d) */ -int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d) +mp_err mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d) { - mp_digit D, r, rr; - int x, res; - mp_int t; + mp_digit D, r, rr; + int x; + mp_err err; + /* if the shift count is <= 0 then we do no work */ + if (b <= 0) { + err = mp_copy(a, c); + if (d != NULL) { + mp_zero(d); + } + return err; + } - /* if the shift count is <= 0 then we do no work */ - if (b <= 0) { - res = mp_copy (a, c); - if (d != NULL) { - mp_zero (d); - } - return res; - } + /* copy */ + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } + /* 'a' should not be used after here - it might be the same as d */ - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + /* get the remainder */ + if (d != NULL) { + if ((err = mp_mod_2d(a, b, d)) != MP_OKAY) { + return err; + } + } - /* get the remainder */ - if (d != NULL) { - if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - } + /* shift by as many digits in the bit count */ + if (b >= MP_DIGIT_BIT) { + mp_rshd(c, b / MP_DIGIT_BIT); + } - /* copy */ - if ((res = mp_copy (a, c)) != MP_OKAY) { - mp_clear (&t); - return res; - } + /* shift any bit count < MP_DIGIT_BIT */ + D = (mp_digit)(b % MP_DIGIT_BIT); + if (D != 0u) { + mp_digit *tmpc, mask, shift; - /* shift by as many digits in the bit count */ - if (b >= (int)DIGIT_BIT) { - mp_rshd (c, b / DIGIT_BIT); - } + /* mask */ + mask = ((mp_digit)1 << D) - 1uL; - /* shift any bit count < DIGIT_BIT */ - D = (mp_digit) (b % DIGIT_BIT); - if (D != 0) { - register mp_digit *tmpc, mask, shift; + /* shift for lsb */ + shift = (mp_digit)MP_DIGIT_BIT - D; - /* mask */ - mask = (((mp_digit)1) << D) - 1; + /* alias */ + tmpc = c->dp + (c->used - 1); - /* shift for lsb */ - shift = DIGIT_BIT - D; + /* carry */ + r = 0; + for (x = c->used - 1; x >= 0; x--) { + /* get the lower bits of this word in a temp */ + rr = *tmpc & mask; - /* alias */ - tmpc = c->dp + (c->used - 1); + /* shift the current word and mix in the carry bits from the previous word */ + *tmpc = (*tmpc >> D) | (r << shift); + --tmpc; - /* carry */ - r = 0; - for (x = c->used - 1; x >= 0; x--) { - /* get the lower bits of this word in a temp */ - rr = *tmpc & mask; - - /* shift the current word and mix in the carry bits from the previous word */ - *tmpc = (*tmpc >> D) | (r << shift); - --tmpc; - - /* set the carry to the carry bits of the current word found above */ - r = rr; - } - } - mp_clamp (c); - if (d != NULL) { - mp_exch (&t, d); - } - mp_clear (&t); - return MP_OKAY; + /* set the carry to the carry bits of the current word found above */ + r = rr; + } + } + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_div_2d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_div_3.c b/lib/hcrypto/libtommath/bn_mp_div_3.c index 78e2381b6..3a23fdff2 100644 --- a/lib/hcrypto/libtommath/bn_mp_div_3.c +++ b/lib/hcrypto/libtommath/bn_mp_div_3.c @@ -1,79 +1,63 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DIV_3_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* divide by three (based on routine from MPI and the GMP manual) */ -int -mp_div_3 (mp_int * a, mp_int *c, mp_digit * d) +mp_err mp_div_3(const mp_int *a, mp_int *c, mp_digit *d) { - mp_int q; - mp_word w, t; - mp_digit b; - int res, ix; + mp_int q; + mp_word w, t; + mp_digit b; + mp_err err; + int ix; - /* b = 2**DIGIT_BIT / 3 */ - b = (((mp_word)1) << ((mp_word)DIGIT_BIT)) / ((mp_word)3); + /* b = 2**MP_DIGIT_BIT / 3 */ + b = ((mp_word)1 << (mp_word)MP_DIGIT_BIT) / (mp_word)3; - if ((res = mp_init_size(&q, a->used)) != MP_OKAY) { - return res; - } + if ((err = mp_init_size(&q, a->used)) != MP_OKAY) { + return err; + } - q.used = a->used; - q.sign = a->sign; - w = 0; - for (ix = a->used - 1; ix >= 0; ix--) { - w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << (mp_word)MP_DIGIT_BIT) | (mp_word)a->dp[ix]; - if (w >= 3) { - /* multiply w by [1/3] */ - t = (w * ((mp_word)b)) >> ((mp_word)DIGIT_BIT); + if (w >= 3u) { + /* multiply w by [1/3] */ + t = (w * (mp_word)b) >> (mp_word)MP_DIGIT_BIT; - /* now subtract 3 * [w/3] from w, to get the remainder */ - w -= t+t+t; + /* now subtract 3 * [w/3] from w, to get the remainder */ + w -= t+t+t; - /* fixup the remainder as required since - * the optimization is not exact. - */ - while (w >= 3) { - t += 1; - w -= 3; - } + /* fixup the remainder as required since + * the optimization is not exact. + */ + while (w >= 3u) { + t += 1u; + w -= 3u; + } } else { - t = 0; + t = 0; } q.dp[ix] = (mp_digit)t; - } + } - /* [optional] store the remainder */ - if (d != NULL) { - *d = (mp_digit)w; - } + /* [optional] store the remainder */ + if (d != NULL) { + *d = (mp_digit)w; + } - /* [optional] store the quotient */ - if (c != NULL) { - mp_clamp(&q); - mp_exch(&q, c); - } - mp_clear(&q); + /* [optional] store the quotient */ + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); - return res; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_div_3.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_div_d.c b/lib/hcrypto/libtommath/bn_mp_div_d.c index 7bd372c20..b9d718bb0 100644 --- a/lib/hcrypto/libtommath/bn_mp_div_d.c +++ b/lib/hcrypto/libtommath/bn_mp_div_d.c @@ -1,115 +1,84 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DIV_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -static int s_is_power_of_two(mp_digit b, int *p) -{ - int x; - - /* fast return if no power of two */ - if ((b==0) || (b & (b-1))) { - return 0; - } - - for (x = 0; x < DIGIT_BIT; x++) { - if (b == (((mp_digit)1)<dp[0] & ((((mp_digit)1)<used)) != MP_OKAY) { - return res; - } - - q.used = a->used; - q.sign = a->sign; - w = 0; - for (ix = a->used - 1; ix >= 0; ix--) { - w = (w << ((mp_word)DIGIT_BIT)) | ((mp_word)a->dp[ix]); - - if (w >= b) { - t = (mp_digit)(w / b); - w -= ((mp_word)t) * ((mp_word)b); - } else { - t = 0; + /* quick outs */ + if ((b == 1u) || MP_IS_ZERO(a)) { + if (d != NULL) { + *d = 0; } - q.dp[ix] = (mp_digit)t; - } + if (c != NULL) { + return mp_copy(a, c); + } + return MP_OKAY; + } - if (d != NULL) { - *d = (mp_digit)w; - } + /* power of two ? */ + if ((b & (b - 1u)) == 0u) { + ix = 1; + while ((ix < MP_DIGIT_BIT) && (b != (((mp_digit)1)<dp[0] & (((mp_digit)1<<(mp_digit)ix) - 1uL); + } + if (c != NULL) { + return mp_div_2d(a, ix, c, NULL); + } + return MP_OKAY; + } - if (c != NULL) { - mp_clamp(&q); - mp_exch(&q, c); - } - mp_clear(&q); + /* three? */ + if (MP_HAS(MP_DIV_3) && (b == 3u)) { + return mp_div_3(a, c, d); + } - return res; + /* no easy answer [c'est la vie]. Just division */ + if ((err = mp_init_size(&q, a->used)) != MP_OKAY) { + return err; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << (mp_word)MP_DIGIT_BIT) | (mp_word)a->dp[ix]; + + if (w >= b) { + t = (mp_digit)(w / b); + w -= (mp_word)t * (mp_word)b; + } else { + t = 0; + } + q.dp[ix] = t; + } + + if (d != NULL) { + *d = (mp_digit)w; + } + + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_div_d.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2007/01/09 04:44:32 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_dr_is_modulus.c b/lib/hcrypto/libtommath/bn_mp_dr_is_modulus.c index 52373440d..83760eacc 100644 --- a/lib/hcrypto/libtommath/bn_mp_dr_is_modulus.c +++ b/lib/hcrypto/libtommath/bn_mp_dr_is_modulus.c @@ -1,43 +1,27 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DR_IS_MODULUS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines if a number is a valid DR modulus */ -int mp_dr_is_modulus(mp_int *a) +mp_bool mp_dr_is_modulus(const mp_int *a) { int ix; /* must be at least two digits */ if (a->used < 2) { - return 0; + return MP_NO; } /* must be of the form b**k - a [a <= b] so all * but the first digit must be equal to -1 (mod b). */ for (ix = 1; ix < a->used; ix++) { - if (a->dp[ix] != MP_MASK) { - return 0; - } + if (a->dp[ix] != MP_MASK) { + return MP_NO; + } } - return 1; + return MP_YES; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_dr_is_modulus.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_dr_reduce.c b/lib/hcrypto/libtommath/bn_mp_dr_reduce.c index e60b5784f..ffc33a6b5 100644 --- a/lib/hcrypto/libtommath/bn_mp_dr_reduce.c +++ b/lib/hcrypto/libtommath/bn_mp_dr_reduce.c @@ -1,19 +1,7 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DR_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* reduce "x" in place modulo "n" using the Diminished Radix algorithm. * @@ -29,66 +17,62 @@ * * Input x must be in the range 0 <= x <= (n-1)**2 */ -int -mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k) +mp_err mp_dr_reduce(mp_int *x, const mp_int *n, mp_digit k) { - int err, i, m; - mp_word r; - mp_digit mu, *tmpx1, *tmpx2; + mp_err err; + int i, m; + mp_word r; + mp_digit mu, *tmpx1, *tmpx2; - /* m = digits in modulus */ - m = n->used; + /* m = digits in modulus */ + m = n->used; - /* ensure that "x" has at least 2m digits */ - if (x->alloc < m + m) { - if ((err = mp_grow (x, m + m)) != MP_OKAY) { - return err; - } - } + /* ensure that "x" has at least 2m digits */ + if (x->alloc < (m + m)) { + if ((err = mp_grow(x, m + m)) != MP_OKAY) { + return err; + } + } -/* top of loop, this is where the code resumes if - * another reduction pass is required. - */ + /* top of loop, this is where the code resumes if + * another reduction pass is required. + */ top: - /* aliases for digits */ - /* alias for lower half of x */ - tmpx1 = x->dp; + /* aliases for digits */ + /* alias for lower half of x */ + tmpx1 = x->dp; - /* alias for upper half of x, or x/B**m */ - tmpx2 = x->dp + m; + /* alias for upper half of x, or x/B**m */ + tmpx2 = x->dp + m; - /* set carry to zero */ - mu = 0; + /* set carry to zero */ + mu = 0; - /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ - for (i = 0; i < m; i++) { - r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu; + /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ + for (i = 0; i < m; i++) { + r = ((mp_word)*tmpx2++ * (mp_word)k) + *tmpx1 + mu; *tmpx1++ = (mp_digit)(r & MP_MASK); - mu = (mp_digit)(r >> ((mp_word)DIGIT_BIT)); - } + mu = (mp_digit)(r >> ((mp_word)MP_DIGIT_BIT)); + } - /* set final carry */ - *tmpx1++ = mu; + /* set final carry */ + *tmpx1++ = mu; - /* zero words above m */ - for (i = m + 1; i < x->used; i++) { - *tmpx1++ = 0; - } + /* zero words above m */ + MP_ZERO_DIGITS(tmpx1, (x->used - m) - 1); - /* clamp, sub and return */ - mp_clamp (x); + /* clamp, sub and return */ + mp_clamp(x); - /* if x >= n then subtract and reduce again - * Each successive "recursion" makes the input smaller and smaller. - */ - if (mp_cmp_mag (x, n) != MP_LT) { - s_mp_sub(x, n, x); - goto top; - } - return MP_OKAY; + /* if x >= n then subtract and reduce again + * Each successive "recursion" makes the input smaller and smaller. + */ + if (mp_cmp_mag(x, n) != MP_LT) { + if ((err = s_mp_sub(x, n, x)) != MP_OKAY) { + return err; + } + goto top; + } + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_dr_reduce.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_dr_setup.c b/lib/hcrypto/libtommath/bn_mp_dr_setup.c index b7d5ed7c0..32d5f3890 100644 --- a/lib/hcrypto/libtommath/bn_mp_dr_setup.c +++ b/lib/hcrypto/libtommath/bn_mp_dr_setup.c @@ -1,32 +1,15 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_DR_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines the setup value */ -void mp_dr_setup(mp_int *a, mp_digit *d) +void mp_dr_setup(const mp_int *a, mp_digit *d) { - /* the casts are required if DIGIT_BIT is one less than - * the number of bits in a mp_digit [e.g. DIGIT_BIT==31] + /* the casts are required if MP_DIGIT_BIT is one less than + * the number of bits in a mp_digit [e.g. MP_DIGIT_BIT==31] */ - *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) - - ((mp_word)a->dp[0])); + *d = (mp_digit)(((mp_word)1 << (mp_word)MP_DIGIT_BIT) - (mp_word)a->dp[0]); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_dr_setup.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_error_to_string.c b/lib/hcrypto/libtommath/bn_mp_error_to_string.c new file mode 100644 index 000000000..2e2adb0f2 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_error_to_string.c @@ -0,0 +1,27 @@ +#include "tommath_private.h" +#ifdef BN_MP_ERROR_TO_STRING_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* return a char * string for a given code */ +const char *mp_error_to_string(mp_err code) +{ + switch (code) { + case MP_OKAY: + return "Successful"; + case MP_ERR: + return "Unknown error"; + case MP_MEM: + return "Out of heap"; + case MP_VAL: + return "Value out of range"; + case MP_ITER: + return "Max. iterations reached"; + case MP_BUF: + return "Buffer overflow"; + default: + return "Invalid error code"; + } +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_exch.c b/lib/hcrypto/libtommath/bn_mp_exch.c index ee551bc3e..552094c6b 100644 --- a/lib/hcrypto/libtommath/bn_mp_exch.c +++ b/lib/hcrypto/libtommath/bn_mp_exch.c @@ -1,34 +1,17 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_EXCH_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* swap the elements of two integers, for cases where you can't simply swap the * mp_int pointers around */ -void -mp_exch (mp_int * a, mp_int * b) +void mp_exch(mp_int *a, mp_int *b) { - mp_int t; + mp_int t; - t = *a; - *a = *b; - *b = t; + t = *a; + *a = *b; + *b = t; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_exch.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_expt_d.c b/lib/hcrypto/libtommath/bn_mp_expt_d.c deleted file mode 100644 index 4bdc2d13a..000000000 --- a/lib/hcrypto/libtommath/bn_mp_expt_d.c +++ /dev/null @@ -1,57 +0,0 @@ -#include -#ifdef BN_MP_EXPT_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* calculate c = a**b using a square-multiply algorithm */ -int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) -{ - int res, x; - mp_int g; - - if ((res = mp_init_copy (&g, a)) != MP_OKAY) { - return res; - } - - /* set initial result */ - mp_set (c, 1); - - for (x = 0; x < (int) DIGIT_BIT; x++) { - /* square */ - if ((res = mp_sqr (c, c)) != MP_OKAY) { - mp_clear (&g); - return res; - } - - /* if the bit is set multiply */ - if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) { - if ((res = mp_mul (c, &g, c)) != MP_OKAY) { - mp_clear (&g); - return res; - } - } - - /* shift to next bit */ - b <<= 1; - } - - mp_clear (&g); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_expt_d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_expt_u32.c b/lib/hcrypto/libtommath/bn_mp_expt_u32.c new file mode 100644 index 000000000..2ab67ba53 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_expt_u32.c @@ -0,0 +1,46 @@ +#include "tommath_private.h" +#ifdef BN_MP_EXPT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* calculate c = a**b using a square-multiply algorithm */ +mp_err mp_expt_u32(const mp_int *a, uint32_t b, mp_int *c) +{ + mp_err err; + + mp_int g; + + if ((err = mp_init_copy(&g, a)) != MP_OKAY) { + return err; + } + + /* set initial result */ + mp_set(c, 1uL); + + while (b > 0u) { + /* if the bit is set multiply */ + if ((b & 1u) != 0u) { + if ((err = mp_mul(c, &g, c)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* square */ + if (b > 1u) { + if ((err = mp_sqr(&g, &g)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* shift to next bit */ + b >>= 1; + } + + err = MP_OKAY; + +LBL_ERR: + mp_clear(&g); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_exptmod.c b/lib/hcrypto/libtommath/bn_mp_exptmod.c index 56d7c11d2..5f811ebd8 100644 --- a/lib/hcrypto/libtommath/bn_mp_exptmod.c +++ b/lib/hcrypto/libtommath/bn_mp_exptmod.c @@ -1,112 +1,76 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_EXPTMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* this is a shell function that calls either the normal or Montgomery * exptmod functions. Originally the call to the montgomery code was * embedded in the normal function but that wasted alot of stack space * for nothing (since 99% of the time the Montgomery code would be called) */ -int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) +mp_err mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y) { - int dr; + int dr; - /* modulus P must be positive */ - if (P->sign == MP_NEG) { - return MP_VAL; - } + /* modulus P must be positive */ + if (P->sign == MP_NEG) { + return MP_VAL; + } - /* if exponent X is negative we have to recurse */ - if (X->sign == MP_NEG) { -#ifdef BN_MP_INVMOD_C - mp_int tmpG, tmpX; - int err; + /* if exponent X is negative we have to recurse */ + if (X->sign == MP_NEG) { + mp_int tmpG, tmpX; + mp_err err; - /* first compute 1/G mod P */ - if ((err = mp_init(&tmpG)) != MP_OKAY) { - return err; - } - if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { - mp_clear(&tmpG); - return err; - } + if (!MP_HAS(MP_INVMOD)) { + return MP_VAL; + } - /* now get |X| */ - if ((err = mp_init(&tmpX)) != MP_OKAY) { - mp_clear(&tmpG); - return err; - } - if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { - mp_clear_multi(&tmpG, &tmpX, NULL); - return err; - } + if ((err = mp_init_multi(&tmpG, &tmpX, NULL)) != MP_OKAY) { + return err; + } - /* and now compute (1/G)**|X| instead of G**X [X < 0] */ - err = mp_exptmod(&tmpG, &tmpX, P, Y); - mp_clear_multi(&tmpG, &tmpX, NULL); - return err; -#else - /* no invmod */ - return MP_VAL; -#endif - } + /* first compute 1/G mod P */ + if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { + goto LBL_ERR; + } -/* modified diminished radix reduction */ -#if defined(BN_MP_REDUCE_IS_2K_L_C) && defined(BN_MP_REDUCE_2K_L_C) && defined(BN_S_MP_EXPTMOD_C) - if (mp_reduce_is_2k_l(P) == MP_YES) { - return s_mp_exptmod(G, X, P, Y, 1); - } -#endif + /* now get |X| */ + if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { + goto LBL_ERR; + } -#ifdef BN_MP_DR_IS_MODULUS_C - /* is it a DR modulus? */ - dr = mp_dr_is_modulus(P); -#else - /* default to no */ - dr = 0; -#endif + /* and now compute (1/G)**|X| instead of G**X [X < 0] */ + err = mp_exptmod(&tmpG, &tmpX, P, Y); +LBL_ERR: + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; + } -#ifdef BN_MP_REDUCE_IS_2K_C - /* if not, is it a unrestricted DR modulus? */ - if (dr == 0) { - dr = mp_reduce_is_2k(P) << 1; - } -#endif + /* modified diminished radix reduction */ + if (MP_HAS(MP_REDUCE_IS_2K_L) && MP_HAS(MP_REDUCE_2K_L) && MP_HAS(S_MP_EXPTMOD) && + (mp_reduce_is_2k_l(P) == MP_YES)) { + return s_mp_exptmod(G, X, P, Y, 1); + } - /* if the modulus is odd or dr != 0 use the montgomery method */ -#ifdef BN_MP_EXPTMOD_FAST_C - if (mp_isodd (P) == 1 || dr != 0) { - return mp_exptmod_fast (G, X, P, Y, dr); - } else { -#endif -#ifdef BN_S_MP_EXPTMOD_C - /* otherwise use the generic Barrett reduction technique */ - return s_mp_exptmod (G, X, P, Y, 0); -#else - /* no exptmod for evens */ - return MP_VAL; -#endif -#ifdef BN_MP_EXPTMOD_FAST_C - } -#endif + /* is it a DR modulus? default to no */ + dr = (MP_HAS(MP_DR_IS_MODULUS) && (mp_dr_is_modulus(P) == MP_YES)) ? 1 : 0; + + /* if not, is it a unrestricted DR modulus? */ + if (MP_HAS(MP_REDUCE_IS_2K) && (dr == 0)) { + dr = (mp_reduce_is_2k(P) == MP_YES) ? 2 : 0; + } + + /* if the modulus is odd or dr != 0 use the montgomery method */ + if (MP_HAS(S_MP_EXPTMOD_FAST) && (MP_IS_ODD(P) || (dr != 0))) { + return s_mp_exptmod_fast(G, X, P, Y, dr); + } else if (MP_HAS(S_MP_EXPTMOD)) { + /* otherwise use the generic Barrett reduction technique */ + return s_mp_exptmod(G, X, P, Y, 0); + } else { + /* no exptmod for evens */ + return MP_VAL; + } } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_exptmod.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_exptmod_fast.c b/lib/hcrypto/libtommath/bn_mp_exptmod_fast.c deleted file mode 100644 index 64fbe7fe2..000000000 --- a/lib/hcrypto/libtommath/bn_mp_exptmod_fast.c +++ /dev/null @@ -1,321 +0,0 @@ -#include -#ifdef BN_MP_EXPTMOD_FAST_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 - * - * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. - * The value of k changes based on the size of the exponent. - * - * Uses Montgomery or Diminished Radix reduction [whichever appropriate] - */ - -#ifdef MP_LOW_MEM - #define TAB_SIZE 32 -#else - #define TAB_SIZE 256 -#endif - -int mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) -{ - mp_int M[TAB_SIZE], res; - mp_digit buf, mp; - int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; - - /* use a pointer to the reduction algorithm. This allows us to use - * one of many reduction algorithms without modding the guts of - * the code with if statements everywhere. - */ - int (*redux)(mp_int*,mp_int*,mp_digit); - - /* find window size */ - x = mp_count_bits (X); - if (x <= 7) { - winsize = 2; - } else if (x <= 36) { - winsize = 3; - } else if (x <= 140) { - winsize = 4; - } else if (x <= 450) { - winsize = 5; - } else if (x <= 1303) { - winsize = 6; - } else if (x <= 3529) { - winsize = 7; - } else { - winsize = 8; - } - -#ifdef MP_LOW_MEM - if (winsize > 5) { - winsize = 5; - } -#endif - - /* init M array */ - /* init first cell */ - if ((err = mp_init(&M[1])) != MP_OKAY) { - return err; - } - - /* now init the second half of the array */ - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - if ((err = mp_init(&M[x])) != MP_OKAY) { - for (y = 1<<(winsize-1); y < x; y++) { - mp_clear (&M[y]); - } - mp_clear(&M[1]); - return err; - } - } - - /* determine and setup reduction code */ - if (redmode == 0) { -#ifdef BN_MP_MONTGOMERY_SETUP_C - /* now setup montgomery */ - if ((err = mp_montgomery_setup (P, &mp)) != MP_OKAY) { - goto LBL_M; - } -#else - err = MP_VAL; - goto LBL_M; -#endif - - /* automatically pick the comba one if available (saves quite a few calls/ifs) */ -#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C - if (((P->used * 2 + 1) < MP_WARRAY) && - P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - redux = fast_mp_montgomery_reduce; - } else -#endif - { -#ifdef BN_MP_MONTGOMERY_REDUCE_C - /* use slower baseline Montgomery method */ - redux = mp_montgomery_reduce; -#else - err = MP_VAL; - goto LBL_M; -#endif - } - } else if (redmode == 1) { -#if defined(BN_MP_DR_SETUP_C) && defined(BN_MP_DR_REDUCE_C) - /* setup DR reduction for moduli of the form B**k - b */ - mp_dr_setup(P, &mp); - redux = mp_dr_reduce; -#else - err = MP_VAL; - goto LBL_M; -#endif - } else { -#if defined(BN_MP_REDUCE_2K_SETUP_C) && defined(BN_MP_REDUCE_2K_C) - /* setup DR reduction for moduli of the form 2**k - b */ - if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) { - goto LBL_M; - } - redux = mp_reduce_2k; -#else - err = MP_VAL; - goto LBL_M; -#endif - } - - /* setup result */ - if ((err = mp_init (&res)) != MP_OKAY) { - goto LBL_M; - } - - /* create M table - * - - * - * The first half of the table is not computed though accept for M[0] and M[1] - */ - - if (redmode == 0) { -#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C - /* now we need R mod m */ - if ((err = mp_montgomery_calc_normalization (&res, P)) != MP_OKAY) { - goto LBL_RES; - } -#else - err = MP_VAL; - goto LBL_RES; -#endif - - /* now set M[1] to G * R mod m */ - if ((err = mp_mulmod (G, &res, P, &M[1])) != MP_OKAY) { - goto LBL_RES; - } - } else { - mp_set(&res, 1); - if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) { - goto LBL_RES; - } - } - - /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ - if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_RES; - } - - for (x = 0; x < (winsize - 1); x++) { - if ((err = mp_sqr (&M[1 << (winsize - 1)], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&M[1 << (winsize - 1)], P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* create upper table */ - for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { - if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&M[x], P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* set initial mode and bit cnt */ - mode = 0; - bitcnt = 1; - buf = 0; - digidx = X->used - 1; - bitcpy = 0; - bitbuf = 0; - - for (;;) { - /* grab next digit as required */ - if (--bitcnt == 0) { - /* if digidx == -1 we are out of digits so break */ - if (digidx == -1) { - break; - } - /* read next digit and reset bitcnt */ - buf = X->dp[digidx--]; - bitcnt = (int)DIGIT_BIT; - } - - /* grab the next msb from the exponent */ - y = (mp_digit)(buf >> (DIGIT_BIT - 1)) & 1; - buf <<= (mp_digit)1; - - /* if the bit is zero and mode == 0 then we ignore it - * These represent the leading zero bits before the first 1 bit - * in the exponent. Technically this opt is not required but it - * does lower the # of trivial squaring/reductions used - */ - if (mode == 0 && y == 0) { - continue; - } - - /* if the bit is zero and mode == 1 then we square */ - if (mode == 1 && y == 0) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - continue; - } - - /* else we add it to the window */ - bitbuf |= (y << (winsize - ++bitcpy)); - mode = 2; - - if (bitcpy == winsize) { - /* ok window is filled so square as required and multiply */ - /* square first */ - for (x = 0; x < winsize; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* then multiply */ - if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - - /* empty window and reset */ - bitcpy = 0; - bitbuf = 0; - mode = 1; - } - } - - /* if bits remain then square/multiply */ - if (mode == 2 && bitcpy > 0) { - /* square then multiply if the bit is set */ - for (x = 0; x < bitcpy; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - - /* get next bit of the window */ - bitbuf <<= 1; - if ((bitbuf & (1 << winsize)) != 0) { - /* then multiply */ - if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - } - } - - if (redmode == 0) { - /* fixup result if Montgomery reduction is used - * recall that any value in a Montgomery system is - * actually multiplied by R mod n. So we have - * to reduce one more time to cancel out the factor - * of R. - */ - if ((err = redux(&res, P, mp)) != MP_OKAY) { - goto LBL_RES; - } - } - - /* swap res with Y */ - mp_exch (&res, Y); - err = MP_OKAY; -LBL_RES:mp_clear (&res); -LBL_M: - mp_clear(&M[1]); - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - mp_clear (&M[x]); - } - return err; -} -#endif - - -/* $Source: /cvs/libtom/libtommath/bn_mp_exptmod_fast.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_exteuclid.c b/lib/hcrypto/libtommath/bn_mp_exteuclid.c index daf0c95ea..faf47ba6a 100644 --- a/lib/hcrypto/libtommath/bn_mp_exteuclid.c +++ b/lib/hcrypto/libtommath/bn_mp_exteuclid.c @@ -1,82 +1,73 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_EXTEUCLID_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* Extended euclidean algorithm of (a, b) produces a*u1 + b*u2 = u3 */ -int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +mp_err mp_exteuclid(const mp_int *a, const mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) { - mp_int u1,u2,u3,v1,v2,v3,t1,t2,t3,q,tmp; - int err; + mp_int u1, u2, u3, v1, v2, v3, t1, t2, t3, q, tmp; + mp_err err; if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { return err; } /* initialize, (u1,u2,u3) = (1,0,a) */ - mp_set(&u1, 1); - if ((err = mp_copy(a, &u3)) != MP_OKAY) { goto _ERR; } + mp_set(&u1, 1uL); + if ((err = mp_copy(a, &u3)) != MP_OKAY) goto LBL_ERR; /* initialize, (v1,v2,v3) = (0,1,b) */ - mp_set(&v2, 1); - if ((err = mp_copy(b, &v3)) != MP_OKAY) { goto _ERR; } + mp_set(&v2, 1uL); + if ((err = mp_copy(b, &v3)) != MP_OKAY) goto LBL_ERR; /* loop while v3 != 0 */ - while (mp_iszero(&v3) == MP_NO) { - /* q = u3/v3 */ - if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) { goto _ERR; } + while (!MP_IS_ZERO(&v3)) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) goto LBL_ERR; - /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ - if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) { goto _ERR; } - if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) { goto _ERR; } + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) goto LBL_ERR; - /* (u1,u2,u3) = (v1,v2,v3) */ - if ((err = mp_copy(&v1, &u1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&v2, &u2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&v3, &u3)) != MP_OKAY) { goto _ERR; } + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) goto LBL_ERR; - /* (v1,v2,v3) = (t1,t2,t3) */ - if ((err = mp_copy(&t1, &v1)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&t2, &v2)) != MP_OKAY) { goto _ERR; } - if ((err = mp_copy(&t3, &v3)) != MP_OKAY) { goto _ERR; } + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) goto LBL_ERR; } /* make sure U3 >= 0 */ if (u3.sign == MP_NEG) { - mp_neg(&u1, &u1); - mp_neg(&u2, &u2); - mp_neg(&u3, &u3); + if ((err = mp_neg(&u1, &u1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_neg(&u2, &u2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_neg(&u3, &u3)) != MP_OKAY) goto LBL_ERR; } /* copy result out */ - if (U1 != NULL) { mp_exch(U1, &u1); } - if (U2 != NULL) { mp_exch(U2, &u2); } - if (U3 != NULL) { mp_exch(U3, &u3); } + if (U1 != NULL) { + mp_exch(U1, &u1); + } + if (U2 != NULL) { + mp_exch(U2, &u2); + } + if (U3 != NULL) { + mp_exch(U3, &u3); + } err = MP_OKAY; -_ERR: mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); +LBL_ERR: + mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_exteuclid.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_find_prime.c b/lib/hcrypto/libtommath/bn_mp_find_prime.c deleted file mode 100644 index 29ef8747d..000000000 --- a/lib/hcrypto/libtommath/bn_mp_find_prime.c +++ /dev/null @@ -1,36 +0,0 @@ -/* TomsFastMath, a fast ISO C bignum library. - * - * This project is public domain and free for all purposes. - * - * Love Hornquist Astrand - */ -#include -#ifdef BN_MP_FIND_PRIME_C -int mp_find_prime(mp_int *a, int t) -{ - int res = MP_NO; - - /* valid value of t? */ - if (t <= 0 || t > PRIME_SIZE) { - return MP_VAL; - } - - if (mp_iseven(a)) - mp_add_d(a, 1, a); - - do { - if (mp_prime_is_prime(a, t, &res) != 0) { - res = MP_VAL; - break; - } - - if (res == MP_NO) { - mp_add_d(a, 2, a); - continue; - } - - } while (res != MP_YES); - - return res; -} -#endif diff --git a/lib/hcrypto/libtommath/bn_mp_fread.c b/lib/hcrypto/libtommath/bn_mp_fread.c index 52f7f32f0..52ea773e4 100644 --- a/lib/hcrypto/libtommath/bn_mp_fread.c +++ b/lib/hcrypto/libtommath/bn_mp_fread.c @@ -1,67 +1,60 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_FREAD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ +#ifndef MP_NO_FILE /* read a bigint from a file stream in ASCII */ -int mp_fread(mp_int *a, int radix, FILE *stream) +mp_err mp_fread(mp_int *a, int radix, FILE *stream) { - int err, ch, neg, y; - - /* clear a */ - mp_zero(a); + mp_err err; + mp_sign neg; /* if first digit is - then set negative */ - ch = fgetc(stream); - if (ch == '-') { + int ch = fgetc(stream); + if (ch == (int)'-') { neg = MP_NEG; ch = fgetc(stream); } else { neg = MP_ZPOS; } - for (;;) { - /* find y in the radix map */ - for (y = 0; y < radix; y++) { - if (mp_s_rmap[y] == ch) { - break; - } + /* no digits, return error */ + if (ch == EOF) { + return MP_ERR; + } + + /* clear a */ + mp_zero(a); + + do { + int y; + unsigned pos = (unsigned)(ch - (int)'('); + if (mp_s_rmap_reverse_sz < pos) { + break; } - if (y == radix) { + + y = (int)mp_s_rmap_reverse[pos]; + + if ((y == 0xff) || (y >= radix)) { break; } /* shift up and add */ - if ((err = mp_mul_d(a, radix, a)) != MP_OKAY) { + if ((err = mp_mul_d(a, (mp_digit)radix, a)) != MP_OKAY) { return err; } - if ((err = mp_add_d(a, y, a)) != MP_OKAY) { + if ((err = mp_add_d(a, (mp_digit)y, a)) != MP_OKAY) { return err; } + } while ((ch = fgetc(stream)) != EOF); - ch = fgetc(stream); - } - if (mp_cmp_d(a, 0) != MP_EQ) { + if (a->used != 0) { a->sign = neg; } return MP_OKAY; } - #endif -/* $Source: /cvs/libtom/libtommath/bn_mp_fread.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_from_sbin.c b/lib/hcrypto/libtommath/bn_mp_from_sbin.c new file mode 100644 index 000000000..20e45971e --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_from_sbin.c @@ -0,0 +1,25 @@ +#include "tommath_private.h" +#ifdef BN_MP_FROM_SBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* read signed bin, big endian, first byte is 0==positive or 1==negative */ +mp_err mp_from_sbin(mp_int *a, const unsigned char *buf, size_t size) +{ + mp_err err; + + /* read magnitude */ + if ((err = mp_from_ubin(a, buf + 1, size - 1u)) != MP_OKAY) { + return err; + } + + /* first byte is 0 for positive, non-zero for negative */ + if (buf[0] == (unsigned char)0) { + a->sign = MP_ZPOS; + } else { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_from_ubin.c b/lib/hcrypto/libtommath/bn_mp_from_ubin.c new file mode 100644 index 000000000..7f73cbccd --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_from_ubin.c @@ -0,0 +1,39 @@ +#include "tommath_private.h" +#ifdef BN_MP_FROM_UBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reads a unsigned char array, assumes the msb is stored first [big endian] */ +mp_err mp_from_ubin(mp_int *a, const unsigned char *buf, size_t size) +{ + mp_err err; + + /* make sure there are at least two digits */ + if (a->alloc < 2) { + if ((err = mp_grow(a, 2)) != MP_OKAY) { + return err; + } + } + + /* zero the int */ + mp_zero(a); + + /* read the bytes in */ + while (size-- > 0u) { + if ((err = mp_mul_2d(a, 8, a)) != MP_OKAY) { + return err; + } + +#ifndef MP_8BIT + a->dp[0] |= *buf++; + a->used += 1; +#else + a->dp[0] = (*buf & MP_MASK); + a->dp[1] |= ((*buf++ >> 7) & 1u); + a->used += 2; +#endif + } + mp_clamp(a); + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_fwrite.c b/lib/hcrypto/libtommath/bn_mp_fwrite.c index dc4529ba2..abe2e6717 100644 --- a/lib/hcrypto/libtommath/bn_mp_fwrite.c +++ b/lib/hcrypto/libtommath/bn_mp_fwrite.c @@ -1,52 +1,45 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_FWRITE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -int mp_fwrite(mp_int *a, int radix, FILE *stream) +#ifndef MP_NO_FILE +mp_err mp_fwrite(const mp_int *a, int radix, FILE *stream) { char *buf; - int err, len, x; + mp_err err; + int len; + size_t written; - if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { - return err; + /* TODO: this function is not in this PR */ + if (MP_HAS(MP_RADIX_SIZE_OVERESTIMATE)) { + /* if ((err = mp_radix_size_overestimate(&t, base, &len)) != MP_OKAY) goto LBL_ERR; */ + } else { + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; + } } - buf = OPT_CAST(char) XMALLOC (len); + buf = (char *) MP_MALLOC((size_t)len); if (buf == NULL) { return MP_MEM; } - if ((err = mp_toradix(a, buf, radix)) != MP_OKAY) { - XFREE (buf); - return err; + if ((err = mp_to_radix(a, buf, (size_t)len, &written, radix)) != MP_OKAY) { + goto LBL_ERR; } - for (x = 0; x < len; x++) { - if (fputc(buf[x], stream) == EOF) { - XFREE (buf); - return MP_VAL; - } + if (fwrite(buf, written, 1uL, stream) != 1uL) { + err = MP_ERR; + goto LBL_ERR; } + err = MP_OKAY; - XFREE (buf); - return MP_OKAY; + +LBL_ERR: + MP_FREE_BUFFER(buf, (size_t)len); + return err; } - #endif -/* $Source: /cvs/libtom/libtommath/bn_mp_fwrite.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_gcd.c b/lib/hcrypto/libtommath/bn_mp_gcd.c index 89795d564..53029baf3 100644 --- a/lib/hcrypto/libtommath/bn_mp_gcd.c +++ b/lib/hcrypto/libtommath/bn_mp_gcd.c @@ -1,105 +1,92 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_GCD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* Greatest Common Divisor using the binary method */ -int mp_gcd (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_gcd(const mp_int *a, const mp_int *b, mp_int *c) { - mp_int u, v; - int k, u_lsb, v_lsb, res; + mp_int u, v; + int k, u_lsb, v_lsb; + mp_err err; - /* either zero than gcd is the largest */ - if (mp_iszero (a) == MP_YES) { - return mp_abs (b, c); - } - if (mp_iszero (b) == MP_YES) { - return mp_abs (a, c); - } + /* either zero than gcd is the largest */ + if (MP_IS_ZERO(a)) { + return mp_abs(b, c); + } + if (MP_IS_ZERO(b)) { + return mp_abs(a, c); + } - /* get copies of a and b we can modify */ - if ((res = mp_init_copy (&u, a)) != MP_OKAY) { - return res; - } + /* get copies of a and b we can modify */ + if ((err = mp_init_copy(&u, a)) != MP_OKAY) { + return err; + } - if ((res = mp_init_copy (&v, b)) != MP_OKAY) { - goto LBL_U; - } + if ((err = mp_init_copy(&v, b)) != MP_OKAY) { + goto LBL_U; + } - /* must be positive for the remainder of the algorithm */ - u.sign = v.sign = MP_ZPOS; + /* must be positive for the remainder of the algorithm */ + u.sign = v.sign = MP_ZPOS; - /* B1. Find the common power of two for u and v */ - u_lsb = mp_cnt_lsb(&u); - v_lsb = mp_cnt_lsb(&v); - k = MIN(u_lsb, v_lsb); + /* B1. Find the common power of two for u and v */ + u_lsb = mp_cnt_lsb(&u); + v_lsb = mp_cnt_lsb(&v); + k = MP_MIN(u_lsb, v_lsb); - if (k > 0) { - /* divide the power of two out */ - if ((res = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { - goto LBL_V; - } + if (k > 0) { + /* divide the power of two out */ + if ((err = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } - if ((res = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } + if ((err = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } - /* divide any remaining factors of two out */ - if (u_lsb != k) { - if ((res = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { - goto LBL_V; - } - } + /* divide any remaining factors of two out */ + if (u_lsb != k) { + if ((err = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + } - if (v_lsb != k) { - if ((res = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } + if (v_lsb != k) { + if ((err = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } - while (mp_iszero(&v) == 0) { - /* make sure v is the largest */ - if (mp_cmp_mag(&u, &v) == MP_GT) { - /* swap u and v to make sure v is >= u */ - mp_exch(&u, &v); - } + while (!MP_IS_ZERO(&v)) { + /* make sure v is the largest */ + if (mp_cmp_mag(&u, &v) == MP_GT) { + /* swap u and v to make sure v is >= u */ + mp_exch(&u, &v); + } - /* subtract smallest from largest */ - if ((res = s_mp_sub(&v, &u, &v)) != MP_OKAY) { - goto LBL_V; - } + /* subtract smallest from largest */ + if ((err = s_mp_sub(&v, &u, &v)) != MP_OKAY) { + goto LBL_V; + } - /* Divide out all factors of two */ - if ((res = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { - goto LBL_V; - } - } + /* Divide out all factors of two */ + if ((err = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } - /* multiply by 2**k which we divided out at the beginning */ - if ((res = mp_mul_2d (&u, k, c)) != MP_OKAY) { - goto LBL_V; - } - c->sign = MP_ZPOS; - res = MP_OKAY; -LBL_V:mp_clear (&u); -LBL_U:mp_clear (&v); - return res; + /* multiply by 2**k which we divided out at the beginning */ + if ((err = mp_mul_2d(&u, k, c)) != MP_OKAY) { + goto LBL_V; + } + c->sign = MP_ZPOS; + err = MP_OKAY; +LBL_V: + mp_clear(&u); +LBL_U: + mp_clear(&v); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_gcd.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_get_double.c b/lib/hcrypto/libtommath/bn_mp_get_double.c new file mode 100644 index 000000000..c9b1b19f4 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_double.c @@ -0,0 +1,18 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_DOUBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +double mp_get_double(const mp_int *a) +{ + int i; + double d = 0.0, fac = 1.0; + for (i = 0; i < MP_DIGIT_BIT; ++i) { + fac *= 2.0; + } + for (i = a->used; i --> 0;) { + d = (d * fac) + (double)a->dp[i]; + } + return (a->sign == MP_NEG) ? -d : d; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_i32.c b/lib/hcrypto/libtommath/bn_mp_get_i32.c new file mode 100644 index 000000000..030b657a3 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_i32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_i32, mp_get_mag_u32, int32_t, uint32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_i64.c b/lib/hcrypto/libtommath/bn_mp_get_i64.c new file mode 100644 index 000000000..969c8d23c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_i64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_i64, mp_get_mag_u64, int64_t, uint64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_int.c b/lib/hcrypto/libtommath/bn_mp_get_int.c deleted file mode 100644 index e8e9b1d44..000000000 --- a/lib/hcrypto/libtommath/bn_mp_get_int.c +++ /dev/null @@ -1,45 +0,0 @@ -#include -#ifdef BN_MP_GET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* get the lower 32-bits of an mp_int */ -unsigned long mp_get_int(mp_int * a) -{ - int i; - unsigned long res; - - if (a->used == 0) { - return 0; - } - - /* get number of digits of the lsb we have to read */ - i = MIN(a->used,(int)((sizeof(unsigned long)*CHAR_BIT+DIGIT_BIT-1)/DIGIT_BIT))-1; - - /* get most significant digit of result */ - res = DIGIT(a,i); - - while (--i >= 0) { - res = (res << DIGIT_BIT) | DIGIT(a,i); - } - - /* force result to 32-bits always so it is consistent on non 32-bit platforms */ - return res & 0xFFFFFFFFUL; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_get_int.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_get_l.c b/lib/hcrypto/libtommath/bn_mp_get_l.c new file mode 100644 index 000000000..55d78ec03 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_l.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_l, mp_get_mag_ul, long, unsigned long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_ll.c b/lib/hcrypto/libtommath/bn_mp_get_ll.c new file mode 100644 index 000000000..268753490 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_ll.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_ll, mp_get_mag_ull, long long, unsigned long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_mag_u32.c b/lib/hcrypto/libtommath/bn_mp_get_mag_u32.c new file mode 100644 index 000000000..d77189be6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_mag_u32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_u32, uint32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_mag_u64.c b/lib/hcrypto/libtommath/bn_mp_get_mag_u64.c new file mode 100644 index 000000000..36dd73f62 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_mag_u64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_u64, uint64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_mag_ul.c b/lib/hcrypto/libtommath/bn_mp_get_mag_ul.c new file mode 100644 index 000000000..e8819aef6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_mag_ul.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_ul, unsigned long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_get_mag_ull.c b/lib/hcrypto/libtommath/bn_mp_get_mag_ull.c new file mode 100644 index 000000000..63a27412c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_get_mag_ull.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_ull, unsigned long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_grow.c b/lib/hcrypto/libtommath/bn_mp_grow.c index cf2b949b2..9e904c547 100644 --- a/lib/hcrypto/libtommath/bn_mp_grow.c +++ b/lib/hcrypto/libtommath/bn_mp_grow.c @@ -1,57 +1,38 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_GROW_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* grow as required */ -int mp_grow (mp_int * a, int size) +mp_err mp_grow(mp_int *a, int size) { - int i; - mp_digit *tmp; + int i; + mp_digit *tmp; - /* if the alloc size is smaller alloc more ram */ - if (a->alloc < size) { - /* ensure there are always at least MP_PREC digits extra on top */ - size += (MP_PREC * 2) - (size % MP_PREC); + /* if the alloc size is smaller alloc more ram */ + if (a->alloc < size) { + /* reallocate the array a->dp + * + * We store the return in a temporary variable + * in case the operation failed we don't want + * to overwrite the dp member of a. + */ + tmp = (mp_digit *) MP_REALLOC(a->dp, + (size_t)a->alloc * sizeof(mp_digit), + (size_t)size * sizeof(mp_digit)); + if (tmp == NULL) { + /* reallocation failed but "a" is still valid [can be freed] */ + return MP_MEM; + } - /* reallocate the array a->dp - * - * We store the return in a temporary variable - * in case the operation failed we don't want - * to overwrite the dp member of a. - */ - tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * size); - if (tmp == NULL) { - /* reallocation failed but "a" is still valid [can be freed] */ - return MP_MEM; - } + /* reallocation succeeded so set a->dp */ + a->dp = tmp; - /* reallocation succeeded so set a->dp */ - a->dp = tmp; - - /* zero excess digits */ - i = a->alloc; - a->alloc = size; - for (; i < a->alloc; i++) { - a->dp[i] = 0; - } - } - return MP_OKAY; + /* zero excess digits */ + i = a->alloc; + a->alloc = size; + MP_ZERO_DIGITS(a->dp + i, a->alloc - i); + } + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_grow.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_incr.c b/lib/hcrypto/libtommath/bn_mp_incr.c new file mode 100644 index 000000000..7695ac73c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_incr.c @@ -0,0 +1,30 @@ +#include "tommath_private.h" +#ifdef BN_MP_INCR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Increment "a" by one like "a++". Changes input! */ +mp_err mp_incr(mp_int *a) +{ + if (MP_IS_ZERO(a)) { + mp_set(a,1uL); + return MP_OKAY; + } else if (a->sign == MP_NEG) { + mp_err err; + a->sign = MP_ZPOS; + if ((err = mp_decr(a)) != MP_OKAY) { + return err; + } + /* There is no -0 in LTM */ + if (!MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + return MP_OKAY; + } else if (a->dp[0] < MP_DIGIT_MAX) { + a->dp[0]++; + return MP_OKAY; + } else { + return mp_add_d(a, 1uL,a); + } +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init.c b/lib/hcrypto/libtommath/bn_mp_init.c index 8be27f569..2eb792409 100644 --- a/lib/hcrypto/libtommath/bn_mp_init.c +++ b/lib/hcrypto/libtommath/bn_mp_init.c @@ -1,46 +1,23 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INIT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* init a new mp_int */ -int mp_init (mp_int * a) +mp_err mp_init(mp_int *a) { - int i; + /* allocate memory required and clear it */ + a->dp = (mp_digit *) MP_CALLOC((size_t)MP_PREC, sizeof(mp_digit)); + if (a->dp == NULL) { + return MP_MEM; + } - /* allocate memory required and clear it */ - a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * MP_PREC); - if (a->dp == NULL) { - return MP_MEM; - } + /* set the used to zero, allocated digits to the default precision + * and sign to positive */ + a->used = 0; + a->alloc = MP_PREC; + a->sign = MP_ZPOS; - /* set the digits to zero */ - for (i = 0; i < MP_PREC; i++) { - a->dp[i] = 0; - } - - /* set the used to zero, allocated digits to the default precision - * and sign to positive */ - a->used = 0; - a->alloc = MP_PREC; - a->sign = MP_ZPOS; - - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_copy.c b/lib/hcrypto/libtommath/bn_mp_init_copy.c index 0160811af..1888203d2 100644 --- a/lib/hcrypto/libtommath/bn_mp_init_copy.c +++ b/lib/hcrypto/libtommath/bn_mp_init_copy.c @@ -1,32 +1,21 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INIT_COPY_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* creates "a" then copies b into it */ -int mp_init_copy (mp_int * a, mp_int * b) +mp_err mp_init_copy(mp_int *a, const mp_int *b) { - int res; + mp_err err; - if ((res = mp_init (a)) != MP_OKAY) { - return res; - } - return mp_copy (b, a); + if ((err = mp_init_size(a, b->used)) != MP_OKAY) { + return err; + } + + if ((err = mp_copy(b, a)) != MP_OKAY) { + mp_clear(a); + } + + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init_copy.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_i32.c b/lib/hcrypto/libtommath/bn_mp_init_i32.c new file mode 100644 index 000000000..bc4de8d50 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_i32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_i32, mp_set_i32, int32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_i64.c b/lib/hcrypto/libtommath/bn_mp_init_i64.c new file mode 100644 index 000000000..2fa1516eb --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_i64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_i64, mp_set_i64, int64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_l.c b/lib/hcrypto/libtommath/bn_mp_init_l.c new file mode 100644 index 000000000..bc380b539 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_l.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_l, mp_set_l, long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_ll.c b/lib/hcrypto/libtommath/bn_mp_init_ll.c new file mode 100644 index 000000000..dc7c4a44e --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_ll.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ll, mp_set_ll, long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_multi.c b/lib/hcrypto/libtommath/bn_mp_init_multi.c index 56e860276..d8390b5a0 100644 --- a/lib/hcrypto/libtommath/bn_mp_init_multi.c +++ b/lib/hcrypto/libtommath/bn_mp_init_multi.c @@ -1,59 +1,41 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INIT_MULTI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + #include -int mp_init_multi(mp_int *mp, ...) +mp_err mp_init_multi(mp_int *mp, ...) { - mp_err res = MP_OKAY; /* Assume ok until proven otherwise */ - int n = 0; /* Number of ok inits */ - mp_int* cur_arg = mp; - va_list args; + mp_err err = MP_OKAY; /* Assume ok until proven otherwise */ + int n = 0; /* Number of ok inits */ + mp_int *cur_arg = mp; + va_list args; - va_start(args, mp); /* init args to next argument from caller */ - while (cur_arg != NULL) { - if (mp_init(cur_arg) != MP_OKAY) { - /* Oops - error! Back-track and mp_clear what we already - succeeded in init-ing, then return error. - */ - va_list clean_args; + va_start(args, mp); /* init args to next argument from caller */ + while (cur_arg != NULL) { + if (mp_init(cur_arg) != MP_OKAY) { + /* Oops - error! Back-track and mp_clear what we already + succeeded in init-ing, then return error. + */ + va_list clean_args; - /* end the current list */ - va_end(args); - - /* now start cleaning up */ - cur_arg = mp; - va_start(clean_args, mp); - while (n--) { - mp_clear(cur_arg); - cur_arg = va_arg(clean_args, mp_int*); - } - va_end(clean_args); - res = MP_MEM; - break; - } - n++; - cur_arg = va_arg(args, mp_int*); - } - va_end(args); - return res; /* Assumed ok, if error flagged above. */ + /* now start cleaning up */ + cur_arg = mp; + va_start(clean_args, mp); + while (n-- != 0) { + mp_clear(cur_arg); + cur_arg = va_arg(clean_args, mp_int *); + } + va_end(clean_args); + err = MP_MEM; + break; + } + n++; + cur_arg = va_arg(args, mp_int *); + } + va_end(args); + return err; /* Assumed ok, if error flagged above. */ } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init_multi.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_set.c b/lib/hcrypto/libtommath/bn_mp_init_set.c index 34edad92f..5068f2bf6 100644 --- a/lib/hcrypto/libtommath/bn_mp_init_set.c +++ b/lib/hcrypto/libtommath/bn_mp_init_set.c @@ -1,32 +1,16 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INIT_SET_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* initialize and set a digit */ -int mp_init_set (mp_int * a, mp_digit b) +mp_err mp_init_set(mp_int *a, mp_digit b) { - int err; - if ((err = mp_init(a)) != MP_OKAY) { - return err; - } - mp_set(a, b); - return err; + mp_err err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + mp_set(a, b); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init_set.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_set_int.c b/lib/hcrypto/libtommath/bn_mp_init_set_int.c deleted file mode 100644 index 5c5599315..000000000 --- a/lib/hcrypto/libtommath/bn_mp_init_set_int.c +++ /dev/null @@ -1,31 +0,0 @@ -#include -#ifdef BN_MP_INIT_SET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* initialize and set a digit */ -int mp_init_set_int (mp_int * a, unsigned long b) -{ - int err; - if ((err = mp_init(a)) != MP_OKAY) { - return err; - } - return mp_set_int(a, b); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init_set_int.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_size.c b/lib/hcrypto/libtommath/bn_mp_init_size.c index 9578ac754..d62268721 100644 --- a/lib/hcrypto/libtommath/bn_mp_init_size.c +++ b/lib/hcrypto/libtommath/bn_mp_init_size.c @@ -1,48 +1,24 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INIT_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* init an mp_init for a given size */ -int mp_init_size (mp_int * a, int size) +mp_err mp_init_size(mp_int *a, int size) { - int x; + size = MP_MAX(MP_MIN_PREC, size); - /* pad size so there are always extra digits */ - size += (MP_PREC * 2) - (size % MP_PREC); + /* alloc mem */ + a->dp = (mp_digit *) MP_CALLOC((size_t)size, sizeof(mp_digit)); + if (a->dp == NULL) { + return MP_MEM; + } - /* alloc mem */ - a->dp = OPT_CAST(mp_digit) XMALLOC (sizeof (mp_digit) * size); - if (a->dp == NULL) { - return MP_MEM; - } + /* set the members */ + a->used = 0; + a->alloc = size; + a->sign = MP_ZPOS; - /* set the members */ - a->used = 0; - a->alloc = size; - a->sign = MP_ZPOS; - - /* zero the digits */ - for (x = 0; x < size; x++) { - a->dp[x] = 0; - } - - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_init_size.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_init_u32.c b/lib/hcrypto/libtommath/bn_mp_init_u32.c new file mode 100644 index 000000000..015d89b90 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_u32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_u32, mp_set_u32, uint32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_u64.c b/lib/hcrypto/libtommath/bn_mp_init_u64.c new file mode 100644 index 000000000..2b35f7ef8 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_u64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_u64, mp_set_u64, uint64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_ul.c b/lib/hcrypto/libtommath/bn_mp_init_ul.c new file mode 100644 index 000000000..5164f7287 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_ul.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ul, mp_set_ul, unsigned long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_init_ull.c b/lib/hcrypto/libtommath/bn_mp_init_ull.c new file mode 100644 index 000000000..84110c002 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_init_ull.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_INIT_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ull, mp_set_ull, unsigned long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_invmod.c b/lib/hcrypto/libtommath/bn_mp_invmod.c index ac1a95231..7b35a2431 100644 --- a/lib/hcrypto/libtommath/bn_mp_invmod.c +++ b/lib/hcrypto/libtommath/bn_mp_invmod.c @@ -1,43 +1,23 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_INVMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* hac 14.61, pp608 */ -int mp_invmod (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) { - /* b cannot be negative */ - if (b->sign == MP_NEG || mp_iszero(b) == 1) { - return MP_VAL; - } + /* b cannot be negative and has to be >1 */ + if ((b->sign == MP_NEG) || (mp_cmp_d(b, 1uL) != MP_GT)) { + return MP_VAL; + } -#ifdef BN_FAST_MP_INVMOD_C - /* if the modulus is odd we can use a faster routine instead */ - if (mp_isodd (b) == 1) { - return fast_mp_invmod (a, b, c); - } -#endif + /* if the modulus is odd we can use a faster routine instead */ + if (MP_HAS(S_MP_INVMOD_FAST) && MP_IS_ODD(b)) { + return s_mp_invmod_fast(a, b, c); + } -#ifdef BN_MP_INVMOD_SLOW_C - return mp_invmod_slow(a, b, c); -#else - return MP_VAL; -#endif + return MP_HAS(S_MP_INVMOD_SLOW) + ? s_mp_invmod_slow(a, b, c) + : MP_VAL; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_invmod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_invmod_slow.c b/lib/hcrypto/libtommath/bn_mp_invmod_slow.c deleted file mode 100644 index 4ec487efa..000000000 --- a/lib/hcrypto/libtommath/bn_mp_invmod_slow.c +++ /dev/null @@ -1,175 +0,0 @@ -#include -#ifdef BN_MP_INVMOD_SLOW_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* hac 14.61, pp608 */ -int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x, y, u, v, A, B, C, D; - int res; - - /* b cannot be negative */ - if (b->sign == MP_NEG || mp_iszero(b) == 1) { - return MP_VAL; - } - - /* init temps */ - if ((res = mp_init_multi(&x, &y, &u, &v, - &A, &B, &C, &D, NULL)) != MP_OKAY) { - return res; - } - - /* x = a, y = b */ - if ((res = mp_mod(a, b, &x)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (b, &y)) != MP_OKAY) { - goto LBL_ERR; - } - - /* 2. [modified] if x,y are both even then return an error! */ - if (mp_iseven (&x) == 1 && mp_iseven (&y) == 1) { - res = MP_VAL; - goto LBL_ERR; - } - - /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ - if ((res = mp_copy (&x, &u)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_copy (&y, &v)) != MP_OKAY) { - goto LBL_ERR; - } - mp_set (&A, 1); - mp_set (&D, 1); - -top: - /* 4. while u is even do */ - while (mp_iseven (&u) == 1) { - /* 4.1 u = u/2 */ - if ((res = mp_div_2 (&u, &u)) != MP_OKAY) { - goto LBL_ERR; - } - /* 4.2 if A or B is odd then */ - if (mp_isodd (&A) == 1 || mp_isodd (&B) == 1) { - /* A = (A+y)/2, B = (B-x)/2 */ - if ((res = mp_add (&A, &y, &A)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_sub (&B, &x, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* A = A/2, B = B/2 */ - if ((res = mp_div_2 (&A, &A)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_div_2 (&B, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 5. while v is even do */ - while (mp_iseven (&v) == 1) { - /* 5.1 v = v/2 */ - if ((res = mp_div_2 (&v, &v)) != MP_OKAY) { - goto LBL_ERR; - } - /* 5.2 if C or D is odd then */ - if (mp_isodd (&C) == 1 || mp_isodd (&D) == 1) { - /* C = (C+y)/2, D = (D-x)/2 */ - if ((res = mp_add (&C, &y, &C)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_sub (&D, &x, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - /* C = C/2, D = D/2 */ - if ((res = mp_div_2 (&C, &C)) != MP_OKAY) { - goto LBL_ERR; - } - if ((res = mp_div_2 (&D, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* 6. if u >= v then */ - if (mp_cmp (&u, &v) != MP_LT) { - /* u = u - v, A = A - C, B = B - D */ - if ((res = mp_sub (&u, &v, &u)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&A, &C, &A)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&B, &D, &B)) != MP_OKAY) { - goto LBL_ERR; - } - } else { - /* v - v - u, C = C - A, D = D - B */ - if ((res = mp_sub (&v, &u, &v)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&C, &A, &C)) != MP_OKAY) { - goto LBL_ERR; - } - - if ((res = mp_sub (&D, &B, &D)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* if not zero goto step 4 */ - if (mp_iszero (&u) == 0) - goto top; - - /* now a = C, b = D, gcd == g*v */ - - /* if v != 1 then there is no inverse */ - if (mp_cmp_d (&v, 1) != MP_EQ) { - res = MP_VAL; - goto LBL_ERR; - } - - /* if its too low */ - while (mp_cmp_d(&C, 0) == MP_LT) { - if ((res = mp_add(&C, b, &C)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* too big */ - while (mp_cmp_mag(&C, b) != MP_LT) { - if ((res = mp_sub(&C, b, &C)) != MP_OKAY) { - goto LBL_ERR; - } - } - - /* C is now the inverse */ - mp_exch (&C, c); - res = MP_OKAY; -LBL_ERR:mp_clear_multi (&x, &y, &u, &v, &A, &B, &C, &D, NULL); - return res; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_invmod_slow.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_is_square.c b/lib/hcrypto/libtommath/bn_mp_is_square.c index 027fcd2f5..69e77a21a 100644 --- a/lib/hcrypto/libtommath/bn_mp_is_square.c +++ b/lib/hcrypto/libtommath/bn_mp_is_square.c @@ -1,109 +1,93 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_IS_SQUARE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* Check if remainders are possible squares - fast exclude non-squares */ static const char rem_128[128] = { - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 }; static const char rem_105[105] = { - 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, - 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, - 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, - 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, - 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, - 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, - 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, + 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, + 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, + 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 }; /* Store non-zero to ret if arg is square, and zero if not */ -int mp_is_square(mp_int *arg,int *ret) +mp_err mp_is_square(const mp_int *arg, mp_bool *ret) { - int res; - mp_digit c; - mp_int t; - unsigned long r; + mp_err err; + mp_digit c; + mp_int t; + unsigned long r; - /* Default to Non-square :) */ - *ret = MP_NO; + /* Default to Non-square :) */ + *ret = MP_NO; - if (arg->sign == MP_NEG) { - return MP_VAL; - } + if (arg->sign == MP_NEG) { + return MP_VAL; + } - /* digits used? (TSD) */ - if (arg->used == 0) { - return MP_OKAY; - } + if (MP_IS_ZERO(arg)) { + return MP_OKAY; + } - /* First check mod 128 (suppose that DIGIT_BIT is at least 7) */ - if (rem_128[127 & DIGIT(arg,0)] == 1) { - return MP_OKAY; - } + /* First check mod 128 (suppose that MP_DIGIT_BIT is at least 7) */ + if (rem_128[127u & arg->dp[0]] == (char)1) { + return MP_OKAY; + } - /* Next check mod 105 (3*5*7) */ - if ((res = mp_mod_d(arg,105,&c)) != MP_OKAY) { - return res; - } - if (rem_105[c] == 1) { - return MP_OKAY; - } + /* Next check mod 105 (3*5*7) */ + if ((err = mp_mod_d(arg, 105uL, &c)) != MP_OKAY) { + return err; + } + if (rem_105[c] == (char)1) { + return MP_OKAY; + } - if ((res = mp_init_set_int(&t,11L*13L*17L*19L*23L*29L*31L)) != MP_OKAY) { - return res; - } - if ((res = mp_mod(arg,&t,&t)) != MP_OKAY) { - goto ERR; - } - r = mp_get_int(&t); - /* Check for other prime modules, note it's not an ERROR but we must - * free "t" so the easiest way is to goto ERR. We know that res - * is already equal to MP_OKAY from the mp_mod call - */ - if ( (1L<<(r%11)) & 0x5C4L ) goto ERR; - if ( (1L<<(r%13)) & 0x9E4L ) goto ERR; - if ( (1L<<(r%17)) & 0x5CE8L ) goto ERR; - if ( (1L<<(r%19)) & 0x4F50CL ) goto ERR; - if ( (1L<<(r%23)) & 0x7ACCA0L ) goto ERR; - if ( (1L<<(r%29)) & 0xC2EDD0CL ) goto ERR; - if ( (1L<<(r%31)) & 0x6DE2B848L ) goto ERR; + if ((err = mp_init_u32(&t, 11u*13u*17u*19u*23u*29u*31u)) != MP_OKAY) { + return err; + } + if ((err = mp_mod(arg, &t, &t)) != MP_OKAY) { + goto LBL_ERR; + } + r = mp_get_u32(&t); + /* Check for other prime modules, note it's not an ERROR but we must + * free "t" so the easiest way is to goto LBL_ERR. We know that err + * is already equal to MP_OKAY from the mp_mod call + */ + if (((1uL<<(r%11uL)) & 0x5C4uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%13uL)) & 0x9E4uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%17uL)) & 0x5CE8uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%19uL)) & 0x4F50CuL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%23uL)) & 0x7ACCA0uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%29uL)) & 0xC2EDD0CuL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%31uL)) & 0x6DE2B848uL) != 0uL) goto LBL_ERR; - /* Final check - is sqr(sqrt(arg)) == arg ? */ - if ((res = mp_sqrt(arg,&t)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sqr(&t,&t)) != MP_OKAY) { - goto ERR; - } + /* Final check - is sqr(sqrt(arg)) == arg ? */ + if ((err = mp_sqrt(arg, &t)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_sqr(&t, &t)) != MP_OKAY) { + goto LBL_ERR; + } - *ret = (mp_cmp_mag(&t,arg) == MP_EQ) ? MP_YES : MP_NO; -ERR:mp_clear(&t); - return res; + *ret = (mp_cmp_mag(&t, arg) == MP_EQ) ? MP_YES : MP_NO; +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_is_square.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_iseven.c b/lib/hcrypto/libtommath/bn_mp_iseven.c new file mode 100644 index 000000000..5cb962284 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_iseven.c @@ -0,0 +1,10 @@ +#include "tommath_private.h" +#ifdef BN_MP_ISEVEN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_bool mp_iseven(const mp_int *a) +{ + return MP_IS_EVEN(a) ? MP_YES : MP_NO; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_isodd.c b/lib/hcrypto/libtommath/bn_mp_isodd.c new file mode 100644 index 000000000..bf17646d7 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_isodd.c @@ -0,0 +1,10 @@ +#include "tommath_private.h" +#ifdef BN_MP_ISODD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_bool mp_isodd(const mp_int *a) +{ + return MP_IS_ODD(a) ? MP_YES : MP_NO; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_jacobi.c b/lib/hcrypto/libtommath/bn_mp_jacobi.c deleted file mode 100644 index 91cfeeade..000000000 --- a/lib/hcrypto/libtommath/bn_mp_jacobi.c +++ /dev/null @@ -1,105 +0,0 @@ -#include -#ifdef BN_MP_JACOBI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* computes the jacobi c = (a | n) (or Legendre if n is prime) - * HAC pp. 73 Algorithm 2.149 - */ -int mp_jacobi (mp_int * a, mp_int * p, int *c) -{ - mp_int a1, p1; - int k, s, r, res; - mp_digit residue; - - /* if p <= 0 return MP_VAL */ - if (mp_cmp_d(p, 0) != MP_GT) { - return MP_VAL; - } - - /* step 1. if a == 0, return 0 */ - if (mp_iszero (a) == 1) { - *c = 0; - return MP_OKAY; - } - - /* step 2. if a == 1, return 1 */ - if (mp_cmp_d (a, 1) == MP_EQ) { - *c = 1; - return MP_OKAY; - } - - /* default */ - s = 0; - - /* step 3. write a = a1 * 2**k */ - if ((res = mp_init_copy (&a1, a)) != MP_OKAY) { - return res; - } - - if ((res = mp_init (&p1)) != MP_OKAY) { - goto LBL_A1; - } - - /* divide out larger power of two */ - k = mp_cnt_lsb(&a1); - if ((res = mp_div_2d(&a1, k, &a1, NULL)) != MP_OKAY) { - goto LBL_P1; - } - - /* step 4. if e is even set s=1 */ - if ((k & 1) == 0) { - s = 1; - } else { - /* else set s=1 if p = 1/7 (mod 8) or s=-1 if p = 3/5 (mod 8) */ - residue = p->dp[0] & 7; - - if (residue == 1 || residue == 7) { - s = 1; - } else if (residue == 3 || residue == 5) { - s = -1; - } - } - - /* step 5. if p == 3 (mod 4) *and* a1 == 3 (mod 4) then s = -s */ - if ( ((p->dp[0] & 3) == 3) && ((a1.dp[0] & 3) == 3)) { - s = -s; - } - - /* if a1 == 1 we're done */ - if (mp_cmp_d (&a1, 1) == MP_EQ) { - *c = s; - } else { - /* n1 = n mod a1 */ - if ((res = mp_mod (p, &a1, &p1)) != MP_OKAY) { - goto LBL_P1; - } - if ((res = mp_jacobi (&p1, &a1, &r)) != MP_OKAY) { - goto LBL_P1; - } - *c = s * r; - } - - /* done */ - res = MP_OKAY; -LBL_P1:mp_clear (&p1); -LBL_A1:mp_clear (&a1); - return res; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_jacobi.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_karatsuba_mul.c b/lib/hcrypto/libtommath/bn_mp_karatsuba_mul.c deleted file mode 100644 index 72a2319c0..000000000 --- a/lib/hcrypto/libtommath/bn_mp_karatsuba_mul.c +++ /dev/null @@ -1,167 +0,0 @@ -#include -#ifdef BN_MP_KARATSUBA_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* c = |a| * |b| using Karatsuba Multiplication using - * three half size multiplications - * - * Let B represent the radix [e.g. 2**DIGIT_BIT] and - * let n represent half of the number of digits in - * the min(a,b) - * - * a = a1 * B**n + a0 - * b = b1 * B**n + b0 - * - * Then, a * b => - a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 - * - * Note that a1b1 and a0b0 are used twice and only need to be - * computed once. So in total three half size (half # of - * digit) multiplications are performed, a0b0, a1b1 and - * (a1+b1)(a0+b0) - * - * Note that a multiplication of half the digits requires - * 1/4th the number of single precision multiplications so in - * total after one call 25% of the single precision multiplications - * are saved. Note also that the call to mp_mul can end up back - * in this function if the a0, a1, b0, or b1 are above the threshold. - * This is known as divide-and-conquer and leads to the famous - * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than - * the standard O(N**2) that the baseline/comba methods use. - * Generally though the overhead of this method doesn't pay off - * until a certain size (N ~ 80) is reached. - */ -int mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c) -{ - mp_int x0, x1, y0, y1, t1, x0y0, x1y1; - int B, err; - - /* default the return code to an error */ - err = MP_MEM; - - /* min # of digits */ - B = MIN (a->used, b->used); - - /* now divide in two */ - B = B >> 1; - - /* init copy all the temps */ - if (mp_init_size (&x0, B) != MP_OKAY) - goto ERR; - if (mp_init_size (&x1, a->used - B) != MP_OKAY) - goto X0; - if (mp_init_size (&y0, B) != MP_OKAY) - goto X1; - if (mp_init_size (&y1, b->used - B) != MP_OKAY) - goto Y0; - - /* init temps */ - if (mp_init_size (&t1, B * 2) != MP_OKAY) - goto Y1; - if (mp_init_size (&x0y0, B * 2) != MP_OKAY) - goto T1; - if (mp_init_size (&x1y1, B * 2) != MP_OKAY) - goto X0Y0; - - /* now shift the digits */ - x0.used = y0.used = B; - x1.used = a->used - B; - y1.used = b->used - B; - - { - register int x; - register mp_digit *tmpa, *tmpb, *tmpx, *tmpy; - - /* we copy the digits directly instead of using higher level functions - * since we also need to shift the digits - */ - tmpa = a->dp; - tmpb = b->dp; - - tmpx = x0.dp; - tmpy = y0.dp; - for (x = 0; x < B; x++) { - *tmpx++ = *tmpa++; - *tmpy++ = *tmpb++; - } - - tmpx = x1.dp; - for (x = B; x < a->used; x++) { - *tmpx++ = *tmpa++; - } - - tmpy = y1.dp; - for (x = B; x < b->used; x++) { - *tmpy++ = *tmpb++; - } - } - - /* only need to clamp the lower words since by definition the - * upper words x1/y1 must have a known number of digits - */ - mp_clamp (&x0); - mp_clamp (&y0); - - /* now calc the products x0y0 and x1y1 */ - /* after this x0 is no longer required, free temp [x0==t2]! */ - if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY) - goto X1Y1; /* x0y0 = x0*y0 */ - if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY) - goto X1Y1; /* x1y1 = x1*y1 */ - - /* now calc x1+x0 and y1+y0 */ - if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = x1 - x0 */ - if (s_mp_add (&y1, &y0, &x0) != MP_OKAY) - goto X1Y1; /* t2 = y1 - y0 */ - if (mp_mul (&t1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ - - /* add x0y0 */ - if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY) - goto X1Y1; /* t2 = x0y0 + x1y1 */ - if (s_mp_sub (&t1, &x0, &t1) != MP_OKAY) - goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ - - /* shift by B */ - if (mp_lshd (&t1, B) != MP_OKAY) - goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))< -#ifdef BN_MP_KARATSUBA_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* Karatsuba squaring, computes b = a*a using three - * half size squarings - * - * See comments of karatsuba_mul for details. It - * is essentially the same algorithm but merely - * tuned to perform recursive squarings. - */ -int mp_karatsuba_sqr (mp_int * a, mp_int * b) -{ - mp_int x0, x1, t1, t2, x0x0, x1x1; - int B, err; - - err = MP_MEM; - - /* min # of digits */ - B = a->used; - - /* now divide in two */ - B = B >> 1; - - /* init copy all the temps */ - if (mp_init_size (&x0, B) != MP_OKAY) - goto ERR; - if (mp_init_size (&x1, a->used - B) != MP_OKAY) - goto X0; - - /* init temps */ - if (mp_init_size (&t1, a->used * 2) != MP_OKAY) - goto X1; - if (mp_init_size (&t2, a->used * 2) != MP_OKAY) - goto T1; - if (mp_init_size (&x0x0, B * 2) != MP_OKAY) - goto T2; - if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY) - goto X0X0; - - { - register int x; - register mp_digit *dst, *src; - - src = a->dp; - - /* now shift the digits */ - dst = x0.dp; - for (x = 0; x < B; x++) { - *dst++ = *src++; - } - - dst = x1.dp; - for (x = B; x < a->used; x++) { - *dst++ = *src++; - } - } - - x0.used = B; - x1.used = a->used - B; - - mp_clamp (&x0); - - /* now calc the products x0*x0 and x1*x1 */ - if (mp_sqr (&x0, &x0x0) != MP_OKAY) - goto X1X1; /* x0x0 = x0*x0 */ - if (mp_sqr (&x1, &x1x1) != MP_OKAY) - goto X1X1; /* x1x1 = x1*x1 */ - - /* now calc (x1+x0)**2 */ - if (s_mp_add (&x1, &x0, &t1) != MP_OKAY) - goto X1X1; /* t1 = x1 - x0 */ - if (mp_sqr (&t1, &t1) != MP_OKAY) - goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ - - /* add x0y0 */ - if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY) - goto X1X1; /* t2 = x0x0 + x1x1 */ - if (s_mp_sub (&t1, &t2, &t1) != MP_OKAY) - goto X1X1; /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */ - - /* shift by B */ - if (mp_lshd (&t1, B) != MP_OKAY) - goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<used == 1) && (a->dp[0] == 1u)) { + *c = 1; + } else { + *c = 0; + } + return MP_OKAY; + } + + if (MP_IS_EVEN(a) && MP_IS_EVEN(p)) { + *c = 0; + return MP_OKAY; + } + + if ((err = mp_init_copy(&a1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_init_copy(&p1, p)) != MP_OKAY) { + goto LBL_KRON_0; + } + + v = mp_cnt_lsb(&p1); + if ((err = mp_div_2d(&p1, v, &p1, NULL)) != MP_OKAY) { + goto LBL_KRON_1; + } + + if ((v & 1) == 0) { + k = 1; + } else { + k = table[a->dp[0] & 7u]; + } + + if (p1.sign == MP_NEG) { + p1.sign = MP_ZPOS; + if (a1.sign == MP_NEG) { + k = -k; + } + } + + if ((err = mp_init(&r)) != MP_OKAY) { + goto LBL_KRON_1; + } + + for (;;) { + if (MP_IS_ZERO(&a1)) { + if (mp_cmp_d(&p1, 1uL) == MP_EQ) { + *c = k; + goto LBL_KRON; + } else { + *c = 0; + goto LBL_KRON; + } + } + + v = mp_cnt_lsb(&a1); + if ((err = mp_div_2d(&a1, v, &a1, NULL)) != MP_OKAY) { + goto LBL_KRON; + } + + if ((v & 1) == 1) { + k = k * table[p1.dp[0] & 7u]; + } + + if (a1.sign == MP_NEG) { + /* + * Compute k = (-1)^((a1)*(p1-1)/4) * k + * a1.dp[0] + 1 cannot overflow because the MSB + * of the type mp_digit is not set by definition + */ + if (((a1.dp[0] + 1u) & p1.dp[0] & 2u) != 0u) { + k = -k; + } + } else { + /* compute k = (-1)^((a1-1)*(p1-1)/4) * k */ + if ((a1.dp[0] & p1.dp[0] & 2u) != 0u) { + k = -k; + } + } + + if ((err = mp_copy(&a1, &r)) != MP_OKAY) { + goto LBL_KRON; + } + r.sign = MP_ZPOS; + if ((err = mp_mod(&p1, &r, &a1)) != MP_OKAY) { + goto LBL_KRON; + } + if ((err = mp_copy(&r, &p1)) != MP_OKAY) { + goto LBL_KRON; + } + } + +LBL_KRON: + mp_clear(&r); +LBL_KRON_1: + mp_clear(&p1); +LBL_KRON_0: + mp_clear(&a1); + + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_lcm.c b/lib/hcrypto/libtommath/bn_mp_lcm.c index 781eef565..c32b269e6 100644 --- a/lib/hcrypto/libtommath/bn_mp_lcm.c +++ b/lib/hcrypto/libtommath/bn_mp_lcm.c @@ -1,60 +1,44 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_LCM_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* computes least common multiple as |a*b|/(a, b) */ -int mp_lcm (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_lcm(const mp_int *a, const mp_int *b, mp_int *c) { - int res; - mp_int t1, t2; + mp_err err; + mp_int t1, t2; - if ((res = mp_init_multi (&t1, &t2, NULL)) != MP_OKAY) { - return res; - } + if ((err = mp_init_multi(&t1, &t2, NULL)) != MP_OKAY) { + return err; + } - /* t1 = get the GCD of the two inputs */ - if ((res = mp_gcd (a, b, &t1)) != MP_OKAY) { - goto LBL_T; - } + /* t1 = get the GCD of the two inputs */ + if ((err = mp_gcd(a, b, &t1)) != MP_OKAY) { + goto LBL_T; + } - /* divide the smallest by the GCD */ - if (mp_cmp_mag(a, b) == MP_LT) { - /* store quotient in t2 such that t2 * b is the LCM */ - if ((res = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { - goto LBL_T; - } - res = mp_mul(b, &t2, c); - } else { - /* store quotient in t2 such that t2 * a is the LCM */ - if ((res = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { - goto LBL_T; - } - res = mp_mul(a, &t2, c); - } + /* divide the smallest by the GCD */ + if (mp_cmp_mag(a, b) == MP_LT) { + /* store quotient in t2 such that t2 * b is the LCM */ + if ((err = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + err = mp_mul(b, &t2, c); + } else { + /* store quotient in t2 such that t2 * a is the LCM */ + if ((err = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + err = mp_mul(a, &t2, c); + } - /* fix the sign to positive */ - c->sign = MP_ZPOS; + /* fix the sign to positive */ + c->sign = MP_ZPOS; LBL_T: - mp_clear_multi (&t1, &t2, NULL); - return res; + mp_clear_multi(&t1, &t2, NULL); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_lcm.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_log_u32.c b/lib/hcrypto/libtommath/bn_mp_log_u32.c new file mode 100644 index 000000000..f7bca01de --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_log_u32.c @@ -0,0 +1,180 @@ +#include "tommath_private.h" +#ifdef BN_MP_LOG_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Compute log_{base}(a) */ +static mp_word s_pow(mp_word base, mp_word exponent) +{ + mp_word result = 1uLL; + while (exponent != 0u) { + if ((exponent & 1u) == 1u) { + result *= base; + } + exponent >>= 1; + base *= base; + } + + return result; +} + +static mp_digit s_digit_ilogb(mp_digit base, mp_digit n) +{ + mp_word bracket_low = 1uLL, bracket_mid, bracket_high, N; + mp_digit ret, high = 1uL, low = 0uL, mid; + + if (n < base) { + return 0uL; + } + if (n == base) { + return 1uL; + } + + bracket_high = (mp_word) base ; + N = (mp_word) n; + + while (bracket_high < N) { + low = high; + bracket_low = bracket_high; + high <<= 1; + bracket_high *= bracket_high; + } + + while (((mp_digit)(high - low)) > 1uL) { + mid = (low + high) >> 1; + bracket_mid = bracket_low * s_pow(base, (mp_word)(mid - low)); + + if (N < bracket_mid) { + high = mid ; + bracket_high = bracket_mid ; + } + if (N > bracket_mid) { + low = mid ; + bracket_low = bracket_mid ; + } + if (N == bracket_mid) { + return (mp_digit) mid; + } + } + + if (bracket_high == N) { + ret = high; + } else { + ret = low; + } + + return ret; +} + +/* TODO: output could be "int" because the output of mp_radix_size is int, too, + as is the output of mp_bitcount. + With the same problem: max size is INT_MAX * MP_DIGIT not INT_MAX only! +*/ +mp_err mp_log_u32(const mp_int *a, uint32_t base, uint32_t *c) +{ + mp_err err; + mp_ord cmp; + uint32_t high, low, mid; + mp_int bracket_low, bracket_high, bracket_mid, t, bi_base; + + err = MP_OKAY; + + if (a->sign == MP_NEG) { + return MP_VAL; + } + + if (MP_IS_ZERO(a)) { + return MP_VAL; + } + + if (base < 2u) { + return MP_VAL; + } + + /* A small shortcut for bases that are powers of two. */ + if ((base & (base - 1u)) == 0u) { + int y, bit_count; + for (y=0; (y < 7) && ((base & 1u) == 0u); y++) { + base >>= 1; + } + bit_count = mp_count_bits(a) - 1; + *c = (uint32_t)(bit_count/y); + return MP_OKAY; + } + + if (a->used == 1) { + *c = (uint32_t)s_digit_ilogb(base, a->dp[0]); + return err; + } + + cmp = mp_cmp_d(a, base); + if ((cmp == MP_LT) || (cmp == MP_EQ)) { + *c = cmp == MP_EQ; + return err; + } + + if ((err = + mp_init_multi(&bracket_low, &bracket_high, + &bracket_mid, &t, &bi_base, NULL)) != MP_OKAY) { + return err; + } + + low = 0u; + mp_set(&bracket_low, 1uL); + high = 1u; + + mp_set(&bracket_high, base); + + /* + A kind of Giant-step/baby-step algorithm. + Idea shamelessly stolen from https://programmingpraxis.com/2010/05/07/integer-logarithms/2/ + The effect is asymptotic, hence needs benchmarks to test if the Giant-step should be skipped + for small n. + */ + while (mp_cmp(&bracket_high, a) == MP_LT) { + low = high; + if ((err = mp_copy(&bracket_high, &bracket_low)) != MP_OKAY) { + goto LBL_ERR; + } + high <<= 1; + if ((err = mp_sqr(&bracket_high, &bracket_high)) != MP_OKAY) { + goto LBL_ERR; + } + } + mp_set(&bi_base, base); + + while ((high - low) > 1u) { + mid = (high + low) >> 1; + + if ((err = mp_expt_u32(&bi_base, (uint32_t)(mid - low), &t)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_mul(&bracket_low, &t, &bracket_mid)) != MP_OKAY) { + goto LBL_ERR; + } + cmp = mp_cmp(a, &bracket_mid); + if (cmp == MP_LT) { + high = mid; + mp_exch(&bracket_mid, &bracket_high); + } + if (cmp == MP_GT) { + low = mid; + mp_exch(&bracket_mid, &bracket_low); + } + if (cmp == MP_EQ) { + *c = mid; + goto LBL_END; + } + } + + *c = (mp_cmp(&bracket_high, a) == MP_EQ) ? high : low; + +LBL_END: +LBL_ERR: + mp_clear_multi(&bracket_low, &bracket_high, &bracket_mid, + &t, &bi_base, NULL); + return err; +} + + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_lshd.c b/lib/hcrypto/libtommath/bn_mp_lshd.c index f118cf1ae..82345809c 100644 --- a/lib/hcrypto/libtommath/bn_mp_lshd.c +++ b/lib/hcrypto/libtommath/bn_mp_lshd.c @@ -1,67 +1,51 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_LSHD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* shift left a certain amount of digits */ -int mp_lshd (mp_int * a, int b) +mp_err mp_lshd(mp_int *a, int b) { - int x, res; + int x; + mp_err err; + mp_digit *top, *bottom; - /* if its less than zero return */ - if (b <= 0) { - return MP_OKAY; - } + /* if its less than zero return */ + if (b <= 0) { + return MP_OKAY; + } + /* no need to shift 0 around */ + if (MP_IS_ZERO(a)) { + return MP_OKAY; + } - /* grow to fit the new digits */ - if (a->alloc < a->used + b) { - if ((res = mp_grow (a, a->used + b)) != MP_OKAY) { - return res; - } - } + /* grow to fit the new digits */ + if (a->alloc < (a->used + b)) { + if ((err = mp_grow(a, a->used + b)) != MP_OKAY) { + return err; + } + } - { - register mp_digit *top, *bottom; + /* increment the used by the shift amount then copy upwards */ + a->used += b; - /* increment the used by the shift amount then copy upwards */ - a->used += b; + /* top */ + top = a->dp + a->used - 1; - /* top */ - top = a->dp + a->used - 1; + /* base */ + bottom = (a->dp + a->used - 1) - b; - /* base */ - bottom = a->dp + a->used - 1 - b; - - /* much like mp_rshd this is implemented using a sliding window - * except the window goes the otherway around. Copying from - * the bottom to the top. see bn_mp_rshd.c for more info. - */ - for (x = a->used - 1; x >= b; x--) { + /* much like mp_rshd this is implemented using a sliding window + * except the window goes the otherway around. Copying from + * the bottom to the top. see bn_mp_rshd.c for more info. + */ + for (x = a->used - 1; x >= b; x--) { *top-- = *bottom--; - } + } - /* zero the lower digits */ - top = a->dp; - for (x = 0; x < b; x++) { - *top++ = 0; - } - } - return MP_OKAY; + /* zero the lower digits */ + MP_ZERO_DIGITS(a->dp, b); + + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_lshd.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mod.c b/lib/hcrypto/libtommath/bn_mp_mod.c index 757335aeb..8fbfe08dc 100644 --- a/lib/hcrypto/libtommath/bn_mp_mod.c +++ b/lib/hcrypto/libtommath/bn_mp_mod.c @@ -1,48 +1,31 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* c = a mod b, 0 <= c < b if b > 0, b < c <= 0 if b < 0 */ -int -mp_mod (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_mod(const mp_int *a, const mp_int *b, mp_int *c) { - mp_int t; - int res; + mp_int t; + mp_err err; - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + if ((err = mp_init_size(&t, b->used)) != MP_OKAY) { + return err; + } - if ((res = mp_div (a, b, NULL, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } + if ((err = mp_div(a, b, NULL, &t)) != MP_OKAY) { + goto LBL_ERR; + } - if (mp_iszero(&t) || t.sign == b->sign) { - res = MP_OKAY; - mp_exch (&t, c); - } else { - res = mp_add (b, &t, c); - } + if (MP_IS_ZERO(&t) || (t.sign == b->sign)) { + err = MP_OKAY; + mp_exch(&t, c); + } else { + err = mp_add(b, &t, c); + } - mp_clear (&t); - return res; +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mod_2d.c b/lib/hcrypto/libtommath/bn_mp_mod_2d.c index e194a0687..5bf57a1a3 100644 --- a/lib/hcrypto/libtommath/bn_mp_mod_2d.c +++ b/lib/hcrypto/libtommath/bn_mp_mod_2d.c @@ -1,55 +1,38 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MOD_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* calc a value mod 2**b */ -int -mp_mod_2d (mp_int * a, int b, mp_int * c) +mp_err mp_mod_2d(const mp_int *a, int b, mp_int *c) { - int x, res; + int x; + mp_err err; - /* if b is <= 0 then zero the int */ - if (b <= 0) { - mp_zero (c); - return MP_OKAY; - } + /* if b is <= 0 then zero the int */ + if (b <= 0) { + mp_zero(c); + return MP_OKAY; + } - /* if the modulus is larger than the value than return */ - if (b >= (int) (a->used * DIGIT_BIT)) { - res = mp_copy (a, c); - return res; - } + /* if the modulus is larger than the value than return */ + if (b >= (a->used * MP_DIGIT_BIT)) { + return mp_copy(a, c); + } - /* copy */ - if ((res = mp_copy (a, c)) != MP_OKAY) { - return res; - } + /* copy */ + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } - /* zero digits above the last digit of the modulus */ - for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x++) { - c->dp[x] = 0; - } - /* clear the digit that is not completely outside/inside the modulus */ - c->dp[b / DIGIT_BIT] &= - (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digit) 1)); - mp_clamp (c); - return MP_OKAY; + /* zero digits above the last digit of the modulus */ + x = (b / MP_DIGIT_BIT) + (((b % MP_DIGIT_BIT) == 0) ? 0 : 1); + MP_ZERO_DIGITS(c->dp + x, c->used - x); + + /* clear the digit that is not completely outside/inside the modulus */ + c->dp[b / MP_DIGIT_BIT] &= + ((mp_digit)1 << (mp_digit)(b % MP_DIGIT_BIT)) - (mp_digit)1; + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mod_2d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mod_d.c b/lib/hcrypto/libtommath/bn_mp_mod_d.c index 9ca37e673..0b6c12a9e 100644 --- a/lib/hcrypto/libtommath/bn_mp_mod_d.c +++ b/lib/hcrypto/libtommath/bn_mp_mod_d.c @@ -1,27 +1,10 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MOD_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -int -mp_mod_d (mp_int * a, mp_digit b, mp_digit * c) +mp_err mp_mod_d(const mp_int *a, mp_digit b, mp_digit *c) { - return mp_div_d(a, b, NULL, c); + return mp_div_d(a, b, NULL, c); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mod_d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_montgomery_calc_normalization.c b/lib/hcrypto/libtommath/bn_mp_montgomery_calc_normalization.c index c669fe0ec..837978925 100644 --- a/lib/hcrypto/libtommath/bn_mp_montgomery_calc_normalization.c +++ b/lib/hcrypto/libtommath/bn_mp_montgomery_calc_normalization.c @@ -1,19 +1,7 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* * shifts with subtractions when the result is greater than b. @@ -21,39 +9,36 @@ * The method is slightly modified to shift B unconditionally upto just under * the leading bit of b. This saves alot of multiple precision shifting. */ -int mp_montgomery_calc_normalization (mp_int * a, mp_int * b) +mp_err mp_montgomery_calc_normalization(mp_int *a, const mp_int *b) { - int x, bits, res; + int x, bits; + mp_err err; - /* how many bits of last digit does b use */ - bits = mp_count_bits (b) % DIGIT_BIT; + /* how many bits of last digit does b use */ + bits = mp_count_bits(b) % MP_DIGIT_BIT; - if (b->used > 1) { - if ((res = mp_2expt (a, (b->used - 1) * DIGIT_BIT + bits - 1)) != MP_OKAY) { - return res; - } - } else { - mp_set(a, 1); - bits = 1; - } - - - /* now compute C = A * B mod b */ - for (x = bits - 1; x < (int)DIGIT_BIT; x++) { - if ((res = mp_mul_2 (a, a)) != MP_OKAY) { - return res; - } - if (mp_cmp_mag (a, b) != MP_LT) { - if ((res = s_mp_sub (a, b, a)) != MP_OKAY) { - return res; + if (b->used > 1) { + if ((err = mp_2expt(a, ((b->used - 1) * MP_DIGIT_BIT) + bits - 1)) != MP_OKAY) { + return err; } - } - } + } else { + mp_set(a, 1uL); + bits = 1; + } - return MP_OKAY; + + /* now compute C = A * B mod b */ + for (x = bits - 1; x < (int)MP_DIGIT_BIT; x++) { + if ((err = mp_mul_2(a, a)) != MP_OKAY) { + return err; + } + if (mp_cmp_mag(a, b) != MP_LT) { + if ((err = s_mp_sub(a, b, a)) != MP_OKAY) { + return err; + } + } + } + + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_montgomery_calc_normalization.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_montgomery_reduce.c b/lib/hcrypto/libtommath/bn_mp_montgomery_reduce.c index b76509051..ffe8341ee 100644 --- a/lib/hcrypto/libtommath/bn_mp_montgomery_reduce.c +++ b/lib/hcrypto/libtommath/bn_mp_montgomery_reduce.c @@ -1,118 +1,102 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MONTGOMERY_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* computes xR**-1 == x (mod N) via Montgomery Reduction */ -int -mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho) +mp_err mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) { - int ix, res, digs; - mp_digit mu; + int ix, digs; + mp_err err; + mp_digit mu; - /* can the fast reduction [comba] method be used? - * - * Note that unlike in mul you're safely allowed *less* - * than the available columns [255 per default] since carries - * are fixed up in the inner loop. - */ - digs = n->used * 2 + 1; - if ((digs < MP_WARRAY) && - n->used < - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_mp_montgomery_reduce (x, n, rho); - } + /* can the fast reduction [comba] method be used? + * + * Note that unlike in mul you're safely allowed *less* + * than the available columns [255 per default] since carries + * are fixed up in the inner loop. + */ + digs = (n->used * 2) + 1; + if ((digs < MP_WARRAY) && + (x->used <= MP_WARRAY) && + (n->used < MP_MAXFAST)) { + return s_mp_montgomery_reduce_fast(x, n, rho); + } - /* grow the input as required */ - if (x->alloc < digs) { - if ((res = mp_grow (x, digs)) != MP_OKAY) { - return res; - } - } - x->used = digs; - - for (ix = 0; ix < n->used; ix++) { - /* mu = ai * rho mod b - * - * The value of rho must be precalculated via - * montgomery_setup() such that - * it equals -1/n0 mod b this allows the - * following inner loop to reduce the - * input one digit at a time - */ - mu = (mp_digit) (((mp_word)x->dp[ix]) * ((mp_word)rho) & MP_MASK); - - /* a = a + mu * m * b**i */ - { - register int iy; - register mp_digit *tmpn, *tmpx, u; - register mp_word r; - - /* alias for digits of the modulus */ - tmpn = n->dp; - - /* alias for the digits of x [the input] */ - tmpx = x->dp + ix; - - /* set the carry to zero */ - u = 0; - - /* Multiply and add in place */ - for (iy = 0; iy < n->used; iy++) { - /* compute product and sum */ - r = ((mp_word)mu) * ((mp_word)*tmpn++) + - ((mp_word) u) + ((mp_word) * tmpx); - - /* get carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - - /* fix digit */ - *tmpx++ = (mp_digit)(r & ((mp_word) MP_MASK)); + /* grow the input as required */ + if (x->alloc < digs) { + if ((err = mp_grow(x, digs)) != MP_OKAY) { + return err; } - /* At this point the ix'th digit of x should be zero */ + } + x->used = digs; + + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * rho mod b + * + * The value of rho must be precalculated via + * montgomery_setup() such that + * it equals -1/n0 mod b this allows the + * following inner loop to reduce the + * input one digit at a time + */ + mu = (mp_digit)(((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK); + + /* a = a + mu * m * b**i */ + { + int iy; + mp_digit *tmpn, *tmpx, u; + mp_word r; + + /* alias for digits of the modulus */ + tmpn = n->dp; + + /* alias for the digits of x [the input] */ + tmpx = x->dp + ix; + + /* set the carry to zero */ + u = 0; + + /* Multiply and add in place */ + for (iy = 0; iy < n->used; iy++) { + /* compute product and sum */ + r = ((mp_word)mu * (mp_word)*tmpn++) + + (mp_word)u + (mp_word)*tmpx; + + /* get carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + + /* fix digit */ + *tmpx++ = (mp_digit)(r & (mp_word)MP_MASK); + } + /* At this point the ix'th digit of x should be zero */ - /* propagate carries upwards as required*/ - while (u) { - *tmpx += u; - u = *tmpx >> DIGIT_BIT; - *tmpx++ &= MP_MASK; + /* propagate carries upwards as required*/ + while (u != 0u) { + *tmpx += u; + u = *tmpx >> MP_DIGIT_BIT; + *tmpx++ &= MP_MASK; + } } - } - } + } - /* at this point the n.used'th least - * significant digits of x are all zero - * which means we can shift x to the - * right by n.used digits and the - * residue is unchanged. - */ + /* at this point the n.used'th least + * significant digits of x are all zero + * which means we can shift x to the + * right by n.used digits and the + * residue is unchanged. + */ - /* x = x/b**n.used */ - mp_clamp(x); - mp_rshd (x, n->used); + /* x = x/b**n.used */ + mp_clamp(x); + mp_rshd(x, n->used); - /* if x >= n then x = x - n */ - if (mp_cmp_mag (x, n) != MP_LT) { - return s_mp_sub (x, n, x); - } + /* if x >= n then x = x - n */ + if (mp_cmp_mag(x, n) != MP_LT) { + return s_mp_sub(x, n, x); + } - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_montgomery_reduce.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_montgomery_setup.c b/lib/hcrypto/libtommath/bn_mp_montgomery_setup.c index f08274936..39f6e9d24 100644 --- a/lib/hcrypto/libtommath/bn_mp_montgomery_setup.c +++ b/lib/hcrypto/libtommath/bn_mp_montgomery_setup.c @@ -1,59 +1,42 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MONTGOMERY_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* setups the montgomery reduction stuff */ -int -mp_montgomery_setup (mp_int * n, mp_digit * rho) +mp_err mp_montgomery_setup(const mp_int *n, mp_digit *rho) { - mp_digit x, b; + mp_digit x, b; -/* fast inversion mod 2**k - * - * Based on the fact that - * - * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) - * => 2*X*A - X*X*A*A = 1 - * => 2*(1) - (1) = 1 - */ - b = n->dp[0]; + /* fast inversion mod 2**k + * + * Based on the fact that + * + * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) + * => 2*X*A - X*X*A*A = 1 + * => 2*(1) - (1) = 1 + */ + b = n->dp[0]; - if ((b & 1) == 0) { - return MP_VAL; - } + if ((b & 1u) == 0u) { + return MP_VAL; + } - x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */ - x *= 2 - b * x; /* here x*a==1 mod 2**8 */ + x = (((b + 2u) & 4u) << 1) + b; /* here x*a==1 mod 2**4 */ + x *= 2u - (b * x); /* here x*a==1 mod 2**8 */ #if !defined(MP_8BIT) - x *= 2 - b * x; /* here x*a==1 mod 2**16 */ + x *= 2u - (b * x); /* here x*a==1 mod 2**16 */ #endif #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) - x *= 2 - b * x; /* here x*a==1 mod 2**32 */ + x *= 2u - (b * x); /* here x*a==1 mod 2**32 */ #endif #ifdef MP_64BIT - x *= 2 - b * x; /* here x*a==1 mod 2**64 */ + x *= 2u - (b * x); /* here x*a==1 mod 2**64 */ #endif - /* rho = -1/m mod b */ - *rho = (unsigned long)(((mp_word)1 << ((mp_word) DIGIT_BIT)) - x) & MP_MASK; + /* rho = -1/m mod b */ + *rho = (mp_digit)(((mp_word)1 << (mp_word)MP_DIGIT_BIT) - x) & MP_MASK; - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_montgomery_setup.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mul.c b/lib/hcrypto/libtommath/bn_mp_mul.c index 816e7b2f0..561913a5e 100644 --- a/lib/hcrypto/libtommath/bn_mp_mul.c +++ b/lib/hcrypto/libtommath/bn_mp_mul.c @@ -1,66 +1,52 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* high level multiplication (handles sign) */ -int mp_mul (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_mul(const mp_int *a, const mp_int *b, mp_int *c) { - int res, neg; - neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + mp_err err; + int min_len = MP_MIN(a->used, b->used), + max_len = MP_MAX(a->used, b->used), + digs = a->used + b->used + 1; + mp_sign neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; - /* use Toom-Cook? */ -#ifdef BN_MP_TOOM_MUL_C - if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) { - res = mp_toom_mul(a, b, c); - } else -#endif -#ifdef BN_MP_KARATSUBA_MUL_C - /* use Karatsuba? */ - if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) { - res = mp_karatsuba_mul (a, b, c); - } else -#endif - { - /* can we use the fast multiplier? - * - * The fast multiplier can be used if the output will - * have less than MP_WARRAY digits and the number of - * digits won't affect carry propagation - */ - int digs = a->used + b->used + 1; - -#ifdef BN_FAST_S_MP_MUL_DIGS_C - if ((digs < MP_WARRAY) && - MIN(a->used, b->used) <= - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - res = fast_s_mp_mul_digs (a, b, c, digs); - } else -#endif -#ifdef BN_S_MP_MUL_DIGS_C - res = s_mp_mul (a, b, c); /* uses s_mp_mul_digs */ -#else - res = MP_VAL; -#endif - - } - c->sign = (c->used > 0) ? neg : MP_ZPOS; - return res; + if (MP_HAS(S_MP_BALANCE_MUL) && + /* Check sizes. The smaller one needs to be larger than the Karatsuba cut-off. + * The bigger one needs to be at least about one MP_KARATSUBA_MUL_CUTOFF bigger + * to make some sense, but it depends on architecture, OS, position of the + * stars... so YMMV. + * Using it to cut the input into slices small enough for fast_s_mp_mul_digs + * was actually slower on the author's machine, but YMMV. + */ + (min_len >= MP_KARATSUBA_MUL_CUTOFF) && + ((max_len / 2) >= MP_KARATSUBA_MUL_CUTOFF) && + /* Not much effect was observed below a ratio of 1:2, but again: YMMV. */ + (max_len >= (2 * min_len))) { + err = s_mp_balance_mul(a,b,c); + } else if (MP_HAS(S_MP_TOOM_MUL) && + (min_len >= MP_TOOM_MUL_CUTOFF)) { + err = s_mp_toom_mul(a, b, c); + } else if (MP_HAS(S_MP_KARATSUBA_MUL) && + (min_len >= MP_KARATSUBA_MUL_CUTOFF)) { + err = s_mp_karatsuba_mul(a, b, c); + } else if (MP_HAS(S_MP_MUL_DIGS_FAST) && + /* can we use the fast multiplier? + * + * The fast multiplier can be used if the output will + * have less than MP_WARRAY digits and the number of + * digits won't affect carry propagation + */ + (digs < MP_WARRAY) && + (min_len <= MP_MAXFAST)) { + err = s_mp_mul_digs_fast(a, b, c, digs); + } else if (MP_HAS(S_MP_MUL_DIGS)) { + err = s_mp_mul_digs(a, b, c, digs); + } else { + err = MP_VAL; + } + c->sign = (c->used > 0) ? neg : MP_ZPOS; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mul.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mul_2.c b/lib/hcrypto/libtommath/bn_mp_mul_2.c index f90654832..bc0691a0d 100644 --- a/lib/hcrypto/libtommath/bn_mp_mul_2.c +++ b/lib/hcrypto/libtommath/bn_mp_mul_2.c @@ -1,82 +1,64 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MUL_2_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* b = a*2 */ -int mp_mul_2(mp_int * a, mp_int * b) +mp_err mp_mul_2(const mp_int *a, mp_int *b) { - int x, res, oldused; + int x, oldused; + mp_err err; - /* grow to accomodate result */ - if (b->alloc < a->used + 1) { - if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) { - return res; - } - } + /* grow to accomodate result */ + if (b->alloc < (a->used + 1)) { + if ((err = mp_grow(b, a->used + 1)) != MP_OKAY) { + return err; + } + } - oldused = b->used; - b->used = a->used; + oldused = b->used; + b->used = a->used; - { - register mp_digit r, rr, *tmpa, *tmpb; + { + mp_digit r, rr, *tmpa, *tmpb; - /* alias for source */ - tmpa = a->dp; + /* alias for source */ + tmpa = a->dp; - /* alias for dest */ - tmpb = b->dp; + /* alias for dest */ + tmpb = b->dp; - /* carry */ - r = 0; - for (x = 0; x < a->used; x++) { + /* carry */ + r = 0; + for (x = 0; x < a->used; x++) { - /* get what will be the *next* carry bit from the - * MSB of the current digit + /* get what will be the *next* carry bit from the + * MSB of the current digit + */ + rr = *tmpa >> (mp_digit)(MP_DIGIT_BIT - 1); + + /* now shift up this digit, add in the carry [from the previous] */ + *tmpb++ = ((*tmpa++ << 1uL) | r) & MP_MASK; + + /* copy the carry that would be from the source + * digit into the next iteration + */ + r = rr; + } + + /* new leading digit? */ + if (r != 0u) { + /* add a MSB which is always 1 at this point */ + *tmpb = 1; + ++(b->used); + } + + /* now zero any excess digits on the destination + * that we didn't write to */ - rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1)); - - /* now shift up this digit, add in the carry [from the previous] */ - *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK; - - /* copy the carry that would be from the source - * digit into the next iteration - */ - r = rr; - } - - /* new leading digit? */ - if (r != 0) { - /* add a MSB which is always 1 at this point */ - *tmpb = 1; - ++(b->used); - } - - /* now zero any excess digits on the destination - * that we didn't write to - */ - tmpb = b->dp + b->used; - for (x = b->used; x < oldused; x++) { - *tmpb++ = 0; - } - } - b->sign = a->sign; - return MP_OKAY; + MP_ZERO_DIGITS(b->dp + b->used, oldused - b->used); + } + b->sign = a->sign; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mul_2.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mul_2d.c b/lib/hcrypto/libtommath/bn_mp_mul_2d.c index d023b382c..87354de20 100644 --- a/lib/hcrypto/libtommath/bn_mp_mul_2d.c +++ b/lib/hcrypto/libtommath/bn_mp_mul_2d.c @@ -1,85 +1,69 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MUL_2D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* shift left by a certain bit count */ -int mp_mul_2d (mp_int * a, int b, mp_int * c) +mp_err mp_mul_2d(const mp_int *a, int b, mp_int *c) { - mp_digit d; - int res; + mp_digit d; + mp_err err; - /* copy */ - if (a != c) { - if ((res = mp_copy (a, c)) != MP_OKAY) { - return res; - } - } + /* copy */ + if (a != c) { + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } + } - if (c->alloc < (int)(c->used + b/DIGIT_BIT + 1)) { - if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 1)) != MP_OKAY) { - return res; - } - } + if (c->alloc < (c->used + (b / MP_DIGIT_BIT) + 1)) { + if ((err = mp_grow(c, c->used + (b / MP_DIGIT_BIT) + 1)) != MP_OKAY) { + return err; + } + } - /* shift by as many digits in the bit count */ - if (b >= (int)DIGIT_BIT) { - if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) { - return res; - } - } + /* shift by as many digits in the bit count */ + if (b >= MP_DIGIT_BIT) { + if ((err = mp_lshd(c, b / MP_DIGIT_BIT)) != MP_OKAY) { + return err; + } + } - /* shift any bit count < DIGIT_BIT */ - d = (mp_digit) (b % DIGIT_BIT); - if (d != 0) { - register mp_digit *tmpc, shift, mask, r, rr; - register int x; + /* shift any bit count < MP_DIGIT_BIT */ + d = (mp_digit)(b % MP_DIGIT_BIT); + if (d != 0u) { + mp_digit *tmpc, shift, mask, r, rr; + int x; - /* bitmask for carries */ - mask = (((mp_digit)1) << d) - 1; + /* bitmask for carries */ + mask = ((mp_digit)1 << d) - (mp_digit)1; - /* shift for msbs */ - shift = DIGIT_BIT - d; + /* shift for msbs */ + shift = (mp_digit)MP_DIGIT_BIT - d; - /* alias */ - tmpc = c->dp; + /* alias */ + tmpc = c->dp; - /* carry */ - r = 0; - for (x = 0; x < c->used; x++) { - /* get the higher bits of the current word */ - rr = (*tmpc >> shift) & mask; + /* carry */ + r = 0; + for (x = 0; x < c->used; x++) { + /* get the higher bits of the current word */ + rr = (*tmpc >> shift) & mask; - /* shift the current word and OR in the carry */ - *tmpc = ((*tmpc << d) | r) & MP_MASK; - ++tmpc; + /* shift the current word and OR in the carry */ + *tmpc = ((*tmpc << d) | r) & MP_MASK; + ++tmpc; - /* set the carry to the carry bits of the current word */ - r = rr; - } + /* set the carry to the carry bits of the current word */ + r = rr; + } - /* set final carry */ - if (r != 0) { - c->dp[(c->used)++] = r; - } - } - mp_clamp (c); - return MP_OKAY; + /* set final carry */ + if (r != 0u) { + c->dp[(c->used)++] = r; + } + } + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mul_2d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mul_d.c b/lib/hcrypto/libtommath/bn_mp_mul_d.c index 00f9a899e..b56dfa3c9 100644 --- a/lib/hcrypto/libtommath/bn_mp_mul_d.c +++ b/lib/hcrypto/libtommath/bn_mp_mul_d.c @@ -1,79 +1,61 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MUL_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* multiply by a digit */ -int -mp_mul_d (mp_int * a, mp_digit b, mp_int * c) +mp_err mp_mul_d(const mp_int *a, mp_digit b, mp_int *c) { - mp_digit u, *tmpa, *tmpc; - mp_word r; - int ix, res, olduse; + mp_digit u, *tmpa, *tmpc; + mp_word r; + mp_err err; + int ix, olduse; - /* make sure c is big enough to hold a*b */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow (c, a->used + 1)) != MP_OKAY) { - return res; - } - } + /* make sure c is big enough to hold a*b */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } - /* get the original destinations used count */ - olduse = c->used; + /* get the original destinations used count */ + olduse = c->used; - /* set the sign */ - c->sign = a->sign; + /* set the sign */ + c->sign = a->sign; - /* alias for a->dp [source] */ - tmpa = a->dp; + /* alias for a->dp [source] */ + tmpa = a->dp; - /* alias for c->dp [dest] */ - tmpc = c->dp; + /* alias for c->dp [dest] */ + tmpc = c->dp; - /* zero carry */ - u = 0; + /* zero carry */ + u = 0; - /* compute columns */ - for (ix = 0; ix < a->used; ix++) { - /* compute product and carry sum for this term */ - r = ((mp_word) u) + ((mp_word)*tmpa++) * ((mp_word)b); + /* compute columns */ + for (ix = 0; ix < a->used; ix++) { + /* compute product and carry sum for this term */ + r = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b); - /* mask off higher bits to get a single digit */ - *tmpc++ = (mp_digit) (r & ((mp_word) MP_MASK)); + /* mask off higher bits to get a single digit */ + *tmpc++ = (mp_digit)(r & (mp_word)MP_MASK); - /* send carry into next iteration */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } + /* send carry into next iteration */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } - /* store final carry [if any] and increment ix offset */ - *tmpc++ = u; - ++ix; + /* store final carry [if any] and increment ix offset */ + *tmpc++ = u; + ++ix; - /* now zero digits above the top */ - while (ix++ < olduse) { - *tmpc++ = 0; - } + /* now zero digits above the top */ + MP_ZERO_DIGITS(tmpc, olduse - ix); - /* set used count */ - c->used = a->used + 1; - mp_clamp(c); + /* set used count */ + c->used = a->used + 1; + mp_clamp(c); - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mul_d.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_mulmod.c b/lib/hcrypto/libtommath/bn_mp_mulmod.c index 003ceb9b9..160d1626a 100644 --- a/lib/hcrypto/libtommath/bn_mp_mulmod.c +++ b/lib/hcrypto/libtommath/bn_mp_mulmod.c @@ -1,40 +1,25 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_MULMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* d = a * b (mod c) */ -int mp_mulmod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +mp_err mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) { - int res; - mp_int t; + mp_err err; + mp_int t; - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + if ((err = mp_init_size(&t, c->used)) != MP_OKAY) { + return err; + } - if ((res = mp_mul (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; + if ((err = mp_mul(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); + +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_mulmod.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_n_root.c b/lib/hcrypto/libtommath/bn_mp_n_root.c deleted file mode 100644 index 85d335cb9..000000000 --- a/lib/hcrypto/libtommath/bn_mp_n_root.c +++ /dev/null @@ -1,132 +0,0 @@ -#include -#ifdef BN_MP_N_ROOT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* find the n'th root of an integer - * - * Result found such that (c)**b <= a and (c+1)**b > a - * - * This algorithm uses Newton's approximation - * x[i+1] = x[i] - f(x[i])/f'(x[i]) - * which will find the root in log(N) time where - * each step involves a fair bit. This is not meant to - * find huge roots [square and cube, etc]. - */ -int mp_n_root (mp_int * a, mp_digit b, mp_int * c) -{ - mp_int t1, t2, t3; - int res, neg; - - /* input must be positive if b is even */ - if ((b & 1) == 0 && a->sign == MP_NEG) { - return MP_VAL; - } - - if ((res = mp_init (&t1)) != MP_OKAY) { - return res; - } - - if ((res = mp_init (&t2)) != MP_OKAY) { - goto LBL_T1; - } - - if ((res = mp_init (&t3)) != MP_OKAY) { - goto LBL_T2; - } - - /* if a is negative fudge the sign but keep track */ - neg = a->sign; - a->sign = MP_ZPOS; - - /* t2 = 2 */ - mp_set (&t2, 2); - - do { - /* t1 = t2 */ - if ((res = mp_copy (&t2, &t1)) != MP_OKAY) { - goto LBL_T3; - } - - /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ - - /* t3 = t1**(b-1) */ - if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) { - goto LBL_T3; - } - - /* numerator */ - /* t2 = t1**b */ - if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - /* t2 = t1**b - a */ - if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - /* denominator */ - /* t3 = t1**(b-1) * b */ - if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { - goto LBL_T3; - } - - /* t3 = (t1**b - a)/(b * t1**(b-1)) */ - if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { - goto LBL_T3; - } - - if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) { - goto LBL_T3; - } - } while (mp_cmp (&t1, &t2) != MP_EQ); - - /* result can be off by a few so check */ - for (;;) { - if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) { - goto LBL_T3; - } - - if (mp_cmp (&t2, a) == MP_GT) { - if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) { - goto LBL_T3; - } - } else { - break; - } - } - - /* reset the sign of a first */ - a->sign = neg; - - /* set the result */ - mp_exch (&t1, c); - - /* set the sign of the result */ - c->sign = neg; - - res = MP_OKAY; - -LBL_T3:mp_clear (&t3); -LBL_T2:mp_clear (&t2); -LBL_T1:mp_clear (&t1); - return res; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_n_root.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_neg.c b/lib/hcrypto/libtommath/bn_mp_neg.c index a7d035ab6..264d90097 100644 --- a/lib/hcrypto/libtommath/bn_mp_neg.c +++ b/lib/hcrypto/libtommath/bn_mp_neg.c @@ -1,40 +1,24 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_NEG_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* b = -a */ -int mp_neg (mp_int * a, mp_int * b) +mp_err mp_neg(const mp_int *a, mp_int *b) { - int res; - if (a != b) { - if ((res = mp_copy (a, b)) != MP_OKAY) { - return res; - } - } + mp_err err; + if (a != b) { + if ((err = mp_copy(a, b)) != MP_OKAY) { + return err; + } + } - if (mp_iszero(b) != MP_YES) { - b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; - } else { - b->sign = MP_ZPOS; - } + if (!MP_IS_ZERO(b)) { + b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; + } else { + b->sign = MP_ZPOS; + } - return MP_OKAY; + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_neg.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_or.c b/lib/hcrypto/libtommath/bn_mp_or.c index bff499548..cdacbfbe5 100644 --- a/lib/hcrypto/libtommath/bn_mp_or.c +++ b/lib/hcrypto/libtommath/bn_mp_or.c @@ -1,50 +1,56 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_OR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* OR two ints together */ -int mp_or (mp_int * a, mp_int * b, mp_int * c) +/* two complement or */ +mp_err mp_or(const mp_int *a, const mp_int *b, mp_int *c) { - int res, ix, px; - mp_int t, *x; + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = ((a->sign == MP_NEG) || (b->sign == MP_NEG)) ? MP_NEG : MP_ZPOS; - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } - for (ix = 0; ix < px; ix++) { - t.dp[ix] |= x->dp[ix]; - } - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; + for (i = 0; i < used; i++) { + mp_digit x, y; + + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } + + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x | y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_or.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_pack.c b/lib/hcrypto/libtommath/bn_mp_pack.c new file mode 100644 index 000000000..6e00b6fc6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_pack.c @@ -0,0 +1,69 @@ +#include "tommath_private.h" +#ifdef BN_MP_PACK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* based on gmp's mpz_export. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +mp_err mp_pack(void *rop, size_t maxcount, size_t *written, mp_order order, size_t size, + mp_endian endian, size_t nails, const mp_int *op) +{ + mp_err err; + size_t odd_nails, nail_bytes, i, j, count; + unsigned char odd_nail_mask; + + mp_int t; + + count = mp_pack_count(op, nails, size); + + if (count > maxcount) { + return MP_BUF; + } + + if ((err = mp_init_copy(&t, op)) != MP_OKAY) { + return err; + } + + if (endian == MP_NATIVE_ENDIAN) { + MP_GET_ENDIANNESS(endian); + } + + odd_nails = (nails % 8u); + odd_nail_mask = 0xff; + for (i = 0u; i < odd_nails; ++i) { + odd_nail_mask ^= (unsigned char)(1u << (7u - i)); + } + nail_bytes = nails / 8u; + + for (i = 0u; i < count; ++i) { + for (j = 0u; j < size; ++j) { + unsigned char *byte = (unsigned char *)rop + + (((order == MP_LSB_FIRST) ? i : ((count - 1u) - i)) * size) + + ((endian == MP_LITTLE_ENDIAN) ? j : ((size - 1u) - j)); + + if (j >= (size - nail_bytes)) { + *byte = 0; + continue; + } + + *byte = (unsigned char)((j == ((size - nail_bytes) - 1u)) ? (t.dp[0] & odd_nail_mask) : (t.dp[0] & 0xFFuL)); + + if ((err = mp_div_2d(&t, (j == ((size - nail_bytes) - 1u)) ? (int)(8u - odd_nails) : 8, &t, NULL)) != MP_OKAY) { + goto LBL_ERR; + } + + } + } + + if (written != NULL) { + *written = count; + } + err = MP_OKAY; + +LBL_ERR: + mp_clear(&t); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_pack_count.c b/lib/hcrypto/libtommath/bn_mp_pack_count.c new file mode 100644 index 000000000..dfecdf98f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_pack_count.c @@ -0,0 +1,12 @@ +#include "tommath_private.h" +#ifdef BN_MP_PACK_COUNT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +size_t mp_pack_count(const mp_int *a, size_t nails, size_t size) +{ + size_t bits = (size_t)mp_count_bits(a); + return ((bits / ((size * 8u) - nails)) + (((bits % ((size * 8u) - nails)) != 0u) ? 1u : 0u)); +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_prime_fermat.c b/lib/hcrypto/libtommath/bn_mp_prime_fermat.c index 8e74a337c..af3e884bb 100644 --- a/lib/hcrypto/libtommath/bn_mp_prime_fermat.c +++ b/lib/hcrypto/libtommath/bn_mp_prime_fermat.c @@ -1,19 +1,7 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_PRIME_FERMAT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* performs one Fermat test. * @@ -23,40 +11,37 @@ * * Sets result to 1 if the congruence holds, or zero otherwise. */ -int mp_prime_fermat (mp_int * a, mp_int * b, int *result) +mp_err mp_prime_fermat(const mp_int *a, const mp_int *b, mp_bool *result) { - mp_int t; - int err; + mp_int t; + mp_err err; - /* default to composite */ - *result = MP_NO; + /* default to composite */ + *result = MP_NO; - /* ensure b > 1 */ - if (mp_cmp_d(b, 1) != MP_GT) { - return MP_VAL; - } + /* ensure b > 1 */ + if (mp_cmp_d(b, 1uL) != MP_GT) { + return MP_VAL; + } - /* init t */ - if ((err = mp_init (&t)) != MP_OKAY) { - return err; - } + /* init t */ + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } - /* compute t = b**a mod a */ - if ((err = mp_exptmod (b, a, a, &t)) != MP_OKAY) { - goto LBL_T; - } + /* compute t = b**a mod a */ + if ((err = mp_exptmod(b, a, a, &t)) != MP_OKAY) { + goto LBL_T; + } - /* is it equal to b? */ - if (mp_cmp (&t, b) == MP_EQ) { - *result = MP_YES; - } + /* is it equal to b? */ + if (mp_cmp(&t, b) == MP_EQ) { + *result = MP_YES; + } - err = MP_OKAY; -LBL_T:mp_clear (&t); - return err; + err = MP_OKAY; +LBL_T: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_fermat.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_frobenius_underwood.c b/lib/hcrypto/libtommath/bn_mp_prime_frobenius_underwood.c new file mode 100644 index 000000000..253e8d53b --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_prime_frobenius_underwood.c @@ -0,0 +1,132 @@ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_FROBENIUS_UNDERWOOD_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + * See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details + */ +#ifndef LTM_USE_ONLY_MR + +#ifdef MP_8BIT +/* + * floor of positive solution of + * (2^16)-1 = (a+4)*(2*a+5) + * TODO: Both values are smaller than N^(1/4), would have to use a bigint + * for a instead but any a biger than about 120 are already so rare that + * it is possible to ignore them and still get enough pseudoprimes. + * But it is still a restriction of the set of available pseudoprimes + * which makes this implementation less secure if used stand-alone. + */ +#define LTM_FROBENIUS_UNDERWOOD_A 177 +#else +#define LTM_FROBENIUS_UNDERWOOD_A 32764 +#endif +mp_err mp_prime_frobenius_underwood(const mp_int *N, mp_bool *result) +{ + mp_int T1z, T2z, Np1z, sz, tz; + + int a, ap2, length, i, j; + mp_err err; + + *result = MP_NO; + + if ((err = mp_init_multi(&T1z, &T2z, &Np1z, &sz, &tz, NULL)) != MP_OKAY) { + return err; + } + + for (a = 0; a < LTM_FROBENIUS_UNDERWOOD_A; a++) { + /* TODO: That's ugly! No, really, it is! */ + if ((a==2) || (a==4) || (a==7) || (a==8) || (a==10) || + (a==14) || (a==18) || (a==23) || (a==26) || (a==28)) { + continue; + } + /* (32764^2 - 4) < 2^31, no bigint for >MP_8BIT needed) */ + mp_set_u32(&T1z, (uint32_t)a); + + if ((err = mp_sqr(&T1z, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if ((err = mp_sub_d(&T1z, 4uL, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if ((err = mp_kronecker(&T1z, N, &j)) != MP_OKAY) goto LBL_FU_ERR; + + if (j == -1) { + break; + } + + if (j == 0) { + /* composite */ + goto LBL_FU_ERR; + } + } + /* Tell it a composite and set return value accordingly */ + if (a >= LTM_FROBENIUS_UNDERWOOD_A) { + err = MP_ITER; + goto LBL_FU_ERR; + } + /* Composite if N and (a+4)*(2*a+5) are not coprime */ + mp_set_u32(&T1z, (uint32_t)((a+4)*((2*a)+5))); + + if ((err = mp_gcd(N, &T1z, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if (!((T1z.used == 1) && (T1z.dp[0] == 1u))) goto LBL_FU_ERR; + + ap2 = a + 2; + if ((err = mp_add_d(N, 1uL, &Np1z)) != MP_OKAY) goto LBL_FU_ERR; + + mp_set(&sz, 1uL); + mp_set(&tz, 2uL); + length = mp_count_bits(&Np1z); + + for (i = length - 2; i >= 0; i--) { + /* + * temp = (sz*(a*sz+2*tz))%N; + * tz = ((tz-sz)*(tz+sz))%N; + * sz = temp; + */ + if ((err = mp_mul_2(&tz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + + /* a = 0 at about 50% of the cases (non-square and odd input) */ + if (a != 0) { + if ((err = mp_mul_d(&sz, (mp_digit)a, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_add(&T1z, &T2z, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + } + + if ((err = mp_mul(&T2z, &sz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_sub(&tz, &sz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_add(&sz, &tz, &sz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mul(&sz, &T2z, &tz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mod(&tz, N, &tz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mod(&T1z, N, &sz)) != MP_OKAY) goto LBL_FU_ERR; + if (s_mp_get_bit(&Np1z, (unsigned int)i) == MP_YES) { + /* + * temp = (a+2) * sz + tz + * tz = 2 * tz - sz + * sz = temp + */ + if (a == 0) { + if ((err = mp_mul_2(&sz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + } else { + if ((err = mp_mul_d(&sz, (mp_digit)ap2, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + } + if ((err = mp_add(&T1z, &tz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mul_2(&tz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_sub(&T2z, &sz, &tz)) != MP_OKAY) goto LBL_FU_ERR; + mp_exch(&sz, &T1z); + } + } + + mp_set_u32(&T1z, (uint32_t)((2 * a) + 5)); + if ((err = mp_mod(&T1z, N, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if (MP_IS_ZERO(&sz) && (mp_cmp(&tz, &T1z) == MP_EQ)) { + *result = MP_YES; + } + +LBL_FU_ERR: + mp_clear_multi(&tz, &sz, &Np1z, &T2z, &T1z, NULL); + return err; +} + +#endif +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_prime_is_divisible.c b/lib/hcrypto/libtommath/bn_mp_prime_is_divisible.c deleted file mode 100644 index 766cde95a..000000000 --- a/lib/hcrypto/libtommath/bn_mp_prime_is_divisible.c +++ /dev/null @@ -1,50 +0,0 @@ -#include -#ifdef BN_MP_PRIME_IS_DIVISIBLE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* determines if an integers is divisible by one - * of the first PRIME_SIZE primes or not - * - * sets result to 0 if not, 1 if yes - */ -int mp_prime_is_divisible (mp_int * a, int *result) -{ - int err, ix; - mp_digit res; - - /* default to not */ - *result = MP_NO; - - for (ix = 0; ix < PRIME_SIZE; ix++) { - /* what is a mod LBL_prime_tab[ix] */ - if ((err = mp_mod_d (a, ltm_prime_tab[ix], &res)) != MP_OKAY) { - return err; - } - - /* is the residue zero? */ - if (res == 0) { - *result = MP_YES; - return MP_OKAY; - } - } - - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_is_divisible.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_is_prime.c b/lib/hcrypto/libtommath/bn_mp_prime_is_prime.c index c316d6210..7f9fc0b45 100644 --- a/lib/hcrypto/libtommath/bn_mp_prime_is_prime.c +++ b/lib/hcrypto/libtommath/bn_mp_prime_is_prime.c @@ -1,83 +1,314 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_PRIME_IS_PRIME_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* performs a variable number of rounds of Miller-Rabin - * - * Probability of error after t rounds is no more than - - * - * Sets result to 1 if probably prime, 0 otherwise - */ -int mp_prime_is_prime (mp_int * a, int t, int *result) +/* portable integer log of two with small footprint */ +static unsigned int s_floor_ilog2(int value) { - mp_int b; - int ix, err, res; + unsigned int r = 0; + while ((value >>= 1) != 0) { + r++; + } + return r; +} - /* default to no */ - *result = MP_NO; - /* valid value of t? */ - if (t <= 0 || t > PRIME_SIZE) { - return MP_VAL; - } +mp_err mp_prime_is_prime(const mp_int *a, int t, mp_bool *result) +{ + mp_int b; + int ix, p_max = 0, size_a, len; + mp_bool res; + mp_err err; + unsigned int fips_rand, mask; - /* is the input equal to one of the primes in the table? */ - for (ix = 0; ix < PRIME_SIZE; ix++) { - if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) { - *result = 1; + /* default to no */ + *result = MP_NO; + + /* Some shortcuts */ + /* N > 3 */ + if (a->used == 1) { + if ((a->dp[0] == 0u) || (a->dp[0] == 1u)) { + *result = MP_NO; return MP_OKAY; } - } + if (a->dp[0] == 2u) { + *result = MP_YES; + return MP_OKAY; + } + } - /* first perform trial division */ - if ((err = mp_prime_is_divisible (a, &res)) != MP_OKAY) { - return err; - } + /* N must be odd */ + if (MP_IS_EVEN(a)) { + return MP_OKAY; + } + /* N is not a perfect square: floor(sqrt(N))^2 != N */ + if ((err = mp_is_square(a, &res)) != MP_OKAY) { + return err; + } + if (res != MP_NO) { + return MP_OKAY; + } - /* return if it was trivially divisible */ - if (res == MP_YES) { - return MP_OKAY; - } - - /* now perform the miller-rabin rounds */ - if ((err = mp_init (&b)) != MP_OKAY) { - return err; - } - - for (ix = 0; ix < t; ix++) { - /* set the prime */ - mp_set (&b, ltm_prime_tab[ix]); - - if ((err = mp_prime_miller_rabin (a, &b, &res)) != MP_OKAY) { - goto LBL_B; - } - - if (res == MP_NO) { - goto LBL_B; - } - } - - /* passed the test */ - *result = MP_YES; -LBL_B:mp_clear (&b); - return err; -} + /* is the input equal to one of the primes in the table? */ + for (ix = 0; ix < PRIVATE_MP_PRIME_TAB_SIZE; ix++) { + if (mp_cmp_d(a, s_mp_prime_tab[ix]) == MP_EQ) { + *result = MP_YES; + return MP_OKAY; + } + } +#ifdef MP_8BIT + /* The search in the loop above was exhaustive in this case */ + if ((a->used == 1) && (PRIVATE_MP_PRIME_TAB_SIZE >= 31)) { + return MP_OKAY; + } #endif -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_is_prime.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ + /* first perform trial division */ + if ((err = s_mp_prime_is_divisible(a, &res)) != MP_OKAY) { + return err; + } + + /* return if it was trivially divisible */ + if (res == MP_YES) { + return MP_OKAY; + } + + /* + Run the Miller-Rabin test with base 2 for the BPSW test. + */ + if ((err = mp_init_set(&b, 2uL)) != MP_OKAY) { + return err; + } + + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + /* + Rumours have it that Mathematica does a second M-R test with base 3. + Other rumours have it that their strong L-S test is slightly different. + It does not hurt, though, beside a bit of extra runtime. + */ + b.dp[0]++; + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + + /* + * Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite + * slow so if speed is an issue, define LTM_USE_ONLY_MR to use M-R tests with + * bases 2, 3 and t random bases. + */ +#ifndef LTM_USE_ONLY_MR + if (t >= 0) { + /* + * Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for + * MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit + * integers but the necesssary analysis is on the todo-list). + */ +#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST) + err = mp_prime_frobenius_underwood(a, &res); + if ((err != MP_OKAY) && (err != MP_ITER)) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } +#else + if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } +#endif + } +#endif + + /* run at least one Miller-Rabin test with a random base */ + if (t == 0) { + t = 1; + } + + /* + Only recommended if the input range is known to be < 3317044064679887385961981 + + It uses the bases necessary for a deterministic M-R test if the input is + smaller than 3317044064679887385961981 + The caller has to check the size. + TODO: can be made a bit finer grained but comparing is not free. + */ + if (t < 0) { + /* + Sorenson, Jonathan; Webster, Jonathan (2015). + "Strong Pseudoprimes to Twelve Prime Bases". + */ + /* 0x437ae92817f9fc85b7e5 = 318665857834031151167461 */ + if ((err = mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) { + goto LBL_B; + } + + if (mp_cmp(a, &b) == MP_LT) { + p_max = 12; + } else { + /* 0x2be6951adc5b22410a5fd = 3317044064679887385961981 */ + if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) { + goto LBL_B; + } + + if (mp_cmp(a, &b) == MP_LT) { + p_max = 13; + } else { + err = MP_VAL; + goto LBL_B; + } + } + + /* we did bases 2 and 3 already, skip them */ + for (ix = 2; ix < p_max; ix++) { + mp_set(&b, s_mp_prime_tab[ix]); + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + } + } + /* + Do "t" M-R tests with random bases between 3 and "a". + See Fips 186.4 p. 126ff + */ + else if (t > 0) { + /* + * The mp_digit's have a defined bit-size but the size of the + * array a.dp is a simple 'int' and this library can not assume full + * compliance to the current C-standard (ISO/IEC 9899:2011) because + * it gets used for small embeded processors, too. Some of those MCUs + * have compilers that one cannot call standard compliant by any means. + * Hence the ugly type-fiddling in the following code. + */ + size_a = mp_count_bits(a); + mask = (1u << s_floor_ilog2(size_a)) - 1u; + /* + Assuming the General Rieman hypothesis (never thought to write that in a + comment) the upper bound can be lowered to 2*(log a)^2. + E. Bach, "Explicit bounds for primality testing and related problems," + Math. Comp. 55 (1990), 355-380. + + size_a = (size_a/10) * 7; + len = 2 * (size_a * size_a); + + E.g.: a number of size 2^2048 would be reduced to the upper limit + + floor(2048/10)*7 = 1428 + 2 * 1428^2 = 4078368 + + (would have been ~4030331.9962 with floats and natural log instead) + That number is smaller than 2^28, the default bit-size of mp_digit. + */ + + /* + How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame + does exactly 1. In words: one. Look at the end of _GMP_is_prime() in + Math-Prime-Util-GMP-0.50/primality.c if you do not believe it. + + The function mp_rand() goes to some length to use a cryptographically + good PRNG. That also means that the chance to always get the same base + in the loop is non-zero, although very low. + If the BPSW test and/or the addtional Frobenious test have been + performed instead of just the Miller-Rabin test with the bases 2 and 3, + a single extra test should suffice, so such a very unlikely event + will not do much harm. + + To preemptivly answer the dangling question: no, a witness does not + need to be prime. + */ + for (ix = 0; ix < t; ix++) { + /* mp_rand() guarantees the first digit to be non-zero */ + if ((err = mp_rand(&b, 1)) != MP_OKAY) { + goto LBL_B; + } + /* + * Reduce digit before casting because mp_digit might be bigger than + * an unsigned int and "mask" on the other side is most probably not. + */ + fips_rand = (unsigned int)(b.dp[0] & (mp_digit) mask); +#ifdef MP_8BIT + /* + * One 8-bit digit is too small, so concatenate two if the size of + * unsigned int allows for it. + */ + if ((MP_SIZEOF_BITS(unsigned int)/2) >= MP_SIZEOF_BITS(mp_digit)) { + if ((err = mp_rand(&b, 1)) != MP_OKAY) { + goto LBL_B; + } + fips_rand <<= MP_SIZEOF_BITS(mp_digit); + fips_rand |= (unsigned int) b.dp[0]; + fips_rand &= mask; + } +#endif + if (fips_rand > (unsigned int)(INT_MAX - MP_DIGIT_BIT)) { + len = INT_MAX / MP_DIGIT_BIT; + } else { + len = (((int)fips_rand + MP_DIGIT_BIT) / MP_DIGIT_BIT); + } + /* Unlikely. */ + if (len < 0) { + ix--; + continue; + } + /* + * As mentioned above, one 8-bit digit is too small and + * although it can only happen in the unlikely case that + * an "unsigned int" is smaller than 16 bit a simple test + * is cheap and the correction even cheaper. + */ +#ifdef MP_8BIT + /* All "a" < 2^8 have been caught before */ + if (len == 1) { + len++; + } +#endif + if ((err = mp_rand(&b, len)) != MP_OKAY) { + goto LBL_B; + } + /* + * That number might got too big and the witness has to be + * smaller than "a" + */ + len = mp_count_bits(&b); + if (len >= size_a) { + len = (len - size_a) + 1; + if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) { + goto LBL_B; + } + } + /* Although the chance for b <= 3 is miniscule, try again. */ + if (mp_cmp_d(&b, 3uL) != MP_GT) { + ix--; + continue; + } + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + } + } + + /* passed the test */ + *result = MP_YES; +LBL_B: + mp_clear(&b); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_prime_miller_rabin.c b/lib/hcrypto/libtommath/bn_mp_prime_miller_rabin.c index 60a8c48ea..96470dba7 100644 --- a/lib/hcrypto/libtommath/bn_mp_prime_miller_rabin.c +++ b/lib/hcrypto/libtommath/bn_mp_prime_miller_rabin.c @@ -1,19 +1,7 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_PRIME_MILLER_RABIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* Miller-Rabin test of "a" to the base of "b" as described in * HAC pp. 139 Algorithm 4.24 @@ -22,82 +10,82 @@ * Randomly the chance of error is no more than 1/4 and often * very much lower. */ -int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) +mp_err mp_prime_miller_rabin(const mp_int *a, const mp_int *b, mp_bool *result) { - mp_int n1, y, r; - int s, j, err; + mp_int n1, y, r; + mp_err err; + int s, j; - /* default */ - *result = MP_NO; + /* default */ + *result = MP_NO; - /* ensure b > 1 */ - if (mp_cmp_d(b, 1) != MP_GT) { - return MP_VAL; - } + /* ensure b > 1 */ + if (mp_cmp_d(b, 1uL) != MP_GT) { + return MP_VAL; + } - /* get n1 = a - 1 */ - if ((err = mp_init_copy (&n1, a)) != MP_OKAY) { - return err; - } - if ((err = mp_sub_d (&n1, 1, &n1)) != MP_OKAY) { - goto LBL_N1; - } + /* get n1 = a - 1 */ + if ((err = mp_init_copy(&n1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_sub_d(&n1, 1uL, &n1)) != MP_OKAY) { + goto LBL_N1; + } - /* set 2**s * r = n1 */ - if ((err = mp_init_copy (&r, &n1)) != MP_OKAY) { - goto LBL_N1; - } + /* set 2**s * r = n1 */ + if ((err = mp_init_copy(&r, &n1)) != MP_OKAY) { + goto LBL_N1; + } - /* count the number of least significant bits - * which are zero - */ - s = mp_cnt_lsb(&r); + /* count the number of least significant bits + * which are zero + */ + s = mp_cnt_lsb(&r); - /* now divide n - 1 by 2**s */ - if ((err = mp_div_2d (&r, s, &r, NULL)) != MP_OKAY) { - goto LBL_R; - } + /* now divide n - 1 by 2**s */ + if ((err = mp_div_2d(&r, s, &r, NULL)) != MP_OKAY) { + goto LBL_R; + } - /* compute y = b**r mod a */ - if ((err = mp_init (&y)) != MP_OKAY) { - goto LBL_R; - } - if ((err = mp_exptmod (b, &r, a, &y)) != MP_OKAY) { - goto LBL_Y; - } - - /* if y != 1 and y != n1 do */ - if (mp_cmp_d (&y, 1) != MP_EQ && mp_cmp (&y, &n1) != MP_EQ) { - j = 1; - /* while j <= s-1 and y != n1 */ - while ((j <= (s - 1)) && mp_cmp (&y, &n1) != MP_EQ) { - if ((err = mp_sqrmod (&y, a, &y)) != MP_OKAY) { - goto LBL_Y; - } - - /* if y == 1 then composite */ - if (mp_cmp_d (&y, 1) == MP_EQ) { - goto LBL_Y; - } - - ++j; - } - - /* if y != n1 then composite */ - if (mp_cmp (&y, &n1) != MP_EQ) { + /* compute y = b**r mod a */ + if ((err = mp_init(&y)) != MP_OKAY) { + goto LBL_R; + } + if ((err = mp_exptmod(b, &r, a, &y)) != MP_OKAY) { goto LBL_Y; - } - } + } - /* probably prime now */ - *result = MP_YES; -LBL_Y:mp_clear (&y); -LBL_R:mp_clear (&r); -LBL_N1:mp_clear (&n1); - return err; + /* if y != 1 and y != n1 do */ + if ((mp_cmp_d(&y, 1uL) != MP_EQ) && (mp_cmp(&y, &n1) != MP_EQ)) { + j = 1; + /* while j <= s-1 and y != n1 */ + while ((j <= (s - 1)) && (mp_cmp(&y, &n1) != MP_EQ)) { + if ((err = mp_sqrmod(&y, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y == 1 then composite */ + if (mp_cmp_d(&y, 1uL) == MP_EQ) { + goto LBL_Y; + } + + ++j; + } + + /* if y != n1 then composite */ + if (mp_cmp(&y, &n1) != MP_EQ) { + goto LBL_Y; + } + } + + /* probably prime now */ + *result = MP_YES; +LBL_Y: + mp_clear(&y); +LBL_R: + mp_clear(&r); +LBL_N1: + mp_clear(&n1); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_miller_rabin.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_next_prime.c b/lib/hcrypto/libtommath/bn_mp_prime_next_prime.c index 8cd3ec2a5..d65656578 100644 --- a/lib/hcrypto/libtommath/bn_mp_prime_next_prime.c +++ b/lib/hcrypto/libtommath/bn_mp_prime_next_prime.c @@ -1,70 +1,42 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_PRIME_NEXT_PRIME_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* finds the next prime after the number "a" using "t" trials * of Miller-Rabin. * * bbs_style = 1 means the prime must be congruent to 3 mod 4 */ -int mp_prime_next_prime(mp_int *a, int t, int bbs_style) +mp_err mp_prime_next_prime(mp_int *a, int t, int bbs_style) { - int err, res = MP_NO, x, y; - mp_digit res_tab[PRIME_SIZE], step, kstep; + int x, y; + mp_ord cmp; + mp_err err; + mp_bool res = MP_NO; + mp_digit res_tab[PRIVATE_MP_PRIME_TAB_SIZE], step, kstep; mp_int b; - /* ensure t is valid */ - if (t <= 0 || t > PRIME_SIZE) { - return MP_VAL; - } - /* force positive */ a->sign = MP_ZPOS; /* simple algo if a is less than the largest prime in the table */ - if (mp_cmp_d(a, ltm_prime_tab[PRIME_SIZE-1]) == MP_LT) { - /* find which prime it is bigger than */ - for (x = PRIME_SIZE - 2; x >= 0; x--) { - if (mp_cmp_d(a, ltm_prime_tab[x]) != MP_LT) { - if (bbs_style == 1) { - /* ok we found a prime smaller or - * equal [so the next is larger] - * - * however, the prime must be - * congruent to 3 mod 4 - */ - if ((ltm_prime_tab[x + 1] & 3) != 3) { - /* scan upwards for a prime congruent to 3 mod 4 */ - for (y = x + 1; y < PRIME_SIZE; y++) { - if ((ltm_prime_tab[y] & 3) == 3) { - mp_set(a, ltm_prime_tab[y]); - return MP_OKAY; - } - } - } - } else { - mp_set(a, ltm_prime_tab[x + 1]); - return MP_OKAY; - } - } - } - /* at this point a maybe 1 */ - if (mp_cmp_d(a, 1) == MP_EQ) { - mp_set(a, 2); - return MP_OKAY; + if (mp_cmp_d(a, s_mp_prime_tab[PRIVATE_MP_PRIME_TAB_SIZE-1]) == MP_LT) { + /* find which prime it is bigger than "a" */ + for (x = 0; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + cmp = mp_cmp_d(a, s_mp_prime_tab[x]); + if (cmp == MP_EQ) { + continue; + } + if (cmp != MP_GT) { + if ((bbs_style == 1) && ((s_mp_prime_tab[x] & 3u) != 3u)) { + /* try again until we get a prime congruent to 3 mod 4 */ + continue; + } else { + mp_set(a, s_mp_prime_tab[x]); + return MP_OKAY; + } + } } /* fall through to the sieve */ } @@ -80,21 +52,23 @@ int mp_prime_next_prime(mp_int *a, int t, int bbs_style) if (bbs_style == 1) { /* if a mod 4 != 3 subtract the correct value to make it so */ - if ((a->dp[0] & 3) != 3) { - if ((err = mp_sub_d(a, (a->dp[0] & 3) + 1, a)) != MP_OKAY) { return err; }; + if ((a->dp[0] & 3u) != 3u) { + if ((err = mp_sub_d(a, (a->dp[0] & 3u) + 1u, a)) != MP_OKAY) { + return err; + } } } else { - if (mp_iseven(a) == 1) { + if (MP_IS_EVEN(a)) { /* force odd */ - if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { + if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) { return err; } } } /* generate the restable */ - for (x = 1; x < PRIME_SIZE; x++) { - if ((err = mp_mod_d(a, ltm_prime_tab[x], res_tab + x)) != MP_OKAY) { + for (x = 1; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + if ((err = mp_mod_d(a, s_mp_prime_tab[x], res_tab + x)) != MP_OKAY) { return err; } } @@ -115,43 +89,35 @@ int mp_prime_next_prime(mp_int *a, int t, int bbs_style) step += kstep; /* compute the new residue without using division */ - for (x = 1; x < PRIME_SIZE; x++) { - /* add the step to each residue */ - res_tab[x] += kstep; + for (x = 1; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + /* add the step to each residue */ + res_tab[x] += kstep; - /* subtract the modulus [instead of using division] */ - if (res_tab[x] >= ltm_prime_tab[x]) { - res_tab[x] -= ltm_prime_tab[x]; - } + /* subtract the modulus [instead of using division] */ + if (res_tab[x] >= s_mp_prime_tab[x]) { + res_tab[x] -= s_mp_prime_tab[x]; + } - /* set flag if zero */ - if (res_tab[x] == 0) { - y = 1; - } + /* set flag if zero */ + if (res_tab[x] == 0u) { + y = 1; + } } - } while (y == 1 && step < ((((mp_digit)1)<= ((((mp_digit)1)<= (((mp_digit)1 << MP_DIGIT_BIT) - kstep))) { continue; } - /* is this prime? */ - for (x = 0; x < t; x++) { - mp_set(&b, ltm_prime_tab[x]); - if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { - goto LBL_ERR; - } - if (res == MP_NO) { - break; - } + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto LBL_ERR; } - if (res == MP_YES) { break; } @@ -164,7 +130,3 @@ LBL_ERR: } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_next_prime.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_rabin_miller_trials.c b/lib/hcrypto/libtommath/bn_mp_prime_rabin_miller_trials.c index 248c2fd2e..8bbaf6cf9 100644 --- a/lib/hcrypto/libtommath/bn_mp_prime_rabin_miller_trials.c +++ b/lib/hcrypto/libtommath/bn_mp_prime_rabin_miller_trials.c @@ -1,32 +1,31 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ static const struct { int k, t; } sizes[] = { -{ 128, 28 }, -{ 256, 16 }, -{ 384, 10 }, -{ 512, 7 }, -{ 640, 6 }, -{ 768, 5 }, -{ 896, 4 }, -{ 1024, 4 } + { 80, -1 }, /* Use deterministic algorithm for size <= 80 bits */ + { 81, 37 }, /* max. error = 2^(-96)*/ + { 96, 32 }, /* max. error = 2^(-96)*/ + { 128, 40 }, /* max. error = 2^(-112)*/ + { 160, 35 }, /* max. error = 2^(-112)*/ + { 256, 27 }, /* max. error = 2^(-128)*/ + { 384, 16 }, /* max. error = 2^(-128)*/ + { 512, 18 }, /* max. error = 2^(-160)*/ + { 768, 11 }, /* max. error = 2^(-160)*/ + { 896, 10 }, /* max. error = 2^(-160)*/ + { 1024, 12 }, /* max. error = 2^(-192)*/ + { 1536, 8 }, /* max. error = 2^(-192)*/ + { 2048, 6 }, /* max. error = 2^(-192)*/ + { 3072, 4 }, /* max. error = 2^(-192)*/ + { 4096, 5 }, /* max. error = 2^(-256)*/ + { 5120, 4 }, /* max. error = 2^(-256)*/ + { 6144, 4 }, /* max. error = 2^(-256)*/ + { 8192, 3 }, /* max. error = 2^(-256)*/ + { 9216, 3 }, /* max. error = 2^(-256)*/ + { 10240, 2 } /* For bigger keysizes use always at least 2 Rounds */ }; /* returns # of RM trials required for a given bit size */ @@ -35,18 +34,14 @@ int mp_prime_rabin_miller_trials(int size) int x; for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) { - if (sizes[x].k == size) { - return sizes[x].t; - } else if (sizes[x].k > size) { - return (x == 0) ? sizes[0].t : sizes[x - 1].t; - } + if (sizes[x].k == size) { + return sizes[x].t; + } else if (sizes[x].k > size) { + return (x == 0) ? sizes[0].t : sizes[x - 1].t; + } } - return sizes[x-1].t + 1; + return sizes[x-1].t; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_rabin_miller_trials.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_rand.c b/lib/hcrypto/libtommath/bn_mp_prime_rand.c new file mode 100644 index 000000000..4530e9a5e --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_prime_rand.c @@ -0,0 +1,141 @@ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_RAND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* makes a truly random prime of a given size (bits), + * + * Flags are as follows: + * + * MP_PRIME_BBS - make prime congruent to 3 mod 4 + * MP_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies MP_PRIME_BBS) + * MP_PRIME_2MSB_ON - make the 2nd highest bit one + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + */ + +/* This is possibly the mother of all prime generation functions, muahahahahaha! */ +mp_err s_mp_prime_random_ex(mp_int *a, int t, int size, int flags, private_mp_prime_callback cb, void *dat) +{ + unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; + int bsize, maskOR_msb_offset; + mp_bool res; + mp_err err; + + /* sanity check the input */ + if ((size <= 1) || (t <= 0)) { + return MP_VAL; + } + + /* MP_PRIME_SAFE implies MP_PRIME_BBS */ + if ((flags & MP_PRIME_SAFE) != 0) { + flags |= MP_PRIME_BBS; + } + + /* calc the byte size */ + bsize = (size>>3) + ((size&7)?1:0); + + /* we need a buffer of bsize bytes */ + tmp = (unsigned char *) MP_MALLOC((size_t)bsize); + if (tmp == NULL) { + return MP_MEM; + } + + /* calc the maskAND value for the MSbyte*/ + maskAND = ((size&7) == 0) ? 0xFFu : (unsigned char)(0xFFu >> (8 - (size & 7))); + + /* calc the maskOR_msb */ + maskOR_msb = 0; + maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0; + if ((flags & MP_PRIME_2MSB_ON) != 0) { + maskOR_msb |= (unsigned char)(0x80 >> ((9 - size) & 7)); + } + + /* get the maskOR_lsb */ + maskOR_lsb = 1u; + if ((flags & MP_PRIME_BBS) != 0) { + maskOR_lsb |= 3u; + } + + do { + /* read the bytes */ + if (cb(tmp, bsize, dat) != bsize) { + err = MP_VAL; + goto error; + } + + /* work over the MSbyte */ + tmp[0] &= maskAND; + tmp[0] |= (unsigned char)(1 << ((size - 1) & 7)); + + /* mix in the maskORs */ + tmp[maskOR_msb_offset] |= maskOR_msb; + tmp[bsize-1] |= maskOR_lsb; + + /* read it in */ + /* TODO: casting only for now until all lengths have been changed to the type "size_t"*/ + if ((err = mp_from_ubin(a, tmp, (size_t)bsize)) != MP_OKAY) { + goto error; + } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto error; + } + if (res == MP_NO) { + continue; + } + + if ((flags & MP_PRIME_SAFE) != 0) { + /* see if (a-1)/2 is prime */ + if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) { + goto error; + } + if ((err = mp_div_2(a, a)) != MP_OKAY) { + goto error; + } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto error; + } + } + } while (res == MP_NO); + + if ((flags & MP_PRIME_SAFE) != 0) { + /* restore a to the original value */ + if ((err = mp_mul_2(a, a)) != MP_OKAY) { + goto error; + } + if ((err = mp_add_d(a, 1uL, a)) != MP_OKAY) { + goto error; + } + } + + err = MP_OKAY; +error: + MP_FREE_BUFFER(tmp, (size_t)bsize); + return err; +} + +static int s_mp_rand_cb(unsigned char *dst, int len, void *dat) +{ + (void)dat; + if (len <= 0) { + return len; + } + if (s_mp_rand_source(dst, (size_t)len) != MP_OKAY) { + return 0; + } + return len; +} + +mp_err mp_prime_rand(mp_int *a, int t, int size, int flags) +{ + return s_mp_prime_random_ex(a, t, size, flags, s_mp_rand_cb, NULL); +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_prime_random_ex.c b/lib/hcrypto/libtommath/bn_mp_prime_random_ex.c deleted file mode 100644 index 7b0d15c94..000000000 --- a/lib/hcrypto/libtommath/bn_mp_prime_random_ex.c +++ /dev/null @@ -1,125 +0,0 @@ -#include -#ifdef BN_MP_PRIME_RANDOM_EX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* makes a truly random prime of a given size (bits), - * - * Flags are as follows: - * - * LTM_PRIME_BBS - make prime congruent to 3 mod 4 - * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) - * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero - * LTM_PRIME_2MSB_ON - make the 2nd highest bit one - * - * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can - * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself - * so it can be NULL - * - */ - -/* This is possibly the mother of all prime generation functions, muahahahahaha! */ -int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat) -{ - unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; - int res, err, bsize, maskOR_msb_offset; - - /* sanity check the input */ - if (size <= 1 || t <= 0) { - return MP_VAL; - } - - /* LTM_PRIME_SAFE implies LTM_PRIME_BBS */ - if (flags & LTM_PRIME_SAFE) { - flags |= LTM_PRIME_BBS; - } - - /* calc the byte size */ - bsize = (size>>3) + ((size&7)?1:0); - - /* we need a buffer of bsize bytes */ - tmp = OPT_CAST(unsigned char) XMALLOC(bsize); - if (tmp == NULL) { - return MP_MEM; - } - - /* calc the maskAND value for the MSbyte*/ - maskAND = ((size&7) == 0) ? 0xFF : (0xFF >> (8 - (size & 7))); - - /* calc the maskOR_msb */ - maskOR_msb = 0; - maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0; - if (flags & LTM_PRIME_2MSB_ON) { - maskOR_msb |= 0x80 >> ((9 - size) & 7); - } - - /* get the maskOR_lsb */ - maskOR_lsb = 1; - if (flags & LTM_PRIME_BBS) { - maskOR_lsb |= 3; - } - - do { - /* read the bytes */ - if (cb(tmp, bsize, dat) != bsize) { - err = MP_VAL; - goto error; - } - - /* work over the MSbyte */ - tmp[0] &= maskAND; - tmp[0] |= 1 << ((size - 1) & 7); - - /* mix in the maskORs */ - tmp[maskOR_msb_offset] |= maskOR_msb; - tmp[bsize-1] |= maskOR_lsb; - - /* read it in */ - if ((err = mp_read_unsigned_bin(a, tmp, bsize)) != MP_OKAY) { goto error; } - - /* is it prime? */ - if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } - if (res == MP_NO) { - continue; - } - - if (flags & LTM_PRIME_SAFE) { - /* see if (a-1)/2 is prime */ - if ((err = mp_sub_d(a, 1, a)) != MP_OKAY) { goto error; } - if ((err = mp_div_2(a, a)) != MP_OKAY) { goto error; } - - /* is it prime? */ - if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { goto error; } - } - } while (res == MP_NO); - - if (flags & LTM_PRIME_SAFE) { - /* restore a to the original value */ - if ((err = mp_mul_2(a, a)) != MP_OKAY) { goto error; } - if ((err = mp_add_d(a, 1, a)) != MP_OKAY) { goto error; } - } - - err = MP_OKAY; -error: - XFREE(tmp); - return err; -} - - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_prime_random_ex.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_prime_strong_lucas_selfridge.c b/lib/hcrypto/libtommath/bn_mp_prime_strong_lucas_selfridge.c new file mode 100644 index 000000000..b50bbcd2f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_prime_strong_lucas_selfridge.c @@ -0,0 +1,289 @@ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_STRONG_LUCAS_SELFRIDGE_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + * See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details + */ +#ifndef LTM_USE_ONLY_MR + +/* + * 8-bit is just too small. You can try the Frobenius test + * but that frobenius test can fail, too, for the same reason. + */ +#ifndef MP_8BIT + +/* + * multiply bigint a with int d and put the result in c + * Like mp_mul_d() but with a signed long as the small input + */ +static mp_err s_mp_mul_si(const mp_int *a, int32_t d, mp_int *c) +{ + mp_int t; + mp_err err; + + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + /* + * mp_digit might be smaller than a long, which excludes + * the use of mp_mul_d() here. + */ + mp_set_i32(&t, d); + err = mp_mul(a, &t, c); + mp_clear(&t); + return err; +} +/* + Strong Lucas-Selfridge test. + returns MP_YES if it is a strong L-S prime, MP_NO if it is composite + + Code ported from Thomas Ray Nicely's implementation of the BPSW test + at http://www.trnicely.net/misc/bpsw.html + + Freeware copyright (C) 2016 Thomas R. Nicely . + Released into the public domain by the author, who disclaims any legal + liability arising from its use + + The multi-line comments are made by Thomas R. Nicely and are copied verbatim. + Additional comments marked "CZ" (without the quotes) are by the code-portist. + + (If that name sounds familiar, he is the guy who found the fdiv bug in the + Pentium (P5x, I think) Intel processor) +*/ +mp_err mp_prime_strong_lucas_selfridge(const mp_int *a, mp_bool *result) +{ + /* CZ TODO: choose better variable names! */ + mp_int Dz, gcd, Np1, Uz, Vz, U2mz, V2mz, Qmz, Q2mz, Qkdz, T1z, T2z, T3z, T4z, Q2kdz; + /* CZ TODO: Some of them need the full 32 bit, hence the (temporary) exclusion of MP_8BIT */ + int32_t D, Ds, J, sign, P, Q, r, s, u, Nbits; + mp_err err; + mp_bool oddness; + + *result = MP_NO; + /* + Find the first element D in the sequence {5, -7, 9, -11, 13, ...} + such that Jacobi(D,N) = -1 (Selfridge's algorithm). Theory + indicates that, if N is not a perfect square, D will "nearly + always" be "small." Just in case, an overflow trap for D is + included. + */ + + if ((err = mp_init_multi(&Dz, &gcd, &Np1, &Uz, &Vz, &U2mz, &V2mz, &Qmz, &Q2mz, &Qkdz, &T1z, &T2z, &T3z, &T4z, &Q2kdz, + NULL)) != MP_OKAY) { + return err; + } + + D = 5; + sign = 1; + + for (;;) { + Ds = sign * D; + sign = -sign; + mp_set_u32(&Dz, (uint32_t)D); + if ((err = mp_gcd(a, &Dz, &gcd)) != MP_OKAY) goto LBL_LS_ERR; + + /* if 1 < GCD < N then N is composite with factor "D", and + Jacobi(D,N) is technically undefined (but often returned + as zero). */ + if ((mp_cmp_d(&gcd, 1uL) == MP_GT) && (mp_cmp(&gcd, a) == MP_LT)) { + goto LBL_LS_ERR; + } + if (Ds < 0) { + Dz.sign = MP_NEG; + } + if ((err = mp_kronecker(&Dz, a, &J)) != MP_OKAY) goto LBL_LS_ERR; + + if (J == -1) { + break; + } + D += 2; + + if (D > (INT_MAX - 2)) { + err = MP_VAL; + goto LBL_LS_ERR; + } + } + + + + P = 1; /* Selfridge's choice */ + Q = (1 - Ds) / 4; /* Required so D = P*P - 4*Q */ + + /* NOTE: The conditions (a) N does not divide Q, and + (b) D is square-free or not a perfect square, are included by + some authors; e.g., "Prime numbers and computer methods for + factorization," Hans Riesel (2nd ed., 1994, Birkhauser, Boston), + p. 130. For this particular application of Lucas sequences, + these conditions were found to be immaterial. */ + + /* Now calculate N - Jacobi(D,N) = N + 1 (even), and calculate the + odd positive integer d and positive integer s for which + N + 1 = 2^s*d (similar to the step for N - 1 in Miller's test). + The strong Lucas-Selfridge test then returns N as a strong + Lucas probable prime (slprp) if any of the following + conditions is met: U_d=0, V_d=0, V_2d=0, V_4d=0, V_8d=0, + V_16d=0, ..., etc., ending with V_{2^(s-1)*d}=V_{(N+1)/2}=0 + (all equalities mod N). Thus d is the highest index of U that + must be computed (since V_2m is independent of U), compared + to U_{N+1} for the standard Lucas-Selfridge test; and no + index of V beyond (N+1)/2 is required, just as in the + standard Lucas-Selfridge test. However, the quantity Q^d must + be computed for use (if necessary) in the latter stages of + the test. The result is that the strong Lucas-Selfridge test + has a running time only slightly greater (order of 10 %) than + that of the standard Lucas-Selfridge test, while producing + only (roughly) 30 % as many pseudoprimes (and every strong + Lucas pseudoprime is also a standard Lucas pseudoprime). Thus + the evidence indicates that the strong Lucas-Selfridge test is + more effective than the standard Lucas-Selfridge test, and a + Baillie-PSW test based on the strong Lucas-Selfridge test + should be more reliable. */ + + if ((err = mp_add_d(a, 1uL, &Np1)) != MP_OKAY) goto LBL_LS_ERR; + s = mp_cnt_lsb(&Np1); + + /* CZ + * This should round towards zero because + * Thomas R. Nicely used GMP's mpz_tdiv_q_2exp() + * and mp_div_2d() is equivalent. Additionally: + * dividing an even number by two does not produce + * any leftovers. + */ + if ((err = mp_div_2d(&Np1, s, &Dz, NULL)) != MP_OKAY) goto LBL_LS_ERR; + /* We must now compute U_d and V_d. Since d is odd, the accumulated + values U and V are initialized to U_1 and V_1 (if the target + index were even, U and V would be initialized instead to U_0=0 + and V_0=2). The values of U_2m and V_2m are also initialized to + U_1 and V_1; the FOR loop calculates in succession U_2 and V_2, + U_4 and V_4, U_8 and V_8, etc. If the corresponding bits + (1, 2, 3, ...) of t are on (the zero bit having been accounted + for in the initialization of U and V), these values are then + combined with the previous totals for U and V, using the + composition formulas for addition of indices. */ + + mp_set(&Uz, 1uL); /* U=U_1 */ + mp_set(&Vz, (mp_digit)P); /* V=V_1 */ + mp_set(&U2mz, 1uL); /* U_1 */ + mp_set(&V2mz, (mp_digit)P); /* V_1 */ + + mp_set_i32(&Qmz, Q); + if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR; + /* Initializes calculation of Q^d */ + mp_set_i32(&Qkdz, Q); + + Nbits = mp_count_bits(&Dz); + + for (u = 1; u < Nbits; u++) { /* zero bit off, already accounted for */ + /* Formulas for doubling of indices (carried out mod N). Note that + * the indices denoted as "2m" are actually powers of 2, specifically + * 2^(ul-1) beginning each loop and 2^ul ending each loop. + * + * U_2m = U_m*V_m + * V_2m = V_m*V_m - 2*Q^m + */ + + if ((err = mp_mul(&U2mz, &V2mz, &U2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&U2mz, a, &U2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sqr(&V2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sub(&V2mz, &Q2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&V2mz, a, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + + /* Must calculate powers of Q for use in V_2m, also for Q^d later */ + if ((err = mp_sqr(&Qmz, &Qmz)) != MP_OKAY) goto LBL_LS_ERR; + + /* prevents overflow */ /* CZ still necessary without a fixed prealloc'd mem.? */ + if ((err = mp_mod(&Qmz, a, &Qmz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR; + + if (s_mp_get_bit(&Dz, (unsigned int)u) == MP_YES) { + /* Formulas for addition of indices (carried out mod N); + * + * U_(m+n) = (U_m*V_n + U_n*V_m)/2 + * V_(m+n) = (V_m*V_n + D*U_m*U_n)/2 + * + * Be careful with division by 2 (mod N)! + */ + if ((err = mp_mul(&U2mz, &Vz, &T1z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&Uz, &V2mz, &T2z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&V2mz, &Vz, &T3z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&U2mz, &Uz, &T4z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = s_mp_mul_si(&T4z, Ds, &T4z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_add(&T1z, &T2z, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ODD(&Uz)) { + if ((err = mp_add(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + } + /* CZ + * This should round towards negative infinity because + * Thomas R. Nicely used GMP's mpz_fdiv_q_2exp(). + * But mp_div_2() does not do so, it is truncating instead. + */ + oddness = MP_IS_ODD(&Uz) ? MP_YES : MP_NO; + if ((err = mp_div_2(&Uz, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if ((Uz.sign == MP_NEG) && (oddness != MP_NO)) { + if ((err = mp_sub_d(&Uz, 1uL, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + } + if ((err = mp_add(&T3z, &T4z, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ODD(&Vz)) { + if ((err = mp_add(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + } + oddness = MP_IS_ODD(&Vz) ? MP_YES : MP_NO; + if ((err = mp_div_2(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((Vz.sign == MP_NEG) && (oddness != MP_NO)) { + if ((err = mp_sub_d(&Vz, 1uL, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + } + if ((err = mp_mod(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + + /* Calculating Q^d for later use */ + if ((err = mp_mul(&Qkdz, &Qmz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + } + } + + /* If U_d or V_d is congruent to 0 mod N, then N is a prime or a + strong Lucas pseudoprime. */ + if (MP_IS_ZERO(&Uz) || MP_IS_ZERO(&Vz)) { + *result = MP_YES; + goto LBL_LS_ERR; + } + + /* NOTE: Ribenboim ("The new book of prime number records," 3rd ed., + 1995/6) omits the condition V0 on p.142, but includes it on + p. 130. The condition is NECESSARY; otherwise the test will + return false negatives---e.g., the primes 29 and 2000029 will be + returned as composite. */ + + /* Otherwise, we must compute V_2d, V_4d, V_8d, ..., V_{2^(s-1)*d} + by repeated use of the formula V_2m = V_m*V_m - 2*Q^m. If any of + these are congruent to 0 mod N, then N is a prime or a strong + Lucas pseudoprime. */ + + /* Initialize 2*Q^(d*2^r) for V_2m */ + if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR; + + for (r = 1; r < s; r++) { + if ((err = mp_sqr(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sub(&Vz, &Q2kdz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ZERO(&Vz)) { + *result = MP_YES; + goto LBL_LS_ERR; + } + /* Calculate Q^{d*2^r} for next r (final iteration irrelevant). */ + if (r < (s - 1)) { + if ((err = mp_sqr(&Qkdz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR; + } + } +LBL_LS_ERR: + mp_clear_multi(&Q2kdz, &T4z, &T3z, &T2z, &T1z, &Qkdz, &Q2mz, &Qmz, &V2mz, &U2mz, &Vz, &Uz, &Np1, &gcd, &Dz, NULL); + return err; +} +#endif +#endif +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_radix_size.c b/lib/hcrypto/libtommath/bn_mp_radix_size.c index af94be867..b96f4874c 100644 --- a/lib/hcrypto/libtommath/bn_mp_radix_size.c +++ b/lib/hcrypto/libtommath/bn_mp_radix_size.c @@ -1,78 +1,65 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_RADIX_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* returns size of ASCII reprensentation */ -int mp_radix_size (mp_int * a, int radix, int *size) +/* returns size of ASCII representation */ +mp_err mp_radix_size(const mp_int *a, int radix, int *size) { - int res, digs; - mp_int t; - mp_digit d; + mp_err err; + int digs; + mp_int t; + mp_digit d; - *size = 0; + *size = 0; - /* special case for binary */ - if (radix == 2) { - *size = mp_count_bits (a) + (a->sign == MP_NEG ? 1 : 0) + 1; - return MP_OKAY; - } + /* make sure the radix is in range */ + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } - /* make sure the radix is in range */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } + if (MP_IS_ZERO(a)) { + *size = 2; + return MP_OKAY; + } - if (mp_iszero(a) == MP_YES) { - *size = 2; - return MP_OKAY; - } + /* special case for binary */ + if (radix == 2) { + *size = (mp_count_bits(a) + ((a->sign == MP_NEG) ? 1 : 0) + 1); + return MP_OKAY; + } - /* digs is the digit count */ - digs = 0; + /* digs is the digit count */ + digs = 0; - /* if it's negative add one for the sign */ - if (a->sign == MP_NEG) { - ++digs; - } + /* if it's negative add one for the sign */ + if (a->sign == MP_NEG) { + ++digs; + } - /* init a copy of the input */ - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } + /* init a copy of the input */ + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } - /* force temp to positive */ - t.sign = MP_ZPOS; + /* force temp to positive */ + t.sign = MP_ZPOS; - /* fetch out all of the digits */ - while (mp_iszero (&t) == MP_NO) { - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - ++digs; - } - mp_clear (&t); + /* fetch out all of the digits */ + while (!MP_IS_ZERO(&t)) { + if ((err = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) { + goto LBL_ERR; + } + ++digs; + } - /* return digs + 1, the 1 is for the NULL byte that would be required. */ - *size = digs + 1; - return MP_OKAY; + /* return digs + 1, the 1 is for the NULL byte that would be required. */ + *size = digs + 1; + err = MP_OKAY; + +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_radix_size.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_radix_smap.c b/lib/hcrypto/libtommath/bn_mp_radix_smap.c index 7d72feb84..a16128d79 100644 --- a/lib/hcrypto/libtommath/bn_mp_radix_smap.c +++ b/lib/hcrypto/libtommath/bn_mp_radix_smap.c @@ -1,24 +1,22 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_RADIX_SMAP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* chars used in radix conversions */ -const char *mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; +const char *const mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; +const uint8_t mp_s_rmap_reverse[] = { + 0xff, 0xff, 0xff, 0x3e, 0xff, 0xff, 0xff, 0x3f, /* ()*+,-./ */ + 0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, /* 01234567 */ + 0x08, 0x09, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* 89:;<=>? */ + 0xff, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f, 0x10, /* @ABCDEFG */ + 0x11, 0x12, 0x13, 0x14, 0x15, 0x16, 0x17, 0x18, /* HIJKLMNO */ + 0x19, 0x1a, 0x1b, 0x1c, 0x1d, 0x1e, 0x1f, 0x20, /* PQRSTUVW */ + 0x21, 0x22, 0x23, 0xff, 0xff, 0xff, 0xff, 0xff, /* XYZ[\]^_ */ + 0xff, 0x24, 0x25, 0x26, 0x27, 0x28, 0x29, 0x2a, /* `abcdefg */ + 0x2b, 0x2c, 0x2d, 0x2e, 0x2f, 0x30, 0x31, 0x32, /* hijklmno */ + 0x33, 0x34, 0x35, 0x36, 0x37, 0x38, 0x39, 0x3a, /* pqrstuvw */ + 0x3b, 0x3c, 0x3d, 0xff, 0xff, 0xff, 0xff, 0xff, /* xyz{|}~. */ +}; +const size_t mp_s_rmap_reverse_sz = sizeof(mp_s_rmap_reverse); #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_radix_smap.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_rand.c b/lib/hcrypto/libtommath/bn_mp_rand.c index 14f5b7503..7e9052c2b 100644 --- a/lib/hcrypto/libtommath/bn_mp_rand.c +++ b/lib/hcrypto/libtommath/bn_mp_rand.c @@ -1,55 +1,46 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_RAND_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* makes a pseudo-random int of a given size */ -int -mp_rand (mp_int * a, int digits) +mp_err(*s_mp_rand_source)(void *out, size_t size) = s_mp_rand_platform; + +void mp_rand_source(mp_err(*source)(void *out, size_t size)) { - int res; - mp_digit d; + s_mp_rand_source = (source == NULL) ? s_mp_rand_platform : source; +} - mp_zero (a); - if (digits <= 0) { - return MP_OKAY; - } +mp_err mp_rand(mp_int *a, int digits) +{ + int i; + mp_err err; - /* first place a random non-zero digit */ - do { - d = ((mp_digit) labs (rand ())) & MP_MASK; - } while (d == 0); + mp_zero(a); - if ((res = mp_add_d (a, d, a)) != MP_OKAY) { - return res; - } + if (digits <= 0) { + return MP_OKAY; + } - while (--digits > 0) { - if ((res = mp_lshd (a, 1)) != MP_OKAY) { - return res; - } + if ((err = mp_grow(a, digits)) != MP_OKAY) { + return err; + } - if ((res = mp_add_d (a, ((mp_digit) labs (rand ())), a)) != MP_OKAY) { - return res; - } - } + if ((err = s_mp_rand_source(a->dp, (size_t)digits * sizeof(mp_digit))) != MP_OKAY) { + return err; + } - return MP_OKAY; + /* TODO: We ensure that the highest digit is nonzero. Should this be removed? */ + while ((a->dp[digits - 1] & MP_MASK) == 0u) { + if ((err = s_mp_rand_source(a->dp + digits - 1, sizeof(mp_digit))) != MP_OKAY) { + return err; + } + } + + a->used = digits; + for (i = 0; i < digits; ++i) { + a->dp[i] &= MP_MASK; + } + + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_rand.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_read_radix.c b/lib/hcrypto/libtommath/bn_mp_read_radix.c index 35ca88673..de18e06fc 100644 --- a/lib/hcrypto/libtommath/bn_mp_read_radix.c +++ b/lib/hcrypto/libtommath/bn_mp_read_radix.c @@ -1,85 +1,79 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_READ_RADIX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#define MP_TOUPPER(c) ((((c) >= 'a') && ((c) <= 'z')) ? (((c) + 'A') - 'a') : (c)) /* read a string [ASCII] in a given radix */ -int mp_read_radix (mp_int * a, const char *str, int radix) +mp_err mp_read_radix(mp_int *a, const char *str, int radix) { - int y, res, neg; - char ch; + mp_err err; + int y; + mp_sign neg; + unsigned pos; + char ch; - /* zero the digit bignum */ - mp_zero(a); + /* zero the digit bignum */ + mp_zero(a); - /* make sure the radix is ok */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } + /* make sure the radix is ok */ + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } - /* if the leading digit is a - * minus set the sign to negative. - */ - if (*str == '-') { - ++str; - neg = MP_NEG; - } else { - neg = MP_ZPOS; - } + /* if the leading digit is a + * minus set the sign to negative. + */ + if (*str == '-') { + ++str; + neg = MP_NEG; + } else { + neg = MP_ZPOS; + } - /* set the integer to the default of zero */ - mp_zero (a); + /* set the integer to the default of zero */ + mp_zero(a); - /* process each digit of the string */ - while (*str) { - /* if the radix < 36 the conversion is case insensitive - * this allows numbers like 1AB and 1ab to represent the same value - * [e.g. in hex] - */ - ch = (char) ((radix < 36) ? toupper (*str) : *str); - for (y = 0; y < 64; y++) { - if (ch == mp_s_rmap[y]) { + /* process each digit of the string */ + while (*str != '\0') { + /* if the radix <= 36 the conversion is case insensitive + * this allows numbers like 1AB and 1ab to represent the same value + * [e.g. in hex] + */ + ch = (radix <= 36) ? (char)MP_TOUPPER((int)*str) : *str; + pos = (unsigned)(ch - '('); + if (mp_s_rmap_reverse_sz < pos) { break; } - } + y = (int)mp_s_rmap_reverse[pos]; - /* if the char was found in the map - * and is less than the given radix add it - * to the number, otherwise exit the loop. - */ - if (y < radix) { - if ((res = mp_mul_d (a, (mp_digit) radix, a)) != MP_OKAY) { - return res; + /* if the char was found in the map + * and is less than the given radix add it + * to the number, otherwise exit the loop. + */ + if ((y == 0xff) || (y >= radix)) { + break; } - if ((res = mp_add_d (a, (mp_digit) y, a)) != MP_OKAY) { - return res; + if ((err = mp_mul_d(a, (mp_digit)radix, a)) != MP_OKAY) { + return err; } - } else { - break; - } - ++str; - } + if ((err = mp_add_d(a, (mp_digit)y, a)) != MP_OKAY) { + return err; + } + ++str; + } - /* set the sign only if a != 0 */ - if (mp_iszero(a) != 1) { - a->sign = neg; - } - return MP_OKAY; + /* if an illegal character was found, fail. */ + if (!((*str == '\0') || (*str == '\r') || (*str == '\n'))) { + mp_zero(a); + return MP_VAL; + } + + /* set the sign only if a != 0 */ + if (!MP_IS_ZERO(a)) { + a->sign = neg; + } + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_read_radix.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_read_signed_bin.c b/lib/hcrypto/libtommath/bn_mp_read_signed_bin.c deleted file mode 100644 index 8da651ce3..000000000 --- a/lib/hcrypto/libtommath/bn_mp_read_signed_bin.c +++ /dev/null @@ -1,41 +0,0 @@ -#include -#ifdef BN_MP_READ_SIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* read signed bin, big endian, first byte is 0==positive or 1==negative */ -int mp_read_signed_bin (mp_int * a, const unsigned char *b, int c) -{ - int res; - - /* read magnitude */ - if ((res = mp_read_unsigned_bin (a, b + 1, c - 1)) != MP_OKAY) { - return res; - } - - /* first byte is 0 for positive, non-zero for negative */ - if (b[0] == 0) { - a->sign = MP_ZPOS; - } else { - a->sign = MP_NEG; - } - - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_read_signed_bin.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_read_unsigned_bin.c b/lib/hcrypto/libtommath/bn_mp_read_unsigned_bin.c deleted file mode 100644 index 1ebba13a0..000000000 --- a/lib/hcrypto/libtommath/bn_mp_read_unsigned_bin.c +++ /dev/null @@ -1,55 +0,0 @@ -#include -#ifdef BN_MP_READ_UNSIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* reads a unsigned char array, assumes the msb is stored first [big endian] */ -int mp_read_unsigned_bin (mp_int * a, const unsigned char *b, int c) -{ - int res; - - /* make sure there are at least two digits */ - if (a->alloc < 2) { - if ((res = mp_grow(a, 2)) != MP_OKAY) { - return res; - } - } - - /* zero the int */ - mp_zero (a); - - /* read the bytes in */ - while (c-- > 0) { - if ((res = mp_mul_2d (a, 8, a)) != MP_OKAY) { - return res; - } - -#ifndef MP_8BIT - a->dp[0] |= *b++; - a->used += 1; -#else - a->dp[0] = (*b & MP_MASK); - a->dp[1] |= ((*b++ >> 7U) & 1); - a->used += 2; -#endif - } - mp_clamp (a); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_read_unsigned_bin.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce.c b/lib/hcrypto/libtommath/bn_mp_reduce.c index ae57a6a00..3c669d491 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce.c @@ -1,100 +1,83 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* reduces x mod m, assumes 0 < x < m**2, mu is * precomputed via mp_reduce_setup. * From HAC pp.604 Algorithm 14.42 */ -int mp_reduce (mp_int * x, mp_int * m, mp_int * mu) +mp_err mp_reduce(mp_int *x, const mp_int *m, const mp_int *mu) { - mp_int q; - int res, um = m->used; + mp_int q; + mp_err err; + int um = m->used; - /* q = x */ - if ((res = mp_init_copy (&q, x)) != MP_OKAY) { - return res; - } + /* q = x */ + if ((err = mp_init_copy(&q, x)) != MP_OKAY) { + return err; + } - /* q1 = x / b**(k-1) */ - mp_rshd (&q, um - 1); + /* q1 = x / b**(k-1) */ + mp_rshd(&q, um - 1); - /* according to HAC this optimization is ok */ - if (((unsigned long) um) > (((mp_digit)1) << (DIGIT_BIT - 1))) { - if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) { + /* according to HAC this optimization is ok */ + if ((mp_digit)um > ((mp_digit)1 << (MP_DIGIT_BIT - 1))) { + if ((err = mp_mul(&q, mu, &q)) != MP_OKAY) { + goto CLEANUP; + } + } else if (MP_HAS(S_MP_MUL_HIGH_DIGS)) { + if ((err = s_mp_mul_high_digs(&q, mu, &q, um)) != MP_OKAY) { + goto CLEANUP; + } + } else if (MP_HAS(S_MP_MUL_HIGH_DIGS_FAST)) { + if ((err = s_mp_mul_high_digs_fast(&q, mu, &q, um)) != MP_OKAY) { + goto CLEANUP; + } + } else { + err = MP_VAL; goto CLEANUP; - } - } else { -#ifdef BN_S_MP_MUL_HIGH_DIGS_C - if ((res = s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { - goto CLEANUP; - } -#elif defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) - if ((res = fast_s_mp_mul_high_digs (&q, mu, &q, um)) != MP_OKAY) { - goto CLEANUP; - } -#else - { - res = MP_VAL; - goto CLEANUP; - } -#endif - } + } - /* q3 = q2 / b**(k+1) */ - mp_rshd (&q, um + 1); + /* q3 = q2 / b**(k+1) */ + mp_rshd(&q, um + 1); - /* x = x mod b**(k+1), quick (no division) */ - if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) { - goto CLEANUP; - } - - /* q = q * m mod b**(k+1), quick (no division) */ - if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) { - goto CLEANUP; - } - - /* x = x - q */ - if ((res = mp_sub (x, &q, x)) != MP_OKAY) { - goto CLEANUP; - } - - /* If x < 0, add b**(k+1) to it */ - if (mp_cmp_d (x, 0) == MP_LT) { - mp_set (&q, 1); - if ((res = mp_lshd (&q, um + 1)) != MP_OKAY) + /* x = x mod b**(k+1), quick (no division) */ + if ((err = mp_mod_2d(x, MP_DIGIT_BIT * (um + 1), x)) != MP_OKAY) { goto CLEANUP; - if ((res = mp_add (x, &q, x)) != MP_OKAY) - goto CLEANUP; - } + } - /* Back off if it's too big */ - while (mp_cmp (x, m) != MP_LT) { - if ((res = s_mp_sub (x, m, x)) != MP_OKAY) { + /* q = q * m mod b**(k+1), quick (no division) */ + if ((err = s_mp_mul_digs(&q, m, &q, um + 1)) != MP_OKAY) { goto CLEANUP; - } - } + } + + /* x = x - q */ + if ((err = mp_sub(x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + + /* If x < 0, add b**(k+1) to it */ + if (mp_cmp_d(x, 0uL) == MP_LT) { + mp_set(&q, 1uL); + if ((err = mp_lshd(&q, um + 1)) != MP_OKAY) { + goto CLEANUP; + } + if ((err = mp_add(x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + } + + /* Back off if it's too big */ + while (mp_cmp(x, m) != MP_LT) { + if ((err = s_mp_sub(x, m, x)) != MP_OKAY) { + goto CLEANUP; + } + } CLEANUP: - mp_clear (&q); + mp_clear(&q); - return res; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_2k.c b/lib/hcrypto/libtommath/bn_mp_reduce_2k.c index 1c4a751dd..1cea6cb21 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_2k.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_2k.c @@ -1,61 +1,48 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_2K_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* reduces a modulo n where n is of the form 2**p - d */ -int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d) +mp_err mp_reduce_2k(mp_int *a, const mp_int *n, mp_digit d) { mp_int q; - int p, res; + mp_err err; + int p; - if ((res = mp_init(&q)) != MP_OKAY) { - return res; + if ((err = mp_init(&q)) != MP_OKAY) { + return err; } p = mp_count_bits(n); top: /* q = a/2**p, a = a mod 2**p */ - if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { - goto ERR; + if ((err = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto LBL_ERR; } - if (d != 1) { + if (d != 1u) { /* q = q * d */ - if ((res = mp_mul_d(&q, d, &q)) != MP_OKAY) { - goto ERR; + if ((err = mp_mul_d(&q, d, &q)) != MP_OKAY) { + goto LBL_ERR; } } /* a = a + q */ - if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { - goto ERR; + if ((err = s_mp_add(a, &q, a)) != MP_OKAY) { + goto LBL_ERR; } if (mp_cmp_mag(a, n) != MP_LT) { - s_mp_sub(a, n, a); + if ((err = s_mp_sub(a, n, a)) != MP_OKAY) { + goto LBL_ERR; + } goto top; } -ERR: +LBL_ERR: mp_clear(&q); - return res; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_2k.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_2k_l.c b/lib/hcrypto/libtommath/bn_mp_reduce_2k_l.c index 71abeaebb..6a9f3d31b 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_2k_l.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_2k_l.c @@ -1,62 +1,49 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_2K_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* reduces a modulo n where n is of the form 2**p - d This differs from reduce_2k since "d" can be larger than a single digit. */ -int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d) +mp_err mp_reduce_2k_l(mp_int *a, const mp_int *n, const mp_int *d) { mp_int q; - int p, res; + mp_err err; + int p; - if ((res = mp_init(&q)) != MP_OKAY) { - return res; + if ((err = mp_init(&q)) != MP_OKAY) { + return err; } p = mp_count_bits(n); top: /* q = a/2**p, a = a mod 2**p */ - if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) { - goto ERR; + if ((err = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto LBL_ERR; } /* q = q * d */ - if ((res = mp_mul(&q, d, &q)) != MP_OKAY) { - goto ERR; + if ((err = mp_mul(&q, d, &q)) != MP_OKAY) { + goto LBL_ERR; } /* a = a + q */ - if ((res = s_mp_add(a, &q, a)) != MP_OKAY) { - goto ERR; + if ((err = s_mp_add(a, &q, a)) != MP_OKAY) { + goto LBL_ERR; } if (mp_cmp_mag(a, n) != MP_LT) { - s_mp_sub(a, n, a); + if ((err = s_mp_sub(a, n, a)) != MP_OKAY) { + goto LBL_ERR; + } goto top; } -ERR: +LBL_ERR: mp_clear(&q); - return res; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_2k_l.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup.c b/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup.c index dca723c81..2eaf7addf 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup.c @@ -1,39 +1,28 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_2K_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines the setup value */ -int mp_reduce_2k_setup(mp_int *a, mp_digit *d) +mp_err mp_reduce_2k_setup(const mp_int *a, mp_digit *d) { - int res, p; + mp_err err; mp_int tmp; + int p; - if ((res = mp_init(&tmp)) != MP_OKAY) { - return res; + if ((err = mp_init(&tmp)) != MP_OKAY) { + return err; } p = mp_count_bits(a); - if ((res = mp_2expt(&tmp, p)) != MP_OKAY) { + if ((err = mp_2expt(&tmp, p)) != MP_OKAY) { mp_clear(&tmp); - return res; + return err; } - if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { + if ((err = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { mp_clear(&tmp); - return res; + return err; } *d = tmp.dp[0]; @@ -41,7 +30,3 @@ int mp_reduce_2k_setup(mp_int *a, mp_digit *d) return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_2k_setup.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup_l.c b/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup_l.c index cc59a6e71..4f9aa14d1 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup_l.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_2k_setup_l.c @@ -1,44 +1,28 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_2K_SETUP_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines the setup value */ -int mp_reduce_2k_setup_l(mp_int *a, mp_int *d) +mp_err mp_reduce_2k_setup_l(const mp_int *a, mp_int *d) { - int res; + mp_err err; mp_int tmp; - if ((res = mp_init(&tmp)) != MP_OKAY) { - return res; + if ((err = mp_init(&tmp)) != MP_OKAY) { + return err; } - if ((res = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { - goto ERR; + if ((err = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { + goto LBL_ERR; } - if ((res = s_mp_sub(&tmp, a, d)) != MP_OKAY) { - goto ERR; + if ((err = s_mp_sub(&tmp, a, d)) != MP_OKAY) { + goto LBL_ERR; } -ERR: +LBL_ERR: mp_clear(&tmp); - return res; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_2k_setup_l.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_is_2k.c b/lib/hcrypto/libtommath/bn_mp_reduce_is_2k.c index c8d25d83e..a9f4f9f2b 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_is_2k.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_is_2k.c @@ -1,22 +1,10 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_IS_2K_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines if mp_reduce_2k can be used */ -int mp_reduce_is_2k(mp_int *a) +mp_bool mp_reduce_is_2k(const mp_int *a) { int ix, iy, iw; mp_digit iz; @@ -31,22 +19,20 @@ int mp_reduce_is_2k(mp_int *a) iw = 1; /* Test every bit from the second digit up, must be 1 */ - for (ix = DIGIT_BIT; ix < iy; ix++) { - if ((a->dp[iw] & iz) == 0) { - return MP_NO; - } - iz <<= 1; - if (iz > (mp_digit)MP_MASK) { - ++iw; - iz = 1; - } + for (ix = MP_DIGIT_BIT; ix < iy; ix++) { + if ((a->dp[iw] & iz) == 0u) { + return MP_NO; + } + iz <<= 1; + if (iz > MP_DIGIT_MAX) { + ++iw; + iz = 1; + } } + return MP_YES; + } else { + return MP_YES; } - return MP_YES; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_is_2k.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_is_2k_l.c b/lib/hcrypto/libtommath/bn_mp_reduce_is_2k_l.c index ad006f39c..4bc69bef8 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_is_2k_l.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_is_2k_l.c @@ -1,22 +1,10 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_IS_2K_L_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* determines if reduce_2k_l can be used */ -int mp_reduce_is_2k_l(mp_int *a) +mp_bool mp_reduce_is_2k_l(const mp_int *a) { int ix, iy; @@ -27,18 +15,14 @@ int mp_reduce_is_2k_l(mp_int *a) } else if (a->used > 1) { /* if more than half of the digits are -1 we're sold */ for (iy = ix = 0; ix < a->used; ix++) { - if (a->dp[ix] == MP_MASK) { - ++iy; - } + if (a->dp[ix] == MP_DIGIT_MAX) { + ++iy; + } } return (iy >= (a->used/2)) ? MP_YES : MP_NO; - + } else { + return MP_NO; } - return MP_NO; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_is_2k_l.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_reduce_setup.c b/lib/hcrypto/libtommath/bn_mp_reduce_setup.c index 035419bf3..f02160fa5 100644 --- a/lib/hcrypto/libtommath/bn_mp_reduce_setup.c +++ b/lib/hcrypto/libtommath/bn_mp_reduce_setup.c @@ -1,34 +1,17 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_REDUCE_SETUP_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* pre-calculate the value required for Barrett reduction * For a given modulus "b" it calulates the value required in "a" */ -int mp_reduce_setup (mp_int * a, mp_int * b) +mp_err mp_reduce_setup(mp_int *a, const mp_int *b) { - int res; - - if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) { - return res; - } - return mp_div (a, b, a, NULL); + mp_err err; + if ((err = mp_2expt(a, b->used * 2 * MP_DIGIT_BIT)) != MP_OKAY) { + return err; + } + return mp_div(a, b, a, NULL); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_reduce_setup.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_root_u32.c b/lib/hcrypto/libtommath/bn_mp_root_u32.c new file mode 100644 index 000000000..ba65549c6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_root_u32.c @@ -0,0 +1,139 @@ +#include "tommath_private.h" +#ifdef BN_MP_ROOT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* find the n'th root of an integer + * + * Result found such that (c)**b <= a and (c+1)**b > a + * + * This algorithm uses Newton's approximation + * x[i+1] = x[i] - f(x[i])/f'(x[i]) + * which will find the root in log(N) time where + * each step involves a fair bit. + */ +mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c) +{ + mp_int t1, t2, t3, a_; + mp_ord cmp; + int ilog2; + mp_err err; + + /* input must be positive if b is even */ + if (((b & 1u) == 0u) && (a->sign == MP_NEG)) { + return MP_VAL; + } + + if ((err = mp_init_multi(&t1, &t2, &t3, NULL)) != MP_OKAY) { + return err; + } + + /* if a is negative fudge the sign but keep track */ + a_ = *a; + a_.sign = MP_ZPOS; + + /* Compute seed: 2^(log_2(n)/b + 2)*/ + ilog2 = mp_count_bits(a); + + /* + If "b" is larger than INT_MAX it is also larger than + log_2(n) because the bit-length of the "n" is measured + with an int and hence the root is always < 2 (two). + */ + if (b > (uint32_t)(INT_MAX/2)) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + + /* "b" is smaller than INT_MAX, we can cast safely */ + if (ilog2 < (int)b) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + ilog2 = ilog2 / ((int)b); + if (ilog2 == 0) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + /* Start value must be larger than root */ + ilog2 += 2; + if ((err = mp_2expt(&t2,ilog2)) != MP_OKAY) goto LBL_ERR; + do { + /* t1 = t2 */ + if ((err = mp_copy(&t2, &t1)) != MP_OKAY) goto LBL_ERR; + + /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ + + /* t3 = t1**(b-1) */ + if ((err = mp_expt_u32(&t1, b - 1u, &t3)) != MP_OKAY) goto LBL_ERR; + + /* numerator */ + /* t2 = t1**b */ + if ((err = mp_mul(&t3, &t1, &t2)) != MP_OKAY) goto LBL_ERR; + + /* t2 = t1**b - a */ + if ((err = mp_sub(&t2, &a_, &t2)) != MP_OKAY) goto LBL_ERR; + + /* denominator */ + /* t3 = t1**(b-1) * b */ + if ((err = mp_mul_d(&t3, b, &t3)) != MP_OKAY) goto LBL_ERR; + + /* t3 = (t1**b - a)/(b * t1**(b-1)) */ + if ((err = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&t1, &t3, &t2)) != MP_OKAY) goto LBL_ERR; + + /* + Number of rounds is at most log_2(root). If it is more it + got stuck, so break out of the loop and do the rest manually. + */ + if (ilog2-- == 0) { + break; + } + } while (mp_cmp(&t1, &t2) != MP_EQ); + + /* result can be off by a few so check */ + /* Loop beneath can overshoot by one if found root is smaller than actual root */ + for (;;) { + if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR; + cmp = mp_cmp(&t2, &a_); + if (cmp == MP_EQ) { + err = MP_OKAY; + goto LBL_ERR; + } + if (cmp == MP_LT) { + if ((err = mp_add_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR; + } else { + break; + } + } + /* correct overshoot from above or from recurrence */ + for (;;) { + if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR; + if (mp_cmp(&t2, &a_) == MP_GT) { + if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR; + } else { + break; + } + } + + /* set the result */ + mp_exch(&t1, c); + + /* set the sign of the result */ + c->sign = a->sign; + + err = MP_OKAY; + +LBL_ERR: + mp_clear_multi(&t1, &t2, &t3, NULL); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_rshd.c b/lib/hcrypto/libtommath/bn_mp_rshd.c index ed13ce59a..bb8743e3b 100644 --- a/lib/hcrypto/libtommath/bn_mp_rshd.c +++ b/lib/hcrypto/libtommath/bn_mp_rshd.c @@ -1,72 +1,51 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_RSHD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* shift right a certain amount of digits */ -void mp_rshd (mp_int * a, int b) +void mp_rshd(mp_int *a, int b) { - int x; + int x; + mp_digit *bottom, *top; - /* if b <= 0 then ignore it */ - if (b <= 0) { - return; - } + /* if b <= 0 then ignore it */ + if (b <= 0) { + return; + } - /* if b > used then simply zero it and return */ - if (a->used <= b) { - mp_zero (a); - return; - } + /* if b > used then simply zero it and return */ + if (a->used <= b) { + mp_zero(a); + return; + } - { - register mp_digit *bottom, *top; + /* shift the digits down */ - /* shift the digits down */ + /* bottom */ + bottom = a->dp; - /* bottom */ - bottom = a->dp; + /* top [offset into digits] */ + top = a->dp + b; - /* top [offset into digits] */ - top = a->dp + b; + /* this is implemented as a sliding window where + * the window is b-digits long and digits from + * the top of the window are copied to the bottom + * + * e.g. - /* this is implemented as a sliding window where - * the window is b-digits long and digits from - * the top of the window are copied to the bottom - * - * e.g. - - b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> - /\ | ----> - \-------------------/ ----> - */ - for (x = 0; x < (a->used - b); x++) { + b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> + /\ | ----> + \-------------------/ ----> + */ + for (x = 0; x < (a->used - b); x++) { *bottom++ = *top++; - } + } - /* zero the top digits */ - for (; x < a->used; x++) { - *bottom++ = 0; - } - } + /* zero the top digits */ + MP_ZERO_DIGITS(bottom, a->used - x); - /* remove excess digits */ - a->used -= b; + /* remove excess digits */ + a->used -= b; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_rshd.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_sbin_size.c b/lib/hcrypto/libtommath/bn_mp_sbin_size.c new file mode 100644 index 000000000..e0993d690 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_sbin_size.c @@ -0,0 +1,11 @@ +#include "tommath_private.h" +#ifdef BN_MP_SBIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* get the size for an signed equivalent */ +size_t mp_sbin_size(const mp_int *a) +{ + return 1u + mp_ubin_size(a); +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set.c b/lib/hcrypto/libtommath/bn_mp_set.c index 174adcbc6..44ac6df57 100644 --- a/lib/hcrypto/libtommath/bn_mp_set.c +++ b/lib/hcrypto/libtommath/bn_mp_set.c @@ -1,29 +1,14 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SET_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* set to a digit */ -void mp_set (mp_int * a, mp_digit b) +void mp_set(mp_int *a, mp_digit b) { - mp_zero (a); - a->dp[0] = b & MP_MASK; - a->used = (a->dp[0] != 0) ? 1 : 0; + a->dp[0] = b & MP_MASK; + a->sign = MP_ZPOS; + a->used = (a->dp[0] != 0u) ? 1 : 0; + MP_ZERO_DIGITS(a->dp + a->used, a->alloc - a->used); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_set.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_set_double.c b/lib/hcrypto/libtommath/bn_mp_set_double.c new file mode 100644 index 000000000..a42fc70d9 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_double.c @@ -0,0 +1,47 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_DOUBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#if defined(__STDC_IEC_559__) || defined(__GCC_IEC_559) +mp_err mp_set_double(mp_int *a, double b) +{ + uint64_t frac; + int exp; + mp_err err; + union { + double dbl; + uint64_t bits; + } cast; + cast.dbl = b; + + exp = (int)((unsigned)(cast.bits >> 52) & 0x7FFu); + frac = (cast.bits & ((1uLL << 52) - 1uLL)) | (1uLL << 52); + + if (exp == 0x7FF) { /* +-inf, NaN */ + return MP_VAL; + } + exp -= 1023 + 52; + + mp_set_u64(a, frac); + + err = (exp < 0) ? mp_div_2d(a, -exp, a, NULL) : mp_mul_2d(a, exp, a); + if (err != MP_OKAY) { + return err; + } + + if (((cast.bits >> 63) != 0uLL) && !MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#else +/* pragma message() not supported by several compilers (in mostly older but still used versions) */ +# ifdef _MSC_VER +# pragma message("mp_set_double implementation is only available on platforms with IEEE754 floating point format") +# else +# warning "mp_set_double implementation is only available on platforms with IEEE754 floating point format" +# endif +#endif +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_i32.c b/lib/hcrypto/libtommath/bn_mp_set_i32.c new file mode 100644 index 000000000..df4513d37 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_i32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_i32, mp_set_u32, int32_t, uint32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_i64.c b/lib/hcrypto/libtommath/bn_mp_set_i64.c new file mode 100644 index 000000000..395103bf5 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_i64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_i64, mp_set_u64, int64_t, uint64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_int.c b/lib/hcrypto/libtommath/bn_mp_set_int.c deleted file mode 100644 index 3072e76e1..000000000 --- a/lib/hcrypto/libtommath/bn_mp_set_int.c +++ /dev/null @@ -1,48 +0,0 @@ -#include -#ifdef BN_MP_SET_INT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* set a 32-bit const */ -int mp_set_int (mp_int * a, unsigned long b) -{ - int x, res; - - mp_zero (a); - - /* set four bits at a time */ - for (x = 0; x < 8; x++) { - /* shift the number up four bits */ - if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) { - return res; - } - - /* OR in the top four bits of the source */ - a->dp[0] |= (b >> 28) & 15; - - /* shift the source up to the next four bits */ - b <<= 4; - - /* ensure that digits are not clamped off */ - a->used += 1; - } - mp_clamp (a); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_set_int.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_set_l.c b/lib/hcrypto/libtommath/bn_mp_set_l.c new file mode 100644 index 000000000..1e445fb62 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_l.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_l, mp_set_ul, long, unsigned long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_ll.c b/lib/hcrypto/libtommath/bn_mp_set_ll.c new file mode 100644 index 000000000..3e2324f32 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_ll.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_ll, mp_set_ull, long long, unsigned long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_u32.c b/lib/hcrypto/libtommath/bn_mp_set_u32.c new file mode 100644 index 000000000..18ba5e14c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_u32.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_u32, uint32_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_u64.c b/lib/hcrypto/libtommath/bn_mp_set_u64.c new file mode 100644 index 000000000..88fab6c54 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_u64.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_u64, uint64_t) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_ul.c b/lib/hcrypto/libtommath/bn_mp_set_ul.c new file mode 100644 index 000000000..adfd85c7f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_ul.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_ul, unsigned long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_set_ull.c b/lib/hcrypto/libtommath/bn_mp_set_ull.c new file mode 100644 index 000000000..8fbc1bd2c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_set_ull.c @@ -0,0 +1,7 @@ +#include "tommath_private.h" +#ifdef BN_MP_SET_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_ull, unsigned long long) +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_shrink.c b/lib/hcrypto/libtommath/bn_mp_shrink.c index c600efc58..cf27ed9ec 100644 --- a/lib/hcrypto/libtommath/bn_mp_shrink.c +++ b/lib/hcrypto/libtommath/bn_mp_shrink.c @@ -1,40 +1,22 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SHRINK_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* shrink a bignum */ -int mp_shrink (mp_int * a) +mp_err mp_shrink(mp_int *a) { - mp_digit *tmp; - int used = 1; - - if(a->used > 0) - used = a->used; - - if (a->alloc != used) { - if ((tmp = OPT_CAST(mp_digit) XREALLOC (a->dp, sizeof (mp_digit) * used)) == NULL) { - return MP_MEM; - } - a->dp = tmp; - a->alloc = used; - } - return MP_OKAY; + mp_digit *tmp; + int alloc = MP_MAX(MP_MIN_PREC, a->used); + if (a->alloc != alloc) { + if ((tmp = (mp_digit *) MP_REALLOC(a->dp, + (size_t)a->alloc * sizeof(mp_digit), + (size_t)alloc * sizeof(mp_digit))) == NULL) { + return MP_MEM; + } + a->dp = tmp; + a->alloc = alloc; + } + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_shrink.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_signed_bin_size.c b/lib/hcrypto/libtommath/bn_mp_signed_bin_size.c deleted file mode 100644 index 6739d19e2..000000000 --- a/lib/hcrypto/libtommath/bn_mp_signed_bin_size.c +++ /dev/null @@ -1,27 +0,0 @@ -#include -#ifdef BN_MP_SIGNED_BIN_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* get the size for an signed equivalent */ -int mp_signed_bin_size (mp_int * a) -{ - return 1 + mp_unsigned_bin_size (a); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_signed_bin_size.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_signed_rsh.c b/lib/hcrypto/libtommath/bn_mp_signed_rsh.c new file mode 100644 index 000000000..8d8d8414d --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_signed_rsh.c @@ -0,0 +1,22 @@ +#include "tommath_private.h" +#ifdef BN_MP_SIGNED_RSH_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift right by a certain bit count with sign extension */ +mp_err mp_signed_rsh(const mp_int *a, int b, mp_int *c) +{ + mp_err res; + if (a->sign == MP_ZPOS) { + return mp_div_2d(a, b, c, NULL); + } + + res = mp_add_d(a, 1uL, c); + if (res != MP_OKAY) { + return res; + } + + res = mp_div_2d(c, b, c, NULL); + return (res == MP_OKAY) ? mp_sub_d(c, 1uL, c) : res; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_sqr.c b/lib/hcrypto/libtommath/bn_mp_sqr.c index 90f4dd6d7..e0d0a73e4 100644 --- a/lib/hcrypto/libtommath/bn_mp_sqr.c +++ b/lib/hcrypto/libtommath/bn_mp_sqr.c @@ -1,58 +1,28 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* computes b = a*a */ -int -mp_sqr (mp_int * a, mp_int * b) +mp_err mp_sqr(const mp_int *a, mp_int *b) { - int res; - -#ifdef BN_MP_TOOM_SQR_C - /* use Toom-Cook? */ - if (a->used >= TOOM_SQR_CUTOFF) { - res = mp_toom_sqr(a, b); - /* Karatsuba? */ - } else -#endif -#ifdef BN_MP_KARATSUBA_SQR_C -if (a->used >= KARATSUBA_SQR_CUTOFF) { - res = mp_karatsuba_sqr (a, b); - } else -#endif - { -#ifdef BN_FAST_S_MP_SQR_C - /* can we use the fast comba multiplier? */ - if ((a->used * 2 + 1) < MP_WARRAY && - a->used < - (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) { - res = fast_s_mp_sqr (a, b); - } else -#endif -#ifdef BN_S_MP_SQR_C - res = s_mp_sqr (a, b); -#else - res = MP_VAL; -#endif - } - b->sign = MP_ZPOS; - return res; + mp_err err; + if (MP_HAS(S_MP_TOOM_SQR) && /* use Toom-Cook? */ + (a->used >= MP_TOOM_SQR_CUTOFF)) { + err = s_mp_toom_sqr(a, b); + } else if (MP_HAS(S_MP_KARATSUBA_SQR) && /* Karatsuba? */ + (a->used >= MP_KARATSUBA_SQR_CUTOFF)) { + err = s_mp_karatsuba_sqr(a, b); + } else if (MP_HAS(S_MP_SQR_FAST) && /* can we use the fast comba multiplier? */ + (((a->used * 2) + 1) < MP_WARRAY) && + (a->used < (MP_MAXFAST / 2))) { + err = s_mp_sqr_fast(a, b); + } else if (MP_HAS(S_MP_SQR)) { + err = s_mp_sqr(a, b); + } else { + err = MP_VAL; + } + b->sign = MP_ZPOS; + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_sqr.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_sqrmod.c b/lib/hcrypto/libtommath/bn_mp_sqrmod.c index 161cbbb30..626ea2c29 100644 --- a/lib/hcrypto/libtommath/bn_mp_sqrmod.c +++ b/lib/hcrypto/libtommath/bn_mp_sqrmod.c @@ -1,41 +1,25 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SQRMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* c = a * a (mod b) */ -int -mp_sqrmod (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_sqrmod(const mp_int *a, const mp_int *b, mp_int *c) { - int res; - mp_int t; + mp_err err; + mp_int t; - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } - if ((res = mp_sqr (a, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, b, c); - mp_clear (&t); - return res; + if ((err = mp_sqr(a, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, b, c); + +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_sqrmod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_sqrt.c b/lib/hcrypto/libtommath/bn_mp_sqrt.c index 8391297f7..82d682467 100644 --- a/lib/hcrypto/libtommath/bn_mp_sqrt.c +++ b/lib/hcrypto/libtommath/bn_mp_sqrt.c @@ -1,81 +1,67 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SQRT_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* this function is less generic than mp_n_root, simpler and faster */ -int mp_sqrt(mp_int *arg, mp_int *ret) +mp_err mp_sqrt(const mp_int *arg, mp_int *ret) { - int res; - mp_int t1,t2; + mp_err err; + mp_int t1, t2; - /* must be positive */ - if (arg->sign == MP_NEG) { - return MP_VAL; - } + /* must be positive */ + if (arg->sign == MP_NEG) { + return MP_VAL; + } - /* easy out */ - if (mp_iszero(arg) == MP_YES) { - mp_zero(ret); - return MP_OKAY; - } + /* easy out */ + if (MP_IS_ZERO(arg)) { + mp_zero(ret); + return MP_OKAY; + } - if ((res = mp_init_copy(&t1, arg)) != MP_OKAY) { - return res; - } + if ((err = mp_init_copy(&t1, arg)) != MP_OKAY) { + return err; + } - if ((res = mp_init(&t2)) != MP_OKAY) { - goto E2; - } + if ((err = mp_init(&t2)) != MP_OKAY) { + goto E2; + } - /* First approx. (not very bad for large arg) */ - mp_rshd (&t1,t1.used/2); + /* First approx. (not very bad for large arg) */ + mp_rshd(&t1, t1.used/2); - /* t1 > 0 */ - if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { - goto E1; - } - if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { - goto E1; - } - if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { - goto E1; - } - /* And now t1 > sqrt(arg) */ - do { - if ((res = mp_div(arg,&t1,&t2,NULL)) != MP_OKAY) { + /* t1 > 0 */ + if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) { goto E1; - } - if ((res = mp_add(&t1,&t2,&t1)) != MP_OKAY) { + } + if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) { goto E1; - } - if ((res = mp_div_2(&t1,&t1)) != MP_OKAY) { + } + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) { goto E1; - } - /* t1 >= sqrt(arg) >= t2 at this point */ - } while (mp_cmp_mag(&t1,&t2) == MP_GT); + } + /* And now t1 > sqrt(arg) */ + do { + if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) { + goto E1; + } + if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) { + goto E1; + } + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) { + goto E1; + } + /* t1 >= sqrt(arg) >= t2 at this point */ + } while (mp_cmp_mag(&t1, &t2) == MP_GT); - mp_exch(&t1,ret); + mp_exch(&t1, ret); -E1: mp_clear(&t2); -E2: mp_clear(&t1); - return res; +E1: + mp_clear(&t2); +E2: + mp_clear(&t1); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_sqrt.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_sqrtmod_prime.c b/lib/hcrypto/libtommath/bn_mp_sqrtmod_prime.c new file mode 100644 index 000000000..a833ed7c1 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_sqrtmod_prime.c @@ -0,0 +1,118 @@ +#include "tommath_private.h" +#ifdef BN_MP_SQRTMOD_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Tonelli-Shanks algorithm + * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm + * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html + * + */ + +mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret) +{ + mp_err err; + int legendre; + mp_int t1, C, Q, S, Z, M, T, R, two; + mp_digit i; + + /* first handle the simple cases */ + if (mp_cmp_d(n, 0uL) == MP_EQ) { + mp_zero(ret); + return MP_OKAY; + } + if (mp_cmp_d(prime, 2uL) == MP_EQ) return MP_VAL; /* prime must be odd */ + if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY) return err; + if (legendre == -1) return MP_VAL; /* quadratic non-residue mod prime */ + + if ((err = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) { + return err; + } + + /* SPECIAL CASE: if prime mod 4 == 3 + * compute directly: err = n^(prime+1)/4 mod prime + * Handbook of Applied Cryptography algorithm 3.36 + */ + if ((err = mp_mod_d(prime, 4uL, &i)) != MP_OKAY) goto cleanup; + if (i == 3u) { + if ((err = mp_add_d(prime, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup; + err = MP_OKAY; + goto cleanup; + } + + /* NOW: Tonelli-Shanks algorithm */ + + /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */ + if ((err = mp_copy(prime, &Q)) != MP_OKAY) goto cleanup; + if ((err = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY) goto cleanup; + /* Q = prime - 1 */ + mp_zero(&S); + /* S = 0 */ + while (MP_IS_EVEN(&Q)) { + if ((err = mp_div_2(&Q, &Q)) != MP_OKAY) goto cleanup; + /* Q = Q / 2 */ + if ((err = mp_add_d(&S, 1uL, &S)) != MP_OKAY) goto cleanup; + /* S = S + 1 */ + } + + /* find a Z such that the Legendre symbol (Z|prime) == -1 */ + mp_set_u32(&Z, 2u); + /* Z = 2 */ + for (;;) { + if ((err = mp_kronecker(&Z, prime, &legendre)) != MP_OKAY) goto cleanup; + if (legendre == -1) break; + if ((err = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY) goto cleanup; + /* Z = Z + 1 */ + } + + if ((err = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY) goto cleanup; + /* C = Z ^ Q mod prime */ + if ((err = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + /* t1 = (Q + 1) / 2 */ + if ((err = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = n ^ ((Q + 1) / 2) mod prime */ + if ((err = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY) goto cleanup; + /* T = n ^ Q mod prime */ + if ((err = mp_copy(&S, &M)) != MP_OKAY) goto cleanup; + /* M = S */ + mp_set_u32(&two, 2u); + + for (;;) { + if ((err = mp_copy(&T, &t1)) != MP_OKAY) goto cleanup; + i = 0; + for (;;) { + if (mp_cmp_d(&t1, 1uL) == MP_EQ) break; + if ((err = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup; + i++; + } + if (i == 0u) { + if ((err = mp_copy(&R, ret)) != MP_OKAY) goto cleanup; + err = MP_OKAY; + goto cleanup; + } + if ((err = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = 2 ^ (M - i - 1) */ + if ((err = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */ + if ((err = mp_sqrmod(&t1, prime, &C)) != MP_OKAY) goto cleanup; + /* C = (t1 * t1) mod prime */ + if ((err = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = (R * t1) mod prime */ + if ((err = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup; + /* T = (T * C) mod prime */ + mp_set(&M, i); + /* M = i */ + } + +cleanup: + mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_sub.c b/lib/hcrypto/libtommath/bn_mp_sub.c index f5015cce4..c1ea39e11 100644 --- a/lib/hcrypto/libtommath/bn_mp_sub.c +++ b/lib/hcrypto/libtommath/bn_mp_sub.c @@ -1,59 +1,40 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SUB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* high level subtraction (handles signs) */ -int -mp_sub (mp_int * a, mp_int * b, mp_int * c) +mp_err mp_sub(const mp_int *a, const mp_int *b, mp_int *c) { - int sa, sb, res; + mp_sign sa = a->sign, sb = b->sign; + mp_err err; - sa = a->sign; - sb = b->sign; - - if (sa != sb) { - /* subtract a negative from a positive, OR */ - /* subtract a positive from a negative. */ - /* In either case, ADD their magnitudes, */ - /* and use the sign of the first number. */ - c->sign = sa; - res = s_mp_add (a, b, c); - } else { - /* subtract a positive from a positive, OR */ - /* subtract a negative from a negative. */ - /* First, take the difference between their */ - /* magnitudes, then... */ - if (mp_cmp_mag (a, b) != MP_LT) { - /* Copy the sign from the first */ + if (sa != sb) { + /* subtract a negative from a positive, OR */ + /* subtract a positive from a negative. */ + /* In either case, ADD their magnitudes, */ + /* and use the sign of the first number. */ c->sign = sa; - /* The first has a larger or equal magnitude */ - res = s_mp_sub (a, b, c); - } else { - /* The result has the *opposite* sign from */ - /* the first number. */ - c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; - /* The second has a larger magnitude */ - res = s_mp_sub (b, a, c); - } - } - return res; + err = s_mp_add(a, b, c); + } else { + /* subtract a positive from a positive, OR */ + /* subtract a negative from a negative. */ + /* First, take the difference between their */ + /* magnitudes, then... */ + if (mp_cmp_mag(a, b) != MP_LT) { + /* Copy the sign from the first */ + c->sign = sa; + /* The first has a larger or equal magnitude */ + err = s_mp_sub(a, b, c); + } else { + /* The result has the *opposite* sign from */ + /* the first number. */ + c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; + /* The second has a larger magnitude */ + err = s_mp_sub(b, a, c); + } + } + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_sub.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_sub_d.c b/lib/hcrypto/libtommath/bn_mp_sub_d.c index 06cdca636..3ebf9b485 100644 --- a/lib/hcrypto/libtommath/bn_mp_sub_d.c +++ b/lib/hcrypto/libtommath/bn_mp_sub_d.c @@ -1,93 +1,74 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SUB_D_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* single digit subtraction */ -int -mp_sub_d (mp_int * a, mp_digit b, mp_int * c) +mp_err mp_sub_d(const mp_int *a, mp_digit b, mp_int *c) { - mp_digit *tmpa, *tmpc, mu; - int res, ix, oldused; + mp_digit *tmpa, *tmpc; + mp_err err; + int ix, oldused; - /* grow c as required */ - if (c->alloc < a->used + 1) { - if ((res = mp_grow(c, a->used + 1)) != MP_OKAY) { - return res; - } - } + /* grow c as required */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } - /* if a is negative just do an unsigned - * addition [with fudged signs] - */ - if (a->sign == MP_NEG) { - a->sign = MP_ZPOS; - res = mp_add_d(a, b, c); - a->sign = c->sign = MP_NEG; + /* if a is negative just do an unsigned + * addition [with fudged signs] + */ + if (a->sign == MP_NEG) { + mp_int a_ = *a; + a_.sign = MP_ZPOS; + err = mp_add_d(&a_, b, c); + c->sign = MP_NEG; - /* clamp */ - mp_clamp(c); + /* clamp */ + mp_clamp(c); - return res; - } + return err; + } - /* setup regs */ - oldused = c->used; - tmpa = a->dp; - tmpc = c->dp; + /* setup regs */ + oldused = c->used; + tmpa = a->dp; + tmpc = c->dp; - /* if a <= b simply fix the single digit */ - if ((a->used == 1 && a->dp[0] <= b) || a->used == 0) { - if (a->used == 1) { - *tmpc++ = b - *tmpa; - } else { - *tmpc++ = b; - } - ix = 1; + /* if a <= b simply fix the single digit */ + if (((a->used == 1) && (a->dp[0] <= b)) || (a->used == 0)) { + if (a->used == 1) { + *tmpc++ = b - *tmpa; + } else { + *tmpc++ = b; + } + ix = 1; - /* negative/1digit */ - c->sign = MP_NEG; - c->used = 1; - } else { - /* positive/size */ - c->sign = MP_ZPOS; - c->used = a->used; + /* negative/1digit */ + c->sign = MP_NEG; + c->used = 1; + } else { + mp_digit mu = b; - /* subtract first digit */ - *tmpc = *tmpa++ - b; - mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); - *tmpc++ &= MP_MASK; + /* positive/size */ + c->sign = MP_ZPOS; + c->used = a->used; - /* handle rest of the digits */ - for (ix = 1; ix < a->used; ix++) { - *tmpc = *tmpa++ - mu; - mu = *tmpc >> (sizeof(mp_digit) * CHAR_BIT - 1); - *tmpc++ &= MP_MASK; - } - } + /* subtract digits, mu is carry */ + for (ix = 0; ix < a->used; ix++) { + *tmpc = *tmpa++ - mu; + mu = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); + *tmpc++ &= MP_MASK; + } + } - /* zero excess digits */ - while (ix++ < oldused) { - *tmpc++ = 0; - } - mp_clamp(c); - return MP_OKAY; + /* zero excess digits */ + MP_ZERO_DIGITS(tmpc, oldused - ix); + + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_sub_d.c,v $ */ -/* $Revision: 1.6 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_submod.c b/lib/hcrypto/libtommath/bn_mp_submod.c index 869e23cde..5ebd37498 100644 --- a/lib/hcrypto/libtommath/bn_mp_submod.c +++ b/lib/hcrypto/libtommath/bn_mp_submod.c @@ -1,42 +1,25 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_SUBMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* d = a - b (mod c) */ -int -mp_submod (mp_int * a, mp_int * b, mp_int * c, mp_int * d) +mp_err mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) { - int res; - mp_int t; + mp_err err; + mp_int t; + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } - if ((res = mp_init (&t)) != MP_OKAY) { - return res; - } + if ((err = mp_sub(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); - if ((res = mp_sub (a, b, &t)) != MP_OKAY) { - mp_clear (&t); - return res; - } - res = mp_mod (&t, c, d); - mp_clear (&t); - return res; +LBL_ERR: + mp_clear(&t); + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_submod.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_to_radix.c b/lib/hcrypto/libtommath/bn_mp_to_radix.c new file mode 100644 index 000000000..7fa86cae1 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_to_radix.c @@ -0,0 +1,84 @@ +#include "tommath_private.h" +#ifdef BN_MP_TO_RADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* stores a bignum as a ASCII string in a given radix (2..64) + * + * Stores upto "size - 1" chars and always a NULL byte, puts the number of characters + * written, including the '\0', in "written". + */ +mp_err mp_to_radix(const mp_int *a, char *str, size_t maxlen, size_t *written, int radix) +{ + size_t digs; + mp_err err; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of radix and size*/ + if (maxlen < 2u) { + return MP_BUF; + } + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } + + /* quick out if its zero */ + if (MP_IS_ZERO(a)) { + *str++ = '0'; + *str = '\0'; + if (written != NULL) { + *written = 2u; + } + return MP_OKAY; + } + + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + /* we have to reverse our digits later... but not the - sign!! */ + ++_s; + + /* store the flag and mark the number as positive */ + *str++ = '-'; + t.sign = MP_ZPOS; + + /* subtract a char */ + --maxlen; + } + digs = 0u; + while (!MP_IS_ZERO(&t)) { + if (--maxlen < 1u) { + /* no more room */ + err = MP_BUF; + goto LBL_ERR; + } + if ((err = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) { + goto LBL_ERR; + } + *str++ = mp_s_rmap[d]; + ++digs; + } + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number + */ + s_mp_reverse((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + digs++; + + if (written != NULL) { + *written = (a->sign == MP_NEG) ? (digs + 1u): digs; + } + +LBL_ERR: + mp_clear(&t); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_to_sbin.c b/lib/hcrypto/libtommath/bn_mp_to_sbin.c new file mode 100644 index 000000000..dbaf53e1f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_to_sbin.c @@ -0,0 +1,22 @@ +#include "tommath_private.h" +#ifdef BN_MP_TO_SBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* store in signed [big endian] format */ +mp_err mp_to_sbin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) +{ + mp_err err; + if (maxlen == 0u) { + return MP_BUF; + } + if ((err = mp_to_ubin(a, buf + 1, maxlen - 1u, written)) != MP_OKAY) { + return err; + } + if (written != NULL) { + (*written)++; + } + buf[0] = (a->sign == MP_ZPOS) ? (unsigned char)0 : (unsigned char)1; + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_to_signed_bin.c b/lib/hcrypto/libtommath/bn_mp_to_signed_bin.c deleted file mode 100644 index 9df83ca52..000000000 --- a/lib/hcrypto/libtommath/bn_mp_to_signed_bin.c +++ /dev/null @@ -1,33 +0,0 @@ -#include -#ifdef BN_MP_TO_SIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* store in signed [big endian] format */ -int mp_to_signed_bin (mp_int * a, unsigned char *b) -{ - int res; - - if ((res = mp_to_unsigned_bin (a, b + 1)) != MP_OKAY) { - return res; - } - b[0] = (unsigned char) ((a->sign == MP_ZPOS) ? 0 : 1); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_to_signed_bin.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_to_signed_bin_n.c b/lib/hcrypto/libtommath/bn_mp_to_signed_bin_n.c deleted file mode 100644 index 677f827d4..000000000 --- a/lib/hcrypto/libtommath/bn_mp_to_signed_bin_n.c +++ /dev/null @@ -1,31 +0,0 @@ -#include -#ifdef BN_MP_TO_SIGNED_BIN_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* store in signed [big endian] format */ -int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) -{ - if (*outlen < (unsigned long)mp_signed_bin_size(a)) { - return MP_VAL; - } - *outlen = mp_signed_bin_size(a); - return mp_to_signed_bin(a, b); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_to_signed_bin_n.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_to_ubin.c b/lib/hcrypto/libtommath/bn_mp_to_ubin.c new file mode 100644 index 000000000..1681ca7ce --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_to_ubin.c @@ -0,0 +1,41 @@ +#include "tommath_private.h" +#ifdef BN_MP_TO_UBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* store in unsigned [big endian] format */ +mp_err mp_to_ubin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) +{ + size_t x, count; + mp_err err; + mp_int t; + + count = mp_ubin_size(a); + if (count > maxlen) { + return MP_BUF; + } + + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } + + for (x = count; x --> 0u;) { +#ifndef MP_8BIT + buf[x] = (unsigned char)(t.dp[0] & 255u); +#else + buf[x] = (unsigned char)(t.dp[0] | ((t.dp[1] & 1u) << 7)); +#endif + if ((err = mp_div_2d(&t, 8, &t, NULL)) != MP_OKAY) { + goto LBL_ERR; + } + } + + if (written != NULL) { + *written = count; + } + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin.c b/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin.c deleted file mode 100644 index c137f104a..000000000 --- a/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin.c +++ /dev/null @@ -1,48 +0,0 @@ -#include -#ifdef BN_MP_TO_UNSIGNED_BIN_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* store in unsigned [big endian] format */ -int mp_to_unsigned_bin (mp_int * a, unsigned char *b) -{ - int x, res; - mp_int t; - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - x = 0; - while (mp_iszero (&t) == 0) { -#ifndef MP_8BIT - b[x++] = (unsigned char) (t.dp[0] & 255); -#else - b[x++] = (unsigned char) (t.dp[0] | ((t.dp[1] & 0x01) << 7)); -#endif - if ((res = mp_div_2d (&t, 8, &t, NULL)) != MP_OKAY) { - mp_clear (&t); - return res; - } - } - bn_reverse (b, x); - mp_clear (&t); - return MP_OKAY; -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_to_unsigned_bin.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin_n.c b/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin_n.c deleted file mode 100644 index 0dc00c623..000000000 --- a/lib/hcrypto/libtommath/bn_mp_to_unsigned_bin_n.c +++ /dev/null @@ -1,31 +0,0 @@ -#include -#ifdef BN_MP_TO_UNSIGNED_BIN_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* store in unsigned [big endian] format */ -int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen) -{ - if (*outlen < (unsigned long)mp_unsigned_bin_size(a)) { - return MP_VAL; - } - *outlen = mp_unsigned_bin_size(a); - return mp_to_unsigned_bin(a, b); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_to_unsigned_bin_n.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_toom_mul.c b/lib/hcrypto/libtommath/bn_mp_toom_mul.c deleted file mode 100644 index b99634246..000000000 --- a/lib/hcrypto/libtommath/bn_mp_toom_mul.c +++ /dev/null @@ -1,284 +0,0 @@ -#include -#ifdef BN_MP_TOOM_MUL_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* multiplication using the Toom-Cook 3-way algorithm - * - * Much more complicated than Karatsuba but has a lower - * asymptotic running time of O(N**1.464). This algorithm is - * only particularly useful on VERY large inputs - * (we're talking 1000s of digits here...). -*/ -int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c) -{ - mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2; - int res, B; - - /* init temps */ - if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, - &a0, &a1, &a2, &b0, &b1, - &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) { - return res; - } - - /* B */ - B = MIN(a->used, b->used) / 3; - - /* a = a2 * B**2 + a1 * B + a0 */ - if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(a, &a1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a1, B); - mp_mod_2d(&a1, DIGIT_BIT * B, &a1); - - if ((res = mp_copy(a, &a2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a2, B*2); - - /* b = b2 * B**2 + b1 * B + b0 */ - if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(b, &b1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&b1, B); - mp_mod_2d(&b1, DIGIT_BIT * B, &b1); - - if ((res = mp_copy(b, &b2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&b2, B*2); - - /* w0 = a0*b0 */ - if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) { - goto ERR; - } - - /* w4 = a2 * b2 */ - if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) { - goto ERR; - } - - /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */ - if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) { - goto ERR; - } - - /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */ - if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) { - goto ERR; - } - - - /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */ - if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) { - goto ERR; - } - - /* now solve the matrix - - 0 0 0 0 1 - 1 2 4 8 16 - 1 1 1 1 1 - 16 8 4 2 1 - 1 0 0 0 0 - - using 12 subtractions, 4 shifts, - 2 small divisions and 1 small multiplication - */ - - /* r1 - r4 */ - if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r0 */ - if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/2 */ - if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3/2 */ - if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { - goto ERR; - } - /* r2 - r0 - r4 */ - if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1 - 8r0 */ - if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - 8r4 */ - if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - /* 3r2 - r1 - r3 */ - if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/3 */ - if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { - goto ERR; - } - /* r3/3 */ - if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { - goto ERR; - } - - /* at this point shift W[n] by B*n */ - if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) { - goto ERR; - } - -ERR: - mp_clear_multi(&w0, &w1, &w2, &w3, &w4, - &a0, &a1, &a2, &b0, &b1, - &b2, &tmp1, &tmp2, NULL); - return res; -} - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_toom_mul.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_toom_sqr.c b/lib/hcrypto/libtommath/bn_mp_toom_sqr.c deleted file mode 100644 index 48880d035..000000000 --- a/lib/hcrypto/libtommath/bn_mp_toom_sqr.c +++ /dev/null @@ -1,226 +0,0 @@ -#include -#ifdef BN_MP_TOOM_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* squaring using Toom-Cook 3-way algorithm */ -int -mp_toom_sqr(mp_int *a, mp_int *b) -{ - mp_int w0, w1, w2, w3, w4, tmp1, a0, a1, a2; - int res, B; - - /* init temps */ - if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL)) != MP_OKAY) { - return res; - } - - /* B */ - B = a->used / 3; - - /* a = a2 * B**2 + a1 * B + a0 */ - if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_copy(a, &a1)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a1, B); - mp_mod_2d(&a1, DIGIT_BIT * B, &a1); - - if ((res = mp_copy(a, &a2)) != MP_OKAY) { - goto ERR; - } - mp_rshd(&a2, B*2); - - /* w0 = a0*a0 */ - if ((res = mp_sqr(&a0, &w0)) != MP_OKAY) { - goto ERR; - } - - /* w4 = a2 * a2 */ - if ((res = mp_sqr(&a2, &w4)) != MP_OKAY) { - goto ERR; - } - - /* w1 = (a2 + 2(a1 + 2a0))**2 */ - if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_sqr(&tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - - /* w3 = (a0 + 2(a1 + 2a2))**2 */ - if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_sqr(&tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - - - /* w2 = (a2 + a1 + a0)**2 */ - if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sqr(&tmp1, &w2)) != MP_OKAY) { - goto ERR; - } - - /* now solve the matrix - - 0 0 0 0 1 - 1 2 4 8 16 - 1 1 1 1 1 - 16 8 4 2 1 - 1 0 0 0 0 - - using 12 subtractions, 4 shifts, 2 small divisions and 1 small multiplication. - */ - - /* r1 - r4 */ - if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r0 */ - if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/2 */ - if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3/2 */ - if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) { - goto ERR; - } - /* r2 - r0 - r4 */ - if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1 - 8r0 */ - if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - 8r4 */ - if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) { - goto ERR; - } - /* 3r2 - r1 - r3 */ - if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) { - goto ERR; - } - /* r1 - r2 */ - if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) { - goto ERR; - } - /* r3 - r2 */ - if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) { - goto ERR; - } - /* r1/3 */ - if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) { - goto ERR; - } - /* r3/3 */ - if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) { - goto ERR; - } - - /* at this point shift W[n] by B*n */ - if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) { - goto ERR; - } - - if ((res = mp_add(&w0, &w1, b)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) { - goto ERR; - } - if ((res = mp_add(&tmp1, b, b)) != MP_OKAY) { - goto ERR; - } - -ERR: - mp_clear_multi(&w0, &w1, &w2, &w3, &w4, &a0, &a1, &a2, &tmp1, NULL); - return res; -} - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_toom_sqr.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_toradix.c b/lib/hcrypto/libtommath/bn_mp_toradix.c deleted file mode 100644 index 0adc28d2f..000000000 --- a/lib/hcrypto/libtommath/bn_mp_toradix.c +++ /dev/null @@ -1,75 +0,0 @@ -#include -#ifdef BN_MP_TORADIX_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* stores a bignum as a ASCII string in a given radix (2..64) */ -int mp_toradix (mp_int * a, char *str, int radix) -{ - int res, digs; - mp_int t; - mp_digit d; - char *_s = str; - - /* check range of the radix */ - if (radix < 2 || radix > 64) { - return MP_VAL; - } - - /* quick out if its zero */ - if (mp_iszero(a) == 1) { - *str++ = '0'; - *str = '\0'; - return MP_OKAY; - } - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - /* if it is negative output a - */ - if (t.sign == MP_NEG) { - ++_s; - *str++ = '-'; - t.sign = MP_ZPOS; - } - - digs = 0; - while (mp_iszero (&t) == 0) { - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - *str++ = mp_s_rmap[d]; - ++digs; - } - - /* reverse the digits of the string. In this case _s points - * to the first digit [exluding the sign] of the number] - */ - bn_reverse ((unsigned char *)_s, digs); - - /* append a NULL so the string is properly terminated */ - *str = '\0'; - - mp_clear (&t); - return MP_OKAY; -} - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_toradix.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_toradix_n.c b/lib/hcrypto/libtommath/bn_mp_toradix_n.c deleted file mode 100644 index 28085124e..000000000 --- a/lib/hcrypto/libtommath/bn_mp_toradix_n.c +++ /dev/null @@ -1,88 +0,0 @@ -#include -#ifdef BN_MP_TORADIX_N_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* stores a bignum as a ASCII string in a given radix (2..64) - * - * Stores upto maxlen-1 chars and always a NULL byte - */ -int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen) -{ - int res, digs; - mp_int t; - mp_digit d; - char *_s = str; - - /* check range of the maxlen, radix */ - if (maxlen < 2 || radix < 2 || radix > 64) { - return MP_VAL; - } - - /* quick out if its zero */ - if (mp_iszero(a) == MP_YES) { - *str++ = '0'; - *str = '\0'; - return MP_OKAY; - } - - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - - /* if it is negative output a - */ - if (t.sign == MP_NEG) { - /* we have to reverse our digits later... but not the - sign!! */ - ++_s; - - /* store the flag and mark the number as positive */ - *str++ = '-'; - t.sign = MP_ZPOS; - - /* subtract a char */ - --maxlen; - } - - digs = 0; - while (mp_iszero (&t) == 0) { - if (--maxlen < 1) { - /* no more room */ - break; - } - if ((res = mp_div_d (&t, (mp_digit) radix, &t, &d)) != MP_OKAY) { - mp_clear (&t); - return res; - } - *str++ = mp_s_rmap[d]; - ++digs; - } - - /* reverse the digits of the string. In this case _s points - * to the first digit [exluding the sign] of the number - */ - bn_reverse ((unsigned char *)_s, digs); - - /* append a NULL so the string is properly terminated */ - *str = '\0'; - - mp_clear (&t); - return MP_OKAY; -} - -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_toradix_n.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_ubin_size.c b/lib/hcrypto/libtommath/bn_mp_ubin_size.c new file mode 100644 index 000000000..21230b48c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_ubin_size.c @@ -0,0 +1,12 @@ +#include "tommath_private.h" +#ifdef BN_MP_UBIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* get the size for an unsigned equivalent */ +size_t mp_ubin_size(const mp_int *a) +{ + size_t size = (size_t)mp_count_bits(a); + return (size / 8u) + (((size & 7u) != 0u) ? 1u : 0u); +} +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_unpack.c b/lib/hcrypto/libtommath/bn_mp_unpack.c new file mode 100644 index 000000000..d4eb90e0c --- /dev/null +++ b/lib/hcrypto/libtommath/bn_mp_unpack.c @@ -0,0 +1,49 @@ +#include "tommath_private.h" +#ifdef BN_MP_UNPACK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* based on gmp's mpz_import. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +mp_err mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, + mp_endian endian, size_t nails, const void *op) +{ + mp_err err; + size_t odd_nails, nail_bytes, i, j; + unsigned char odd_nail_mask; + + mp_zero(rop); + + if (endian == MP_NATIVE_ENDIAN) { + MP_GET_ENDIANNESS(endian); + } + + odd_nails = (nails % 8u); + odd_nail_mask = 0xff; + for (i = 0; i < odd_nails; ++i) { + odd_nail_mask ^= (unsigned char)(1u << (7u - i)); + } + nail_bytes = nails / 8u; + + for (i = 0; i < count; ++i) { + for (j = 0; j < (size - nail_bytes); ++j) { + unsigned char byte = *((const unsigned char *)op + + (((order == MP_MSB_FIRST) ? i : ((count - 1u) - i)) * size) + + ((endian == MP_BIG_ENDIAN) ? (j + nail_bytes) : (((size - 1u) - j) - nail_bytes))); + + if ((err = mp_mul_2d(rop, (j == 0u) ? (int)(8u - odd_nails) : 8, rop)) != MP_OKAY) { + return err; + } + + rop->dp[0] |= (j == 0u) ? (mp_digit)(byte & odd_nail_mask) : (mp_digit)byte; + rop->used += 1; + } + } + + mp_clamp(rop); + + return MP_OKAY; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_mp_unsigned_bin_size.c b/lib/hcrypto/libtommath/bn_mp_unsigned_bin_size.c deleted file mode 100644 index 6dc3bd5fc..000000000 --- a/lib/hcrypto/libtommath/bn_mp_unsigned_bin_size.c +++ /dev/null @@ -1,28 +0,0 @@ -#include -#ifdef BN_MP_UNSIGNED_BIN_SIZE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* get the size for an unsigned equivalent */ -int mp_unsigned_bin_size (mp_int * a) -{ - int size = mp_count_bits (a); - return (size / 8 + ((size & 7) != 0 ? 1 : 0)); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_unsigned_bin_size.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_xor.c b/lib/hcrypto/libtommath/bn_mp_xor.c index 59ff2e183..71e7ca187 100644 --- a/lib/hcrypto/libtommath/bn_mp_xor.c +++ b/lib/hcrypto/libtommath/bn_mp_xor.c @@ -1,51 +1,56 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_XOR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ -/* XOR two ints together */ -int -mp_xor (mp_int * a, mp_int * b, mp_int * c) +/* two complement xor */ +mp_err mp_xor(const mp_int *a, const mp_int *b, mp_int *c) { - int res, ix, px; - mp_int t, *x; + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = (a->sign != b->sign) ? MP_NEG : MP_ZPOS; - if (a->used > b->used) { - if ((res = mp_init_copy (&t, a)) != MP_OKAY) { - return res; - } - px = b->used; - x = b; - } else { - if ((res = mp_init_copy (&t, b)) != MP_OKAY) { - return res; - } - px = a->used; - x = a; - } + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } - for (ix = 0; ix < px; ix++) { - t.dp[ix] ^= x->dp[ix]; - } - mp_clamp (&t); - mp_exch (c, &t); - mp_clear (&t); - return MP_OKAY; + for (i = 0; i < used; i++) { + mp_digit x, y; + + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } + + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x ^ y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_xor.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_zero.c b/lib/hcrypto/libtommath/bn_mp_zero.c index b0977d443..72a255efc 100644 --- a/lib/hcrypto/libtommath/bn_mp_zero.c +++ b/lib/hcrypto/libtommath/bn_mp_zero.c @@ -1,36 +1,13 @@ -#include +#include "tommath_private.h" #ifdef BN_MP_ZERO_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* set to zero */ -void mp_zero (mp_int * a) +void mp_zero(mp_int *a) { - int n; - mp_digit *tmp; - - a->sign = MP_ZPOS; - a->used = 0; - - tmp = a->dp; - for (n = 0; n < a->alloc; n++) { - *tmp++ = 0; - } + a->sign = MP_ZPOS; + a->used = 0; + MP_ZERO_DIGITS(a->dp, a->alloc); } #endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_zero.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_mp_zero_multi.c b/lib/hcrypto/libtommath/bn_mp_zero_multi.c deleted file mode 100644 index 339a75fbf..000000000 --- a/lib/hcrypto/libtommath/bn_mp_zero_multi.c +++ /dev/null @@ -1,35 +0,0 @@ -#include -#ifdef BN_MP_ZERO_MULTI_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ -#include - -/* set to zero */ -void mp_zero_multi (mp_int * mp, ...) -{ - mp_int* next_mp = mp; - va_list args; - va_start(args, mp); - while (next_mp != NULL) { - mp_zero(next_mp); - next_mp = va_arg(args, mp_int*); - } - va_end(args); -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_mp_zero.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_prime_tab.c b/lib/hcrypto/libtommath/bn_prime_tab.c index bd252477e..a6c07f8da 100644 --- a/lib/hcrypto/libtommath/bn_prime_tab.c +++ b/lib/hcrypto/libtommath/bn_prime_tab.c @@ -1,61 +1,61 @@ -#include +#include "tommath_private.h" #ifdef BN_PRIME_TAB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + const mp_digit ltm_prime_tab[] = { - 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, - 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, - 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, - 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, + 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, + 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, + 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, + 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, #ifndef MP_8BIT - 0x0083, - 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, - 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, - 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, - 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, + 0x0083, + 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, + 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, + 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, + 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, - 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, - 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, - 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, - 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, - 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, - 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, - 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, - 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, + 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, + 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, + 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, + 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, + 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, + 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, + 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, + 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, - 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, - 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, - 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, - 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, - 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, - 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, - 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, - 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, + 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, + 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, + 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, + 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, + 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, + 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, + 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, + 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, - 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, - 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, - 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, - 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, - 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, - 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, - 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, - 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 + 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, + 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, + 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, + 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, + 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, + 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, + 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, + 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 #endif }; + +#if defined(__GNUC__) && __GNUC__ >= 4 +#pragma GCC diagnostic push +#pragma GCC diagnostic ignored "-Wdeprecated-declarations" +const mp_digit *s_mp_prime_tab = ltm_prime_tab; +#pragma GCC diagnostic pop +#elif defined(_MSC_VER) && _MSC_VER >= 1500 +#pragma warning(push) +#pragma warning(disable: 4996) +const mp_digit *s_mp_prime_tab = ltm_prime_tab; +#pragma warning(pop) +#else +const mp_digit *s_mp_prime_tab = ltm_prime_tab; #endif -/* $Source: /cvs/libtom/libtommath/bn_prime_tab.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ +#endif diff --git a/lib/hcrypto/libtommath/bn_reverse.c b/lib/hcrypto/libtommath/bn_reverse.c deleted file mode 100644 index ddfa827a0..000000000 --- a/lib/hcrypto/libtommath/bn_reverse.c +++ /dev/null @@ -1,39 +0,0 @@ -#include -#ifdef BN_REVERSE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* reverse an array, used for radix code */ -void -bn_reverse (unsigned char *s, int len) -{ - int ix, iy; - unsigned char t; - - ix = 0; - iy = len - 1; - while (ix < iy) { - t = s[ix]; - s[ix] = s[iy]; - s[iy] = t; - ++ix; - --iy; - } -} -#endif - -/* $Source: /cvs/libtom/libtommath/bn_reverse.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_add.c b/lib/hcrypto/libtommath/bn_s_mp_add.c index e7f54f4cf..c946aa80d 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_add.c +++ b/lib/hcrypto/libtommath/bn_s_mp_add.c @@ -1,109 +1,91 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_ADD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* low level addition, based on HAC pp.594, Algorithm 14.7 */ -int -s_mp_add (mp_int * a, mp_int * b, mp_int * c) +mp_err s_mp_add(const mp_int *a, const mp_int *b, mp_int *c) { - mp_int *x; - int olduse, res, min, max; + const mp_int *x; + mp_err err; + int olduse, min, max; - /* find sizes, we let |a| <= |b| which means we have to sort - * them. "x" will point to the input with the most digits - */ - if (a->used > b->used) { - min = b->used; - max = a->used; - x = a; - } else { - min = a->used; - max = b->used; - x = b; - } + /* find sizes, we let |a| <= |b| which means we have to sort + * them. "x" will point to the input with the most digits + */ + if (a->used > b->used) { + min = b->used; + max = a->used; + x = a; + } else { + min = a->used; + max = b->used; + x = b; + } - /* init result */ - if (c->alloc < max + 1) { - if ((res = mp_grow (c, max + 1)) != MP_OKAY) { - return res; - } - } - - /* get old used digit count and set new one */ - olduse = c->used; - c->used = max + 1; - - { - register mp_digit u, *tmpa, *tmpb, *tmpc; - register int i; - - /* alias for digit pointers */ - - /* first input */ - tmpa = a->dp; - - /* second input */ - tmpb = b->dp; - - /* destination */ - tmpc = c->dp; - - /* zero the carry */ - u = 0; - for (i = 0; i < min; i++) { - /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ - *tmpc = *tmpa++ + *tmpb++ + u; - - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)DIGIT_BIT); - - /* take away carry bit from T[i] */ - *tmpc++ &= MP_MASK; - } - - /* now copy higher words if any, that is in A+B - * if A or B has more digits add those in - */ - if (min != max) { - for (; i < max; i++) { - /* T[i] = X[i] + U */ - *tmpc = x->dp[i] + u; - - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)DIGIT_BIT); - - /* take away carry bit from T[i] */ - *tmpc++ &= MP_MASK; + /* init result */ + if (c->alloc < (max + 1)) { + if ((err = mp_grow(c, max + 1)) != MP_OKAY) { + return err; } - } + } - /* add carry */ - *tmpc++ = u; + /* get old used digit count and set new one */ + olduse = c->used; + c->used = max + 1; - /* clear digits above oldused */ - for (i = c->used; i < olduse; i++) { - *tmpc++ = 0; - } - } + { + mp_digit u, *tmpa, *tmpb, *tmpc; + int i; - mp_clamp (c); - return MP_OKAY; + /* alias for digit pointers */ + + /* first input */ + tmpa = a->dp; + + /* second input */ + tmpb = b->dp; + + /* destination */ + tmpc = c->dp; + + /* zero the carry */ + u = 0; + for (i = 0; i < min; i++) { + /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ + *tmpc = *tmpa++ + *tmpb++ + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> (mp_digit)MP_DIGIT_BIT; + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, that is in A+B + * if A or B has more digits add those in + */ + if (min != max) { + for (; i < max; i++) { + /* T[i] = X[i] + U */ + *tmpc = x->dp[i] + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> (mp_digit)MP_DIGIT_BIT; + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + } + + /* add carry */ + *tmpc++ = u; + + /* clear digits above oldused */ + MP_ZERO_DIGITS(tmpc, olduse - c->used); + } + + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_add.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_balance_mul.c b/lib/hcrypto/libtommath/bn_s_mp_balance_mul.c new file mode 100644 index 000000000..7ece5d794 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_balance_mul.c @@ -0,0 +1,81 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_BALANCE_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* single-digit multiplication with the smaller number as the single-digit */ +mp_err s_mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + int count, len_a, len_b, nblocks, i, j, bsize; + mp_int a0, tmp, A, B, r; + mp_err err; + + len_a = a->used; + len_b = b->used; + + nblocks = MP_MAX(a->used, b->used) / MP_MIN(a->used, b->used); + bsize = MP_MIN(a->used, b->used) ; + + if ((err = mp_init_size(&a0, bsize + 2)) != MP_OKAY) { + return err; + } + if ((err = mp_init_multi(&tmp, &r, NULL)) != MP_OKAY) { + mp_clear(&a0); + return err; + } + + /* Make sure that A is the larger one*/ + if (len_a < len_b) { + B = *a; + A = *b; + } else { + A = *a; + B = *b; + } + + for (i = 0, j=0; i < nblocks; i++) { + /* Cut a slice off of a */ + a0.used = 0; + for (count = 0; count < bsize; count++) { + a0.dp[count] = A.dp[ j++ ]; + a0.used++; + } + mp_clamp(&a0); + /* Multiply with b */ + if ((err = mp_mul(&a0, &B, &tmp)) != MP_OKAY) { + goto LBL_ERR; + } + /* Shift tmp to the correct position */ + if ((err = mp_lshd(&tmp, bsize * i)) != MP_OKAY) { + goto LBL_ERR; + } + /* Add to output. No carry needed */ + if ((err = mp_add(&r, &tmp, &r)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* The left-overs; there are always left-overs */ + if (j < A.used) { + a0.used = 0; + for (count = 0; j < A.used; count++) { + a0.dp[count] = A.dp[ j++ ]; + a0.used++; + } + mp_clamp(&a0); + if ((err = mp_mul(&a0, &B, &tmp)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_lshd(&tmp, bsize * i)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_add(&r, &tmp, &r)) != MP_OKAY) { + goto LBL_ERR; + } + } + + mp_exch(&r,c); +LBL_ERR: + mp_clear_multi(&a0, &tmp, &r,NULL); + return err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_exptmod.c b/lib/hcrypto/libtommath/bn_s_mp_exptmod.c index deb4b4ddb..c3bfa95e8 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_exptmod.c +++ b/lib/hcrypto/libtommath/bn_s_mp_exptmod.c @@ -1,252 +1,198 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_EXPTMOD_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + #ifdef MP_LOW_MEM - #define TAB_SIZE 32 +# define TAB_SIZE 32 +# define MAX_WINSIZE 5 #else - #define TAB_SIZE 256 +# define TAB_SIZE 256 +# define MAX_WINSIZE 0 #endif -int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode) +mp_err s_mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) { - mp_int M[TAB_SIZE], res, mu; - mp_digit buf; - int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; - int (*redux)(mp_int*,mp_int*,mp_int*); + mp_int M[TAB_SIZE], res, mu; + mp_digit buf; + mp_err err; + int bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + mp_err(*redux)(mp_int *x, const mp_int *m, const mp_int *mu); - /* find window size */ - x = mp_count_bits (X); - if (x <= 7) { - winsize = 2; - } else if (x <= 36) { - winsize = 3; - } else if (x <= 140) { - winsize = 4; - } else if (x <= 450) { - winsize = 5; - } else if (x <= 1303) { - winsize = 6; - } else if (x <= 3529) { - winsize = 7; - } else { - winsize = 8; - } + /* find window size */ + x = mp_count_bits(X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } -#ifdef MP_LOW_MEM - if (winsize > 5) { - winsize = 5; - } -#endif + winsize = MAX_WINSIZE ? MP_MIN(MAX_WINSIZE, winsize) : winsize; - /* init M array */ - /* init first cell */ - if ((err = mp_init(&M[1])) != MP_OKAY) { - return err; - } - - /* now init the second half of the array */ - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - if ((err = mp_init(&M[x])) != MP_OKAY) { - for (y = 1<<(winsize-1); y < x; y++) { - mp_clear (&M[y]); - } - mp_clear(&M[1]); + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { return err; - } - } + } - /* create mu, used for Barrett reduction */ - if ((err = mp_init (&mu)) != MP_OKAY) { - goto LBL_M; - } - - if (redmode == 0) { - if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) { - goto LBL_MU; - } - redux = mp_reduce; - } else { - if ((err = mp_reduce_2k_setup_l (P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - redux = mp_reduce_2k_l; - } - - /* create M table - * - * The M table contains powers of the base, - * e.g. M[x] = G**x mod P - * - * The first half of the table is not - * computed though accept for M[0] and M[1] - */ - if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) { - goto LBL_MU; - } - - /* compute the value at M[1<<(winsize-1)] by squaring - * M[1] (winsize-1) times - */ - if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_MU; - } - - for (x = 0; x < (winsize - 1); x++) { - /* square it */ - if ((err = mp_sqr (&M[1 << (winsize - 1)], - &M[1 << (winsize - 1)])) != MP_OKAY) { - goto LBL_MU; - } - - /* reduce modulo P */ - if ((err = redux (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - } - - /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) - * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) - */ - for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { - if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) { - goto LBL_MU; - } - if ((err = redux (&M[x], P, &mu)) != MP_OKAY) { - goto LBL_MU; - } - } - - /* setup result */ - if ((err = mp_init (&res)) != MP_OKAY) { - goto LBL_MU; - } - mp_set (&res, 1); - - /* set initial mode and bit cnt */ - mode = 0; - bitcnt = 1; - buf = 0; - digidx = X->used - 1; - bitcpy = 0; - bitbuf = 0; - - for (;;) { - /* grab next digit as required */ - if (--bitcnt == 0) { - /* if digidx == -1 we are out of digits */ - if (digidx == -1) { - break; + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear(&M[y]); + } + mp_clear(&M[1]); + return err; } - /* read next digit and reset the bitcnt */ - buf = X->dp[digidx--]; - bitcnt = (int) DIGIT_BIT; - } + } - /* grab the next msb from the exponent */ - y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1; - buf <<= (mp_digit)1; + /* create mu, used for Barrett reduction */ + if ((err = mp_init(&mu)) != MP_OKAY) goto LBL_M; - /* if the bit is zero and mode == 0 then we ignore it - * These represent the leading zero bits before the first 1 bit - * in the exponent. Technically this opt is not required but it - * does lower the # of trivial squaring/reductions used - */ - if (mode == 0 && y == 0) { - continue; - } + if (redmode == 0) { + if ((err = mp_reduce_setup(&mu, P)) != MP_OKAY) goto LBL_MU; + redux = mp_reduce; + } else { + if ((err = mp_reduce_2k_setup_l(P, &mu)) != MP_OKAY) goto LBL_MU; + redux = mp_reduce_2k_l; + } - /* if the bit is zero and mode == 1 then we square */ - if (mode == 1 && y == 0) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - continue; - } + /* create M table + * + * The M table contains powers of the base, + * e.g. M[x] = G**x mod P + * + * The first half of the table is not + * computed though accept for M[0] and M[1] + */ + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) goto LBL_MU; - /* else we add it to the window */ - bitbuf |= (y << (winsize - ++bitcpy)); - mode = 2; + /* compute the value at M[1<<(winsize-1)] by squaring + * M[1] (winsize-1) times + */ + if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_MU; - if (bitcpy == winsize) { - /* ok window is filled so square as required and multiply */ - /* square first */ - for (x = 0; x < winsize; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } + for (x = 0; x < (winsize - 1); x++) { + /* square it */ + if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)], + &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_MU; + + /* reduce modulo P */ + if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, &mu)) != MP_OKAY) goto LBL_MU; + } + + /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) + * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) + */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) goto LBL_MU; + if ((err = redux(&M[x], P, &mu)) != MP_OKAY) goto LBL_MU; + } + + /* setup result */ + if ((err = mp_init(&res)) != MP_OKAY) goto LBL_MU; + mp_set(&res, 1uL); + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits */ + if (digidx == -1) { + break; + } + /* read next digit and reset the bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)MP_DIGIT_BIT; } - /* then multiply */ - if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; + /* grab the next msb from the exponent */ + y = (buf >> (mp_digit)(MP_DIGIT_BIT - 1)) & 1uL; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if ((mode == 0) && (y == 0)) { + continue; } - /* empty window and reset */ - bitcpy = 0; - bitbuf = 0; - mode = 1; - } - } - - /* if bits remain then square/multiply */ - if (mode == 2 && bitcpy > 0) { - /* square then multiply if the bit is set */ - for (x = 0; x < bitcpy; x++) { - if ((err = mp_sqr (&res, &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; + /* if the bit is zero and mode == 1 then we square */ + if ((mode == 1) && (y == 0)) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + continue; } - bitbuf <<= 1; - if ((bitbuf & (1 << winsize)) != 0) { - /* then multiply */ - if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) { - goto LBL_RES; - } - if ((err = redux (&res, P, &mu)) != MP_OKAY) { - goto LBL_RES; - } - } - } - } + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; - mp_exch (&res, Y); - err = MP_OKAY; -LBL_RES:mp_clear (&res); -LBL_MU:mp_clear (&mu); + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + } + + /* then multiply */ + if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if ((mode == 2) && (bitcpy > 0)) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + } + } + } + + mp_exch(&res, Y); + err = MP_OKAY; +LBL_RES: + mp_clear(&res); +LBL_MU: + mp_clear(&mu); LBL_M: - mp_clear(&M[1]); - for (x = 1<<(winsize-1); x < (1 << winsize); x++) { - mp_clear (&M[x]); - } - return err; + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear(&M[x]); + } + return err; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_exptmod.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_exptmod_fast.c b/lib/hcrypto/libtommath/bn_s_mp_exptmod_fast.c new file mode 100644 index 000000000..682ded845 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_exptmod_fast.c @@ -0,0 +1,254 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_EXPTMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 + * + * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. + * The value of k changes based on the size of the exponent. + * + * Uses Montgomery or Diminished Radix reduction [whichever appropriate] + */ + +#ifdef MP_LOW_MEM +# define TAB_SIZE 32 +# define MAX_WINSIZE 5 +#else +# define TAB_SIZE 256 +# define MAX_WINSIZE 0 +#endif + +mp_err s_mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) +{ + mp_int M[TAB_SIZE], res; + mp_digit buf, mp; + int bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + mp_err err; + + /* use a pointer to the reduction algorithm. This allows us to use + * one of many reduction algorithms without modding the guts of + * the code with if statements everywhere. + */ + mp_err(*redux)(mp_int *x, const mp_int *n, mp_digit rho); + + /* find window size */ + x = mp_count_bits(X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + + winsize = MAX_WINSIZE ? MP_MIN(MAX_WINSIZE, winsize) : winsize; + + /* init M array */ + /* init first cell */ + if ((err = mp_init_size(&M[1], P->alloc)) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init_size(&M[x], P->alloc)) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear(&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* determine and setup reduction code */ + if (redmode == 0) { + if (MP_HAS(MP_MONTGOMERY_SETUP)) { + /* now setup montgomery */ + if ((err = mp_montgomery_setup(P, &mp)) != MP_OKAY) goto LBL_M; + } else { + err = MP_VAL; + goto LBL_M; + } + + /* automatically pick the comba one if available (saves quite a few calls/ifs) */ + if (MP_HAS(S_MP_MONTGOMERY_REDUCE_FAST) && + (((P->used * 2) + 1) < MP_WARRAY) && + (P->used < MP_MAXFAST)) { + redux = s_mp_montgomery_reduce_fast; + } else if (MP_HAS(MP_MONTGOMERY_REDUCE)) { + /* use slower baseline Montgomery method */ + redux = mp_montgomery_reduce; + } else { + err = MP_VAL; + goto LBL_M; + } + } else if (redmode == 1) { + if (MP_HAS(MP_DR_SETUP) && MP_HAS(MP_DR_REDUCE)) { + /* setup DR reduction for moduli of the form B**k - b */ + mp_dr_setup(P, &mp); + redux = mp_dr_reduce; + } else { + err = MP_VAL; + goto LBL_M; + } + } else if (MP_HAS(MP_REDUCE_2K_SETUP) && MP_HAS(MP_REDUCE_2K)) { + /* setup DR reduction for moduli of the form 2**k - b */ + if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) goto LBL_M; + redux = mp_reduce_2k; + } else { + err = MP_VAL; + goto LBL_M; + } + + /* setup result */ + if ((err = mp_init_size(&res, P->alloc)) != MP_OKAY) goto LBL_M; + + /* create M table + * + + * + * The first half of the table is not computed though accept for M[0] and M[1] + */ + + if (redmode == 0) { + if (MP_HAS(MP_MONTGOMERY_CALC_NORMALIZATION)) { + /* now we need R mod m */ + if ((err = mp_montgomery_calc_normalization(&res, P)) != MP_OKAY) goto LBL_RES; + + /* now set M[1] to G * R mod m */ + if ((err = mp_mulmod(G, &res, P, &M[1])) != MP_OKAY) goto LBL_RES; + } else { + err = MP_VAL; + goto LBL_RES; + } + } else { + mp_set(&res, 1uL); + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) goto LBL_RES; + } + + /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ + if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_RES; + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_RES; + if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* create upper table */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) goto LBL_RES; + if ((err = redux(&M[x], P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits so break */ + if (digidx == -1) { + break; + } + /* read next digit and reset bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)MP_DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (mp_digit)(buf >> (MP_DIGIT_BIT - 1)) & 1uL; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if ((mode == 0) && (y == 0)) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if ((mode == 1) && (y == 0)) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* then multiply */ + if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if ((mode == 2) && (bitcpy > 0)) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + + /* get next bit of the window */ + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + } + } + + if (redmode == 0) { + /* fixup result if Montgomery reduction is used + * recall that any value in a Montgomery system is + * actually multiplied by R mod n. So we have + * to reduce one more time to cancel out the factor + * of R. + */ + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* swap res with Y */ + mp_exch(&res, Y); + err = MP_OKAY; +LBL_RES: + mp_clear(&res); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear(&M[x]); + } + return err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_get_bit.c b/lib/hcrypto/libtommath/bn_s_mp_get_bit.c new file mode 100644 index 000000000..28598dfec --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_get_bit.c @@ -0,0 +1,21 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_GET_BIT_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Get bit at position b and return MP_YES if the bit is 1, MP_NO if it is 0 */ +mp_bool s_mp_get_bit(const mp_int *a, unsigned int b) +{ + mp_digit bit; + int limb = (int)(b / MP_DIGIT_BIT); + + if (limb >= a->used) { + return MP_NO; + } + + bit = (mp_digit)1 << (b % MP_DIGIT_BIT); + return ((a->dp[limb] & bit) != 0u) ? MP_YES : MP_NO; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_invmod_fast.c b/lib/hcrypto/libtommath/bn_s_mp_invmod_fast.c new file mode 100644 index 000000000..677d7ab69 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_invmod_fast.c @@ -0,0 +1,118 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_INVMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes the modular inverse via binary extended euclidean algorithm, + * that is c = 1/a mod b + * + * Based on slow invmod except this is optimized for the case where b is + * odd as per HAC Note 14.64 on pp. 610 + */ +mp_err s_mp_invmod_fast(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x, y, u, v, B, D; + mp_sign neg; + mp_err err; + + /* 2. [modified] b must be odd */ + if (MP_IS_EVEN(b)) { + return MP_VAL; + } + + /* init all our temps */ + if ((err = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { + return err; + } + + /* x == modulus, y == value to invert */ + if ((err = mp_copy(b, &x)) != MP_OKAY) goto LBL_ERR; + + /* we need y = |a| */ + if ((err = mp_mod(a, b, &y)) != MP_OKAY) goto LBL_ERR; + + /* if one of x,y is zero return an error! */ + if (MP_IS_ZERO(&x) || MP_IS_ZERO(&y)) { + err = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((err = mp_copy(&x, &u)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&y, &v)) != MP_OKAY) goto LBL_ERR; + mp_set(&D, 1uL); + +top: + /* 4. while u is even do */ + while (MP_IS_EVEN(&u)) { + /* 4.1 u = u/2 */ + if ((err = mp_div_2(&u, &u)) != MP_OKAY) goto LBL_ERR; + + /* 4.2 if B is odd then */ + if (MP_IS_ODD(&B)) { + if ((err = mp_sub(&B, &x, &B)) != MP_OKAY) goto LBL_ERR; + } + /* B = B/2 */ + if ((err = mp_div_2(&B, &B)) != MP_OKAY) goto LBL_ERR; + } + + /* 5. while v is even do */ + while (MP_IS_EVEN(&v)) { + /* 5.1 v = v/2 */ + if ((err = mp_div_2(&v, &v)) != MP_OKAY) goto LBL_ERR; + + /* 5.2 if D is odd then */ + if (MP_IS_ODD(&D)) { + /* D = (D-x)/2 */ + if ((err = mp_sub(&D, &x, &D)) != MP_OKAY) goto LBL_ERR; + } + /* D = D/2 */ + if ((err = mp_div_2(&D, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* 6. if u >= v then */ + if (mp_cmp(&u, &v) != MP_LT) { + /* u = u - v, B = B - D */ + if ((err = mp_sub(&u, &v, &u)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&B, &D, &B)) != MP_OKAY) goto LBL_ERR; + } else { + /* v - v - u, D = D - B */ + if ((err = mp_sub(&v, &u, &v)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&D, &B, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* if not zero goto step 4 */ + if (!MP_IS_ZERO(&u)) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d(&v, 1uL) != MP_EQ) { + err = MP_VAL; + goto LBL_ERR; + } + + /* b is now the inverse */ + neg = a->sign; + while (D.sign == MP_NEG) { + if ((err = mp_add(&D, b, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* too big */ + while (mp_cmp_mag(&D, b) != MP_LT) { + if ((err = mp_sub(&D, b, &D)) != MP_OKAY) goto LBL_ERR; + } + + mp_exch(&D, c); + c->sign = neg; + err = MP_OKAY; + +LBL_ERR: + mp_clear_multi(&x, &y, &u, &v, &B, &D, NULL); + return err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_invmod_slow.c b/lib/hcrypto/libtommath/bn_s_mp_invmod_slow.c new file mode 100644 index 000000000..4c5db3300 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_invmod_slow.c @@ -0,0 +1,119 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_INVMOD_SLOW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* hac 14.61, pp608 */ +mp_err s_mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x, y, u, v, A, B, C, D; + mp_err err; + + /* b cannot be negative */ + if ((b->sign == MP_NEG) || MP_IS_ZERO(b)) { + return MP_VAL; + } + + /* init temps */ + if ((err = mp_init_multi(&x, &y, &u, &v, + &A, &B, &C, &D, NULL)) != MP_OKAY) { + return err; + } + + /* x = a, y = b */ + if ((err = mp_mod(a, b, &x)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(b, &y)) != MP_OKAY) goto LBL_ERR; + + /* 2. [modified] if x,y are both even then return an error! */ + if (MP_IS_EVEN(&x) && MP_IS_EVEN(&y)) { + err = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((err = mp_copy(&x, &u)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&y, &v)) != MP_OKAY) goto LBL_ERR; + mp_set(&A, 1uL); + mp_set(&D, 1uL); + +top: + /* 4. while u is even do */ + while (MP_IS_EVEN(&u)) { + /* 4.1 u = u/2 */ + if ((err = mp_div_2(&u, &u)) != MP_OKAY) goto LBL_ERR; + + /* 4.2 if A or B is odd then */ + if (MP_IS_ODD(&A) || MP_IS_ODD(&B)) { + /* A = (A+y)/2, B = (B-x)/2 */ + if ((err = mp_add(&A, &y, &A)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&B, &x, &B)) != MP_OKAY) goto LBL_ERR; + } + /* A = A/2, B = B/2 */ + if ((err = mp_div_2(&A, &A)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2(&B, &B)) != MP_OKAY) goto LBL_ERR; + } + + /* 5. while v is even do */ + while (MP_IS_EVEN(&v)) { + /* 5.1 v = v/2 */ + if ((err = mp_div_2(&v, &v)) != MP_OKAY) goto LBL_ERR; + + /* 5.2 if C or D is odd then */ + if (MP_IS_ODD(&C) || MP_IS_ODD(&D)) { + /* C = (C+y)/2, D = (D-x)/2 */ + if ((err = mp_add(&C, &y, &C)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&D, &x, &D)) != MP_OKAY) goto LBL_ERR; + } + /* C = C/2, D = D/2 */ + if ((err = mp_div_2(&C, &C)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2(&D, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* 6. if u >= v then */ + if (mp_cmp(&u, &v) != MP_LT) { + /* u = u - v, A = A - C, B = B - D */ + if ((err = mp_sub(&u, &v, &u)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&A, &C, &A)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&B, &D, &B)) != MP_OKAY) goto LBL_ERR; + } else { + /* v - v - u, C = C - A, D = D - B */ + if ((err = mp_sub(&v, &u, &v)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&C, &A, &C)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&D, &B, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* if not zero goto step 4 */ + if (!MP_IS_ZERO(&u)) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d(&v, 1uL) != MP_EQ) { + err = MP_VAL; + goto LBL_ERR; + } + + /* if its too low */ + while (mp_cmp_d(&C, 0uL) == MP_LT) { + if ((err = mp_add(&C, b, &C)) != MP_OKAY) goto LBL_ERR; + } + + /* too big */ + while (mp_cmp_mag(&C, b) != MP_LT) { + if ((err = mp_sub(&C, b, &C)) != MP_OKAY) goto LBL_ERR; + } + + /* C is now the inverse */ + mp_exch(&C, c); + err = MP_OKAY; +LBL_ERR: + mp_clear_multi(&x, &y, &u, &v, &A, &B, &C, &D, NULL); + return err; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_karatsuba_mul.c b/lib/hcrypto/libtommath/bn_s_mp_karatsuba_mul.c new file mode 100644 index 000000000..85899fb87 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_karatsuba_mul.c @@ -0,0 +1,174 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_KARATSUBA_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* c = |a| * |b| using Karatsuba Multiplication using + * three half size multiplications + * + * Let B represent the radix [e.g. 2**MP_DIGIT_BIT] and + * let n represent half of the number of digits in + * the min(a,b) + * + * a = a1 * B**n + a0 + * b = b1 * B**n + b0 + * + * Then, a * b => + a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 + * + * Note that a1b1 and a0b0 are used twice and only need to be + * computed once. So in total three half size (half # of + * digit) multiplications are performed, a0b0, a1b1 and + * (a1+b1)(a0+b0) + * + * Note that a multiplication of half the digits requires + * 1/4th the number of single precision multiplications so in + * total after one call 25% of the single precision multiplications + * are saved. Note also that the call to mp_mul can end up back + * in this function if the a0, a1, b0, or b1 are above the threshold. + * This is known as divide-and-conquer and leads to the famous + * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than + * the standard O(N**2) that the baseline/comba methods use. + * Generally though the overhead of this method doesn't pay off + * until a certain size (N ~ 80) is reached. + */ +mp_err s_mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x0, x1, y0, y1, t1, x0y0, x1y1; + int B; + mp_err err = MP_MEM; /* default the return code to an error */ + + /* min # of digits */ + B = MP_MIN(a->used, b->used); + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size(&x0, B) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_init_size(&x1, a->used - B) != MP_OKAY) { + goto X0; + } + if (mp_init_size(&y0, B) != MP_OKAY) { + goto X1; + } + if (mp_init_size(&y1, b->used - B) != MP_OKAY) { + goto Y0; + } + + /* init temps */ + if (mp_init_size(&t1, B * 2) != MP_OKAY) { + goto Y1; + } + if (mp_init_size(&x0y0, B * 2) != MP_OKAY) { + goto T1; + } + if (mp_init_size(&x1y1, B * 2) != MP_OKAY) { + goto X0Y0; + } + + /* now shift the digits */ + x0.used = y0.used = B; + x1.used = a->used - B; + y1.used = b->used - B; + + { + int x; + mp_digit *tmpa, *tmpb, *tmpx, *tmpy; + + /* we copy the digits directly instead of using higher level functions + * since we also need to shift the digits + */ + tmpa = a->dp; + tmpb = b->dp; + + tmpx = x0.dp; + tmpy = y0.dp; + for (x = 0; x < B; x++) { + *tmpx++ = *tmpa++; + *tmpy++ = *tmpb++; + } + + tmpx = x1.dp; + for (x = B; x < a->used; x++) { + *tmpx++ = *tmpa++; + } + + tmpy = y1.dp; + for (x = B; x < b->used; x++) { + *tmpy++ = *tmpb++; + } + } + + /* only need to clamp the lower words since by definition the + * upper words x1/y1 must have a known number of digits + */ + mp_clamp(&x0); + mp_clamp(&y0); + + /* now calc the products x0y0 and x1y1 */ + /* after this x0 is no longer required, free temp [x0==t2]! */ + if (mp_mul(&x0, &y0, &x0y0) != MP_OKAY) { + goto X1Y1; /* x0y0 = x0*y0 */ + } + if (mp_mul(&x1, &y1, &x1y1) != MP_OKAY) { + goto X1Y1; /* x1y1 = x1*y1 */ + } + + /* now calc x1+x0 and y1+y0 */ + if (s_mp_add(&x1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = x1 - x0 */ + } + if (s_mp_add(&y1, &y0, &x0) != MP_OKAY) { + goto X1Y1; /* t2 = y1 - y0 */ + } + if (mp_mul(&t1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ + } + + /* add x0y0 */ + if (mp_add(&x0y0, &x1y1, &x0) != MP_OKAY) { + goto X1Y1; /* t2 = x0y0 + x1y1 */ + } + if (s_mp_sub(&t1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ + } + + /* shift by B */ + if (mp_lshd(&t1, B) != MP_OKAY) { + goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<used; + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size(&x0, B) != MP_OKAY) + goto LBL_ERR; + if (mp_init_size(&x1, a->used - B) != MP_OKAY) + goto X0; + + /* init temps */ + if (mp_init_size(&t1, a->used * 2) != MP_OKAY) + goto X1; + if (mp_init_size(&t2, a->used * 2) != MP_OKAY) + goto T1; + if (mp_init_size(&x0x0, B * 2) != MP_OKAY) + goto T2; + if (mp_init_size(&x1x1, (a->used - B) * 2) != MP_OKAY) + goto X0X0; + + { + int x; + mp_digit *dst, *src; + + src = a->dp; + + /* now shift the digits */ + dst = x0.dp; + for (x = 0; x < B; x++) { + *dst++ = *src++; + } + + dst = x1.dp; + for (x = B; x < a->used; x++) { + *dst++ = *src++; + } + } + + x0.used = B; + x1.used = a->used - B; + + mp_clamp(&x0); + + /* now calc the products x0*x0 and x1*x1 */ + if (mp_sqr(&x0, &x0x0) != MP_OKAY) + goto X1X1; /* x0x0 = x0*x0 */ + if (mp_sqr(&x1, &x1x1) != MP_OKAY) + goto X1X1; /* x1x1 = x1*x1 */ + + /* now calc (x1+x0)**2 */ + if (s_mp_add(&x1, &x0, &t1) != MP_OKAY) + goto X1X1; /* t1 = x1 - x0 */ + if (mp_sqr(&t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ + + /* add x0y0 */ + if (s_mp_add(&x0x0, &x1x1, &t2) != MP_OKAY) + goto X1X1; /* t2 = x0x0 + x1x1 */ + if (s_mp_sub(&t1, &t2, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */ + + /* shift by B */ + if (mp_lshd(&t1, B) != MP_OKAY) + goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<used > MP_WARRAY) { + return MP_VAL; + } + + /* get old used count */ + olduse = x->used; + + /* grow a as required */ + if (x->alloc < (n->used + 1)) { + if ((err = mp_grow(x, n->used + 1)) != MP_OKAY) { + return err; + } + } + + /* first we have to get the digits of the input into + * an array of double precision words W[...] + */ + { + mp_word *_W; + mp_digit *tmpx; + + /* alias for the W[] array */ + _W = W; + + /* alias for the digits of x*/ + tmpx = x->dp; + + /* copy the digits of a into W[0..a->used-1] */ + for (ix = 0; ix < x->used; ix++) { + *_W++ = *tmpx++; + } + + /* zero the high words of W[a->used..m->used*2] */ + if (ix < ((n->used * 2) + 1)) { + MP_ZERO_BUFFER(_W, sizeof(mp_word) * (size_t)(((n->used * 2) + 1) - ix)); + } + } + + /* now we proceed to zero successive digits + * from the least significant upwards + */ + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * m' mod b + * + * We avoid a double precision multiplication (which isn't required) + * by casting the value down to a mp_digit. Note this requires + * that W[ix-1] have the carry cleared (see after the inner loop) + */ + mp_digit mu; + mu = ((W[ix] & MP_MASK) * rho) & MP_MASK; + + /* a = a + mu * m * b**i + * + * This is computed in place and on the fly. The multiplication + * by b**i is handled by offseting which columns the results + * are added to. + * + * Note the comba method normally doesn't handle carries in the + * inner loop In this case we fix the carry from the previous + * column since the Montgomery reduction requires digits of the + * result (so far) [see above] to work. This is + * handled by fixing up one carry after the inner loop. The + * carry fixups are done in order so after these loops the + * first m->used words of W[] have the carries fixed + */ + { + int iy; + mp_digit *tmpn; + mp_word *_W; + + /* alias for the digits of the modulus */ + tmpn = n->dp; + + /* Alias for the columns set by an offset of ix */ + _W = W + ix; + + /* inner loop */ + for (iy = 0; iy < n->used; iy++) { + *_W++ += (mp_word)mu * (mp_word)*tmpn++; + } + } + + /* now fix carry for next digit, W[ix+1] */ + W[ix + 1] += W[ix] >> (mp_word)MP_DIGIT_BIT; + } + + /* now we have to propagate the carries and + * shift the words downward [all those least + * significant digits we zeroed]. + */ + { + mp_digit *tmpx; + mp_word *_W, *_W1; + + /* nox fix rest of carries */ + + /* alias for current word */ + _W1 = W + ix; + + /* alias for next word, where the carry goes */ + _W = W + ++ix; + + for (; ix < ((n->used * 2) + 1); ix++) { + *_W++ += *_W1++ >> (mp_word)MP_DIGIT_BIT; + } + + /* copy out, A = A/b**n + * + * The result is A/b**n but instead of converting from an + * array of mp_word to mp_digit than calling mp_rshd + * we just copy them in the right order + */ + + /* alias for destination word */ + tmpx = x->dp; + + /* alias for shifted double precision result */ + _W = W + n->used; + + for (ix = 0; ix < (n->used + 1); ix++) { + *tmpx++ = *_W++ & (mp_word)MP_MASK; + } + + /* zero oldused digits, if the input a was larger than + * m->used+1 we'll have to clear the digits + */ + MP_ZERO_DIGITS(tmpx, olduse - ix); + } + + /* set the max used and clamp */ + x->used = n->used + 1; + mp_clamp(x); + + /* if A >= m then A = A - m */ + if (mp_cmp_mag(x, n) != MP_LT) { + return s_mp_sub(x, n, x); + } + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_mul_digs.c b/lib/hcrypto/libtommath/bn_s_mp_mul_digs.c index c5892181f..64509d4cb 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_mul_digs.c +++ b/lib/hcrypto/libtommath/bn_s_mp_mul_digs.c @@ -1,90 +1,74 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_MUL_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* multiplies |a| * |b| and only computes upto digs digits of result * HAC pp. 595, Algorithm 14.12 Modified so you can control how * many digits of output are created. */ -int s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +mp_err s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) { - mp_int t; - int res, pa, pb, ix, iy; - mp_digit u; - mp_word r; - mp_digit tmpx, *tmpt, *tmpy; + mp_int t; + mp_err err; + int pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; - /* can we use the fast multiplier? */ - if (((digs) < MP_WARRAY) && - MIN (a->used, b->used) < - (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_s_mp_mul_digs (a, b, c, digs); - } + /* can we use the fast multiplier? */ + if ((digs < MP_WARRAY) && + (MP_MIN(a->used, b->used) < MP_MAXFAST)) { + return s_mp_mul_digs_fast(a, b, c, digs); + } - if ((res = mp_init_size (&t, digs)) != MP_OKAY) { - return res; - } - t.used = digs; + if ((err = mp_init_size(&t, digs)) != MP_OKAY) { + return err; + } + t.used = digs; - /* compute the digits of the product directly */ - pa = a->used; - for (ix = 0; ix < pa; ix++) { - /* set the carry to zero */ - u = 0; + /* compute the digits of the product directly */ + pa = a->used; + for (ix = 0; ix < pa; ix++) { + /* set the carry to zero */ + u = 0; - /* limit ourselves to making digs digits of output */ - pb = MIN (b->used, digs - ix); + /* limit ourselves to making digs digits of output */ + pb = MP_MIN(b->used, digs - ix); - /* setup some aliases */ - /* copy of the digit from a used within the nested loop */ - tmpx = a->dp[ix]; + /* setup some aliases */ + /* copy of the digit from a used within the nested loop */ + tmpx = a->dp[ix]; - /* an alias for the destination shifted ix places */ - tmpt = t.dp + ix; + /* an alias for the destination shifted ix places */ + tmpt = t.dp + ix; - /* an alias for the digits of b */ - tmpy = b->dp; + /* an alias for the digits of b */ + tmpy = b->dp; - /* compute the columns of the output and propagate the carry */ - for (iy = 0; iy < pb; iy++) { - /* compute the column as a mp_word */ - r = ((mp_word)*tmpt) + - ((mp_word)tmpx) * ((mp_word)*tmpy++) + - ((mp_word) u); + /* compute the columns of the output and propagate the carry */ + for (iy = 0; iy < pb; iy++) { + /* compute the column as a mp_word */ + r = (mp_word)*tmpt + + ((mp_word)tmpx * (mp_word)*tmpy++) + + (mp_word)u; - /* the new column is the lower part of the result */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + /* the new column is the lower part of the result */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); - /* get the carry word from the result */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } - /* set carry if it is placed below digs */ - if (ix + iy < digs) { - *tmpt = u; - } - } + /* get the carry word from the result */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + /* set carry if it is placed below digs */ + if ((ix + iy) < digs) { + *tmpt = u; + } + } - mp_clamp (&t); - mp_exch (&t, c); + mp_clamp(&t); + mp_exch(&t, c); - mp_clear (&t); - return MP_OKAY; + mp_clear(&t); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_mul_digs.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_mul_digs_fast.c b/lib/hcrypto/libtommath/bn_s_mp_mul_digs_fast.c new file mode 100644 index 000000000..b2a287b02 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_mul_digs_fast.c @@ -0,0 +1,90 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_DIGS_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Fast (comba) multiplier + * + * This is the fast column-array [comba] multiplier. It is + * designed to compute the columns of the product first + * then handle the carries afterwards. This has the effect + * of making the nested loops that compute the columns very + * simple and schedulable on super-scalar processors. + * + * This has been modified to produce a variable number of + * digits of output so if say only a half-product is required + * you don't have to compute the upper half (a feature + * required for fast Barrett reduction). + * + * Based on Algorithm 14.12 on pp.595 of HAC. + * + */ +mp_err s_mp_mul_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + int olduse, pa, ix, iz; + mp_err err; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + if (c->alloc < digs) { + if ((err = mp_grow(c, digs)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + pa = MP_MIN(digs, a->used + b->used); + + /* clear the carry */ + _W = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty; + int iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MP_MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; ++iz) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + + } + + /* store term */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + _W = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + mp_digit *tmpc; + tmpc = c->dp; + for (ix = 0; ix < pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpc, olduse - ix); + } + mp_clamp(c); + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs.c b/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs.c index 2b718f23c..2bb2a5098 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs.c +++ b/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs.c @@ -1,81 +1,64 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_MUL_HIGH_DIGS_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* multiplies |a| * |b| and does not compute the lower digs digits * [meant to get the higher part of the product] */ -int -s_mp_mul_high_digs (mp_int * a, mp_int * b, mp_int * c, int digs) +mp_err s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) { - mp_int t; - int res, pa, pb, ix, iy; - mp_digit u; - mp_word r; - mp_digit tmpx, *tmpt, *tmpy; + mp_int t; + int pa, pb, ix, iy; + mp_err err; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; - /* can we use the fast multiplier? */ -#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C - if (((a->used + b->used + 1) < MP_WARRAY) - && MIN (a->used, b->used) < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) { - return fast_s_mp_mul_high_digs (a, b, c, digs); - } -#endif + /* can we use the fast multiplier? */ + if (MP_HAS(S_MP_MUL_HIGH_DIGS_FAST) + && ((a->used + b->used + 1) < MP_WARRAY) + && (MP_MIN(a->used, b->used) < MP_MAXFAST)) { + return s_mp_mul_high_digs_fast(a, b, c, digs); + } - if ((res = mp_init_size (&t, a->used + b->used + 1)) != MP_OKAY) { - return res; - } - t.used = a->used + b->used + 1; + if ((err = mp_init_size(&t, a->used + b->used + 1)) != MP_OKAY) { + return err; + } + t.used = a->used + b->used + 1; - pa = a->used; - pb = b->used; - for (ix = 0; ix < pa; ix++) { - /* clear the carry */ - u = 0; + pa = a->used; + pb = b->used; + for (ix = 0; ix < pa; ix++) { + /* clear the carry */ + u = 0; - /* left hand side of A[ix] * B[iy] */ - tmpx = a->dp[ix]; + /* left hand side of A[ix] * B[iy] */ + tmpx = a->dp[ix]; - /* alias to the address of where the digits will be stored */ - tmpt = &(t.dp[digs]); + /* alias to the address of where the digits will be stored */ + tmpt = &(t.dp[digs]); - /* alias for where to read the right hand side from */ - tmpy = b->dp + (digs - ix); + /* alias for where to read the right hand side from */ + tmpy = b->dp + (digs - ix); - for (iy = digs - ix; iy < pb; iy++) { - /* calculate the double precision result */ - r = ((mp_word)*tmpt) + - ((mp_word)tmpx) * ((mp_word)*tmpy++) + - ((mp_word) u); + for (iy = digs - ix; iy < pb; iy++) { + /* calculate the double precision result */ + r = (mp_word)*tmpt + + ((mp_word)tmpx * (mp_word)*tmpy++) + + (mp_word)u; - /* get the lower part */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + /* get the lower part */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); - /* carry the carry */ - u = (mp_digit) (r >> ((mp_word) DIGIT_BIT)); - } - *tmpt = u; - } - mp_clamp (&t); - mp_exch (&t, c); - mp_clear (&t); - return MP_OKAY; + /* carry the carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + *tmpt = u; + } + mp_clamp(&t); + mp_exch(&t, c); + mp_clear(&t); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_mul_high_digs.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs_fast.c b/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs_fast.c new file mode 100644 index 000000000..a2c4fb692 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_mul_high_digs_fast.c @@ -0,0 +1,81 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_HIGH_DIGS_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* this is a modified version of fast_s_mul_digs that only produces + * output digits *above* digs. See the comments for fast_s_mul_digs + * to see how it works. + * + * This is used in the Barrett reduction since for one of the multiplications + * only the higher digits were needed. This essentially halves the work. + * + * Based on Algorithm 14.12 on pp.595 of HAC. + */ +mp_err s_mp_mul_high_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + int olduse, pa, ix, iz; + mp_err err; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + pa = a->used + b->used; + if (c->alloc < pa) { + if ((err = mp_grow(c, pa)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + pa = a->used + b->used; + _W = 0; + for (ix = digs; ix < pa; ix++) { + int tx, ty, iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MP_MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + } + + /* store term */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + _W = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + mp_digit *tmpc; + + tmpc = c->dp + digs; + for (ix = digs; ix < pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpc, olduse - ix); + } + mp_clamp(c); + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_prime_is_divisible.c b/lib/hcrypto/libtommath/bn_s_mp_prime_is_divisible.c new file mode 100644 index 000000000..ffd5093e6 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_prime_is_divisible.c @@ -0,0 +1,35 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_PRIME_IS_DIVISIBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines if an integers is divisible by one + * of the first PRIME_SIZE primes or not + * + * sets result to 0 if not, 1 if yes + */ +mp_err s_mp_prime_is_divisible(const mp_int *a, mp_bool *result) +{ + int ix; + mp_err err; + mp_digit res; + + /* default to not */ + *result = MP_NO; + + for (ix = 0; ix < PRIVATE_MP_PRIME_TAB_SIZE; ix++) { + /* what is a mod LBL_prime_tab[ix] */ + if ((err = mp_mod_d(a, s_mp_prime_tab[ix], &res)) != MP_OKAY) { + return err; + } + + /* is the residue zero? */ + if (res == 0u) { + *result = MP_YES; + return MP_OKAY; + } + } + + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_rand_jenkins.c b/lib/hcrypto/libtommath/bn_s_mp_rand_jenkins.c new file mode 100644 index 000000000..da0771c5f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_rand_jenkins.c @@ -0,0 +1,52 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_RAND_JENKINS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Bob Jenkins' http://burtleburtle.net/bob/rand/smallprng.html */ +/* Chosen for speed and a good "mix" */ +typedef struct { + uint64_t a; + uint64_t b; + uint64_t c; + uint64_t d; +} ranctx; + +static ranctx jenkins_x; + +#define rot(x,k) (((x)<<(k))|((x)>>(64-(k)))) +static uint64_t s_rand_jenkins_val(void) +{ + uint64_t e = jenkins_x.a - rot(jenkins_x.b, 7); + jenkins_x.a = jenkins_x.b ^ rot(jenkins_x.c, 13); + jenkins_x.b = jenkins_x.c + rot(jenkins_x.d, 37); + jenkins_x.c = jenkins_x.d + e; + jenkins_x.d = e + jenkins_x.a; + return jenkins_x.d; +} + +void s_mp_rand_jenkins_init(uint64_t seed) +{ + uint64_t i; + jenkins_x.a = 0xf1ea5eedULL; + jenkins_x.b = jenkins_x.c = jenkins_x.d = seed; + for (i = 0uLL; i < 20uLL; ++i) { + (void)s_rand_jenkins_val(); + } +} + +mp_err s_mp_rand_jenkins(void *p, size_t n) +{ + char *q = (char *)p; + while (n > 0u) { + int i; + uint64_t x = s_rand_jenkins_val(); + for (i = 0; (i < 8) && (n > 0u); ++i, --n) { + *q++ = (char)(x & 0xFFuLL); + x >>= 8; + } + } + return MP_OKAY; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_rand_platform.c b/lib/hcrypto/libtommath/bn_s_mp_rand_platform.c new file mode 100644 index 000000000..27339bff8 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_rand_platform.c @@ -0,0 +1,148 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_RAND_PLATFORM_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* First the OS-specific special cases + * - *BSD + * - Windows + */ +#if defined(__FreeBSD__) || defined(__OpenBSD__) || defined(__NetBSD__) || defined(__DragonFly__) +#define BN_S_READ_ARC4RANDOM_C +static mp_err s_read_arc4random(void *p, size_t n) +{ + arc4random_buf(p, n); + return MP_OKAY; +} +#endif + +#if defined(_WIN32) || defined(_WIN32_WCE) +#define BN_S_READ_WINCSP_C + +#ifndef _WIN32_WINNT +#define _WIN32_WINNT 0x0400 +#endif +#ifdef _WIN32_WCE +#define UNDER_CE +#define ARM +#endif + +#define WIN32_LEAN_AND_MEAN +#include +#include + +static mp_err s_read_wincsp(void *p, size_t n) +{ + static HCRYPTPROV hProv = 0; + if (hProv == 0) { + HCRYPTPROV h = 0; + if (!CryptAcquireContext(&h, NULL, MS_DEF_PROV, PROV_RSA_FULL, + (CRYPT_VERIFYCONTEXT | CRYPT_MACHINE_KEYSET)) && + !CryptAcquireContext(&h, NULL, MS_DEF_PROV, PROV_RSA_FULL, + CRYPT_VERIFYCONTEXT | CRYPT_MACHINE_KEYSET | CRYPT_NEWKEYSET)) { + return MP_ERR; + } + hProv = h; + } + return CryptGenRandom(hProv, (DWORD)n, (BYTE *)p) == TRUE ? MP_OKAY : MP_ERR; +} +#endif /* WIN32 */ + +#if !defined(BN_S_READ_WINCSP_C) && defined(__linux__) && defined(__GLIBC_PREREQ) +#if __GLIBC_PREREQ(2, 25) +#define BN_S_READ_GETRANDOM_C +#include +#include + +static mp_err s_read_getrandom(void *p, size_t n) +{ + char *q = (char *)p; + while (n > 0u) { + ssize_t ret = getrandom(q, n, 0); + if (ret < 0) { + if (errno == EINTR) { + continue; + } + return MP_ERR; + } + q += ret; + n -= (size_t)ret; + } + return MP_OKAY; +} +#endif +#endif + +/* We assume all platforms besides windows provide "/dev/urandom". + * In case yours doesn't, define MP_NO_DEV_URANDOM at compile-time. + */ +#if !defined(BN_S_READ_WINCSP_C) && !defined(MP_NO_DEV_URANDOM) +#define BN_S_READ_URANDOM_C +#ifndef MP_DEV_URANDOM +#define MP_DEV_URANDOM "/dev/urandom" +#endif +#include +#include +#include + +static mp_err s_read_urandom(void *p, size_t n) +{ + int fd; + char *q = (char *)p; + + do { + fd = open(MP_DEV_URANDOM, O_RDONLY); + } while ((fd == -1) && (errno == EINTR)); + if (fd == -1) return MP_ERR; + + while (n > 0u) { + ssize_t ret = read(fd, p, n); + if (ret < 0) { + if (errno == EINTR) { + continue; + } + close(fd); + return MP_ERR; + } + q += ret; + n -= (size_t)ret; + } + + close(fd); + return MP_OKAY; +} +#endif + +#if defined(MP_PRNG_ENABLE_LTM_RNG) +#define BN_S_READ_LTM_RNG +unsigned long (*ltm_rng)(unsigned char *out, unsigned long outlen, void (*callback)(void)); +void (*ltm_rng_callback)(void); + +static mp_err s_read_ltm_rng(void *p, size_t n) +{ + unsigned long res; + if (ltm_rng == NULL) return MP_ERR; + res = ltm_rng(p, n, ltm_rng_callback); + if (res != n) return MP_ERR; + return MP_OKAY; +} +#endif + +mp_err s_read_arc4random(void *p, size_t n); +mp_err s_read_wincsp(void *p, size_t n); +mp_err s_read_getrandom(void *p, size_t n); +mp_err s_read_urandom(void *p, size_t n); +mp_err s_read_ltm_rng(void *p, size_t n); + +mp_err s_mp_rand_platform(void *p, size_t n) +{ + mp_err err = MP_ERR; + if ((err != MP_OKAY) && MP_HAS(S_READ_ARC4RANDOM)) err = s_read_arc4random(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_WINCSP)) err = s_read_wincsp(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_GETRANDOM)) err = s_read_getrandom(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_URANDOM)) err = s_read_urandom(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_LTM_RNG)) err = s_read_ltm_rng(p, n); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_reverse.c b/lib/hcrypto/libtommath/bn_s_mp_reverse.c new file mode 100644 index 000000000..c549e605e --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_reverse.c @@ -0,0 +1,22 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_REVERSE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reverse an array, used for radix code */ +void s_mp_reverse(unsigned char *s, size_t len) +{ + size_t ix, iy; + unsigned char t; + + ix = 0u; + iy = len - 1u; + while (ix < iy) { + t = s[ix]; + s[ix] = s[iy]; + s[iy] = t; + ++ix; + --iy; + } +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_sqr.c b/lib/hcrypto/libtommath/bn_s_mp_sqr.c index c1c3826db..505c9f053 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_sqr.c +++ b/lib/hcrypto/libtommath/bn_s_mp_sqr.c @@ -1,84 +1,69 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_SQR_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ -int s_mp_sqr (mp_int * a, mp_int * b) +mp_err s_mp_sqr(const mp_int *a, mp_int *b) { - mp_int t; - int res, ix, iy, pa; - mp_word r; - mp_digit u, tmpx, *tmpt; + mp_int t; + int ix, iy, pa; + mp_err err; + mp_word r; + mp_digit u, tmpx, *tmpt; - pa = a->used; - if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) { - return res; - } + pa = a->used; + if ((err = mp_init_size(&t, (2 * pa) + 1)) != MP_OKAY) { + return err; + } - /* default used is maximum possible size */ - t.used = 2*pa + 1; + /* default used is maximum possible size */ + t.used = (2 * pa) + 1; - for (ix = 0; ix < pa; ix++) { - /* first calculate the digit at 2*ix */ - /* calculate double precision result */ - r = ((mp_word) t.dp[2*ix]) + - ((mp_word)a->dp[ix])*((mp_word)a->dp[ix]); + for (ix = 0; ix < pa; ix++) { + /* first calculate the digit at 2*ix */ + /* calculate double precision result */ + r = (mp_word)t.dp[2*ix] + + ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]); - /* store lower part in result */ - t.dp[ix+ix] = (mp_digit) (r & ((mp_word) MP_MASK)); + /* store lower part in result */ + t.dp[ix+ix] = (mp_digit)(r & (mp_word)MP_MASK); - /* get the carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); + /* get the carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); - /* left hand side of A[ix] * A[iy] */ - tmpx = a->dp[ix]; + /* left hand side of A[ix] * A[iy] */ + tmpx = a->dp[ix]; - /* alias for where to store the results */ - tmpt = t.dp + (2*ix + 1); + /* alias for where to store the results */ + tmpt = t.dp + ((2 * ix) + 1); - for (iy = ix + 1; iy < pa; iy++) { - /* first calculate the product */ - r = ((mp_word)tmpx) * ((mp_word)a->dp[iy]); + for (iy = ix + 1; iy < pa; iy++) { + /* first calculate the product */ + r = (mp_word)tmpx * (mp_word)a->dp[iy]; - /* now calculate the double precision result, note we use - * addition instead of *2 since it's easier to optimize - */ - r = ((mp_word) *tmpt) + r + r + ((mp_word) u); + /* now calculate the double precision result, note we use + * addition instead of *2 since it's easier to optimize + */ + r = (mp_word)*tmpt + r + r + (mp_word)u; - /* store lower part */ - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); + /* store lower part */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); - /* get carry */ - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - } - /* propagate upwards */ - while (u != ((mp_digit) 0)) { - r = ((mp_word) *tmpt) + ((mp_word) u); - *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK)); - u = (mp_digit)(r >> ((mp_word) DIGIT_BIT)); - } - } + /* get carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + /* propagate upwards */ + while (u != 0uL) { + r = (mp_word)*tmpt + (mp_word)u; + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + } - mp_clamp (&t); - mp_exch (&t, b); - mp_clear (&t); - return MP_OKAY; + mp_clamp(&t); + mp_exch(&t, b); + mp_clear(&t); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_sqr.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_sqr_fast.c b/lib/hcrypto/libtommath/bn_s_mp_sqr_fast.c new file mode 100644 index 000000000..4a8a8912f --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_sqr_fast.c @@ -0,0 +1,97 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_SQR_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* the jist of squaring... + * you do like mult except the offset of the tmpx [one that + * starts closer to zero] can't equal the offset of tmpy. + * So basically you set up iy like before then you min it with + * (ty-tx) so that it never happens. You double all those + * you add in the inner loop + +After that loop you do the squares and add them in. +*/ + +mp_err s_mp_sqr_fast(const mp_int *a, mp_int *b) +{ + int olduse, pa, ix, iz; + mp_digit W[MP_WARRAY], *tmpx; + mp_word W1; + mp_err err; + + /* grow the destination as required */ + pa = a->used + a->used; + if (b->alloc < pa) { + if ((err = mp_grow(b, pa)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + W1 = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty, iy; + mp_word _W; + mp_digit *tmpy; + + /* clear counter */ + _W = 0; + + /* get offsets into the two bignums */ + ty = MP_MIN(a->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = a->dp + ty; + + /* this is the number of times the loop will iterrate, essentially + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* now for squaring tx can never equal ty + * we halve the distance since they approach at a rate of 2x + * and we have to round because odd cases need to be executed + */ + iy = MP_MIN(iy, ((ty-tx)+1)>>1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + } + + /* double the inner product and add carry */ + _W = _W + _W + W1; + + /* even columns have the square term in them */ + if (((unsigned)ix & 1u) == 0u) { + _W += (mp_word)a->dp[ix>>1] * (mp_word)a->dp[ix>>1]; + } + + /* store it */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + W1 = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = b->used; + b->used = a->used+a->used; + + { + mp_digit *tmpb; + tmpb = b->dp; + for (ix = 0; ix < pa; ix++) { + *tmpb++ = W[ix] & MP_MASK; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpb, olduse - ix); + } + mp_clamp(b); + return MP_OKAY; +} +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_sub.c b/lib/hcrypto/libtommath/bn_s_mp_sub.c index 6a60c3932..5672dab51 100644 --- a/lib/hcrypto/libtommath/bn_s_mp_sub.c +++ b/lib/hcrypto/libtommath/bn_s_mp_sub.c @@ -1,89 +1,71 @@ -#include +#include "tommath_private.h" #ifdef BN_S_MP_SUB_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ -int -s_mp_sub (mp_int * a, mp_int * b, mp_int * c) +mp_err s_mp_sub(const mp_int *a, const mp_int *b, mp_int *c) { - int olduse, res, min, max; + int olduse, min, max; + mp_err err; - /* find sizes */ - min = b->used; - max = a->used; + /* find sizes */ + min = b->used; + max = a->used; - /* init result */ - if (c->alloc < max) { - if ((res = mp_grow (c, max)) != MP_OKAY) { - return res; - } - } - olduse = c->used; - c->used = max; + /* init result */ + if (c->alloc < max) { + if ((err = mp_grow(c, max)) != MP_OKAY) { + return err; + } + } + olduse = c->used; + c->used = max; - { - register mp_digit u, *tmpa, *tmpb, *tmpc; - register int i; + { + mp_digit u, *tmpa, *tmpb, *tmpc; + int i; - /* alias for digit pointers */ - tmpa = a->dp; - tmpb = b->dp; - tmpc = c->dp; + /* alias for digit pointers */ + tmpa = a->dp; + tmpb = b->dp; + tmpc = c->dp; - /* set carry to zero */ - u = 0; - for (i = 0; i < min; i++) { - /* T[i] = A[i] - B[i] - U */ - *tmpc = *tmpa++ - *tmpb++ - u; + /* set carry to zero */ + u = 0; + for (i = 0; i < min; i++) { + /* T[i] = A[i] - B[i] - U */ + *tmpc = (*tmpa++ - *tmpb++) - u; - /* U = carry bit of T[i] - * Note this saves performing an AND operation since - * if a carry does occur it will propagate all the way to the - * MSB. As a result a single shift is enough to get the carry - */ - u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + /* U = carry bit of T[i] + * Note this saves performing an AND operation since + * if a carry does occur it will propagate all the way to the + * MSB. As a result a single shift is enough to get the carry + */ + u = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); - /* Clear carry from T[i] */ - *tmpc++ &= MP_MASK; - } + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } - /* now copy higher words if any, e.g. if A has more digits than B */ - for (; i < max; i++) { - /* T[i] = A[i] - U */ - *tmpc = *tmpa++ - u; + /* now copy higher words if any, e.g. if A has more digits than B */ + for (; i < max; i++) { + /* T[i] = A[i] - U */ + *tmpc = *tmpa++ - u; - /* U = carry bit of T[i] */ - u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1)); + /* U = carry bit of T[i] */ + u = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); - /* Clear carry from T[i] */ - *tmpc++ &= MP_MASK; - } + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } - /* clear digits above used (since we may not have grown result above) */ - for (i = c->used; i < olduse; i++) { - *tmpc++ = 0; - } - } + /* clear digits above used (since we may not have grown result above) */ + MP_ZERO_DIGITS(tmpc, olduse - c->used); + } - mp_clamp (c); - return MP_OKAY; + mp_clamp(c); + return MP_OKAY; } #endif - -/* $Source: /cvs/libtom/libtommath/bn_s_mp_sub.c,v $ */ -/* $Revision: 1.4 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/bn_s_mp_toom_mul.c b/lib/hcrypto/libtommath/bn_s_mp_toom_mul.c new file mode 100644 index 000000000..86901b074 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_toom_mul.c @@ -0,0 +1,215 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_TOOM_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* multiplication using the Toom-Cook 3-way algorithm + * + * Much more complicated than Karatsuba but has a lower + * asymptotic running time of O(N**1.464). This algorithm is + * only particularly useful on VERY large inputs + * (we're talking 1000s of digits here...). +*/ + +/* + This file contains code from J. Arndt's book "Matters Computational" + and the accompanying FXT-library with permission of the author. +*/ + +/* + Setup from + + Chung, Jaewook, and M. Anwar Hasan. "Asymmetric squaring formulae." + 18th IEEE Symposium on Computer Arithmetic (ARITH'07). IEEE, 2007. + + The interpolation from above needed one temporary variable more + than the interpolation here: + + Bodrato, Marco, and Alberto Zanoni. "What about Toom-Cook matrices optimality." + Centro Vito Volterra Universita di Roma Tor Vergata (2006) +*/ + +mp_err s_mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int S1, S2, T1, a0, a1, a2, b0, b1, b2; + int B, count; + mp_err err; + + /* init temps */ + if ((err = mp_init_multi(&S1, &S2, &T1, NULL)) != MP_OKAY) { + return err; + } + + /* B */ + B = MP_MIN(a->used, b->used) / 3; + + /** a = a2 * x^2 + a1 * x + a0; */ + if ((err = mp_init_size(&a0, B)) != MP_OKAY) goto LBL_ERRa0; + + for (count = 0; count < B; count++) { + a0.dp[count] = a->dp[count]; + a0.used++; + } + mp_clamp(&a0); + if ((err = mp_init_size(&a1, B)) != MP_OKAY) goto LBL_ERRa1; + for (; count < (2 * B); count++) { + a1.dp[count - B] = a->dp[count]; + a1.used++; + } + mp_clamp(&a1); + if ((err = mp_init_size(&a2, B + (a->used - (3 * B)))) != MP_OKAY) goto LBL_ERRa2; + for (; count < a->used; count++) { + a2.dp[count - (2 * B)] = a->dp[count]; + a2.used++; + } + mp_clamp(&a2); + + /** b = b2 * x^2 + b1 * x + b0; */ + if ((err = mp_init_size(&b0, B)) != MP_OKAY) goto LBL_ERRb0; + for (count = 0; count < B; count++) { + b0.dp[count] = b->dp[count]; + b0.used++; + } + mp_clamp(&b0); + if ((err = mp_init_size(&b1, B)) != MP_OKAY) goto LBL_ERRb1; + for (; count < (2 * B); count++) { + b1.dp[count - B] = b->dp[count]; + b1.used++; + } + mp_clamp(&b1); + if ((err = mp_init_size(&b2, B + (b->used - (3 * B)))) != MP_OKAY) goto LBL_ERRb2; + for (; count < b->used; count++) { + b2.dp[count - (2 * B)] = b->dp[count]; + b2.used++; + } + mp_clamp(&b2); + + /** \\ S1 = (a2+a1+a0) * (b2+b1+b0); */ + /** T1 = a2 + a1; */ + if ((err = mp_add(&a2, &a1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = T1 + a0; */ + if ((err = mp_add(&T1, &a0, &S2)) != MP_OKAY) goto LBL_ERR; + + /** c = b2 + b1; */ + if ((err = mp_add(&b2, &b1, c)) != MP_OKAY) goto LBL_ERR; + + /** S1 = c + b0; */ + if ((err = mp_add(c, &b0, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 * S2; */ + if ((err = mp_mul(&S1, &S2, &S1)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = (4*a2+2*a1+a0) * (4*b2+2*b1+b0); */ + /** T1 = T1 + a2; */ + if ((err = mp_add(&T1, &a2, &T1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = T1 << 1; */ + if ((err = mp_mul_2(&T1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = T1 + a0; */ + if ((err = mp_add(&T1, &a0, &T1)) != MP_OKAY) goto LBL_ERR; + + /** c = c + b2; */ + if ((err = mp_add(c, &b2, c)) != MP_OKAY) goto LBL_ERR; + + /** c = c << 1; */ + if ((err = mp_mul_2(c, c)) != MP_OKAY) goto LBL_ERR; + + /** c = c + b0; */ + if ((err = mp_add(c, &b0, c)) != MP_OKAY) goto LBL_ERR; + + /** S2 = T1 * c; */ + if ((err = mp_mul(&T1, c, &S2)) != MP_OKAY) goto LBL_ERR; + + /** \\S3 = (a2-a1+a0) * (b2-b1+b0); */ + /** a1 = a2 - a1; */ + if ((err = mp_sub(&a2, &a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 + a0; */ + if ((err = mp_add(&a1, &a0, &a1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = b2 - b1; */ + if ((err = mp_sub(&b2, &b1, &b1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = b1 + b0; */ + if ((err = mp_add(&b1, &b0, &b1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 * b1; */ + if ((err = mp_mul(&a1, &b1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = a2 * b2; */ + if ((err = mp_mul(&a2, &b2, &b1)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = (S2 - S3)/3; */ + /** S2 = S2 - a1; */ + if ((err = mp_sub(&S2, &a1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 / 3; \\ this is an exact division */ + if ((err = mp_div_3(&S2, &S2, NULL)) != MP_OKAY) goto LBL_ERR; + + /** a1 = S1 - a1; */ + if ((err = mp_sub(&S1, &a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 >> 1; */ + if ((err = mp_div_2(&a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a0 = a0 * b0; */ + if ((err = mp_mul(&a0, &b0, &a0)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - a0; */ + if ((err = mp_sub(&S1, &a0, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 - S1; */ + if ((err = mp_sub(&S2, &S1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 >> 1; */ + if ((err = mp_div_2(&S2, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - a1; */ + if ((err = mp_sub(&S1, &a1, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - b1; */ + if ((err = mp_sub(&S1, &b1, &S1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = b1 << 1; */ + if ((err = mp_mul_2(&b1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 - T1; */ + if ((err = mp_sub(&S2, &T1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 - S2; */ + if ((err = mp_sub(&a1, &S2, &a1)) != MP_OKAY) goto LBL_ERR; + + + /** P = b1*x^4+ S2*x^3+ S1*x^2+ a1*x + a0; */ + if ((err = mp_lshd(&b1, 4 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&S2, 3 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &S2, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&S1, 2 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &S1, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a1, 1 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &a1, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &a0, c)) != MP_OKAY) goto LBL_ERR; + + /** a * b - P */ + + +LBL_ERR: + mp_clear(&b2); +LBL_ERRb2: + mp_clear(&b1); +LBL_ERRb1: + mp_clear(&b0); +LBL_ERRb0: + mp_clear(&a2); +LBL_ERRa2: + mp_clear(&a1); +LBL_ERRa1: + mp_clear(&a0); +LBL_ERRa0: + mp_clear_multi(&S1, &S2, &T1, NULL); + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bn_s_mp_toom_sqr.c b/lib/hcrypto/libtommath/bn_s_mp_toom_sqr.c new file mode 100644 index 000000000..f2ffb30c2 --- /dev/null +++ b/lib/hcrypto/libtommath/bn_s_mp_toom_sqr.c @@ -0,0 +1,147 @@ +#include "tommath_private.h" +#ifdef BN_S_MP_TOOM_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* squaring using Toom-Cook 3-way algorithm */ + +/* + This file contains code from J. Arndt's book "Matters Computational" + and the accompanying FXT-library with permission of the author. +*/ + +/* squaring using Toom-Cook 3-way algorithm */ +/* + Setup and interpolation from algorithm SQR_3 in + + Chung, Jaewook, and M. Anwar Hasan. "Asymmetric squaring formulae." + 18th IEEE Symposium on Computer Arithmetic (ARITH'07). IEEE, 2007. + +*/ +mp_err s_mp_toom_sqr(const mp_int *a, mp_int *b) +{ + mp_int S0, a0, a1, a2; + mp_digit *tmpa, *tmpc; + int B, count; + mp_err err; + + + /* init temps */ + if ((err = mp_init(&S0)) != MP_OKAY) { + return err; + } + + /* B */ + B = a->used / 3; + + /** a = a2 * x^2 + a1 * x + a0; */ + if ((err = mp_init_size(&a0, B)) != MP_OKAY) goto LBL_ERRa0; + + a0.used = B; + if ((err = mp_init_size(&a1, B)) != MP_OKAY) goto LBL_ERRa1; + a1.used = B; + if ((err = mp_init_size(&a2, B + (a->used - (3 * B)))) != MP_OKAY) goto LBL_ERRa2; + + tmpa = a->dp; + tmpc = a0.dp; + for (count = 0; count < B; count++) { + *tmpc++ = *tmpa++; + } + tmpc = a1.dp; + for (; count < (2 * B); count++) { + *tmpc++ = *tmpa++; + } + tmpc = a2.dp; + for (; count < a->used; count++) { + *tmpc++ = *tmpa++; + a2.used++; + } + mp_clamp(&a0); + mp_clamp(&a1); + mp_clamp(&a2); + + /** S0 = a0^2; */ + if ((err = mp_sqr(&a0, &S0)) != MP_OKAY) goto LBL_ERR; + + /** \\S1 = (a2 + a1 + a0)^2 */ + /** \\S2 = (a2 - a1 + a0)^2 */ + /** \\S1 = a0 + a2; */ + /** a0 = a0 + a2; */ + if ((err = mp_add(&a0, &a2, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S1 - a1; */ + /** b = a0 - a1; */ + if ((err = mp_sub(&a0, &a1, b)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1 + a1; */ + /** a0 = a0 + a1; */ + if ((err = mp_add(&a0, &a1, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1^2; */ + /** a0 = a0^2; */ + if ((err = mp_sqr(&a0, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S2^2; */ + /** b = b^2; */ + if ((err = mp_sqr(b, b)) != MP_OKAY) goto LBL_ERR; + + /** \\ S3 = 2 * a1 * a2 */ + /** \\S3 = a1 * a2; */ + /** a1 = a1 * a2; */ + if ((err = mp_mul(&a1, &a2, &a1)) != MP_OKAY) goto LBL_ERR; + /** \\S3 = S3 << 1; */ + /** a1 = a1 << 1; */ + if ((err = mp_mul_2(&a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** \\S4 = a2^2; */ + /** a2 = a2^2; */ + if ((err = mp_sqr(&a2, &a2)) != MP_OKAY) goto LBL_ERR; + + /** \\ tmp = (S1 + S2)/2 */ + /** \\tmp = S1 + S2; */ + /** b = a0 + b; */ + if ((err = mp_add(&a0, b, b)) != MP_OKAY) goto LBL_ERR; + /** \\tmp = tmp >> 1; */ + /** b = b >> 1; */ + if ((err = mp_div_2(b, b)) != MP_OKAY) goto LBL_ERR; + + /** \\ S1 = S1 - tmp - S3 */ + /** \\S1 = S1 - tmp; */ + /** a0 = a0 - b; */ + if ((err = mp_sub(&a0, b, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1 - S3; */ + /** a0 = a0 - a1; */ + if ((err = mp_sub(&a0, &a1, &a0)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = tmp - S4 -S0 */ + /** \\S2 = tmp - S4; */ + /** b = b - a2; */ + if ((err = mp_sub(b, &a2, b)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S2 - S0; */ + /** b = b - S0; */ + if ((err = mp_sub(b, &S0, b)) != MP_OKAY) goto LBL_ERR; + + + /** \\P = S4*x^4 + S3*x^3 + S2*x^2 + S1*x + S0; */ + /** P = a2*x^4 + a1*x^3 + b*x^2 + a0*x + S0; */ + + if ((err = mp_lshd(&a2, 4 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a1, 3 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(b, 2 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a0, 1 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&a2, &a1, &a2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&a2, b, b)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(b, &a0, b)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(b, &S0, b)) != MP_OKAY) goto LBL_ERR; + /** a^2 - P */ + + +LBL_ERR: + mp_clear(&a2); +LBL_ERRa2: + mp_clear(&a1); +LBL_ERRa1: + mp_clear(&a0); +LBL_ERRa0: + mp_clear(&S0); + + return err; +} + +#endif diff --git a/lib/hcrypto/libtommath/bncore.c b/lib/hcrypto/libtommath/bncore.c deleted file mode 100644 index 919e3b33b..000000000 --- a/lib/hcrypto/libtommath/bncore.c +++ /dev/null @@ -1,36 +0,0 @@ -#include -#ifdef BNCORE_C -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://libtom.org - */ - -/* Known optimal configurations - - CPU /Compiler /MUL CUTOFF/SQR CUTOFF -------------------------------------------------------------- - Intel P4 Northwood /GCC v3.4.1 / 88/ 128/LTM 0.32 ;-) - AMD Athlon64 /GCC v3.4.4 / 80/ 120/LTM 0.35 - -*/ - -int KARATSUBA_MUL_CUTOFF = 80, /* Min. number of digits before Karatsuba multiplication is used. */ - KARATSUBA_SQR_CUTOFF = 120, /* Min. number of digits before Karatsuba squaring is used. */ - - TOOM_MUL_CUTOFF = 350, /* no optimal values of these are known yet so set em high */ - TOOM_SQR_CUTOFF = 400; -#endif - -/* $Source: /cvs/libtom/libtommath/bncore.c,v $ */ -/* $Revision: 1.5 $ */ -/* $Date: 2006/12/28 01:25:13 $ */ diff --git a/lib/hcrypto/libtommath/booker.pl b/lib/hcrypto/libtommath/booker.pl deleted file mode 100644 index 49f1889d1..000000000 --- a/lib/hcrypto/libtommath/booker.pl +++ /dev/null @@ -1,265 +0,0 @@ -#!/bin/perl -# -#Used to prepare the book "tommath.src" for LaTeX by pre-processing it into a .tex file -# -#Essentially you write the "tommath.src" as normal LaTex except where you want code snippets you put -# -#EXAM,file -# -#This preprocessor will then open "file" and insert it as a verbatim copy. -# -#Tom St Denis - -#get graphics type -if (shift =~ /PDF/) { - $graph = ""; -} else { - $graph = ".ps"; -} - -open(IN,"tommath.tex") or die "Can't open destination file"; - -print "Scanning for sections\n"; -$chapter = $section = $subsection = 0; -$x = 0; -while () { - print "."; - if (!(++$x % 80)) { print "\n"; } - #update the headings - if (~($_ =~ /\*/)) { - if ($_ =~ /\\chapter{.+}/) { - ++$chapter; - $section = $subsection = 0; - } elsif ($_ =~ /\\section{.+}/) { - ++$section; - $subsection = 0; - } elsif ($_ =~ /\\subsection{.+}/) { - ++$subsection; - } - } - - if ($_ =~ m/MARK/) { - @m = split(",",$_); - chomp(@m[1]); - $index1{@m[1]} = $chapter; - $index2{@m[1]} = $section; - $index3{@m[1]} = $subsection; - } -} -close(IN); - -open(IN,") { - ++$readline; - ++$srcline; - - if ($_ =~ m/MARK/) { - } elsif ($_ =~ m/EXAM/ || $_ =~ m/LIST/) { - if ($_ =~ m/EXAM/) { - $skipheader = 1; - } else { - $skipheader = 0; - } - - # EXAM,file - chomp($_); - @m = split(",",$_); - open(SRC,"<$m[1]") or die "Error:$srcline:Can't open source file $m[1]"; - - print "$srcline:Inserting $m[1]:"; - - $line = 0; - $tmp = $m[1]; - $tmp =~ s/_/"\\_"/ge; - print OUT "\\vspace{+3mm}\\begin{small}\n\\hspace{-5.1mm}{\\bf File}: $tmp\n\\vspace{-3mm}\n\\begin{alltt}\n"; - $wroteline += 5; - - if ($skipheader == 1) { - # scan till next end of comment, e.g. skip license - while () { - $text[$line++] = $_; - last if ($_ =~ /math\.libtomcrypt\.com/); - } - ; - } - - $inline = 0; - while () { - next if ($_ =~ /\$Source/); - next if ($_ =~ /\$Revision/); - next if ($_ =~ /\$Date/); - $text[$line++] = $_; - ++$inline; - chomp($_); - $_ =~ s/\t/" "/ge; - $_ =~ s/{/"^{"/ge; - $_ =~ s/}/"^}"/ge; - $_ =~ s/\\/'\symbol{92}'/ge; - $_ =~ s/\^/"\\"/ge; - - printf OUT ("%03d ", $line); - for ($x = 0; $x < length($_); $x++) { - print OUT chr(vec($_, $x, 8)); - if ($x == 75) { - print OUT "\n "; - ++$wroteline; - } - } - print OUT "\n"; - ++$wroteline; - } - $totlines = $line; - print OUT "\\end{alltt}\n\\end{small}\n"; - close(SRC); - print "$inline lines\n"; - $wroteline += 2; - } elsif ($_ =~ m/@\d+,.+@/) { - # line contains [number,text] - # e.g. @14,for (ix = 0)@ - $txt = $_; - while ($txt =~ m/@\d+,.+@/) { - @m = split("@",$txt); # splits into text, one, two - @parms = split(",",$m[1]); # splits one,two into two elements - - # now search from $parms[0] down for $parms[1] - $found1 = 0; - $found2 = 0; - for ($i = $parms[0]; $i < $totlines && $found1 == 0; $i++) { - if ($text[$i] =~ m/\Q$parms[1]\E/) { - $foundline1 = $i + 1; - $found1 = 1; - } - } - - # now search backwards - for ($i = $parms[0] - 1; $i >= 0 && $found2 == 0; $i--) { - if ($text[$i] =~ m/\Q$parms[1]\E/) { - $foundline2 = $i + 1; - $found2 = 1; - } - } - - # now use the closest match or the first if tied - if ($found1 == 1 && $found2 == 0) { - $found = 1; - $foundline = $foundline1; - } elsif ($found1 == 0 && $found2 == 1) { - $found = 1; - $foundline = $foundline2; - } elsif ($found1 == 1 && $found2 == 1) { - $found = 1; - if (($foundline1 - $parms[0]) <= ($parms[0] - $foundline2)) { - $foundline = $foundline1; - } else { - $foundline = $foundline2; - } - } else { - $found = 0; - } - - # if found replace - if ($found == 1) { - $delta = $parms[0] - $foundline; - print "Found replacement tag for \"$parms[1]\" on line $srcline which refers to line $foundline (delta $delta)\n"; - $_ =~ s/@\Q$m[1]\E@/$foundline/; - } else { - print "ERROR: The tag \"$parms[1]\" on line $srcline was not found in the most recently parsed source!\n"; - } - - # remake the rest of the line - $cnt = @m; - $txt = ""; - for ($i = 2; $i < $cnt; $i++) { - $txt = $txt . $m[$i] . "@"; - } - } - print OUT $_; - ++$wroteline; - } elsif ($_ =~ /~.+~/) { - # line contains a ~text~ pair used to refer to indexing :-) - $txt = $_; - while ($txt =~ /~.+~/) { - @m = split("~", $txt); - - # word is the second position - $word = @m[1]; - $a = $index1{$word}; - $b = $index2{$word}; - $c = $index3{$word}; - - # if chapter (a) is zero it wasn't found - if ($a == 0) { - print "ERROR: the tag \"$word\" on line $srcline was not found previously marked.\n"; - } else { - # format the tag as x, x.y or x.y.z depending on the values - $str = $a; - $str = $str . ".$b" if ($b != 0); - $str = $str . ".$c" if ($c != 0); - - if ($b == 0 && $c == 0) { - # its a chapter - if ($a <= 10) { - if ($a == 1) { - $str = "chapter one"; - } elsif ($a == 2) { - $str = "chapter two"; - } elsif ($a == 3) { - $str = "chapter three"; - } elsif ($a == 4) { - $str = "chapter four"; - } elsif ($a == 5) { - $str = "chapter five"; - } elsif ($a == 6) { - $str = "chapter six"; - } elsif ($a == 7) { - $str = "chapter seven"; - } elsif ($a == 8) { - $str = "chapter eight"; - } elsif ($a == 9) { - $str = "chapter nine"; - } elsif ($a == 10) { - $str = "chapter ten"; - } - } else { - $str = "chapter " . $str; - } - } else { - $str = "section " . $str if ($b != 0 && $c == 0); - $str = "sub-section " . $str if ($b != 0 && $c != 0); - } - - #substitute - $_ =~ s/~\Q$word\E~/$str/; - - print "Found replacement tag for marker \"$word\" on line $srcline which refers to $str\n"; - } - - # remake rest of the line - $cnt = @m; - $txt = ""; - for ($i = 2; $i < $cnt; $i++) { - $txt = $txt . $m[$i] . "~"; - } - } - print OUT $_; - ++$wroteline; - } elsif ($_ =~ m/FIGU/) { - # FIGU,file,caption - chomp($_); - @m = split(",", $_); - print OUT "\\begin{center}\n\\begin{figure}[here]\n\\includegraphics{pics/$m[1]$graph}\n"; - print OUT "\\caption{$m[2]}\n\\label{pic:$m[1]}\n\\end{figure}\n\\end{center}\n"; - $wroteline += 4; - } else { - print OUT $_; - ++$wroteline; - } -} -print "Read $readline lines, wrote $wroteline lines\n"; - -close (OUT); -close (IN); diff --git a/lib/hcrypto/libtommath/callgraph.txt b/lib/hcrypto/libtommath/callgraph.txt deleted file mode 100644 index 2efcf245b..000000000 --- a/lib/hcrypto/libtommath/callgraph.txt +++ /dev/null @@ -1,11913 +0,0 @@ -BN_PRIME_TAB_C - - -BN_MP_SQRT_C -+--->BN_MP_N_ROOT_C -| +--->BN_MP_INIT_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_EXPT_D_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_MP_SUB_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_ZERO_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -+--->BN_MP_DIV_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_SET_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_CMP_D_C - - -BN_MP_EXCH_C - - -BN_MP_IS_SQUARE_C -+--->BN_MP_MOD_D_C -| +--->BN_MP_DIV_D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_SET_INT_C -| +--->BN_MP_INIT_C -| +--->BN_MP_SET_INT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_MOD_C -| +--->BN_MP_INIT_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_GET_INT_C -+--->BN_MP_SQRT_C -| +--->BN_MP_N_ROOT_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_EXPT_D_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_ABS_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CMP_C -| | | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_SUB_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_CLEAR_C - - -BN_MP_NEG_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C - - -BN_MP_EXPTMOD_C -+--->BN_MP_INIT_C -+--->BN_MP_INVMOD_C -| +--->BN_FAST_MP_INVMOD_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_ABS_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_CMP_D_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_INVMOD_SLOW_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_ABS_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_CMP_D_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C -+--->BN_MP_ABS_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CLEAR_MULTI_C -+--->BN_MP_REDUCE_IS_2K_L_C -+--->BN_S_MP_EXPTMOD_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_REDUCE_SETUP_C -| | +--->BN_MP_2EXPT_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_REDUCE_C -| | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_D_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_REDUCE_2K_SETUP_L_C -| | +--->BN_MP_2EXPT_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_REDUCE_2K_L_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_SQR_C -| | +--->BN_MP_TOOM_SQR_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_KARATSUBA_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | +--->BN_FAST_S_MP_SQR_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_EXCH_C -+--->BN_MP_DR_IS_MODULUS_C -+--->BN_MP_REDUCE_IS_2K_C -| +--->BN_MP_REDUCE_2K_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_COUNT_BITS_C -+--->BN_MP_EXPTMOD_FAST_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_MONTGOMERY_SETUP_C -| +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| +--->BN_MP_MONTGOMERY_REDUCE_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| +--->BN_MP_DR_SETUP_C -| +--->BN_MP_DR_REDUCE_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| +--->BN_MP_REDUCE_2K_SETUP_C -| | +--->BN_MP_2EXPT_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_REDUCE_2K_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | +--->BN_MP_2EXPT_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_MULMOD_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_SQR_C -| | +--->BN_MP_TOOM_SQR_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_KARATSUBA_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | +--->BN_FAST_S_MP_SQR_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C - - -BN_MP_OR_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_ZERO_C - - -BN_MP_GROW_C - - -BN_MP_COUNT_BITS_C - - -BN_MP_PRIME_FERMAT_C -+--->BN_MP_CMP_D_C -+--->BN_MP_INIT_C -+--->BN_MP_EXPTMOD_C -| +--->BN_MP_INVMOD_C -| | +--->BN_FAST_MP_INVMOD_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_ABS_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2D_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_INIT_COPY_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INVMOD_SLOW_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_ABS_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2D_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_INIT_COPY_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ABS_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_REDUCE_IS_2K_L_C -| +--->BN_S_MP_EXPTMOD_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_REDUCE_SETUP_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_C -| | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_SETUP_L_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_L_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DR_IS_MODULUS_C -| +--->BN_MP_REDUCE_IS_2K_C -| | +--->BN_MP_REDUCE_2K_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_EXPTMOD_FAST_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_MONTGOMERY_SETUP_C -| | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_MONTGOMERY_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_DR_SETUP_C -| | +--->BN_MP_DR_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_REDUCE_2K_SETUP_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MULMOD_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2D_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_INIT_COPY_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_COPY_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_MP_CLEAR_C - - -BN_MP_SUBMOD_C -+--->BN_MP_INIT_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C - - -BN_MP_MOD_2D_C -+--->BN_MP_ZERO_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_MP_TORADIX_N_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_DIV_D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C - - -BN_MP_CMP_C -+--->BN_MP_CMP_MAG_C - - -BNCORE_C - - -BN_MP_TORADIX_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_DIV_D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C - - -BN_MP_ADD_D_C -+--->BN_MP_GROW_C -+--->BN_MP_SUB_D_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLAMP_C - - -BN_MP_DIV_3_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_FAST_S_MP_MUL_DIGS_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_MP_SQRMOD_C -+--->BN_MP_INIT_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C - - -BN_MP_INVMOD_C -+--->BN_FAST_MP_INVMOD_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_ABS_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_DIV_2_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_MP_CMP_D_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_INVMOD_SLOW_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_ABS_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_DIV_2_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_MP_CMP_D_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C - - -BN_MP_AND_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_MUL_D_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_FAST_MP_INVMOD_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_MOD_C -| +--->BN_MP_INIT_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_MP_CMP_D_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_FWRITE_C -+--->BN_MP_RADIX_SIZE_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_TORADIX_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C - - -BN_S_MP_SQR_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_N_ROOT_C -+--->BN_MP_INIT_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_EXPT_D_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_SQR_C -| | +--->BN_MP_TOOM_SQR_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_KARATSUBA_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_FAST_S_MP_SQR_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_MP_SUB_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ADD_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_PRIME_RABIN_MILLER_TRIALS_C - - -BN_MP_RADIX_SIZE_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_DIV_D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C - - -BN_MP_READ_SIGNED_BIN_C -+--->BN_MP_READ_UNSIGNED_BIN_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C - - -BN_MP_PRIME_RANDOM_EX_C -+--->BN_MP_READ_UNSIGNED_BIN_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_PRIME_IS_PRIME_C -| +--->BN_MP_CMP_D_C -| +--->BN_MP_PRIME_IS_DIVISIBLE_C -| | +--->BN_MP_MOD_D_C -| | | +--->BN_MP_DIV_D_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_DIV_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_INIT_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_INIT_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_PRIME_MILLER_RABIN_C -| | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_SUB_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CNT_LSB_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXPTMOD_C -| | | +--->BN_MP_INVMOD_C -| | | | +--->BN_FAST_MP_INVMOD_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_MOD_C -| | | | | | +--->BN_MP_DIV_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | | +--->BN_MP_ABS_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | | +--->BN_MP_CLEAR_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_INVMOD_SLOW_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_MOD_C -| | | | | | +--->BN_MP_DIV_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | | +--->BN_MP_ABS_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | | +--->BN_MP_CLEAR_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_ABS_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_REDUCE_IS_2K_L_C -| | | +--->BN_S_MP_EXPTMOD_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_REDUCE_SETUP_C -| | | | | +--->BN_MP_2EXPT_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_REDUCE_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_C -| | | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | | | +--->BN_MP_COPY_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_MUL_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_3_C -| | | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_REDUCE_2K_SETUP_L_C -| | | | | +--->BN_MP_2EXPT_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_REDUCE_2K_L_C -| | | | | +--->BN_MP_MUL_C -| | | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | | | +--->BN_MP_COPY_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_MUL_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_3_C -| | | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_SQR_C -| | | | | +--->BN_MP_TOOM_SQR_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_FAST_S_MP_SQR_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SQR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_DR_IS_MODULUS_C -| | | +--->BN_MP_REDUCE_IS_2K_C -| | | | +--->BN_MP_REDUCE_2K_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_EXPTMOD_FAST_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_MONTGOMERY_SETUP_C -| | | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_MONTGOMERY_REDUCE_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_DR_SETUP_C -| | | | +--->BN_MP_DR_REDUCE_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_REDUCE_2K_SETUP_C -| | | | | +--->BN_MP_2EXPT_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_REDUCE_2K_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | | | | +--->BN_MP_2EXPT_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MULMOD_C -| | | | | +--->BN_MP_MUL_C -| | | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | | | +--->BN_MP_COPY_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_MUL_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_2_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_DIV_3_C -| | | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_MOD_C -| | | | | | +--->BN_MP_DIV_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_C -| | | | | | | +--->BN_MP_SUB_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_ADD_C -| | | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_MUL_D_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_SQR_C -| | | | | +--->BN_MP_TOOM_SQR_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_FAST_S_MP_SQR_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SQR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_CMP_C -| | | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_SQRMOD_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_ABS_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_SUB_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ADD_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_2_C -| +--->BN_MP_GROW_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C - - -BN_MP_KARATSUBA_SQR_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -+--->BN_S_MP_ADD_C -| +--->BN_MP_GROW_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -+--->BN_MP_ADD_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CLEAR_C - - -BN_MP_INIT_COPY_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C - - -BN_MP_CLAMP_C - - -BN_MP_TOOM_SQR_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_MOD_2D_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C -+--->BN_MP_SQR_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_MUL_2_C -| +--->BN_MP_GROW_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_3_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_MOD_C -+--->BN_MP_INIT_C -+--->BN_MP_DIV_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_SET_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C - - -BN_MP_INIT_C - - -BN_MP_TOOM_MUL_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_MOD_2D_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C -+--->BN_MP_MUL_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_MUL_2_C -| +--->BN_MP_GROW_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_3_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_PRIME_IS_PRIME_C -+--->BN_MP_CMP_D_C -+--->BN_MP_PRIME_IS_DIVISIBLE_C -| +--->BN_MP_MOD_D_C -| | +--->BN_MP_DIV_D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_INIT_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_PRIME_MILLER_RABIN_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_SUB_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CNT_LSB_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXPTMOD_C -| | +--->BN_MP_INVMOD_C -| | | +--->BN_FAST_MP_INVMOD_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | +--->BN_MP_ABS_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INVMOD_SLOW_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | +--->BN_MP_ABS_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_ABS_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_REDUCE_IS_2K_L_C -| | +--->BN_S_MP_EXPTMOD_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_REDUCE_SETUP_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_SETUP_L_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_L_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DR_IS_MODULUS_C -| | +--->BN_MP_REDUCE_IS_2K_C -| | | +--->BN_MP_REDUCE_2K_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_EXPTMOD_FAST_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_MONTGOMERY_SETUP_C -| | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_MONTGOMERY_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_DR_SETUP_C -| | | +--->BN_MP_DR_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_REDUCE_2K_SETUP_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MULMOD_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_MP_SQRMOD_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_ABS_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C - - -BN_MP_COPY_C -+--->BN_MP_GROW_C - - -BN_S_MP_SUB_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_MP_READ_UNSIGNED_BIN_C -+--->BN_MP_GROW_C -+--->BN_MP_ZERO_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLAMP_C - - -BN_MP_EXPTMOD_FAST_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_INIT_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MONTGOMERY_SETUP_C -+--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -+--->BN_MP_MONTGOMERY_REDUCE_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -+--->BN_MP_DR_SETUP_C -+--->BN_MP_DR_REDUCE_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -+--->BN_MP_REDUCE_2K_SETUP_C -| +--->BN_MP_2EXPT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_GROW_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_REDUCE_2K_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| +--->BN_MP_2EXPT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_MULMOD_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_ABS_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2D_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_EXCH_C - - -BN_MP_TO_UNSIGNED_BIN_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_SET_INT_C -+--->BN_MP_ZERO_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLAMP_C - - -BN_MP_MOD_D_C -+--->BN_MP_DIV_D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C - - -BN_MP_SQR_C -+--->BN_MP_TOOM_SQR_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_KARATSUBA_SQR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| +--->BN_MP_ADD_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_CLEAR_C -+--->BN_FAST_S_MP_SQR_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_S_MP_SQR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C - - -BN_MP_MULMOD_C -+--->BN_MP_INIT_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C - - -BN_MP_DIV_2D_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_ZERO_C -+--->BN_MP_INIT_C -+--->BN_MP_MOD_2D_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C -+--->BN_MP_RSHD_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C - - -BN_S_MP_ADD_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_FAST_S_MP_SQR_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_S_MP_MUL_DIGS_C -+--->BN_FAST_S_MP_MUL_DIGS_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_XOR_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_RADIX_SMAP_C - - -BN_MP_DR_IS_MODULUS_C - - -BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_2EXPT_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_GROW_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_MUL_2_C -| +--->BN_MP_GROW_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C - - -BN_MP_SUB_C -+--->BN_S_MP_ADD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C - - -BN_MP_INIT_MULTI_C -+--->BN_MP_INIT_C -+--->BN_MP_CLEAR_C - - -BN_S_MP_MUL_HIGH_DIGS_C -+--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_PRIME_NEXT_PRIME_C -+--->BN_MP_CMP_D_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_SUB_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ADD_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MOD_D_C -| +--->BN_MP_DIV_D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_PRIME_MILLER_RABIN_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_CNT_LSB_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXPTMOD_C -| | +--->BN_MP_INVMOD_C -| | | +--->BN_FAST_MP_INVMOD_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | +--->BN_MP_ABS_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_INVMOD_SLOW_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COUNT_BITS_C -| | | | | | +--->BN_MP_ABS_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | | +--->BN_MP_CLEAR_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_ABS_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_REDUCE_IS_2K_L_C -| | +--->BN_S_MP_EXPTMOD_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_REDUCE_SETUP_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_SETUP_L_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_L_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DR_IS_MODULUS_C -| | +--->BN_MP_REDUCE_IS_2K_C -| | | +--->BN_MP_REDUCE_2K_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_EXPTMOD_FAST_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_MONTGOMERY_SETUP_C -| | | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_MONTGOMERY_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_DR_SETUP_C -| | | +--->BN_MP_DR_REDUCE_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_REDUCE_2K_SETUP_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_REDUCE_2K_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | | | +--->BN_MP_2EXPT_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MULMOD_C -| | | | +--->BN_MP_MUL_C -| | | | | +--->BN_MP_TOOM_MUL_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MOD_2D_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | | +--->BN_MP_COPY_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_MUL_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_2_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_DIV_3_C -| | | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_MOD_C -| | | | | +--->BN_MP_DIV_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_INIT_MULTI_C -| | | | | | +--->BN_MP_MUL_2D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_LSHD_C -| | | | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_C -| | | | | | +--->BN_MP_SUB_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_ADD_C -| | | | | | | +--->BN_S_MP_ADD_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | | +--->BN_S_MP_SUB_C -| | | | | | | | +--->BN_MP_GROW_C -| | | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_MUL_D_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SQR_C -| | | | +--->BN_MP_TOOM_SQR_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_SQR_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SQR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_MP_SQRMOD_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_COUNT_BITS_C -| | | | +--->BN_MP_ABS_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C - - -BN_MP_SIGNED_BIN_SIZE_C -+--->BN_MP_UNSIGNED_BIN_SIZE_C -| +--->BN_MP_COUNT_BITS_C - - -BN_MP_INVMOD_SLOW_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_MOD_C -| +--->BN_MP_INIT_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_DIV_2_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_MP_CMP_D_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_LCM_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_GCD_C -| +--->BN_MP_ABS_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_CNT_LSB_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_EXCH_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_DIV_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_SET_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_INIT_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_REDUCE_2K_L_C -+--->BN_MP_INIT_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_S_MP_ADD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_REVERSE_C - - -BN_MP_PRIME_IS_DIVISIBLE_C -+--->BN_MP_MOD_D_C -| +--->BN_MP_DIV_D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C - - -BN_MP_SET_C -+--->BN_MP_ZERO_C - - -BN_MP_GCD_C -+--->BN_MP_ABS_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_ZERO_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CNT_LSB_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_EXCH_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_MP_REDUCE_2K_SETUP_L_C -+--->BN_MP_INIT_C -+--->BN_MP_2EXPT_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_GROW_C -+--->BN_MP_COUNT_BITS_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_MP_READ_RADIX_C -+--->BN_MP_ZERO_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_SUB_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C - - -BN_FAST_S_MP_MUL_HIGH_DIGS_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_FAST_MP_MONTGOMERY_REDUCE_C -+--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C -+--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C - - -BN_MP_DIV_D_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_DIV_3_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_REDUCE_2K_SETUP_C -+--->BN_MP_INIT_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_2EXPT_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_GROW_C -+--->BN_MP_CLEAR_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C - - -BN_MP_INIT_SET_C -+--->BN_MP_INIT_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C - - -BN_MP_REDUCE_2K_C -+--->BN_MP_INIT_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_S_MP_ADD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_ERROR_C - - -BN_MP_EXPT_D_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C - - -BN_S_MP_EXPTMOD_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_INIT_C -+--->BN_MP_CLEAR_C -+--->BN_MP_REDUCE_SETUP_C -| +--->BN_MP_2EXPT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_REDUCE_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_S_MP_MUL_HIGH_DIGS_C -| | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_D_C -| +--->BN_MP_SET_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| | +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_REDUCE_2K_SETUP_L_C -| +--->BN_MP_2EXPT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_GROW_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_REDUCE_2K_L_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_C -| | +--->BN_MP_TOOM_MUL_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_KARATSUBA_MUL_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_MUL_DIGS_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_SQR_C -| +--->BN_MP_TOOM_SQR_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_KARATSUBA_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_ADD_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| +--->BN_FAST_S_MP_SQR_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SQR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_EXCH_C - - -BN_MP_ABS_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C - - -BN_MP_INIT_SET_INT_C -+--->BN_MP_INIT_C -+--->BN_MP_SET_INT_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C - - -BN_MP_SUB_D_C -+--->BN_MP_GROW_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLAMP_C - - -BN_MP_TO_SIGNED_BIN_C -+--->BN_MP_TO_UNSIGNED_BIN_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C - - -BN_MP_DIV_2_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C - - -BN_MP_REDUCE_IS_2K_C -+--->BN_MP_REDUCE_2K_C -| +--->BN_MP_INIT_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_COUNT_BITS_C - - -BN_MP_INIT_SIZE_C -+--->BN_MP_INIT_C - - -BN_MP_DIV_C -+--->BN_MP_CMP_MAG_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_ZERO_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_SET_C -+--->BN_MP_COUNT_BITS_C -+--->BN_MP_ABS_C -+--->BN_MP_MUL_2D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_INIT_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_INIT_C -+--->BN_MP_INIT_COPY_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -+--->BN_MP_RSHD_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_MP_CLEAR_C - - -BN_MP_MONTGOMERY_REDUCE_C -+--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C - - -BN_MP_MUL_2_C -+--->BN_MP_GROW_C - - -BN_MP_UNSIGNED_BIN_SIZE_C -+--->BN_MP_COUNT_BITS_C - - -BN_MP_ADDMOD_C -+--->BN_MP_INIT_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C - - -BN_MP_ADD_C -+--->BN_S_MP_ADD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C - - -BN_MP_TO_SIGNED_BIN_N_C -+--->BN_MP_SIGNED_BIN_SIZE_C -| +--->BN_MP_UNSIGNED_BIN_SIZE_C -| | +--->BN_MP_COUNT_BITS_C -+--->BN_MP_TO_SIGNED_BIN_C -| +--->BN_MP_TO_UNSIGNED_BIN_C -| | +--->BN_MP_INIT_COPY_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C - - -BN_MP_REDUCE_IS_2K_L_C - - -BN_MP_RAND_C -+--->BN_MP_ZERO_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_SUB_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C - - -BN_MP_CNT_LSB_C - - -BN_MP_2EXPT_C -+--->BN_MP_ZERO_C -+--->BN_MP_GROW_C - - -BN_MP_RSHD_C -+--->BN_MP_ZERO_C - - -BN_MP_SHRINK_C - - -BN_MP_TO_UNSIGNED_BIN_N_C -+--->BN_MP_UNSIGNED_BIN_SIZE_C -| +--->BN_MP_COUNT_BITS_C -+--->BN_MP_TO_UNSIGNED_BIN_C -| +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C - - -BN_MP_REDUCE_C -+--->BN_MP_REDUCE_SETUP_C -| +--->BN_MP_2EXPT_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2D_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_INIT_COPY_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_S_MP_MUL_HIGH_DIGS_C -| +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_MOD_2D_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_S_MP_MUL_DIGS_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_INIT_SIZE_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_D_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CLEAR_C - - -BN_MP_MUL_2D_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_GROW_C -+--->BN_MP_LSHD_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -+--->BN_MP_CLAMP_C - - -BN_MP_GET_INT_C - - -BN_MP_JACOBI_C -+--->BN_MP_CMP_D_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_CNT_LSB_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_MOD_C -| +--->BN_MP_DIV_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_SET_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_ABS_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_CLEAR_MULTI_C -+--->BN_MP_CLEAR_C - - -BN_MP_MUL_C -+--->BN_MP_TOOM_MUL_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_ZERO_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -| +--->BN_MP_MUL_2_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_3_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_KARATSUBA_MUL_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CMP_MAG_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| +--->BN_MP_CLEAR_C -+--->BN_FAST_S_MP_MUL_DIGS_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_S_MP_MUL_DIGS_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_C - - -BN_MP_EXTEUCLID_C -+--->BN_MP_INIT_MULTI_C -| +--->BN_MP_INIT_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_SET_C -| +--->BN_MP_ZERO_C -+--->BN_MP_COPY_C -| +--->BN_MP_GROW_C -+--->BN_MP_DIV_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_INIT_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_KARATSUBA_MUL_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -+--->BN_MP_NEG_C -+--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_CLEAR_C - - -BN_MP_DR_REDUCE_C -+--->BN_MP_GROW_C -+--->BN_MP_CLAMP_C -+--->BN_MP_CMP_MAG_C -+--->BN_S_MP_SUB_C - - -BN_MP_FREAD_C -+--->BN_MP_ZERO_C -+--->BN_MP_MUL_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_ADD_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_SUB_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CMP_D_C - - -BN_MP_REDUCE_SETUP_C -+--->BN_MP_2EXPT_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_GROW_C -+--->BN_MP_DIV_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_INIT_MULTI_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_SET_C -| +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_ABS_C -| +--->BN_MP_MUL_2D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CMP_C -| +--->BN_MP_SUB_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_ADD_C -| | +--->BN_S_MP_ADD_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SUB_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| +--->BN_MP_DIV_2D_C -| | +--->BN_MP_INIT_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_CLEAR_C -| | +--->BN_MP_RSHD_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_EXCH_C -| +--->BN_MP_CLEAR_MULTI_C -| | +--->BN_MP_CLEAR_C -| +--->BN_MP_INIT_SIZE_C -| | +--->BN_MP_INIT_C -| +--->BN_MP_INIT_C -| +--->BN_MP_INIT_COPY_C -| +--->BN_MP_LSHD_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_MUL_D_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C - - -BN_MP_MONTGOMERY_SETUP_C - - -BN_MP_KARATSUBA_MUL_C -+--->BN_MP_MUL_C -| +--->BN_MP_TOOM_MUL_C -| | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_MOD_2D_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_RSHD_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MUL_2_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_SUB_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_2_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_2D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MUL_D_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_DIV_3_C -| | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_INIT_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_LSHD_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_FAST_S_MP_MUL_DIGS_C -| | +--->BN_MP_GROW_C -| | +--->BN_MP_CLAMP_C -| +--->BN_S_MP_MUL_DIGS_C -| | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_INIT_C -| | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -| | +--->BN_MP_CLEAR_C -+--->BN_MP_INIT_SIZE_C -| +--->BN_MP_INIT_C -+--->BN_MP_CLAMP_C -+--->BN_MP_SUB_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -+--->BN_MP_ADD_C -| +--->BN_S_MP_ADD_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_CMP_MAG_C -| +--->BN_S_MP_SUB_C -| | +--->BN_MP_GROW_C -+--->BN_MP_LSHD_C -| +--->BN_MP_GROW_C -| +--->BN_MP_RSHD_C -| | +--->BN_MP_ZERO_C -+--->BN_MP_CLEAR_C - - -BN_MP_LSHD_C -+--->BN_MP_GROW_C -+--->BN_MP_RSHD_C -| +--->BN_MP_ZERO_C - - -BN_MP_PRIME_MILLER_RABIN_C -+--->BN_MP_CMP_D_C -+--->BN_MP_INIT_COPY_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -+--->BN_MP_SUB_D_C -| +--->BN_MP_GROW_C -| +--->BN_MP_ADD_D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLAMP_C -+--->BN_MP_CNT_LSB_C -+--->BN_MP_DIV_2D_C -| +--->BN_MP_COPY_C -| | +--->BN_MP_GROW_C -| +--->BN_MP_ZERO_C -| +--->BN_MP_MOD_2D_C -| | +--->BN_MP_CLAMP_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_RSHD_C -| +--->BN_MP_CLAMP_C -| +--->BN_MP_EXCH_C -+--->BN_MP_EXPTMOD_C -| +--->BN_MP_INVMOD_C -| | +--->BN_FAST_MP_INVMOD_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_ABS_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_INVMOD_SLOW_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_COUNT_BITS_C -| | | | | +--->BN_MP_ABS_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_CLEAR_MULTI_C -| | | | | | +--->BN_MP_CLEAR_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_CLEAR_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_ABS_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| +--->BN_MP_CLEAR_MULTI_C -| +--->BN_MP_REDUCE_IS_2K_L_C -| +--->BN_S_MP_EXPTMOD_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_REDUCE_SETUP_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_S_MP_MUL_HIGH_DIGS_C -| | | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_FAST_S_MP_MUL_HIGH_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_SETUP_L_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_L_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_SET_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_EXCH_C -| +--->BN_MP_DR_IS_MODULUS_C -| +--->BN_MP_REDUCE_IS_2K_C -| | +--->BN_MP_REDUCE_2K_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_COUNT_BITS_C -| +--->BN_MP_EXPTMOD_FAST_C -| | +--->BN_MP_COUNT_BITS_C -| | +--->BN_MP_MONTGOMERY_SETUP_C -| | +--->BN_FAST_MP_MONTGOMERY_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_MONTGOMERY_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_DR_SETUP_C -| | +--->BN_MP_DR_REDUCE_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | +--->BN_MP_REDUCE_2K_SETUP_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_REDUCE_2K_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -| | | +--->BN_MP_2EXPT_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_SET_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_MULMOD_C -| | | +--->BN_MP_MUL_C -| | | | +--->BN_MP_TOOM_MUL_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_MOD_2D_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | | +--->BN_MP_COPY_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_MUL_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_2_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_DIV_3_C -| | | | | | +--->BN_MP_INIT_SIZE_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_KARATSUBA_MUL_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CMP_MAG_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_MUL_DIGS_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_MOD_C -| | | | +--->BN_MP_DIV_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_MP_COPY_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_INIT_MULTI_C -| | | | | +--->BN_MP_SET_C -| | | | | +--->BN_MP_MUL_2D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_LSHD_C -| | | | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_C -| | | | | +--->BN_MP_SUB_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_ADD_C -| | | | | | +--->BN_S_MP_ADD_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | | +--->BN_S_MP_SUB_C -| | | | | | | +--->BN_MP_GROW_C -| | | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_MUL_D_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_SET_C -| | | +--->BN_MP_ZERO_C -| | +--->BN_MP_MOD_C -| | | +--->BN_MP_DIV_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | +--->BN_MP_COPY_C -| | | +--->BN_MP_GROW_C -| | +--->BN_MP_SQR_C -| | | +--->BN_MP_TOOM_SQR_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | +--->BN_FAST_S_MP_SQR_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_SQR_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_MUL_C -| | | +--->BN_MP_TOOM_MUL_C -| | | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_MOD_2D_C -| | | | | +--->BN_MP_ZERO_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_MUL_2_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_2_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_2D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_MUL_D_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_DIV_3_C -| | | | | +--->BN_MP_INIT_SIZE_C -| | | | | +--->BN_MP_CLAMP_C -| | | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_KARATSUBA_MUL_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_SUB_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_ADD_C -| | | | | +--->BN_S_MP_ADD_C -| | | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CMP_MAG_C -| | | | | +--->BN_S_MP_SUB_C -| | | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_RSHD_C -| | | | | | +--->BN_MP_ZERO_C -| | | +--->BN_FAST_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_S_MP_MUL_DIGS_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_CMP_C -| +--->BN_MP_CMP_MAG_C -+--->BN_MP_SQRMOD_C -| +--->BN_MP_SQR_C -| | +--->BN_MP_TOOM_SQR_C -| | | +--->BN_MP_INIT_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_MOD_2D_C -| | | | +--->BN_MP_ZERO_C -| | | | +--->BN_MP_COPY_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_MUL_2_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_2_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_DIV_3_C -| | | | +--->BN_MP_INIT_SIZE_C -| | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_MP_EXCH_C -| | | | +--->BN_MP_CLEAR_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | | +--->BN_MP_CLEAR_C -| | +--->BN_MP_KARATSUBA_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_MP_CMP_MAG_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLEAR_C -| | +--->BN_FAST_S_MP_SQR_C -| | | +--->BN_MP_GROW_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_S_MP_SQR_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_C -| +--->BN_MP_CLEAR_C -| +--->BN_MP_MOD_C -| | +--->BN_MP_DIV_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_MP_COPY_C -| | | | +--->BN_MP_GROW_C -| | | +--->BN_MP_ZERO_C -| | | +--->BN_MP_INIT_MULTI_C -| | | +--->BN_MP_SET_C -| | | +--->BN_MP_COUNT_BITS_C -| | | +--->BN_MP_ABS_C -| | | +--->BN_MP_MUL_2D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_LSHD_C -| | | | | +--->BN_MP_RSHD_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_SUB_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_ADD_C -| | | | +--->BN_S_MP_ADD_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | | +--->BN_S_MP_SUB_C -| | | | | +--->BN_MP_GROW_C -| | | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_EXCH_C -| | | +--->BN_MP_CLEAR_MULTI_C -| | | +--->BN_MP_INIT_SIZE_C -| | | +--->BN_MP_LSHD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_RSHD_C -| | | +--->BN_MP_MUL_D_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_ADD_C -| | | +--->BN_S_MP_ADD_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | | +--->BN_MP_CMP_MAG_C -| | | +--->BN_S_MP_SUB_C -| | | | +--->BN_MP_GROW_C -| | | | +--->BN_MP_CLAMP_C -| | +--->BN_MP_EXCH_C -+--->BN_MP_CLEAR_C - - -BN_MP_DR_SETUP_C - - -BN_MP_CMP_MAG_C - - diff --git a/lib/hcrypto/libtommath/changes.txt b/lib/hcrypto/libtommath/changes.txt index b0da4da4f..ebf7382a0 100644 --- a/lib/hcrypto/libtommath/changes.txt +++ b/lib/hcrypto/libtommath/changes.txt @@ -1,5 +1,105 @@ +XXX XXth, 2019 +v1.2.0 + -- A huge refactoring of the library happened - renaming, + deprecating and replacing existing functions by improved API's. + + All deprecated functions, macros and symbols are only marked as such + so this version is still API and ABI compatible to v1.x. + + -- Daniel Mendler was pushing for those changes and contributing a load of patches, + refactorings, code reviews and whatnotelse. + -- Christoph Zurnieden re-worked internals of the library, improved the performance, + did code reviews and wrote documentation. + -- Francois Perrad did some refactoring and took again care of linting the sources and + provided all fixes. + -- Jan Nijtmans, Karel Miko and Joachim Breitner contributed various patches. + + -- Private symbols can now be hidden for the shared library builds, disabled by default. + -- All API's follow a single code style, are prefixed the same etc. + -- Unified, safer and improved API's + -- Less magic numbers - return values (where appropriate) and most flags are now enums, + this was implemented in a backwards compatible way where return values were int. + -- API's with return values are now by default marked as "warn on unsused result", this + can be disabled if required (which will most likely hide bugs), c.f. MP_WUR in tommath.h + -- Provide a whole set of setters&getters for different primitive types (long, uint32_t, etc.) + -- All those primitive setters are now optimized. + -- It's possible to automatically tune the cutoff values for Karatsuba&Toom-Cook + -- The custom allocators which were formerly known as XMALLOC(), XFREE() etc. are now available + as MP_MALLOC(), MP_REALLOC(), MP_CALLOC() and MP_FREE(). MP_REALLOC() and MP_FREE() now also + provide the allocated size to ease the usage of simple allocators without tracking. + -- Building is now also possible with MSVC 2015, 2017 and 2019 (use makefile.msvc) + -- Added mp_decr() and mp_incr() + -- Added mp_log_u32() + -- Improved prime-checking + -- Improved Toom-Cook multiplication + -- Removed the LTM book (`make docs` now builds the user manual) + + +Jan 28th, 2019 +v1.1.0 + -- Christoph Zurnieden contributed FIPS 186.4 compliant + prime-checking (PR #113), several other fixes and a load of documentation + -- Daniel Mendler provided two's-complement functions (PR #124) + and mp_{set,get}_double() (PR #123) + -- Francois Perrad took care of linting the sources, provided all fixes and + a astylerc to auto-format the sources. + -- A bunch of patches by Kevin B Kenny have been back-ported from TCL + -- Jan Nijtmans provided the patches to `const`ify all API + function arguments (also from TCL) + -- mp_rand() has now several native random provider implementations + and doesn't rely on `rand()` anymore + -- Karel Miko provided fixes when building for MS Windows + and re-worked the makefile generating process + -- The entire environment and build logic has been extended and improved + regarding auto-detection of platforms, libtool and a lot more + -- Prevent some potential BOF cases + -- Improved/fixed mp_lshd() and mp_invmod() + -- A load more bugs were fixed by various contributors + + +Aug 29th, 2017 +v1.0.1 + -- Dmitry Kovalenko provided fixes to mp_add_d() and mp_init_copy() + -- Matt Johnston contributed some improvements to mp_div_2d(), + mp_exptmod_fast(), mp_mod() and mp_mulmod() + -- Julien Nabet provided a fix to the error handling in mp_init_multi() + -- Ben Gardner provided a fix regarding usage of reserved keywords + -- Fixed mp_rand() to fill the correct number of bits + -- Fixed mp_invmod() + -- Use the same 64-bit detection code as in libtomcrypt + -- Correct usage of DESTDIR, PREFIX, etc. when installing the library + -- Francois Perrad updated all the perl scripts to an actual perl version + + +Feb 5th, 2016 +v1.0 + -- Bump to 1.0 + -- Dirkjan Bussink provided a faster version of mp_expt_d() + -- Moritz Lenz contributed a fix to mp_mod() + and provided mp_get_long() and mp_set_long() + -- Fixed bugs in mp_read_radix(), mp_radix_size + Thanks to shameister, Gerhard R, + -- Christopher Brown provided mp_export() and mp_import() + -- Improvements in the code of mp_init_copy() + Thanks to ramkumarkoppu, + -- lomereiter provided mp_balance_mul() + -- Alexander Boström from the heimdal project contributed patches to + mp_prime_next_prime() and mp_invmod() and added a mp_isneg() macro + -- Fix build issues for Linux x32 ABI + -- Added mp_get_long_long() and mp_set_long_long() + -- Carlin provided a patch to use arc4random() instead of rand() + on platforms where it is supported + -- Karel Miko provided mp_sqrtmod_prime() + + +July 23rd, 2010 +v0.42.0 + -- Fix for mp_prime_next_prime() bug when checking generated prime + -- allow mp_shrink to shrink initialized, but empty MPI's + -- Added project and solution files for Visual Studio 2005 and Visual Studio 2008. + March 10th, 2007 -v0.41 -- Wolfgang Ehrhardt suggested a quick fix to mp_div_d() which makes the detection of powers of two quicker. +v0.41 -- Wolfgang Ehrhardt suggested a quick fix to mp_div_d() which makes the detection of powers of two quicker. -- [CRI] Added libtommath.dsp for Visual C++ users. December 24th, 2006 @@ -16,11 +116,11 @@ v0.39 -- Jim Wigginton pointed out my Montgomery examples in figures 6.4 and 6. Jan 26th, 2006 v0.38 -- broken makefile.shared fixed -- removed some carry stores that were not required [updated text] - + November 18th, 2005 v0.37 -- [Don Porter] reported on a TCL list [HEY SEND ME BUGREPORTS ALREADY!!!] that mp_add_d() would compute -0 with some inputs. Fixed. -- [rinick@gmail.com] reported the makefile.bcc was messed up. Fixed. - -- [Kevin Kenny] reported some issues with mp_toradix_n(). Now it doesn't require a min of 3 chars of output. + -- [Kevin Kenny] reported some issues with mp_toradix_n(). Now it doesn't require a min of 3 chars of output. -- Made the make command renamable. Wee August 1st, 2005 @@ -30,8 +130,8 @@ v0.36 -- LTM_PRIME_2MSB_ON was fixed and the "OFF" flag was removed. -- Ported LTC patch to fix the prime_random_ex() function to get the bitsize correct [and the maskOR flags] -- Kevin Kenny pointed out a stray // -- David Hulton pointed out a typo in the textbook [mp_montgomery_setup() pseudo-code] - -- Neal Hamilton (Elliptic Semiconductor) pointed out that my Karatsuba notation was backwards and that I could use - unsigned operations in the routine. + -- Neal Hamilton (Elliptic Semiconductor) pointed out that my Karatsuba notation was backwards and that I could use + unsigned operations in the routine. -- Paul Schmidt pointed out a linking error in mp_exptmod() when BN_S_MP_EXPTMOD_C is undefined (and another for read_radix) -- Updated makefiles to be way more flexible @@ -42,7 +142,7 @@ v0.35 -- Stupid XOR function missing line again... oops. -- [Wolfgang Ehrhardt] Suggested a fix for mp_reduce() which avoided underruns. ;-) -- mp_rand() would emit one too many digits and it was possible to get a 0 out of it ... oops -- Added montgomery to the testing to make sure it handles 1..10 digit moduli correctly - -- Fixed bug in comba that would lead to possible erroneous outputs when "pa < digs" + -- Fixed bug in comba that would lead to possible erroneous outputs when "pa < digs" -- Fixed bug in mp_toradix_size for "0" [Kevin Kenny] -- Updated chapters 1-5 of the textbook ;-) It now talks about the new comba code! @@ -53,7 +153,7 @@ v0.34 -- Fixed two more small errors in mp_prime_random_ex() -- Added "large" diminished radix support. Speeds up things like DSA where the moduli is of the form 2^k - P for some P < 2^(k/2) or so Actually is faster than Montgomery on my AMD64 (and probably much faster on a P4) -- Updated the manual a bit - -- Ok so I haven't done the textbook work yet... My current freelance gig has landed me in France till the + -- Ok so I haven't done the textbook work yet... My current freelance gig has landed me in France till the end of Feb/05. Once I get back I'll have tons of free time and I plan to go to town on the book. As of this release the API will freeze. At least until the book catches up with all the changes. I welcome bug reports but new algorithms will have to wait. @@ -70,7 +170,7 @@ v0.33 -- Fixed "small" variant for mp_div() which would munge with negative div October 29th, 2004 v0.32 -- Added "makefile.shared" for shared object support -- Added more to the build options/configs in the manual - -- Started the Depends framework, wrote dep.pl to scan deps and + -- Started the Depends framework, wrote dep.pl to scan deps and produce "callgraph.txt" ;-) -- Wrote SC_RSA_1 which will enable close to the minimum required to perform RSA on 32-bit [or 64-bit] platforms with LibTomCrypt @@ -78,7 +178,7 @@ v0.32 -- Added "makefile.shared" for shared object support you want to use as your mp_div() at build time. Saves roughly 8KB or so. -- Renamed a few files and changed some comments to make depends system work better. (No changes to function names) - -- Merged in new Combas that perform 2 reads per inner loop instead of the older + -- Merged in new Combas that perform 2 reads per inner loop instead of the older 3reads/2writes per inner loop of the old code. Really though if you want speed learn to use TomsFastMath ;-) @@ -107,8 +207,8 @@ v0.30 -- Added "mp_toradix_n" which stores upto "n-1" least significant digits call. -- Removed /etclib directory [um LibTomPoly deprecates this]. -- Fixed mp_mod() so the sign of the result agrees with the sign of the modulus. - ++ N.B. My semester is almost up so expect updates to the textbook to be posted to the libtomcrypt.org - website. + ++ N.B. My semester is almost up so expect updates to the textbook to be posted to the libtomcrypt.org + website. Jan 25th, 2004 v0.29 ++ Note: "Henrik" from the v0.28 changelog refers to Henrik Goldman ;-) diff --git a/lib/hcrypto/libtommath/demo/demo.c b/lib/hcrypto/libtommath/demo/demo.c deleted file mode 100644 index 2442611f9..000000000 --- a/lib/hcrypto/libtommath/demo/demo.c +++ /dev/null @@ -1,740 +0,0 @@ -#include - -#ifdef IOWNANATHLON -#include -#define SLEEP sleep(4) -#else -#define SLEEP -#endif - -#include "tommath.h" - -void ndraw(mp_int * a, char *name) -{ - char buf[16000]; - - printf("%s: ", name); - mp_toradix(a, buf, 10); - printf("%s\n", buf); -} - -static void draw(mp_int * a) -{ - ndraw(a, ""); -} - - -unsigned long lfsr = 0xAAAAAAAAUL; - -int lbit(void) -{ - if (lfsr & 0x80000000UL) { - lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL; - return 1; - } else { - lfsr <<= 1; - return 0; - } -} - -int myrng(unsigned char *dst, int len, void *dat) -{ - int x; - - for (x = 0; x < len; x++) - dst[x] = rand() & 0xFF; - return len; -} - - - -char cmd[4096], buf[4096]; -int main(void) -{ - mp_int a, b, c, d, e, f; - unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, - gcd_n, lcm_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n, t; - unsigned rr; - int i, n, err, cnt, ix, old_kara_m, old_kara_s; - mp_digit mp; - - - mp_init(&a); - mp_init(&b); - mp_init(&c); - mp_init(&d); - mp_init(&e); - mp_init(&f); - - srand(time(NULL)); - -#if 0 - // test montgomery - printf("Testing montgomery...\n"); - for (i = 1; i < 10; i++) { - printf("Testing digit size: %d\n", i); - for (n = 0; n < 1000; n++) { - mp_rand(&a, i); - a.dp[0] |= 1; - - // let's see if R is right - mp_montgomery_calc_normalization(&b, &a); - mp_montgomery_setup(&a, &mp); - - // now test a random reduction - for (ix = 0; ix < 100; ix++) { - mp_rand(&c, 1 + abs(rand()) % (2*i)); - mp_copy(&c, &d); - mp_copy(&c, &e); - - mp_mod(&d, &a, &d); - mp_montgomery_reduce(&c, &a, mp); - mp_mulmod(&c, &b, &a, &c); - - if (mp_cmp(&c, &d) != MP_EQ) { -printf("d = e mod a, c = e MOD a\n"); -mp_todecimal(&a, buf); printf("a = %s\n", buf); -mp_todecimal(&e, buf); printf("e = %s\n", buf); -mp_todecimal(&d, buf); printf("d = %s\n", buf); -mp_todecimal(&c, buf); printf("c = %s\n", buf); -printf("compare no compare!\n"); exit(EXIT_FAILURE); } - } - } - } - printf("done\n"); - - // test mp_get_int - printf("Testing: mp_get_int\n"); - for (i = 0; i < 1000; ++i) { - t = ((unsigned long) rand() * rand() + 1) & 0xFFFFFFFF; - mp_set_int(&a, t); - if (t != mp_get_int(&a)) { - printf("mp_get_int() bad result!\n"); - return 1; - } - } - mp_set_int(&a, 0); - if (mp_get_int(&a) != 0) { - printf("mp_get_int() bad result!\n"); - return 1; - } - mp_set_int(&a, 0xffffffff); - if (mp_get_int(&a) != 0xffffffff) { - printf("mp_get_int() bad result!\n"); - return 1; - } - // test mp_sqrt - printf("Testing: mp_sqrt\n"); - for (i = 0; i < 1000; ++i) { - printf("%6d\r", i); - fflush(stdout); - n = (rand() & 15) + 1; - mp_rand(&a, n); - if (mp_sqrt(&a, &b) != MP_OKAY) { - printf("mp_sqrt() error!\n"); - return 1; - } - mp_n_root(&a, 2, &a); - if (mp_cmp_mag(&b, &a) != MP_EQ) { - printf("mp_sqrt() bad result!\n"); - return 1; - } - } - - printf("\nTesting: mp_is_square\n"); - for (i = 0; i < 1000; ++i) { - printf("%6d\r", i); - fflush(stdout); - - /* test mp_is_square false negatives */ - n = (rand() & 7) + 1; - mp_rand(&a, n); - mp_sqr(&a, &a); - if (mp_is_square(&a, &n) != MP_OKAY) { - printf("fn:mp_is_square() error!\n"); - return 1; - } - if (n == 0) { - printf("fn:mp_is_square() bad result!\n"); - return 1; - } - - /* test for false positives */ - mp_add_d(&a, 1, &a); - if (mp_is_square(&a, &n) != MP_OKAY) { - printf("fp:mp_is_square() error!\n"); - return 1; - } - if (n == 1) { - printf("fp:mp_is_square() bad result!\n"); - return 1; - } - - } - printf("\n\n"); - - /* test for size */ - for (ix = 10; ix < 128; ix++) { - printf("Testing (not safe-prime): %9d bits \r", ix); - fflush(stdout); - err = - mp_prime_random_ex(&a, 8, ix, - (rand() & 1) ? LTM_PRIME_2MSB_OFF : - LTM_PRIME_2MSB_ON, myrng, NULL); - if (err != MP_OKAY) { - printf("failed with err code %d\n", err); - return EXIT_FAILURE; - } - if (mp_count_bits(&a) != ix) { - printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); - return EXIT_FAILURE; - } - } - - for (ix = 16; ix < 128; ix++) { - printf("Testing ( safe-prime): %9d bits \r", ix); - fflush(stdout); - err = - mp_prime_random_ex(&a, 8, ix, - ((rand() & 1) ? LTM_PRIME_2MSB_OFF : - LTM_PRIME_2MSB_ON) | LTM_PRIME_SAFE, myrng, - NULL); - if (err != MP_OKAY) { - printf("failed with err code %d\n", err); - return EXIT_FAILURE; - } - if (mp_count_bits(&a) != ix) { - printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); - return EXIT_FAILURE; - } - /* let's see if it's really a safe prime */ - mp_sub_d(&a, 1, &a); - mp_div_2(&a, &a); - mp_prime_is_prime(&a, 8, &cnt); - if (cnt != MP_YES) { - printf("sub is not prime!\n"); - return EXIT_FAILURE; - } - } - - printf("\n\n"); - - mp_read_radix(&a, "123456", 10); - mp_toradix_n(&a, buf, 10, 3); - printf("a == %s\n", buf); - mp_toradix_n(&a, buf, 10, 4); - printf("a == %s\n", buf); - mp_toradix_n(&a, buf, 10, 30); - printf("a == %s\n", buf); - - -#if 0 - for (;;) { - fgets(buf, sizeof(buf), stdin); - mp_read_radix(&a, buf, 10); - mp_prime_next_prime(&a, 5, 1); - mp_toradix(&a, buf, 10); - printf("%s, %lu\n", buf, a.dp[0] & 3); - } -#endif - - /* test mp_cnt_lsb */ - printf("testing mp_cnt_lsb...\n"); - mp_set(&a, 1); - for (ix = 0; ix < 1024; ix++) { - if (mp_cnt_lsb(&a) != ix) { - printf("Failed at %d, %d\n", ix, mp_cnt_lsb(&a)); - return 0; - } - mp_mul_2(&a, &a); - } - -/* test mp_reduce_2k */ - printf("Testing mp_reduce_2k...\n"); - for (cnt = 3; cnt <= 128; ++cnt) { - mp_digit tmp; - - mp_2expt(&a, cnt); - mp_sub_d(&a, 2, &a); /* a = 2**cnt - 2 */ - - - printf("\nTesting %4d bits", cnt); - printf("(%d)", mp_reduce_is_2k(&a)); - mp_reduce_2k_setup(&a, &tmp); - printf("(%d)", tmp); - for (ix = 0; ix < 1000; ix++) { - if (!(ix & 127)) { - printf("."); - fflush(stdout); - } - mp_rand(&b, (cnt / DIGIT_BIT + 1) * 2); - mp_copy(&c, &b); - mp_mod(&c, &a, &c); - mp_reduce_2k(&b, &a, 2); - if (mp_cmp(&c, &b)) { - printf("FAILED\n"); - exit(0); - } - } - } - -/* test mp_div_3 */ - printf("Testing mp_div_3...\n"); - mp_set(&d, 3); - for (cnt = 0; cnt < 10000;) { - mp_digit r1, r2; - - if (!(++cnt & 127)) - printf("%9d\r", cnt); - mp_rand(&a, abs(rand()) % 128 + 1); - mp_div(&a, &d, &b, &e); - mp_div_3(&a, &c, &r2); - - if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) { - printf("\n\nmp_div_3 => Failure\n"); - } - } - printf("\n\nPassed div_3 testing\n"); - -/* test the DR reduction */ - printf("testing mp_dr_reduce...\n"); - for (cnt = 2; cnt < 32; cnt++) { - printf("%d digit modulus\n", cnt); - mp_grow(&a, cnt); - mp_zero(&a); - for (ix = 1; ix < cnt; ix++) { - a.dp[ix] = MP_MASK; - } - a.used = cnt; - a.dp[0] = 3; - - mp_rand(&b, cnt - 1); - mp_copy(&b, &c); - - rr = 0; - do { - if (!(rr & 127)) { - printf("%9lu\r", rr); - fflush(stdout); - } - mp_sqr(&b, &b); - mp_add_d(&b, 1, &b); - mp_copy(&b, &c); - - mp_mod(&b, &a, &b); - mp_dr_reduce(&c, &a, (((mp_digit) 1) << DIGIT_BIT) - a.dp[0]); - - if (mp_cmp(&b, &c) != MP_EQ) { - printf("Failed on trial %lu\n", rr); - exit(-1); - - } - } while (++rr < 500); - printf("Passed DR test for %d digits\n", cnt); - } - -#endif - -/* test the mp_reduce_2k_l code */ -#if 0 -#if 0 -/* first load P with 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF */ - mp_2expt(&a, 1024); - mp_read_radix(&b, "2A434B9FDEC95D8F9D550FFFFFFFFFFFFFFFF", 16); - mp_sub(&a, &b, &a); -#elif 1 -/* p = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F */ - mp_2expt(&a, 2048); - mp_read_radix(&b, - "1000000000000000000000000000000004945DDBF8EA2A91D5776399BB83E188F", - 16); - mp_sub(&a, &b, &a); -#endif - - mp_todecimal(&a, buf); - printf("p==%s\n", buf); -/* now mp_reduce_is_2k_l() should return */ - if (mp_reduce_is_2k_l(&a) != 1) { - printf("mp_reduce_is_2k_l() return 0, should be 1\n"); - return EXIT_FAILURE; - } - mp_reduce_2k_setup_l(&a, &d); - /* now do a million square+1 to see if it varies */ - mp_rand(&b, 64); - mp_mod(&b, &a, &b); - mp_copy(&b, &c); - printf("testing mp_reduce_2k_l..."); - fflush(stdout); - for (cnt = 0; cnt < (1UL << 20); cnt++) { - mp_sqr(&b, &b); - mp_add_d(&b, 1, &b); - mp_reduce_2k_l(&b, &a, &d); - mp_sqr(&c, &c); - mp_add_d(&c, 1, &c); - mp_mod(&c, &a, &c); - if (mp_cmp(&b, &c) != MP_EQ) { - printf("mp_reduce_2k_l() failed at step %lu\n", cnt); - mp_tohex(&b, buf); - printf("b == %s\n", buf); - mp_tohex(&c, buf); - printf("c == %s\n", buf); - return EXIT_FAILURE; - } - } - printf("...Passed\n"); -#endif - - div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n = - sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = cnt = add_d_n = - sub_d_n = 0; - - /* force KARA and TOOM to enable despite cutoffs */ - KARATSUBA_SQR_CUTOFF = KARATSUBA_MUL_CUTOFF = 8; - TOOM_SQR_CUTOFF = TOOM_MUL_CUTOFF = 16; - - for (;;) { - /* randomly clear and re-init one variable, this has the affect of triming the alloc space */ - switch (abs(rand()) % 7) { - case 0: - mp_clear(&a); - mp_init(&a); - break; - case 1: - mp_clear(&b); - mp_init(&b); - break; - case 2: - mp_clear(&c); - mp_init(&c); - break; - case 3: - mp_clear(&d); - mp_init(&d); - break; - case 4: - mp_clear(&e); - mp_init(&e); - break; - case 5: - mp_clear(&f); - mp_init(&f); - break; - case 6: - break; /* don't clear any */ - } - - - printf - ("%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu ", - add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n, - expt_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n); - fgets(cmd, 4095, stdin); - cmd[strlen(cmd) - 1] = 0; - printf("%s ]\r", cmd); - fflush(stdout); - if (!strcmp(cmd, "mul2d")) { - ++mul2d_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - sscanf(buf, "%d", &rr); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - - mp_mul_2d(&a, rr, &a); - a.sign = b.sign; - if (mp_cmp(&a, &b) != MP_EQ) { - printf("mul2d failed, rr == %d\n", rr); - draw(&a); - draw(&b); - return 0; - } - } else if (!strcmp(cmd, "div2d")) { - ++div2d_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - sscanf(buf, "%d", &rr); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - - mp_div_2d(&a, rr, &a, &e); - a.sign = b.sign; - if (a.used == b.used && a.used == 0) { - a.sign = b.sign = MP_ZPOS; - } - if (mp_cmp(&a, &b) != MP_EQ) { - printf("div2d failed, rr == %d\n", rr); - draw(&a); - draw(&b); - return 0; - } - } else if (!strcmp(cmd, "add")) { - ++add_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_copy(&a, &d); - mp_add(&d, &b, &d); - if (mp_cmp(&c, &d) != MP_EQ) { - printf("add %lu failure!\n", add_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - return 0; - } - - /* test the sign/unsigned storage functions */ - - rr = mp_signed_bin_size(&c); - mp_to_signed_bin(&c, (unsigned char *) cmd); - memset(cmd + rr, rand() & 255, sizeof(cmd) - rr); - mp_read_signed_bin(&d, (unsigned char *) cmd, rr); - if (mp_cmp(&c, &d) != MP_EQ) { - printf("mp_signed_bin failure!\n"); - draw(&c); - draw(&d); - return 0; - } - - - rr = mp_unsigned_bin_size(&c); - mp_to_unsigned_bin(&c, (unsigned char *) cmd); - memset(cmd + rr, rand() & 255, sizeof(cmd) - rr); - mp_read_unsigned_bin(&d, (unsigned char *) cmd, rr); - if (mp_cmp_mag(&c, &d) != MP_EQ) { - printf("mp_unsigned_bin failure!\n"); - draw(&c); - draw(&d); - return 0; - } - - } else if (!strcmp(cmd, "sub")) { - ++sub_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_copy(&a, &d); - mp_sub(&d, &b, &d); - if (mp_cmp(&c, &d) != MP_EQ) { - printf("sub %lu failure!\n", sub_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - return 0; - } - } else if (!strcmp(cmd, "mul")) { - ++mul_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_copy(&a, &d); - mp_mul(&d, &b, &d); - if (mp_cmp(&c, &d) != MP_EQ) { - printf("mul %lu failure!\n", mul_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - return 0; - } - } else if (!strcmp(cmd, "div")) { - ++div_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&d, buf, 64); - - mp_div(&a, &b, &e, &f); - if (mp_cmp(&c, &e) != MP_EQ || mp_cmp(&d, &f) != MP_EQ) { - printf("div %lu %d, %d, failure!\n", div_n, mp_cmp(&c, &e), - mp_cmp(&d, &f)); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - draw(&e); - draw(&f); - return 0; - } - - } else if (!strcmp(cmd, "sqr")) { - ++sqr_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - mp_copy(&a, &c); - mp_sqr(&c, &c); - if (mp_cmp(&b, &c) != MP_EQ) { - printf("sqr %lu failure!\n", sqr_n); - draw(&a); - draw(&b); - draw(&c); - return 0; - } - } else if (!strcmp(cmd, "gcd")) { - ++gcd_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_copy(&a, &d); - mp_gcd(&d, &b, &d); - d.sign = c.sign; - if (mp_cmp(&c, &d) != MP_EQ) { - printf("gcd %lu failure!\n", gcd_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - return 0; - } - } else if (!strcmp(cmd, "lcm")) { - ++lcm_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_copy(&a, &d); - mp_lcm(&d, &b, &d); - d.sign = c.sign; - if (mp_cmp(&c, &d) != MP_EQ) { - printf("lcm %lu failure!\n", lcm_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - return 0; - } - } else if (!strcmp(cmd, "expt")) { - ++expt_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&d, buf, 64); - mp_copy(&a, &e); - mp_exptmod(&e, &b, &c, &e); - if (mp_cmp(&d, &e) != MP_EQ) { - printf("expt %lu failure!\n", expt_n); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - draw(&e); - return 0; - } - } else if (!strcmp(cmd, "invmod")) { - ++inv_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&c, buf, 64); - mp_invmod(&a, &b, &d); - mp_mulmod(&d, &a, &b, &e); - if (mp_cmp_d(&e, 1) != MP_EQ) { - printf("inv [wrong value from MPI?!] failure\n"); - draw(&a); - draw(&b); - draw(&c); - draw(&d); - mp_gcd(&a, &b, &e); - draw(&e); - return 0; - } - - } else if (!strcmp(cmd, "div2")) { - ++div2_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - mp_div_2(&a, &c); - if (mp_cmp(&c, &b) != MP_EQ) { - printf("div_2 %lu failure\n", div2_n); - draw(&a); - draw(&b); - draw(&c); - return 0; - } - } else if (!strcmp(cmd, "mul2")) { - ++mul2_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - mp_mul_2(&a, &c); - if (mp_cmp(&c, &b) != MP_EQ) { - printf("mul_2 %lu failure\n", mul2_n); - draw(&a); - draw(&b); - draw(&c); - return 0; - } - } else if (!strcmp(cmd, "add_d")) { - ++add_d_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - sscanf(buf, "%d", &ix); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - mp_add_d(&a, ix, &c); - if (mp_cmp(&b, &c) != MP_EQ) { - printf("add_d %lu failure\n", add_d_n); - draw(&a); - draw(&b); - draw(&c); - printf("d == %d\n", ix); - return 0; - } - } else if (!strcmp(cmd, "sub_d")) { - ++sub_d_n; - fgets(buf, 4095, stdin); - mp_read_radix(&a, buf, 64); - fgets(buf, 4095, stdin); - sscanf(buf, "%d", &ix); - fgets(buf, 4095, stdin); - mp_read_radix(&b, buf, 64); - mp_sub_d(&a, ix, &c); - if (mp_cmp(&b, &c) != MP_EQ) { - printf("sub_d %lu failure\n", sub_d_n); - draw(&a); - draw(&b); - draw(&c); - printf("d == %d\n", ix); - return 0; - } - } - } - return 0; -} - -/* $Source: /cvs/libtom/libtommath/demo/demo.c,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2005/06/24 11:32:07 $ */ diff --git a/lib/hcrypto/libtommath/demo/mtest_opponent.c b/lib/hcrypto/libtommath/demo/mtest_opponent.c new file mode 100644 index 000000000..7fbd35ebc --- /dev/null +++ b/lib/hcrypto/libtommath/demo/mtest_opponent.c @@ -0,0 +1,402 @@ +#include "shared.h" + +#ifdef LTM_MTEST_REAL_RAND +#define LTM_MTEST_RAND_SEED time(NULL) +#else +#define LTM_MTEST_RAND_SEED 23 +#endif + +static void draw(mp_int *a) +{ + ndraw(a, ""); +} + +#define FGETS(str, size, stream) \ + { \ + char *ret = fgets(str, size, stream); \ + if (!ret) { fprintf(stderr, "\n%d: fgets failed\n", __LINE__); goto LBL_ERR; } \ + } + +static int mtest_opponent(void) +{ + char cmd[4096]; + char buf[4096]; + int ix; + unsigned rr; + mp_int a, b, c, d, e, f; + unsigned long expt_n, add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, + gcd_n, lcm_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n; + + srand(LTM_MTEST_RAND_SEED); + + if (mp_init_multi(&a, &b, &c, &d, &e, &f, NULL)!= MP_OKAY) + return EXIT_FAILURE; + + div2_n = mul2_n = inv_n = expt_n = lcm_n = gcd_n = add_n = + sub_n = mul_n = div_n = sqr_n = mul2d_n = div2d_n = add_d_n = sub_d_n = 0; + +#ifndef MP_FIXED_CUTOFFS + /* force KARA and TOOM to enable despite cutoffs */ + KARATSUBA_SQR_CUTOFF = KARATSUBA_MUL_CUTOFF = 8; + TOOM_SQR_CUTOFF = TOOM_MUL_CUTOFF = 16; +#endif + + for (;;) { + /* randomly clear and re-init one variable, this has the affect of triming the alloc space */ + switch (abs(rand()) % 7) { + case 0: + mp_clear(&a); + mp_init(&a); + break; + case 1: + mp_clear(&b); + mp_init(&b); + break; + case 2: + mp_clear(&c); + mp_init(&c); + break; + case 3: + mp_clear(&d); + mp_init(&d); + break; + case 4: + mp_clear(&e); + mp_init(&e); + break; + case 5: + mp_clear(&f); + mp_init(&f); + break; + case 6: + break; /* don't clear any */ + } + + + printf("%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu/%4lu ", + add_n, sub_n, mul_n, div_n, sqr_n, mul2d_n, div2d_n, gcd_n, lcm_n, + expt_n, inv_n, div2_n, mul2_n, add_d_n, sub_d_n); + FGETS(cmd, 4095, stdin); + cmd[strlen(cmd) - 1u] = '\0'; + printf("%-6s ]\r", cmd); + fflush(stdout); + if (strcmp(cmd, "mul2d") == 0) { + ++mul2d_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + sscanf(buf, "%u", &rr); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + + mp_mul_2d(&a, (int)rr, &a); + a.sign = b.sign; + if (mp_cmp(&a, &b) != MP_EQ) { + printf("mul2d failed, rr == %u\n", rr); + draw(&a); + draw(&b); + goto LBL_ERR; + } + } else if (strcmp(cmd, "div2d") == 0) { + ++div2d_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + sscanf(buf, "%u", &rr); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + + mp_div_2d(&a, (int)rr, &a, &e); + a.sign = b.sign; + if ((a.used == b.used) && (a.used == 0)) { + a.sign = b.sign = MP_ZPOS; + } + if (mp_cmp(&a, &b) != MP_EQ) { + printf("div2d failed, rr == %u\n", rr); + draw(&a); + draw(&b); + goto LBL_ERR; + } + } else if (strcmp(cmd, "add") == 0) { + ++add_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_add(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("add %lu failure!\n", add_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + goto LBL_ERR; + } + + /* test the sign/unsigned storage functions */ + + rr = (unsigned)mp_sbin_size(&c); + mp_to_sbin(&c, (unsigned char *) cmd, (size_t)rr, NULL); + memset(cmd + rr, rand() & 0xFF, sizeof(cmd) - rr); + mp_from_sbin(&d, (unsigned char *) cmd, (size_t)rr); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("mp_signed_bin failure!\n"); + draw(&c); + draw(&d); + goto LBL_ERR; + } + + rr = (unsigned)mp_ubin_size(&c); + mp_to_ubin(&c, (unsigned char *) cmd, (size_t)rr, NULL); + memset(cmd + rr, rand() & 0xFF, sizeof(cmd) - rr); + mp_from_ubin(&d, (unsigned char *) cmd, (size_t)rr); + if (mp_cmp_mag(&c, &d) != MP_EQ) { + printf("mp_unsigned_bin failure!\n"); + draw(&c); + draw(&d); + goto LBL_ERR; + } + + } else if (strcmp(cmd, "sub") == 0) { + ++sub_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_sub(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("sub %lu failure!\n", sub_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + goto LBL_ERR; + } + } else if (strcmp(cmd, "mul") == 0) { + ++mul_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_mul(&d, &b, &d); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("mul %lu failure!\n", mul_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + goto LBL_ERR; + } + } else if (strcmp(cmd, "div") == 0) { + ++div_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&d, buf, 64); + + mp_div(&a, &b, &e, &f); + if ((mp_cmp(&c, &e) != MP_EQ) || (mp_cmp(&d, &f) != MP_EQ)) { + printf("div %lu %d, %d, failure!\n", div_n, mp_cmp(&c, &e), + mp_cmp(&d, &f)); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + draw(&e); + draw(&f); + goto LBL_ERR; + } + + } else if (strcmp(cmd, "sqr") == 0) { + ++sqr_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + mp_copy(&a, &c); + mp_sqr(&c, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("sqr %lu failure!\n", sqr_n); + draw(&a); + draw(&b); + draw(&c); + goto LBL_ERR; + } + } else if (strcmp(cmd, "gcd") == 0) { + ++gcd_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_gcd(&d, &b, &d); + d.sign = c.sign; + if (mp_cmp(&c, &d) != MP_EQ) { + printf("gcd %lu failure!\n", gcd_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + goto LBL_ERR; + } + } else if (strcmp(cmd, "lcm") == 0) { + ++lcm_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_copy(&a, &d); + mp_lcm(&d, &b, &d); + d.sign = c.sign; + if (mp_cmp(&c, &d) != MP_EQ) { + printf("lcm %lu failure!\n", lcm_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + goto LBL_ERR; + } + } else if (strcmp(cmd, "expt") == 0) { + ++expt_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&d, buf, 64); + mp_copy(&a, &e); + mp_exptmod(&e, &b, &c, &e); + if (mp_cmp(&d, &e) != MP_EQ) { + printf("expt %lu failure!\n", expt_n); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + draw(&e); + goto LBL_ERR; + } + } else if (strcmp(cmd, "invmod") == 0) { + ++inv_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&c, buf, 64); + mp_invmod(&a, &b, &d); + mp_mulmod(&d, &a, &b, &e); + if (mp_cmp_d(&e, 1uL) != MP_EQ) { + printf("inv [wrong value from MPI?!] failure\n"); + draw(&a); + draw(&b); + draw(&c); + draw(&d); + draw(&e); + mp_gcd(&a, &b, &e); + draw(&e); + goto LBL_ERR; + } + + } else if (strcmp(cmd, "div2") == 0) { + ++div2_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + mp_div_2(&a, &c); + if (mp_cmp(&c, &b) != MP_EQ) { + printf("div_2 %lu failure\n", div2_n); + draw(&a); + draw(&b); + draw(&c); + goto LBL_ERR; + } + } else if (strcmp(cmd, "mul2") == 0) { + ++mul2_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + mp_mul_2(&a, &c); + if (mp_cmp(&c, &b) != MP_EQ) { + printf("mul_2 %lu failure\n", mul2_n); + draw(&a); + draw(&b); + draw(&c); + goto LBL_ERR; + } + } else if (strcmp(cmd, "add_d") == 0) { + ++add_d_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + sscanf(buf, "%d", &ix); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + mp_add_d(&a, (mp_digit)ix, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("add_d %lu failure\n", add_d_n); + draw(&a); + draw(&b); + draw(&c); + printf("d == %d\n", ix); + goto LBL_ERR; + } + } else if (strcmp(cmd, "sub_d") == 0) { + ++sub_d_n; + FGETS(buf, 4095, stdin); + mp_read_radix(&a, buf, 64); + FGETS(buf, 4095, stdin); + sscanf(buf, "%d", &ix); + FGETS(buf, 4095, stdin); + mp_read_radix(&b, buf, 64); + mp_sub_d(&a, (mp_digit)ix, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("sub_d %lu failure\n", sub_d_n); + draw(&a); + draw(&b); + draw(&c); + printf("d == %d\n", ix); + goto LBL_ERR; + } + } else if (strcmp(cmd, "exit") == 0) { + printf("\nokay, exiting now\n"); + break; + } + } + + mp_clear_multi(&a, &b, &c, &d, &e, &f, NULL); + printf("\n"); + return 0; + +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, &e, &f, NULL); + printf("\n"); + return EXIT_FAILURE; +} + +int main(void) +{ + print_header(); + + return mtest_opponent(); +} diff --git a/lib/hcrypto/libtommath/demo/shared.c b/lib/hcrypto/libtommath/demo/shared.c new file mode 100644 index 000000000..dc8e05a6a --- /dev/null +++ b/lib/hcrypto/libtommath/demo/shared.c @@ -0,0 +1,42 @@ +#include "shared.h" + +void ndraw(mp_int *a, const char *name) +{ + char *buf = NULL; + int size; + + mp_radix_size(a, 10, &size); + buf = (char *)malloc((size_t) size); + if (buf == NULL) { + fprintf(stderr, "\nndraw: malloc(%d) failed\n", size); + exit(EXIT_FAILURE); + } + + printf("%s: ", name); + mp_to_decimal(a, buf, (size_t) size); + printf("%s\n", buf); + mp_to_hex(a, buf, (size_t) size); + printf("0x%s\n", buf); + + free(buf); +} + +void print_header(void) +{ +#ifdef MP_8BIT + printf("Digit size 8 Bit \n"); +#endif +#ifdef MP_16BIT + printf("Digit size 16 Bit \n"); +#endif +#ifdef MP_32BIT + printf("Digit size 32 Bit \n"); +#endif +#ifdef MP_64BIT + printf("Digit size 64 Bit \n"); +#endif + printf("Size of mp_digit: %u\n", (unsigned int)sizeof(mp_digit)); + printf("Size of mp_word: %u\n", (unsigned int)sizeof(mp_word)); + printf("MP_DIGIT_BIT: %d\n", MP_DIGIT_BIT); + printf("MP_PREC: %d\n", MP_PREC); +} diff --git a/lib/hcrypto/libtommath/demo/shared.h b/lib/hcrypto/libtommath/demo/shared.h new file mode 100644 index 000000000..4d5eb53b2 --- /dev/null +++ b/lib/hcrypto/libtommath/demo/shared.h @@ -0,0 +1,21 @@ +#include +#include +#include + +/* + * Configuration + */ +#ifndef LTM_DEMO_TEST_REDUCE_2K_L +/* This test takes a moment so we disable it by default, but it can be: + * 0 to disable testing + * 1 to make the test with P = 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF + * 2 to make the test with P = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F + */ +#define LTM_DEMO_TEST_REDUCE_2K_L 0 +#endif + +#define MP_WUR /* TODO: result checks disabled for now */ +#include "tommath_private.h" + +extern void ndraw(mp_int* a, const char* name); +extern void print_header(void); diff --git a/lib/hcrypto/libtommath/demo/test.c b/lib/hcrypto/libtommath/demo/test.c new file mode 100644 index 000000000..7b29a4ce9 --- /dev/null +++ b/lib/hcrypto/libtommath/demo/test.c @@ -0,0 +1,2522 @@ +#include +#include "shared.h" + +static long rand_long(void) +{ + long x; + if (s_mp_rand_source(&x, sizeof(x)) != MP_OKAY) { + fprintf(stderr, "s_mp_rand_source failed\n"); + exit(EXIT_FAILURE); + } + return x; +} + +static int rand_int(void) +{ + int x; + if (s_mp_rand_source(&x, sizeof(x)) != MP_OKAY) { + fprintf(stderr, "s_mp_rand_source failed\n"); + exit(EXIT_FAILURE); + } + return x; +} + +static int32_t rand_int32(void) +{ + int32_t x; + if (s_mp_rand_source(&x, sizeof(x)) != MP_OKAY) { + fprintf(stderr, "s_mp_rand_source failed\n"); + exit(EXIT_FAILURE); + } + return x; +} + +static int64_t rand_int64(void) +{ + int64_t x; + if (s_mp_rand_source(&x, sizeof(x)) != MP_OKAY) { + fprintf(stderr, "s_mp_rand_source failed\n"); + exit(EXIT_FAILURE); + } + return x; +} + +static uint32_t uabs32(int32_t x) +{ + return x > 0 ? (uint32_t)x : -(uint32_t)x; +} + +static uint64_t uabs64(int64_t x) +{ + return x > 0 ? (uint64_t)x : -(uint64_t)x; +} + +/* This function prototype is needed + * to test dead code elimination + * which is used for feature detection. + * + * If the feature detection does not + * work as desired we will get a linker error. + */ +void does_not_exist(void); + +static int test_feature_detection(void) +{ +#define BN_TEST_FEATURE1_C + if (!MP_HAS(TEST_FEATURE1)) { + does_not_exist(); + return EXIT_FAILURE; + } + +#define BN_TEST_FEATURE2_C 1 + if (MP_HAS(TEST_FEATURE2)) { + does_not_exist(); + return EXIT_FAILURE; + } + +#define BN_TEST_FEATURE3_C 0 + if (MP_HAS(TEST_FEATURE3)) { + does_not_exist(); + return EXIT_FAILURE; + } + +#define BN_TEST_FEATURE4_C something + if (MP_HAS(TEST_FEATURE4)) { + does_not_exist(); + return EXIT_FAILURE; + } + + if (MP_HAS(TEST_FEATURE5)) { + does_not_exist(); + return EXIT_FAILURE; + } + + return EXIT_SUCCESS; +} + +static int test_trivial_stuff(void) +{ + mp_int a, b, c, d; + mp_err e; + if ((e = mp_init_multi(&a, &b, &c, &d, NULL)) != MP_OKAY) { + return EXIT_FAILURE; + } + (void)mp_error_to_string(e); + + /* a: 0->5 */ + mp_set(&a, 5u); + /* a: 5-> b: -5 */ + mp_neg(&a, &b); + if (mp_cmp(&a, &b) != MP_GT) { + goto LBL_ERR; + } + if (mp_cmp(&b, &a) != MP_LT) { + goto LBL_ERR; + } + /* a: 5-> a: -5 */ + mp_neg(&a, &a); + if (mp_cmp(&b, &a) != MP_EQ) { + goto LBL_ERR; + } + /* a: -5-> b: 5 */ + mp_abs(&a, &b); + if (mp_isneg(&b) != MP_NO) { + goto LBL_ERR; + } + /* a: -5-> b: -4 */ + mp_add_d(&a, 1uL, &b); + if (mp_isneg(&b) != MP_YES) { + goto LBL_ERR; + } + if (mp_get_i32(&b) != -4) { + goto LBL_ERR; + } + if (mp_get_u32(&b) != (uint32_t)-4) { + goto LBL_ERR; + } + if (mp_get_mag_u32(&b) != 4) { + goto LBL_ERR; + } + /* a: -5-> b: 1 */ + mp_add_d(&a, 6uL, &b); + if (mp_get_u32(&b) != 1) { + goto LBL_ERR; + } + /* a: -5-> a: 1 */ + mp_add_d(&a, 6uL, &a); + if (mp_get_u32(&a) != 1) { + goto LBL_ERR; + } + mp_zero(&a); + /* a: 0-> a: 6 */ + mp_add_d(&a, 6uL, &a); + if (mp_get_u32(&a) != 6) { + goto LBL_ERR; + } + + mp_set(&a, 42u); + mp_set(&b, 1u); + mp_neg(&b, &b); + mp_set(&c, 1u); + mp_exptmod(&a, &b, &c, &d); + + mp_set(&c, 7u); + mp_exptmod(&a, &b, &c, &d); + + if (mp_iseven(&a) == mp_isodd(&a)) { + goto LBL_ERR; + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int check_get_set_i32(mp_int *a, int32_t b) +{ + mp_clear(a); + if (mp_shrink(a) != MP_OKAY) return EXIT_FAILURE; + + mp_set_i32(a, b); + if (mp_shrink(a) != MP_OKAY) return EXIT_FAILURE; + if (mp_get_i32(a) != b) return EXIT_FAILURE; + if (mp_get_u32(a) != (uint32_t)b) return EXIT_FAILURE; + if (mp_get_mag_u32(a) != uabs32(b)) return EXIT_FAILURE; + + mp_set_u32(a, (uint32_t)b); + if (mp_get_u32(a) != (uint32_t)b) return EXIT_FAILURE; + if (mp_get_i32(a) != (int32_t)(uint32_t)b) return EXIT_FAILURE; + + return EXIT_SUCCESS; +} + +static int test_mp_get_set_i32(void) +{ + int i; + mp_int a; + + if (mp_init(&a) != MP_OKAY) { + return EXIT_FAILURE; + } + + check_get_set_i32(&a, 0); + check_get_set_i32(&a, -1); + check_get_set_i32(&a, 1); + check_get_set_i32(&a, INT32_MIN); + check_get_set_i32(&a, INT32_MAX); + + for (i = 0; i < 1000; ++i) { + int32_t b = rand_int32(); + if (check_get_set_i32(&a, b) != EXIT_SUCCESS) { + goto LBL_ERR; + } + } + + mp_clear(&a); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear(&a); + return EXIT_FAILURE; +} + +static int check_get_set_i64(mp_int *a, int64_t b) +{ + mp_clear(a); + if (mp_shrink(a) != MP_OKAY) return EXIT_FAILURE; + + mp_set_i64(a, b); + if (mp_shrink(a) != MP_OKAY) return EXIT_FAILURE; + if (mp_get_i64(a) != b) return EXIT_FAILURE; + if (mp_get_u64(a) != (uint64_t)b) return EXIT_FAILURE; + if (mp_get_mag_u64(a) != uabs64(b)) return EXIT_FAILURE; + + mp_set_u64(a, (uint64_t)b); + if (mp_get_u64(a) != (uint64_t)b) return EXIT_FAILURE; + if (mp_get_i64(a) != (int64_t)(uint64_t)b) return EXIT_FAILURE; + + return EXIT_SUCCESS; +} + +static int test_mp_get_set_i64(void) +{ + int i; + mp_int a; + + if (mp_init(&a) != MP_OKAY) { + return EXIT_FAILURE; + } + + check_get_set_i64(&a, 0); + check_get_set_i64(&a, -1); + check_get_set_i64(&a, 1); + check_get_set_i64(&a, INT64_MIN); + check_get_set_i64(&a, INT64_MAX); + + for (i = 0; i < 1000; ++i) { + int64_t b = rand_int64(); + if (check_get_set_i64(&a, b) != EXIT_SUCCESS) { + goto LBL_ERR; + } + } + + mp_clear(&a); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear(&a); + return EXIT_FAILURE; +} + +static int test_mp_fread_fwrite(void) +{ + mp_int a, b; + mp_err e; + FILE *tmp = NULL; + if ((e = mp_init_multi(&a, &b, NULL)) != MP_OKAY) { + return EXIT_FAILURE; + } + + mp_set_ul(&a, 123456uL); + tmp = tmpfile(); + if ((e = mp_fwrite(&a, 64, tmp)) != MP_OKAY) { + goto LBL_ERR; + } + rewind(tmp); + if ((e = mp_fread(&b, 64, tmp)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_get_u32(&b) != 123456uL) { + goto LBL_ERR; + } + fclose(tmp); + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + if (tmp != NULL) fclose(tmp); + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static mp_err very_random_source(void *out, size_t size) +{ + memset(out, 0xff, size); + return MP_OKAY; +} + +static int test_mp_rand(void) +{ + mp_int a, b; + int n; + mp_err err; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + mp_rand_source(very_random_source); + for (n = 1; n < 1024; ++n) { + if ((err = mp_rand(&a, n)) != MP_OKAY) { + printf("Failed mp_rand() %s.\n", mp_error_to_string(err)); + break; + } + if ((err = mp_incr(&a)) != MP_OKAY) { + printf("Failed mp_incr() %s.\n", mp_error_to_string(err)); + break; + } + if ((err = mp_div_2d(&a, n * MP_DIGIT_BIT, &b, NULL)) != MP_OKAY) { + printf("Failed mp_div_2d() %s.\n", mp_error_to_string(err)); + break; + } + if (mp_cmp_d(&b, 1) != MP_EQ) { + ndraw(&a, "mp_rand() a"); + ndraw(&b, "mp_rand() b"); + err = MP_ERR; + break; + } + } + mp_rand_source(s_mp_rand_jenkins); + mp_clear_multi(&a, &b, NULL); + return err == MP_OKAY ? EXIT_SUCCESS : EXIT_FAILURE; +} + +static int test_mp_kronecker(void) +{ + struct mp_kronecker_st { + long n; + int c[21]; + }; + static struct mp_kronecker_st kronecker[] = { + /*-10, -9, -8, -7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10*/ + { -10, { 0, -1, 0, -1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0 } }, + { -9, { -1, 0, -1, 1, 0, -1, -1, 0, -1, -1, 0, 1, 1, 0, 1, 1, 0, -1, 1, 0, 1 } }, + { -8, { 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0 } }, + { -7, { 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1 } }, + { -6, { 0, 0, 0, -1, 0, -1, 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0 } }, + { -5, { 0, -1, 1, -1, 1, 0, -1, -1, 1, -1, 0, 1, -1, 1, 1, 0, -1, 1, -1, 1, 0 } }, + { -4, { 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0 } }, + { -3, { -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1, -1, 0, 1 } }, + { -2, { 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0 } }, + { -1, { -1, -1, -1, 1, 1, -1, -1, 1, -1, -1, 1, 1, 1, -1, 1, 1, -1, -1, 1, 1, 1 } }, + { 0, { 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0 } }, + { 1, { 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 } }, + { 2, { 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0 } }, + { 3, { 1, 0, -1, -1, 0, -1, 1, 0, -1, 1, 0, 1, -1, 0, 1, -1, 0, -1, -1, 0, 1 } }, + { 4, { 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0 } }, + { 5, { 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0, 1, -1, -1, 1, 0 } }, + { 6, { 0, 0, 0, -1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 0, 0 } }, + { 7, { -1, 1, 1, 0, 1, -1, 1, 1, 1, 1, 0, 1, 1, 1, 1, -1, 1, 0, 1, 1, -1 } }, + { 8, { 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0 } }, + { 9, { 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1 } }, + { 10, { 0, 1, 0, -1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, -1, 0, 1, 0 } } + }; + + long k, m; + int i, cnt; + mp_err err; + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + mp_set_ul(&a, 0uL); + mp_set_ul(&b, 1uL); + if ((err = mp_kronecker(&a, &b, &i)) != MP_OKAY) { + printf("Failed executing mp_kronecker(0 | 1) %s.\n", mp_error_to_string(err)); + goto LBL_ERR; + } + if (i != 1) { + printf("Failed trivial mp_kronecker(0 | 1) %d != 1\n", i); + goto LBL_ERR; + } + for (cnt = 0; cnt < (int)(sizeof(kronecker)/sizeof(kronecker[0])); ++cnt) { + k = kronecker[cnt].n; + mp_set_l(&a, k); + /* only test positive values of a */ + for (m = -10; m <= 10; m++) { + mp_set_l(&b, m); + if ((err = mp_kronecker(&a, &b, &i)) != MP_OKAY) { + printf("Failed executing mp_kronecker(%ld | %ld) %s.\n", kronecker[cnt].n, m, mp_error_to_string(err)); + goto LBL_ERR; + } + if ((err == MP_OKAY) && (i != kronecker[cnt].c[m + 10])) { + printf("Failed trivial mp_kronecker(%ld | %ld) %d != %d\n", kronecker[cnt].n, m, i, kronecker[cnt].c[m + 10]); + goto LBL_ERR; + } + } + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_complement(void) +{ + int i; + + mp_int a, b, c; + if (mp_init_multi(&a, &b, &c, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + long l = rand_long(); + mp_set_l(&a, l); + mp_complement(&a, &b); + + l = ~l; + mp_set_l(&c, l); + + if (mp_cmp(&b, &c) != MP_EQ) { + printf("\nmp_complement() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_signed_rsh(void) +{ + int i; + + mp_int a, b, d; + if (mp_init_multi(&a, &b, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + long l; + int em; + + l = rand_long(); + mp_set_l(&a, l); + + em = abs(rand_int()) % 32; + + mp_set_l(&d, l >> em); + + mp_signed_rsh(&a, em, &b); + if (mp_cmp(&b, &d) != MP_EQ) { + printf("\nmp_signed_rsh() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &d, NULL); + return EXIT_FAILURE; + +} + +static int test_mp_xor(void) +{ + int i; + + mp_int a, b, c, d; + if (mp_init_multi(&a, &b, &c, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + long l, em; + + l = rand_long(); + mp_set_l(&a,l); + + em = rand_long(); + mp_set_l(&b, em); + + mp_set_l(&d, l ^ em); + + mp_xor(&a, &b, &c); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("\nmp_xor() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; + +} + +static int test_mp_or(void) +{ + int i; + + mp_int a, b, c, d; + if (mp_init_multi(&a, &b, &c, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + long l, em; + + l = rand_long(); + mp_set_l(&a, l); + + em = rand_long(); + mp_set_l(&b, em); + + mp_set_l(&d, l | em); + + mp_or(&a, &b, &c); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("\nmp_or() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int test_mp_and(void) +{ + int i; + + mp_int a, b, c, d; + if (mp_init_multi(&a, &b, &c, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + long l, em; + + l = rand_long(); + mp_set_l(&a, l); + + em = rand_long(); + mp_set_l(&b, em); + + mp_set_l(&d, l & em); + + mp_and(&a, &b, &c); + if (mp_cmp(&c, &d) != MP_EQ) { + printf("\nmp_and() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int test_mp_invmod(void) +{ + mp_int a, b, c, d; + if (mp_init_multi(&a, &b, &c, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* mp_invmod corner-case of https://github.com/libtom/libtommath/issues/118 */ + { + const char *a_ = "47182BB8DF0FFE9F61B1F269BACC066B48BA145D35137D426328DC3F88A5EA44"; + const char *b_ = "FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFF"; + const char *should_ = "0521A82E10376F8E4FDEF9A32A427AC2A0FFF686E00290D39E3E4B5522409596"; + + if (mp_read_radix(&a, a_, 16) != MP_OKAY) { + printf("\nmp_read_radix(a) failed!"); + goto LBL_ERR; + } + if (mp_read_radix(&b, b_, 16) != MP_OKAY) { + printf("\nmp_read_radix(b) failed!"); + goto LBL_ERR; + } + if (mp_read_radix(&c, should_, 16) != MP_OKAY) { + printf("\nmp_read_radix(should) failed!"); + goto LBL_ERR; + } + + if (mp_invmod(&a, &b, &d) != MP_OKAY) { + printf("\nmp_invmod() failed!"); + goto LBL_ERR; + } + + if (mp_cmp(&c, &d) != MP_EQ) { + printf("\nmp_invmod() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; + +} + +#if defined(__STDC_IEC_559__) || defined(__GCC_IEC_559) +static int test_mp_set_double(void) +{ + int i; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test mp_get_double/mp_set_double */ + if (mp_set_double(&a, +1.0/0.0) != MP_VAL) { + printf("\nmp_set_double should return MP_VAL for +inf"); + goto LBL_ERR; + } + if (mp_set_double(&a, -1.0/0.0) != MP_VAL) { + printf("\nmp_set_double should return MP_VAL for -inf"); + goto LBL_ERR; + } + if (mp_set_double(&a, +0.0/0.0) != MP_VAL) { + printf("\nmp_set_double should return MP_VAL for NaN"); + goto LBL_ERR; + } + if (mp_set_double(&a, -0.0/0.0) != MP_VAL) { + printf("\nmp_set_double should return MP_VAL for NaN"); + goto LBL_ERR; + } + + for (i = 0; i < 1000; ++i) { + int tmp = rand_int(); + double dbl = (double)tmp * rand_int() + 1; + if (mp_set_double(&a, dbl) != MP_OKAY) { + printf("\nmp_set_double() failed"); + goto LBL_ERR; + } + if (dbl != mp_get_double(&a)) { + printf("\nmp_get_double() bad result!"); + goto LBL_ERR; + } + if (mp_set_double(&a, -dbl) != MP_OKAY) { + printf("\nmp_set_double() failed"); + goto LBL_ERR; + } + if (-dbl != mp_get_double(&a)) { + printf("\nmp_get_double() bad result!"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; + +} +#endif + +static int test_mp_get_u32(void) +{ + unsigned long t; + int i; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + t = (unsigned long)rand_long() & 0xFFFFFFFFuL; + mp_set_ul(&a, t); + if (t != mp_get_u32(&a)) { + printf("\nmp_get_u32() bad result!"); + goto LBL_ERR; + } + } + mp_set_ul(&a, 0uL); + if (mp_get_u32(&a) != 0) { + printf("\nmp_get_u32() bad result!"); + goto LBL_ERR; + } + mp_set_ul(&a, 0xFFFFFFFFuL); + if (mp_get_u32(&a) != 0xFFFFFFFFuL) { + printf("\nmp_get_u32() bad result!"); + goto LBL_ERR; + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_get_ul(void) +{ + unsigned long s, t; + int i; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < ((int)MP_SIZEOF_BITS(unsigned long) - 1); ++i) { + t = (1UL << (i+1)) - 1; + if (!t) + t = ~0UL; + printf(" t = 0x%lx i = %d\r", t, i); + do { + mp_set_ul(&a, t); + s = mp_get_ul(&a); + if (s != t) { + printf("\nmp_get_ul() bad result! 0x%lx != 0x%lx", s, t); + goto LBL_ERR; + } + t <<= 1; + } while (t != 0uL); + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_get_u64(void) +{ + unsigned long long q, r; + int i; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < (int)(MP_SIZEOF_BITS(unsigned long long) - 1); ++i) { + r = (1ULL << (i+1)) - 1; + if (!r) + r = ~0ULL; + printf(" r = 0x%llx i = %d\r", r, i); + do { + mp_set_u64(&a, r); + q = mp_get_u64(&a); + if (q != r) { + printf("\nmp_get_u64() bad result! 0x%llx != 0x%llx", q, r); + goto LBL_ERR; + } + r <<= 1; + } while (r != 0uLL); + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; + +} + +static int test_mp_sqrt(void) +{ + int i, n; + + mp_int a, b, c; + if (mp_init_multi(&a, &b, &c, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + printf("%6d\r", i); + fflush(stdout); + n = (rand_int() & 15) + 1; + mp_rand(&a, n); + if (mp_sqrt(&a, &b) != MP_OKAY) { + printf("\nmp_sqrt() error!"); + goto LBL_ERR; + } + mp_root_u32(&a, 2uL, &c); + if (mp_cmp_mag(&b, &c) != MP_EQ) { + printf("mp_sqrt() bad result!\n"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_is_square(void) +{ + int i, n; + + mp_int a, b; + mp_bool res; + + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + for (i = 0; i < 1000; ++i) { + printf("%6d\r", i); + fflush(stdout); + + /* test mp_is_square false negatives */ + n = (rand_int() & 7) + 1; + mp_rand(&a, n); + mp_sqr(&a, &a); + if (mp_is_square(&a, &res) != MP_OKAY) { + printf("\nfn:mp_is_square() error!"); + goto LBL_ERR; + } + if (res == MP_NO) { + printf("\nfn:mp_is_square() bad result!"); + goto LBL_ERR; + } + + /* test for false positives */ + mp_add_d(&a, 1uL, &a); + if (mp_is_square(&a, &res) != MP_OKAY) { + printf("\nfp:mp_is_square() error!"); + goto LBL_ERR; + } + if (res == MP_YES) { + printf("\nfp:mp_is_square() bad result!"); + goto LBL_ERR; + } + + } + printf("\n\n"); + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_sqrtmod_prime(void) +{ + struct mp_sqrtmod_prime_st { + unsigned long p; + unsigned long n; + mp_digit r; + }; + + static struct mp_sqrtmod_prime_st sqrtmod_prime[] = { + { 5, 14, 3 }, + { 7, 9, 4 }, + { 113, 2, 62 } + }; + int i; + + mp_int a, b, c; + if (mp_init_multi(&a, &b, &c, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* r^2 = n (mod p) */ + for (i = 0; i < (int)(sizeof(sqrtmod_prime)/sizeof(sqrtmod_prime[0])); ++i) { + mp_set_ul(&a, sqrtmod_prime[i].p); + mp_set_ul(&b, sqrtmod_prime[i].n); + if (mp_sqrtmod_prime(&b, &a, &c) != MP_OKAY) { + printf("Failed executing %d. mp_sqrtmod_prime\n", (i+1)); + goto LBL_ERR; + } + if (mp_cmp_d(&c, sqrtmod_prime[i].r) != MP_EQ) { + printf("Failed %d. trivial mp_sqrtmod_prime\n", (i+1)); + ndraw(&c, "r"); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_prime_rand(void) +{ + int ix; + mp_err err; + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test for size */ + for (ix = 10; ix < 128; ix++) { + printf("Testing (not safe-prime): %9d bits \r", ix); + fflush(stdout); + err = mp_prime_rand(&a, 8, ix, (rand_int() & 1) ? 0 : MP_PRIME_2MSB_ON); + if (err != MP_OKAY) { + printf("\nfailed with error: %s\n", mp_error_to_string(err)); + goto LBL_ERR; + } + if (mp_count_bits(&a) != ix) { + printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); + goto LBL_ERR; + } + } + printf("\n"); + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_prime_is_prime(void) +{ + int ix; + mp_err err; + mp_bool cnt, fu; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* strong Miller-Rabin pseudoprime to the first 200 primes (F. Arnault) */ + puts("Testing mp_prime_is_prime() with Arnault's pseudoprime 803...901 \n"); + mp_read_radix(&a, + "91xLNF3roobhzgTzoFIG6P13ZqhOVYSN60Fa7Cj2jVR1g0k89zdahO9/kAiRprpfO1VAp1aBHucLFV/qLKLFb+zonV7R2Vxp1K13ClwUXStpV0oxTNQVjwybmFb5NBEHImZ6V7P6+udRJuH8VbMEnS0H8/pSqQrg82OoQQ2fPpAk6G1hkjqoCv5s/Yr", + 64); + mp_prime_is_prime(&a, mp_prime_rabin_miller_trials(mp_count_bits(&a)), &cnt); + if (cnt == MP_YES) { + printf("Arnault's pseudoprime is not prime but mp_prime_is_prime says it is.\n"); + goto LBL_ERR; + } + /* About the same size as Arnault's pseudoprime */ + puts("Testing mp_prime_is_prime() with certified prime 2^1119 + 53\n"); + mp_set(&a, 1uL); + mp_mul_2d(&a,1119,&a); + mp_add_d(&a, 53uL, &a); + err = mp_prime_is_prime(&a, mp_prime_rabin_miller_trials(mp_count_bits(&a)), &cnt); + /* small problem */ + if (err != MP_OKAY) { + printf("\nfailed with error: %s\n", mp_error_to_string(err)); + } + /* large problem */ + if (cnt == MP_NO) { + printf("A certified prime is a prime but mp_prime_is_prime says it is not.\n"); + } + if ((err != MP_OKAY) || (cnt == MP_NO)) { + printf("prime tested was: 0x"); + mp_fwrite(&a,16,stdout); + putchar('\n'); + goto LBL_ERR; + } + for (ix = 16; ix < 128; ix++) { + printf("Testing ( safe-prime): %9d bits \r", ix); + fflush(stdout); + err = mp_prime_rand(&a, 8, ix, ((rand_int() & 1) ? 0 : MP_PRIME_2MSB_ON) | MP_PRIME_SAFE); + if (err != MP_OKAY) { + printf("\nfailed with error: %s\n", mp_error_to_string(err)); + goto LBL_ERR; + } + if (mp_count_bits(&a) != ix) { + printf("Prime is %d not %d bits!!!\n", mp_count_bits(&a), ix); + goto LBL_ERR; + } + /* let's see if it's really a safe prime */ + mp_sub_d(&a, 1uL, &b); + mp_div_2(&b, &b); + err = mp_prime_is_prime(&b, mp_prime_rabin_miller_trials(mp_count_bits(&b)), &cnt); + /* small problem */ + if (err != MP_OKAY) { + printf("\nfailed with error: %s\n", mp_error_to_string(err)); + } + /* large problem */ + if (cnt == MP_NO) { + printf("\nsub is not prime!\n"); + } + mp_prime_frobenius_underwood(&b, &fu); + if (fu == MP_NO) { + printf("\nfrobenius-underwood says sub is not prime!\n"); + } + if ((err != MP_OKAY) || (cnt == MP_NO)) { + printf("prime tested was: 0x"); + mp_fwrite(&a,16,stdout); + putchar('\n'); + printf("sub tested was: 0x"); + mp_fwrite(&b,16,stdout); + putchar('\n'); + goto LBL_ERR; + } + + } + /* Check regarding problem #143 */ +#ifndef MP_8BIT + mp_read_radix(&a, + "FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F14374FE1356D6D51C245E485B576625E7EC6F44C42E9A63A3620FFFFFFFFFFFFFFFF", + 16); + err = mp_prime_strong_lucas_selfridge(&a, &cnt); + /* small problem */ + if (err != MP_OKAY) { + printf("\nmp_prime_strong_lucas_selfridge failed with error: %s\n", mp_error_to_string(err)); + } + /* large problem */ + if (cnt == MP_NO) { + printf("\n\nissue #143 - mp_prime_strong_lucas_selfridge FAILED!\n"); + } + if ((err != MP_OKAY) || (cnt == MP_NO)) { + printf("prime tested was: 0x"); + mp_fwrite(&a,16,stdout); + putchar('\n'); + goto LBL_ERR; + } +#endif + + printf("\n\n"); + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; + +} + + +static int test_mp_prime_next_prime(void) +{ + mp_err err; + mp_int a, b, c; + + mp_init_multi(&a, &b, &c, NULL); + + + /* edge cases */ + mp_set(&a, 0u); + if ((err = mp_prime_next_prime(&a, 5, 0)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp_d(&a, 2u) != MP_EQ) { + printf("mp_prime_next_prime: output should have been 2 but was: "); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + + mp_set(&a, 0u); + if ((err = mp_prime_next_prime(&a, 5, 1)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp_d(&a, 3u) != MP_EQ) { + printf("mp_prime_next_prime: output should have been 3 but was: "); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + + mp_set(&a, 2u); + if ((err = mp_prime_next_prime(&a, 5, 0)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp_d(&a, 3u) != MP_EQ) { + printf("mp_prime_next_prime: output should have been 3 but was: "); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + + mp_set(&a, 2u); + if ((err = mp_prime_next_prime(&a, 5, 1)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp_d(&a, 3u) != MP_EQ) { + printf("mp_prime_next_prime: output should have been 3 but was: "); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + mp_set(&a, 8); + if ((err = mp_prime_next_prime(&a, 5, 1)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp_d(&a, 11u) != MP_EQ) { + printf("mp_prime_next_prime: output should have been 11 but was: "); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + /* 2^300 + 157 is a 300 bit large prime to guarantee a multi-limb bigint */ + if ((err = mp_2expt(&a, 300)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set_u32(&b, 157); + if ((err = mp_add(&a, &b, &a)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_copy(&a, &b)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 2^300 + 385 is the next prime */ + mp_set_u32(&c, 228); + if ((err = mp_add(&b, &c, &b)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_prime_next_prime(&a, 5, 0)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp(&a, &b) != MP_EQ) { + printf("mp_prime_next_prime: output should have been\n"); + mp_fwrite(&b,10,stdout); + putchar('\n'); + printf("but was:\n"); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + + /* Use another temporary variable or recompute? Mmh... */ + if ((err = mp_2expt(&a, 300)) != MP_OKAY) { + goto LBL_ERR; + } + mp_set_u32(&b, 157); + if ((err = mp_add(&a, &b, &a)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_copy(&a, &b)) != MP_OKAY) { + goto LBL_ERR; + } + + /* 2^300 + 631 is the next prime congruent to 3 mod 4*/ + mp_set_u32(&c, 474); + if ((err = mp_add(&b, &c, &b)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_prime_next_prime(&a, 5, 1)) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_cmp(&a, &b) != MP_EQ) { + printf("mp_prime_next_prime (bbs): output should have been\n"); + mp_fwrite(&b,10,stdout); + putchar('\n'); + printf("but was:\n"); + mp_fwrite(&a,10,stdout); + putchar('\n'); + goto LBL_ERR; + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_montgomery_reduce(void) +{ + mp_digit mp; + int ix, i, n; + char buf[4096]; + + /* size_t written; */ + + mp_int a, b, c, d, e; + if (mp_init_multi(&a, &b, &c, &d, &e, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test montgomery */ + for (i = 1; i <= 10; i++) { + if (i == 10) + i = 1000; + printf(" digit size: %2d\r", i); + fflush(stdout); + for (n = 0; n < 1000; n++) { + mp_rand(&a, i); + a.dp[0] |= 1; + + /* let's see if R is right */ + mp_montgomery_calc_normalization(&b, &a); + mp_montgomery_setup(&a, &mp); + + /* now test a random reduction */ + for (ix = 0; ix < 100; ix++) { + mp_rand(&c, 1 + abs(rand_int()) % (2*i)); + mp_copy(&c, &d); + mp_copy(&c, &e); + + mp_mod(&d, &a, &d); + mp_montgomery_reduce(&c, &a, mp); + mp_mulmod(&c, &b, &a, &c); + + if (mp_cmp(&c, &d) != MP_EQ) { +/* *INDENT-OFF* */ + printf("d = e mod a, c = e MOD a\n"); + mp_to_decimal(&a, buf, sizeof(buf)); printf("a = %s\n", buf); + mp_to_decimal(&e, buf, sizeof(buf)); printf("e = %s\n", buf); + mp_to_decimal(&d, buf, sizeof(buf)); printf("d = %s\n", buf); + mp_to_decimal(&c, buf, sizeof(buf)); printf("c = %s\n", buf); + + printf("compare no compare!\n"); goto LBL_ERR; +/* *INDENT-ON* */ + } + /* only one big montgomery reduction */ + if (i > 10) { + n = 1000; + ix = 100; + } + } + } + } + + printf("\n\n"); + + mp_clear_multi(&a, &b, &c, &d, &e, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, &e, NULL); + return EXIT_FAILURE; + +} + +static int test_mp_read_radix(void) +{ + char buf[4096]; + size_t written; + mp_err err; + + mp_int a; + if (mp_init_multi(&a, NULL)!= MP_OKAY) goto LTM_ERR; + + if ((err = mp_read_radix(&a, "123456", 10)) != MP_OKAY) goto LTM_ERR; + + if ((err = mp_to_radix(&a, buf, SIZE_MAX, &written, 10)) != MP_OKAY) goto LTM_ERR; + printf(" '123456' a == %s, length = %zu\n", buf, written); + + /* See comment in bn_mp_to_radix.c */ + /* + if( (err = mp_to_radix(&a, buf, 3u, &written, 10) ) != MP_OKAY) goto LTM_ERR; + printf(" '56' a == %s, length = %zu\n", buf, written); + + if( (err = mp_to_radix(&a, buf, 4u, &written, 10) ) != MP_OKAY) goto LTM_ERR; + printf(" '456' a == %s, length = %zu\n", buf, written); + if( (err = mp_to_radix(&a, buf, 30u, &written, 10) ) != MP_OKAY) goto LTM_ERR; + printf(" '123456' a == %s, length = %zu, error = %s\n", + buf, written, mp_error_to_string(err)); + */ + if ((err = mp_read_radix(&a, "-123456", 10)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_to_radix(&a, buf, SIZE_MAX, &written, 10)) != MP_OKAY) goto LTM_ERR; + printf(" '-123456' a == %s, length = %zu\n", buf, written); + + if ((err = mp_read_radix(&a, "0", 10)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_to_radix(&a, buf, SIZE_MAX, &written, 10)) != MP_OKAY) goto LTM_ERR; + printf(" '0' a == %s, length = %zu\n", buf, written); + + + + /* Although deprecated it needs to function as long as it isn't dropped */ + /* + printf("Testing deprecated mp_toradix_n\n"); + if( (err = mp_read_radix(&a, "-123456", 10) ) != MP_OKAY) goto LTM_ERR; + if( (err = mp_toradix_n(&a, buf, 10, 3) ) != MP_OKAY) goto LTM_ERR; + printf("a == %s\n", buf); + if( (err = mp_toradix_n(&a, buf, 10, 4) ) != MP_OKAY) goto LTM_ERR; + printf("a == %s\n", buf); + if( (err = mp_toradix_n(&a, buf, 10, 30) ) != MP_OKAY) goto LTM_ERR; + printf("a == %s\n", buf); + */ + + + while (0) { + char *s = fgets(buf, sizeof(buf), stdin); + if (s != buf) break; + mp_read_radix(&a, buf, 10); + mp_prime_next_prime(&a, 5, 1); + mp_to_radix(&a, buf, sizeof(buf), NULL, 10); + printf("%s, %lu\n", buf, (unsigned long)a.dp[0] & 3uL); + } + + mp_clear(&a); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear(&a); + return EXIT_FAILURE; +} + +static int test_mp_cnt_lsb(void) +{ + int ix; + + mp_int a, b; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + mp_set(&a, 1uL); + for (ix = 0; ix < 1024; ix++) { + if (mp_cnt_lsb(&a) != ix) { + printf("Failed at %d, %d\n", ix, mp_cnt_lsb(&a)); + goto LBL_ERR; + } + mp_mul_2(&a, &a); + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; + +} + +static int test_mp_reduce_2k(void) +{ + int ix, cnt; + + mp_int a, b, c, d; + if (mp_init_multi(&a, &b, &c, &d, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test mp_reduce_2k */ + for (cnt = 3; cnt <= 128; ++cnt) { + mp_digit tmp; + + mp_2expt(&a, cnt); + mp_sub_d(&a, 2uL, &a); /* a = 2**cnt - 2 */ + + printf("\r %4d bits", cnt); + printf("(%d)", mp_reduce_is_2k(&a)); + mp_reduce_2k_setup(&a, &tmp); + printf("(%lu)", (unsigned long) tmp); + for (ix = 0; ix < 1000; ix++) { + if (!(ix & 127)) { + printf("."); + fflush(stdout); + } + mp_rand(&b, (cnt / MP_DIGIT_BIT + 1) * 2); + mp_copy(&c, &b); + mp_mod(&c, &a, &c); + mp_reduce_2k(&b, &a, 2uL); + if (mp_cmp(&c, &b) != MP_EQ) { + printf("FAILED\n"); + goto LBL_ERR; + } + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int test_mp_div_3(void) +{ + int cnt; + + mp_int a, b, c, d, e; + if (mp_init_multi(&a, &b, &c, &d, &e, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test mp_div_3 */ + mp_set(&d, 3uL); + for (cnt = 0; cnt < 10000;) { + mp_digit r2; + + if (!(++cnt & 127)) { + printf("%9d\r", cnt); + fflush(stdout); + } + mp_rand(&a, abs(rand_int()) % 128 + 1); + mp_div(&a, &d, &b, &e); + mp_div_3(&a, &c, &r2); + + if (mp_cmp(&b, &c) || mp_cmp_d(&e, r2)) { + printf("\nmp_div_3 => Failure\n"); + goto LBL_ERR; + } + } + printf("\nPassed div_3 testing"); + + mp_clear_multi(&a, &b, &c, &d, &e, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, &d, &e, NULL); + return EXIT_FAILURE; +} + +static int test_mp_dr_reduce(void) +{ + mp_digit mp; + int cnt; + unsigned rr; + int ix; + + mp_int a, b, c; + if (mp_init_multi(&a, &b, &c, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + + /* test the DR reduction */ + for (cnt = 2; cnt < 32; cnt++) { + printf("\r%d digit modulus", cnt); + mp_grow(&a, cnt); + mp_zero(&a); + for (ix = 1; ix < cnt; ix++) { + a.dp[ix] = MP_MASK; + } + a.used = cnt; + a.dp[0] = 3; + + mp_rand(&b, cnt - 1); + mp_copy(&b, &c); + + rr = 0; + do { + if (!(rr & 127)) { + printf("."); + fflush(stdout); + } + mp_sqr(&b, &b); + mp_add_d(&b, 1uL, &b); + mp_copy(&b, &c); + + mp_mod(&b, &a, &b); + mp_dr_setup(&a, &mp); + mp_dr_reduce(&c, &a, mp); + + if (mp_cmp(&b, &c) != MP_EQ) { + printf("Failed on trial %u\n", rr); + goto LBL_ERR; + } + } while (++rr < 500); + printf(" passed"); + fflush(stdout); + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_reduce_2k_l(void) +{ +# if LTM_DEMO_TEST_REDUCE_2K_L + mp_int a, b, c, d; + int cnt; + char buf[4096]; + size_t length[1]; + if (mp_init_multi(&a, &b, NULL)!= MP_OKAY) { + return EXIT_FAILURE; + } + /* test the mp_reduce_2k_l code */ +# if LTM_DEMO_TEST_REDUCE_2K_L == 1 + /* first load P with 2^1024 - 0x2A434 B9FDEC95 D8F9D550 FFFFFFFF FFFFFFFF */ + mp_2expt(&a, 1024); + mp_read_radix(&b, "2A434B9FDEC95D8F9D550FFFFFFFFFFFFFFFF", 16); + mp_sub(&a, &b, &a); +# elif LTM_DEMO_TEST_REDUCE_2K_L == 2 + /* p = 2^2048 - 0x1 00000000 00000000 00000000 00000000 4945DDBF 8EA2A91D 5776399B B83E188F */ + mp_2expt(&a, 2048); + mp_read_radix(&b, + "1000000000000000000000000000000004945DDBF8EA2A91D5776399BB83E188F", + 16); + mp_sub(&a, &b, &a); +# else +# error oops +# endif + *length = sizeof(buf); + mp_to_radix(&a, buf, length, 10); + printf("\n\np==%s, length = %zu\n", buf, *length); + /* now mp_reduce_is_2k_l() should return */ + if (mp_reduce_is_2k_l(&a) != 1) { + printf("mp_reduce_is_2k_l() return 0, should be 1\n"); + goto LBL_ERR; + } + mp_reduce_2k_setup_l(&a, &d); + /* now do a million square+1 to see if it varies */ + mp_rand(&b, 64); + mp_mod(&b, &a, &b); + mp_copy(&b, &c); + printf("Testing: mp_reduce_2k_l..."); + fflush(stdout); + for (cnt = 0; cnt < (int)(1uL << 20); cnt++) { + mp_sqr(&b, &b); + mp_add_d(&b, 1uL, &b); + mp_reduce_2k_l(&b, &a, &d); + mp_sqr(&c, &c); + mp_add_d(&c, 1uL, &c); + mp_mod(&c, &a, &c); + if (mp_cmp(&b, &c) != MP_EQ) { + printf("mp_reduce_2k_l() failed at step %d\n", cnt); + mp_to_hex(&b, buf, sizeof(buf)); + printf("b == %s\n", buf); + mp_to_hex(&c, buf, sizeof(buf)); + printf("c == %s\n", buf); + goto LBL_ERR; + } + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +#else + return EXIT_SUCCESS; +# endif /* LTM_DEMO_TEST_REDUCE_2K_L */ +} +/* stripped down version of mp_radix_size. The faster version can be off by up t +o +3 */ +/* TODO: This function should be removed, replaced by mp_radix_size, mp_radix_size_overestimate in 2.0 */ +static mp_err s_rs(const mp_int *a, int radix, uint32_t *size) +{ + mp_err res; + uint32_t digs = 0u; + mp_int t; + mp_digit d; + *size = 0u; + if (mp_iszero(a) == MP_YES) { + *size = 2u; + return MP_OKAY; + } + if (radix == 2) { + *size = (uint32_t)mp_count_bits(a) + 1u; + return MP_OKAY; + } + if ((res = mp_init_copy(&t, a)) != MP_OKAY) { + return res; + } + t.sign = MP_ZPOS; + while (mp_iszero(&t) == MP_NO) { + if ((res = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) { + mp_clear(&t); + return res; + } + ++digs; + } + mp_clear(&t); + *size = digs + 1; + return MP_OKAY; +} +static int test_mp_log_u32(void) +{ + mp_int a; + mp_digit d; + uint32_t base, lb, size; + const uint32_t max_base = MP_MIN(UINT32_MAX, MP_DIGIT_MAX); + + if (mp_init(&a) != MP_OKAY) { + goto LBL_ERR; + } + + /* + base a result + 0 x MP_VAL + 1 x MP_VAL + */ + mp_set(&a, 42uL); + base = 0u; + if (mp_log_u32(&a, base, &lb) != MP_VAL) { + goto LBL_ERR; + } + base = 1u; + if (mp_log_u32(&a, base, &lb) != MP_VAL) { + goto LBL_ERR; + } + /* + base a result + 2 0 MP_VAL + 2 1 0 + 2 2 1 + 2 3 1 + */ + base = 2u; + mp_zero(&a); + if (mp_log_u32(&a, base, &lb) != MP_VAL) { + goto LBL_ERR; + } + + for (d = 1; d < 4; d++) { + mp_set(&a, d); + if (mp_log_u32(&a, base, &lb) != MP_OKAY) { + goto LBL_ERR; + } + if (lb != ((d == 1)?0uL:1uL)) { + goto LBL_ERR; + } + } + /* + base a result + 3 0 MP_VAL + 3 1 0 + 3 2 0 + 3 3 1 + */ + base = 3u; + mp_zero(&a); + if (mp_log_u32(&a, base, &lb) != MP_VAL) { + goto LBL_ERR; + } + for (d = 1; d < 4; d++) { + mp_set(&a, d); + if (mp_log_u32(&a, base, &lb) != MP_OKAY) { + goto LBL_ERR; + } + if (lb != ((d < base)?0uL:1uL)) { + goto LBL_ERR; + } + } + + /* + bases 2..64 with "a" a random large constant. + The range of bases tested allows to check with + radix_size. + */ + if (mp_rand(&a, 10) != MP_OKAY) { + goto LBL_ERR; + } + for (base = 2u; base < 65u; base++) { + if (mp_log_u32(&a, base, &lb) != MP_OKAY) { + goto LBL_ERR; + } + if (s_rs(&a,(int)base, &size) != MP_OKAY) { + goto LBL_ERR; + } + /* radix_size includes the memory needed for '\0', too*/ + size -= 2; + if (lb != size) { + goto LBL_ERR; + } + } + + /* + bases 2..64 with "a" a random small constant to + test the part of mp_ilogb that uses native types. + */ + if (mp_rand(&a, 1) != MP_OKAY) { + goto LBL_ERR; + } + for (base = 2u; base < 65u; base++) { + if (mp_log_u32(&a, base, &lb) != MP_OKAY) { + goto LBL_ERR; + } + if (s_rs(&a,(int)base, &size) != MP_OKAY) { + goto LBL_ERR; + } + size -= 2; + if (lb != size) { + goto LBL_ERR; + } + } + + /*Test upper edgecase with base UINT32_MAX and number (UINT32_MAX/2)*UINT32_MAX^10 */ + mp_set(&a, max_base); + if (mp_expt_u32(&a, 10uL, &a) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_add_d(&a, max_base / 2, &a) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_log_u32(&a, max_base, &lb) != MP_OKAY) { + goto LBL_ERR; + } + if (lb != 10u) { + goto LBL_ERR; + } + + mp_clear(&a); + return EXIT_SUCCESS; +LBL_ERR: + mp_clear(&a); + return EXIT_FAILURE; +} + +static int test_mp_incr(void) +{ + mp_int a, b; + mp_err e = MP_OKAY; + + if ((e = mp_init_multi(&a, &b, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + + /* Does it increment inside the limits of a MP_xBIT limb? */ + mp_set(&a, MP_MASK/2); + if ((e = mp_incr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp_d(&a, (MP_MASK/2uL) + 1uL) != MP_EQ) { + goto LTM_ERR; + } + + /* Does it increment outside of the limits of a MP_xBIT limb? */ + mp_set(&a, MP_MASK); + mp_set(&b, MP_MASK); + if ((e = mp_incr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if ((e = mp_add_d(&b, 1uL, &b)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&a, &b) != MP_EQ) { + goto LTM_ERR; + } + + /* Does it increment from -1 to 0? */ + mp_set(&a, 1uL); + a.sign = MP_NEG; + if ((e = mp_incr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp_d(&a, 0uL) != MP_EQ) { + goto LTM_ERR; + } + + /* Does it increment from -(MP_MASK + 1) to -MP_MASK? */ + mp_set(&a, MP_MASK); + if ((e = mp_add_d(&a, 1uL, &a)) != MP_OKAY) { + goto LTM_ERR; + } + a.sign = MP_NEG; + if ((e = mp_incr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (a.sign != MP_NEG) { + goto LTM_ERR; + } + a.sign = MP_ZPOS; + if (mp_cmp_d(&a, MP_MASK) != MP_EQ) { + goto LTM_ERR; + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int test_mp_decr(void) +{ + mp_int a, b; + mp_err e = MP_OKAY; + + if ((e = mp_init_multi(&a, &b, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + + /* Does it decrement inside the limits of a MP_xBIT limb? */ + mp_set(&a, MP_MASK/2); + if ((e = mp_decr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp_d(&a, (MP_MASK/2uL) - 1uL) != MP_EQ) { + goto LTM_ERR; + } + + /* Does it decrement outside of the limits of a MP_xBIT limb? */ + mp_set(&a, MP_MASK); + if ((e = mp_add_d(&a, 1uL, &a)) != MP_OKAY) { + goto LTM_ERR; + } + if ((e = mp_decr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp_d(&a, MP_MASK) != MP_EQ) { + goto LTM_ERR; + } + + /* Does it decrement from 0 to -1? */ + mp_zero(&a); + if ((e = mp_decr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (a.sign == MP_NEG) { + a.sign = MP_ZPOS; + if (mp_cmp_d(&a, 1uL) != MP_EQ) { + goto LTM_ERR; + } + } else { + goto LTM_ERR; + } + + + /* Does it decrement from -MP_MASK to -(MP_MASK + 1)? */ + mp_set(&a, MP_MASK); + a.sign = MP_NEG; + mp_set(&b, MP_MASK); + b.sign = MP_NEG; + if ((e = mp_sub_d(&b, 1uL, &b)) != MP_OKAY) { + goto LTM_ERR; + } + if ((e = mp_decr(&a)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&a, &b) != MP_EQ) { + goto LTM_ERR; + } + + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +/* + Cannot test mp_exp(_d) without mp_root and vice versa. + So one of the two has to be tested from scratch. + + Numbers generated by + for i in {1..10} + do + seed=$(head -c 10000 /dev/urandom | tr -dc '[:digit:]' | head -c 120); + echo $seed; + convertbase $seed 10 64; + done + + (The program "convertbase" uses libtommath's to/from_radix functions) + + Roots were precalculated with Pari/GP + + default(realprecision,1000); + for(n=3,100,r = floor(a^(1/n));printf("\"" r "\", ")) + + All numbers as strings to simplifiy things, especially for the + low-mp branch. +*/ + +static int test_mp_root_u32(void) +{ + mp_int a, c, r; + mp_err e; + int i, j; + + const char *input[] = { + "4n9cbk886QtLQmofprid3l2Q0GD8Yv979Lh8BdZkFE8g2pDUUSMBET/+M/YFyVZ3mBp", + "5NlgzHhmIX05O5YoW5yW5reAlVNtRAlIcN2dfoATnNdc1Cw5lHZUTwNthmK6/ZLKfY6", + "3gweiHDX+ji5utraSe46IJX+uuh7iggs63xIpMP5MriU4Np+LpHI5are8RzS9pKh9xP", + "5QOJUSKMrfe7LkeyJOlupS8h7bjT+TXmZkDzOjZtfj7mdA7cbg0lRX3CuafhjIrpK8S", + "4HtYFldVkyVbrlg/s7kmaA7j45PvLQm+1bbn6ehgP8tVoBmGbv2yDQI1iQQze4AlHyN", + "3bwCUx79NAR7c68OPSp5ZabhZ9aBEr7rWNTO2oMY7zhbbbw7p6shSMxqE9K9nrTNucf", + "4j5RGb78TfuYSzrXn0z6tiAoWiRI81hGY3el9AEa9S+gN4x/AmzotHT2Hvj6lyBpE7q", + "4lwg30SXqZhEHNsl5LIXdyu7UNt0VTWebP3m7+WUL+hsnFW9xJe7UnzYngZsvWh14IE", + "1+tcqFeRuGqjRADRoRUJ8gL4UUSFQVrVVoV6JpwVcKsuBq5G0pABn0dLcQQQMViiVRj", + "hXwxuFySNSFcmbrs/coz4FUAaUYaOEt+l4V5V8vY71KyBvQPxRq/6lsSrG2FHvWDax" + }; + /* roots 3-100 of the above */ + const char *root[10][100] = { + { + "9163694094944489658600517465135586130944", + "936597377180979771960755204040", "948947857956884030956907", + "95727185767390496595", "133844854039712620", "967779611885360", + "20926191452627", "974139547476", "79203891950", "9784027073", + "1667309744", "365848129", "98268452", "31109156", "11275351", + "4574515", "2040800", "986985", "511525", "281431", "163096", + "98914", "62437", "40832", "27556", "19127", "13614", "9913", + "7367", "5577", "4294", "3357", "2662", "2138", "1738", "1428", + "1185", "993", "839", "715", "613", "530", "461", "403", "355", + "314", "279", "249", "224", "202", "182", "166", "151", "138", + "126", "116", "107", "99", "92", "85", "79", "74", "69", "65", "61", + "57", "54", "51", "48", "46", "43", "41", "39", "37", "36", "34", + "32", "31", "30", "28", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "9534798256755061606359588498764080011382", + "964902943621813525741417593772", "971822399862464674540423", + "97646291566833512831", "136141536090599560", "982294733581430", + "21204945933335", "985810529393", "80066084985", "9881613813", + "1682654547", "368973625", "99051783", "31341581", "11354620", + "4604882", "2053633", "992879", "514434", "282959", "163942", + "99406", "62736", "41020", "27678", "19208", "13670", "9952", + "7395", "5598", "4310", "3369", "2671", "2145", "1744", "1433", + "1189", "996", "842", "717", "615", "531", "462", "404", "356", + "315", "280", "250", "224", "202", "183", "166", "151", "138", + "127", "116", "107", "99", "92", "85", "80", "74", "70", "65", "61", + "58", "54", "51", "48", "46", "43", "41", "39", "37", "36", "34", + "32", "31", "30", "29", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "8398539113202579297642815367509019445624", + "877309458945432597462853440936", "900579899458998599215071", + "91643543761699761637", "128935656335800903", "936647990947203", + "20326748623514", "948988882684", "77342677787", "9573063447", + "1634096832", "359076114", "96569670", "30604705", "11103188", + "4508519", "2012897", "974160", "505193", "278105", "161251", + "97842", "61788", "40423", "27291", "18949", "13492", "9826", + "7305", "5532", "4260", "3332", "2642", "2123", "1726", "1418", + "1177", "986", "834", "710", "610", "527", "458", "401", "353", + "312", "278", "248", "223", "201", "181", "165", "150", "137", + "126", "116", "107", "99", "91", "85", "79", "74", "69", "65", "61", + "57", "54", "51", "48", "46", "43", "41", "39", "37", "35", "34", + "32", "31", "30", "28", "27", "26", "25", "24", "23", "22", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "9559098494021810340217797724866627755195", + "966746709063325235560830083787", "973307706084821682248292", + "97770642291138756434", "136290128605981259", "983232784778520", + "21222944848922", "986563584410", "80121684894", "9887903837", + "1683643206", "369174929", "99102220", "31356542", "11359721", + "4606836", "2054458", "993259", "514621", "283057", "163997", + "99437", "62755", "41032", "27686", "19213", "13674", "9955", + "7397", "5599", "4311", "3370", "2672", "2146", "1744", "1433", + "1189", "996", "842", "717", "615", "532", "462", "404", "356", + "315", "280", "250", "224", "202", "183", "166", "151", "138", + "127", "116", "107", "99", "92", "86", "80", "74", "70", "65", "61", + "58", "54", "51", "48", "46", "43", "41", "39", "37", "36", "34", + "32", "31", "30", "29", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "8839202025813295923132694443541993309220", + "911611499784863252820288596270", "928640961450376817534853", + "94017030509441723821", "131792686685970629", "954783483196511", + "20676214073400", "963660189823", "78428929840", "9696237956", + "1653495486", "363032624", "97562430", "30899570", "11203842", + "4547110", "2029216", "981661", "508897", "280051", "162331", + "98469", "62168", "40663", "27446", "19053", "13563", "9877", + "7341", "5558", "4280", "3347", "2654", "2132", "1733", "1424", + "1182", "990", "837", "713", "612", "529", "460", "402", "354", + "313", "279", "249", "223", "201", "182", "165", "150", "138", + "126", "116", "107", "99", "92", "85", "79", "74", "69", "65", "61", + "57", "54", "51", "48", "46", "43", "41", "39", "37", "36", "34", + "32", "31", "30", "28", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "8338442683973420410660145045849076963795", + "872596990706967613912664152945", "896707843885562730147307", + "91315073695274540969", "128539440806486007", "934129001105825", + "20278149285734", "946946589774", "77191347471", "9555892093", + "1631391010", "358523975", "96431070", "30563524", "11089126", + "4503126", "2010616", "973111", "504675", "277833", "161100", + "97754", "61734", "40390", "27269", "18934", "13482", "9819", + "7300", "5528", "4257", "3330", "2641", "2122", "1725", "1417", + "1177", "986", "833", "710", "609", "527", "458", "401", "353", + "312", "278", "248", "222", "200", "181", "165", "150", "137", + "126", "116", "107", "99", "91", "85", "79", "74", "69", "65", "61", + "57", "54", "51", "48", "46", "43", "41", "39", "37", "35", "34", + "32", "31", "30", "28", "27", "26", "25", "24", "23", "22", "22", + "21", "20", "20", "19", "18", "18", "17", "17", "16", "16", "15" + }, { + "9122818552483814953977703257848970704164", + "933462289569511464780529972314", "946405863353935713909178", + "95513446972056321834", "133588658082928446", + "966158521967027", "20895030642048", "972833934108", + "79107381638", "9773098125", "1665590516", "365497822", + "98180628", "31083090", "11266459", "4571108", "2039360", + "986323", "511198", "281260", "163001", "98858", + "62404", "40811", "27543", "19117", "13608", "9908", + "7363", "5575", "4292", "3356", "2661", "2138", + "1737", "1428", "1185", "993", "839", "714", "613", + "530", "461", "403", "355", "314", "279", "249", + "224", "202", "182", "165", "151", "138", "126", + "116", "107", "99", "92", "85", "79", "74", "69", + "65", "61", "57", "54", "51", "48", "46", "43", + "41", "39", "37", "36", "34", "32", "31", "30", + "28", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", + "16", "16", "15" + }, { + "9151329724083804100369546479681933027521", + "935649419557299174433860420387", "948179413831316112751907", + "95662582675170358900", "133767426788182384", + "967289728859610", "20916775466497", "973745045600", + "79174731802", "9780725058", "1666790321", "365742295", + "98241919", "31101281", "11272665", "4573486", "2040365", + "986785", "511426", "281380", "163067", "98897", + "62427", "40826", "27552", "19124", "13612", "9911", + "7366", "5576", "4294", "3357", "2662", "2138", + "1738", "1428", "1185", "993", "839", "715", "613", + "530", "461", "403", "355", "314", "279", "249", + "224", "202", "182", "165", "151", "138", "126", + "116", "107", "99", "92", "85", "79", "74", "69", + "65", "61", "57", "54", "51", "48", "46", "43", + "41", "39", "37", "36", "34", "32", "31", "30", + "28", "27", "26", "25", "24", "23", "23", "22", + "21", "20", "20", "19", "18", "18", "17", "17", + "16", "16", "15" + }, { + "6839396355168045468586008471269923213531", + "752078770083218822016981965090", "796178899357307807726034", + "82700643015444840424", "118072966296549115", + "867224751770392", "18981881485802", "892288574037", + "73130030771", "9093989389", "1558462688", "343617470", + "92683740", "29448679", "10708016", "4356820", "1948676", + "944610", "490587", "270425", "156989", "95362", + "60284", "39477", "26675", "18536", "13208", "9627", + "7161", "5426", "4181", "3272", "2596", "2087", + "1697", "1395", "1159", "971", "821", "700", "601", + "520", "452", "396", "348", "308", "274", "245", + "220", "198", "179", "163", "148", "136", "124", + "114", "106", "98", "91", "84", "78", "73", "68", + "64", "60", "57", "53", "50", "48", "45", "43", + "41", "39", "37", "35", "34", "32", "31", "29", + "28", "27", "26", "25", "24", "23", "22", "22", + "21", "20", "19", "19", "18", "18", "17", "17", + "16", "16", "15" + }, { + "4788090721380022347683138981782307670424", + "575601315594614059890185238256", "642831903229558719812840", + "69196031110028430211", "101340693763170691", + "758683936560287", "16854690815260", "801767985909", + "66353290503", "8318415180", "1435359033", "318340531", + "86304307", "27544217", "10054988", "4105446", "1841996", + "895414", "466223", "257591", "149855", "91205", + "57758", "37886", "25639", "17842", "12730", "9290", + "6918", "5248", "4048", "3170", "2518", "2026", + "1649", "1357", "1128", "946", "800", "682", "586", + "507", "441", "387", "341", "302", "268", "240", + "215", "194", "176", "160", "146", "133", "122", + "112", "104", "96", "89", "83", "77", "72", "67", + "63", "59", "56", "53", "50", "47", "45", "42", + "40", "38", "36", "35", "33", "32", "30", "29", + "28", "27", "26", "25", "24", "23", "22", "21", + "21", "20", "19", "19", "18", "17", "17", "16", + "16", "15", "15" + } + }; + + if ((e = mp_init_multi(&a, &c, &r, NULL)) != MP_OKAY) { + return EXIT_FAILURE; + } +#ifdef MP_8BIT + for (i = 0; i < 1; i++) { +#else + for (i = 0; i < 10; i++) { +#endif + mp_read_radix(&a, input[i], 64); +#ifdef MP_8BIT + for (j = 3; j < 10; j++) { +#else + for (j = 3; j < 100; j++) { +#endif + mp_root_u32(&a, (uint32_t)j, &c); + mp_read_radix(&r, root[i][j-3], 10); + if (mp_cmp(&r, &c) != MP_EQ) { + fprintf(stderr, "mp_root_u32 failed at input #%d, root #%d\n", i, j); + goto LTM_ERR; + } + } + } + mp_clear_multi(&a, &c, &r, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &c, &r, NULL); + return EXIT_FAILURE; +} + +static int test_s_mp_balance_mul(void) +{ + mp_int a, b, c; + mp_err e = MP_OKAY; + + const char *na = + "4b0I5uMTujCysw+1OOuOyH2FX2WymrHUqi8BBDb7XpkV/4i7vXTbEYUy/kdIfCKu5jT5JEqYkdmnn3jAYo8XShPzNLxZx9yoLjxYRyptSuOI2B1DspvbIVYXY12sxPZ4/HCJ4Usm2MU5lO/006KnDMxuxiv1rm6YZJZ0eZU"; + const char *nb = "3x9vs0yVi4hIq7poAeVcggC3WoRt0zRLKO"; + const char *nc = + "HzrSq9WVt1jDTVlwUxSKqxctu2GVD+N8+SVGaPFRqdxyld6IxDBbj27BPJzYUdR96k3sWpkO8XnDBvupGPnehpQe4KlO/KmN1PjFov/UTZYM+LYzkFcBPyV6hkkL8ePC1rlFLAHzgJMBCXVp4mRqtkQrDsZXXlcqlbTFu69wF6zDEysiX2cAtn/kP9ldblJiwYPCD8hG"; + + if ((e = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((e = mp_read_radix(&a, na, 64)) != MP_OKAY) { + goto LTM_ERR; + } + if ((e = mp_read_radix(&b, nb, 64)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((e = s_mp_balance_mul(&a, &b, &c)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((e = mp_read_radix(&b, nc, 64)) != MP_OKAY) { + goto LTM_ERR; + } + + if (mp_cmp(&b, &c) != MP_EQ) { + goto LTM_ERR; + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) +static int test_s_mp_karatsuba_mul(void) +{ + mp_int a, b, c, d; + int size, err; + + if ((err = mp_init_multi(&a, &b, &c, &d, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + for (size = MP_KARATSUBA_MUL_CUTOFF; size < MP_KARATSUBA_MUL_CUTOFF + 20; size++) { + if ((err = mp_rand(&a, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_rand(&b, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_karatsuba_mul(&a, &b, &c)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_mul(&a,&b,&d)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&c, &d) != MP_EQ) { + fprintf(stderr, "Karatsuba multiplication failed at size %d\n", size); + goto LTM_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int test_s_mp_karatsuba_sqr(void) +{ + mp_int a, b, c; + int size, err; + + if ((err = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + for (size = MP_KARATSUBA_SQR_CUTOFF; size < MP_KARATSUBA_SQR_CUTOFF + 20; size++) { + if ((err = mp_rand(&a, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_karatsuba_sqr(&a, &b)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_sqr(&a, &c)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&b, &c) != MP_EQ) { + fprintf(stderr, "Karatsuba squaring failed at size %d\n", size); + goto LTM_ERR; + } + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_s_mp_toom_mul(void) +{ + mp_int a, b, c, d; + int size, err; + +#if (MP_DIGIT_BIT == 60) + int tc_cutoff; +#endif + + if ((err = mp_init_multi(&a, &b, &c, &d, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + /* This number construction is limb-size specific */ +#if (MP_DIGIT_BIT == 60) + if ((err = mp_rand(&a, 1196)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_mul_2d(&a,71787 - mp_count_bits(&a), &a)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((err = mp_rand(&b, 1338)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_mul_2d(&b, 80318 - mp_count_bits(&b), &b)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_mul_2d(&b, 6310, &b)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_2expt(&c, 99000 - 1000)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_add(&b, &c, &b)) != MP_OKAY) { + goto LTM_ERR; + } + + tc_cutoff = TOOM_MUL_CUTOFF; + TOOM_MUL_CUTOFF = INT_MAX; + if ((err = mp_mul(&a, &b, &c)) != MP_OKAY) { + goto LTM_ERR; + } + TOOM_MUL_CUTOFF = tc_cutoff; + if ((err = mp_mul(&a, &b, &d)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&c, &d) != MP_EQ) { + fprintf(stderr, "Toom-Cook 3-way multiplication failed for edgecase f1 * f2\n"); + goto LTM_ERR; + } +#endif + + for (size = MP_TOOM_MUL_CUTOFF; size < MP_TOOM_MUL_CUTOFF + 20; size++) { + if ((err = mp_rand(&a, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = mp_rand(&b, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_toom_mul(&a, &b, &c)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_mul(&a,&b,&d)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&c, &d) != MP_EQ) { + fprintf(stderr, "Toom-Cook 3-way multiplication failed at size %d\n", size); + goto LTM_ERR; + } + } + + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return EXIT_FAILURE; +} + +static int test_s_mp_toom_sqr(void) +{ + mp_int a, b, c; + int size, err; + + if ((err = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + for (size = MP_TOOM_SQR_CUTOFF; size < MP_TOOM_SQR_CUTOFF + 20; size++) { + if ((err = mp_rand(&a, size)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_toom_sqr(&a, &b)) != MP_OKAY) { + goto LTM_ERR; + } + if ((err = s_mp_sqr(&a, &c)) != MP_OKAY) { + goto LTM_ERR; + } + if (mp_cmp(&b, &c) != MP_EQ) { + fprintf(stderr, "Toom-Cook 3-way squaring failed at size %d\n", size); + goto LTM_ERR; + } + } + + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LTM_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_read_write_ubin(void) +{ + mp_int a, b, c; + int err; + size_t size, len; + unsigned char *buf = NULL; + + if ((err = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((err = mp_rand(&a, 15)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_neg(&a, &b)) != MP_OKAY) goto LTM_ERR; + + size = mp_ubin_size(&a); + printf("mp_to_ubin_size %zu\n", size); + buf = malloc(sizeof(*buf) * size); + if (buf == NULL) { + fprintf(stderr, "test_read_write_binaries (u) failed to allocate %zu bytes\n", + sizeof(*buf) * size); + goto LTM_ERR; + } + + if ((err = mp_to_ubin(&a, buf, size, &len)) != MP_OKAY) goto LTM_ERR; + printf("mp_to_ubin len = %zu\n", len); + + if ((err = mp_from_ubin(&c, buf, len)) != MP_OKAY) goto LTM_ERR; + + if (mp_cmp(&a, &c) != MP_EQ) { + fprintf(stderr, "to/from ubin cycle failed\n"); + goto LTM_ERR; + } + free(buf); + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LTM_ERR: + free(buf); + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_read_write_sbin(void) +{ + mp_int a, b, c; + int err; + size_t size, len; + unsigned char *buf = NULL; + + if ((err = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + goto LTM_ERR; + } + + if ((err = mp_rand(&a, 15)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_neg(&a, &b)) != MP_OKAY) goto LTM_ERR; + + size = mp_sbin_size(&a); + printf("mp_to_sbin_size %zu\n", size); + buf = malloc(sizeof(*buf) * size); + if (buf == NULL) { + fprintf(stderr, "test_read_write_binaries (s) failed to allocate %zu bytes\n", + sizeof(*buf) * size); + goto LTM_ERR; + } + + if ((err = mp_to_sbin(&b, buf, size, &len)) != MP_OKAY) goto LTM_ERR; + printf("mp_to_sbin len = %zu\n", len); + + if ((err = mp_from_sbin(&c, buf, len)) != MP_OKAY) goto LTM_ERR; + + if (mp_cmp(&b, &c) != MP_EQ) { + fprintf(stderr, "to/from ubin cycle failed\n"); + goto LTM_ERR; + } + + free(buf); + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_SUCCESS; +LTM_ERR: + free(buf); + mp_clear_multi(&a, &b, &c, NULL); + return EXIT_FAILURE; +} + +static int test_mp_pack_unpack(void) +{ + mp_int a, b; + int err; + size_t written, count; + unsigned char *buf = NULL; + + mp_order order = MP_LSB_FIRST; + mp_endian endianess = MP_NATIVE_ENDIAN; + + if ((err = mp_init_multi(&a, &b, NULL)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_rand(&a, 15)) != MP_OKAY) goto LTM_ERR; + + count = mp_pack_count(&a, 0, 1); + + buf = malloc(count); + if (buf == NULL) { + fprintf(stderr, "test_pack_unpack failed to allocate\n"); + goto LTM_ERR; + } + + if ((err = mp_pack((void *)buf, count, &written, order, 1, + endianess, 0, &a)) != MP_OKAY) goto LTM_ERR; + if ((err = mp_unpack(&b, count, order, 1, + endianess, 0, (const void *)buf)) != MP_OKAY) goto LTM_ERR; + + if (mp_cmp(&a, &b) != MP_EQ) { + fprintf(stderr, "pack/unpack cycle failed\n"); + goto LTM_ERR; + } + + free(buf); + mp_clear_multi(&a, &b, NULL); + return EXIT_SUCCESS; +LTM_ERR: + free(buf); + mp_clear_multi(&a, &b, NULL); + return EXIT_FAILURE; +} + +static int unit_tests(int argc, char **argv) +{ + static const struct { + const char *name; + int (*fn)(void); + } test[] = { +#define T0(n) { #n, test_##n } +#define T1(n, o) { #n, MP_HAS(o) ? test_##n : NULL } +#define T2(n, o1, o2) { #n, MP_HAS(o1) && MP_HAS(o2) ? test_##n : NULL } + T0(feature_detection), + T0(trivial_stuff), + T2(mp_get_set_i32, MP_GET_I32, MP_GET_MAG_U32), + T2(mp_get_set_i64, MP_GET_I64, MP_GET_MAG_U64), + T1(mp_and, MP_AND), + T1(mp_cnt_lsb, MP_CNT_LSB), + T1(mp_complement, MP_COMPLEMENT), + T1(mp_decr, MP_DECR), + T1(mp_div_3, MP_DIV_3), + T1(mp_dr_reduce, MP_DR_REDUCE), + T2(mp_pack_unpack,MP_PACK, MP_UNPACK), + T2(mp_fread_fwrite, MP_FREAD, MP_FWRITE), + T1(mp_get_u32, MP_GET_I32), + T1(mp_get_u64, MP_GET_I64), + T1(mp_get_ul, MP_GET_L), + T1(mp_log_u32, MP_LOG_U32), + T1(mp_incr, MP_INCR), + T1(mp_invmod, MP_INVMOD), + T1(mp_is_square, MP_IS_SQUARE), + T1(mp_kronecker, MP_KRONECKER), + T1(mp_montgomery_reduce, MP_MONTGOMERY_REDUCE), + T1(mp_root_u32, MP_ROOT_U32), + T1(mp_or, MP_OR), + T1(mp_prime_is_prime, MP_PRIME_IS_PRIME), + T1(mp_prime_next_prime, MP_PRIME_NEXT_PRIME), + T1(mp_prime_rand, MP_PRIME_RAND), + T1(mp_rand, MP_RAND), + T1(mp_read_radix, MP_READ_RADIX), + T1(mp_read_write_ubin, MP_TO_UBIN), + T1(mp_read_write_sbin, MP_TO_SBIN), + T1(mp_reduce_2k, MP_REDUCE_2K), + T1(mp_reduce_2k_l, MP_REDUCE_2K_L), +#if defined(__STDC_IEC_559__) || defined(__GCC_IEC_559) + T1(mp_set_double, MP_SET_DOUBLE), +#endif + T1(mp_signed_rsh, MP_SIGNED_RSH), + T1(mp_sqrt, MP_SQRT), + T1(mp_sqrtmod_prime, MP_SQRTMOD_PRIME), + T1(mp_xor, MP_XOR), + T1(s_mp_balance_mul, S_MP_BALANCE_MUL), + T1(s_mp_karatsuba_mul, S_MP_KARATSUBA_MUL), + T1(s_mp_karatsuba_sqr, S_MP_KARATSUBA_SQR), + T1(s_mp_toom_mul, S_MP_TOOM_MUL), + T1(s_mp_toom_sqr, S_MP_TOOM_SQR) +#undef T2 +#undef T1 + }; + unsigned long i, ok, fail, nop; + uint64_t t; + int j; + + ok = fail = nop = 0; + + t = (uint64_t)time(NULL); + printf("SEED: 0x%"PRIx64"\n\n", t); + s_mp_rand_jenkins_init(t); + mp_rand_source(s_mp_rand_jenkins); + + for (i = 0; i < sizeof(test) / sizeof(test[0]); ++i) { + if (argc > 1) { + for (j = 1; j < argc; ++j) { + if (strstr(test[i].name, argv[j]) != NULL) { + break; + } + } + if (j == argc) continue; + } + printf("TEST %s\n\n", test[i].name); + if (test[i].fn == NULL) { + nop++; + printf("NOP %s\n\n", test[i].name); + } else if (test[i].fn() == EXIT_SUCCESS) { + ok++; + printf("\n\n"); + } else { + fail++; + printf("\n\nFAIL %s\n\n", test[i].name); + } + } + printf("Tests OK/NOP/FAIL: %lu/%lu/%lu\n", ok, nop, fail); + + if (fail != 0) return EXIT_FAILURE; + else return EXIT_SUCCESS; +} + +int main(int argc, char **argv) +{ + print_header(); + + return unit_tests(argc, argv); +} diff --git a/lib/hcrypto/libtommath/demo/timing.c b/lib/hcrypto/libtommath/demo/timing.c index d4660a9b5..f620b8cd4 100644 --- a/lib/hcrypto/libtommath/demo/timing.c +++ b/lib/hcrypto/libtommath/demo/timing.c @@ -1,7 +1,11 @@ -#include #include +#include +#include +#include +#include -ulong64 _tt; +#define MP_WUR +#include #ifdef IOWNANATHLON #include @@ -10,44 +14,52 @@ ulong64 _tt; #define SLEEP #endif +#ifdef LTM_TIMING_REAL_RAND +#define LTM_TIMING_RAND_SEED time(NULL) +#else +#define LTM_TIMING_RAND_SEED 23 +#endif -void ndraw(mp_int * a, char *name) + +static void ndraw(mp_int *a, const char *name) { char buf[4096]; printf("%s: ", name); - mp_toradix(a, buf, 64); + mp_to_radix(a, buf, sizeof(buf), NULL, 64); printf("%s\n", buf); } -static void draw(mp_int * a) +static void draw(mp_int *a) { ndraw(a, ""); } -unsigned long lfsr = 0xAAAAAAAAUL; +static unsigned long lfsr = 0xAAAAAAAAuL; -int lbit(void) +static unsigned int lbit(void) { - if (lfsr & 0x80000000UL) { - lfsr = ((lfsr << 1) ^ 0x8000001BUL) & 0xFFFFFFFFUL; - return 1; + if ((lfsr & 0x80000000uL) != 0uL) { + lfsr = ((lfsr << 1) ^ 0x8000001BuL) & 0xFFFFFFFFuL; + return 1u; } else { lfsr <<= 1; - return 0; + return 0u; } } /* RDTSC from Scott Duplichan */ -static ulong64 TIMFUNC(void) +static uint64_t TIMFUNC(void) { #if defined __GNUC__ #if defined(__i386__) || defined(__x86_64__) - unsigned long long a; - __asm__ __volatile__("rdtsc\nmovl %%eax,%0\nmovl %%edx,4+%0\n":: - "m"(a):"%eax", "%edx"); - return a; + /* version from http://www.mcs.anl.gov/~kazutomo/rdtsc.html + * the old code always got a warning issued by gcc, clang did not complain... + */ + unsigned hi, lo; + __asm__ __volatile__("rdtsc" : "=a"(lo), "=d"(hi)); + return ((uint64_t)lo)|(((uint64_t)hi)<<32); #else /* gcc-IA64 version */ unsigned long result; __asm__ __volatile__("mov %0=ar.itc":"=r"(result)::"memory"); @@ -58,7 +70,7 @@ static ulong64 TIMFUNC(void) return result; #endif - // Microsoft and Intel Windows compilers + /* Microsoft and Intel Windows compilers */ #elif defined _M_IX86 __asm rdtsc #elif defined _M_AMD64 @@ -73,17 +85,51 @@ static ulong64 TIMFUNC(void) #endif } +#if 1 #define DO(x) x; x; -//#define DO4(x) DO2(x); DO2(x); -//#define DO8(x) DO4(x); DO4(x); -//#define DO(x) DO8(x); DO8(x); +#else +#define DO2(x) x; x; +#define DO4(x) DO2(x); DO2(x); +#define DO8(x) DO4(x); DO4(x); +#define DO(x) DO8(x); DO8(x); +#endif -int main(void) +#ifdef TIMING_NO_LOGS +#define FOPEN(a, b) NULL +#define FPRINTF(a,b,c,d) +#define FFLUSH(a) +#define FCLOSE(a) (void)(a) +#else +#define FOPEN(a,b) fopen(a,b) +#define FPRINTF(a,b,c,d) fprintf(a,b,c,d) +#define FFLUSH(a) fflush(a) +#define FCLOSE(a) fclose(a) +#endif + +static int should_test(const char *test, int argc, char **argv) { - ulong64 tt, gg, CLK_PER_SEC; + int j; + if (argc > 1) { + for (j = 1; j < argc; ++j) { + if (strstr(test, argv[j]) != NULL) { + return 1; + } + } + if (j == argc) return 0; + } + return 1; +} + +int main(int argc, char **argv) +{ + uint64_t tt, gg, CLK_PER_SEC; FILE *log, *logb, *logc, *logd; mp_int a, b, c, d, e, f; - int n, cnt, ix, old_kara_m, old_kara_s; +#ifdef LTM_TIMING_PRIME_IS_PRIME + const char *name; + int m; +#endif + int n, cnt, ix, old_kara_m, old_kara_s, old_toom_m, old_toom_s; unsigned rr; mp_init(&a); @@ -93,227 +139,268 @@ int main(void) mp_init(&e); mp_init(&f); - srand(time(NULL)); + srand(LTM_TIMING_RAND_SEED); - /* temp. turn off TOOM */ - TOOM_MUL_CUTOFF = TOOM_SQR_CUTOFF = 100000; - CLK_PER_SEC = TIMFUNC(); sleep(1); CLK_PER_SEC = TIMFUNC() - CLK_PER_SEC; - printf("CLK_PER_SEC == %llu\n", CLK_PER_SEC); - goto exptmod; - log = fopen("logs/add.log", "w"); - for (cnt = 8; cnt <= 128; cnt += 8) { - SLEEP; - mp_rand(&a, cnt); - mp_rand(&b, cnt); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_add(&a, &b, &c)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 100000); - printf("Adding\t\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt); - fflush(log); - } - fclose(log); + printf("CLK_PER_SEC == %" PRIu64 "\n", CLK_PER_SEC); - log = fopen("logs/sub.log", "w"); - for (cnt = 8; cnt <= 128; cnt += 8) { - SLEEP; - mp_rand(&a, cnt); - mp_rand(&b, cnt); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_sub(&a, &b, &c)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 100000); - - printf("Subtracting\t\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt); - fflush(log); - } - fclose(log); - - /* do mult/square twice, first without karatsuba and second with */ - multtest: - old_kara_m = KARATSUBA_MUL_CUTOFF; - old_kara_s = KARATSUBA_SQR_CUTOFF; - for (ix = 0; ix < 2; ix++) { - printf("With%s Karatsuba\n", (ix == 0) ? "out" : ""); - - KARATSUBA_MUL_CUTOFF = (ix == 0) ? 9999 : old_kara_m; - KARATSUBA_SQR_CUTOFF = (ix == 0) ? 9999 : old_kara_s; - - log = fopen((ix == 0) ? "logs/mult.log" : "logs/mult_kara.log", "w"); - for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) { - SLEEP; - mp_rand(&a, cnt); - mp_rand(&b, cnt); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_mul(&a, &b, &c)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 100); - printf("Multiplying\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt); - fflush(log); +#ifdef LTM_TIMING_PRIME_IS_PRIME + if (should_test("prime", argc, argv)) { + for (m = 0; m < 2; ++m) { + if (m == 0) { + name = " Arnault"; + mp_read_radix(&a, + "91xLNF3roobhzgTzoFIG6P13ZqhOVYSN60Fa7Cj2jVR1g0k89zdahO9/kAiRprpfO1VAp1aBHucLFV/qLKLFb+zonV7R2Vxp1K13ClwUXStpV0oxTNQVjwybmFb5NBEHImZ6V7P6+udRJuH8VbMEnS0H8/pSqQrg82OoQQ2fPpAk6G1hkjqoCv5s/Yr", + 64); + } else { + name = "2^1119 + 53"; + mp_set(&a,1u); + mp_mul_2d(&a,1119,&a); + mp_add_d(&a,53,&a); + } + cnt = mp_prime_rabin_miller_trials(mp_count_bits(&a)); + ix = -cnt; + for (; cnt >= ix; cnt += ix) { + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_prime_is_prime(&a, cnt, &n)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + if ((m == 0) && (n == MP_YES)) { + printf("Arnault's pseudoprime is not prime but mp_prime_is_prime says it is.\n"); + return EXIT_FAILURE; + } + } while (++rr < 100u); + printf("Prime-check\t%s(%2d) => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + name, cnt, CLK_PER_SEC / tt, tt); + } } - fclose(log); - - log = fopen((ix == 0) ? "logs/sqr.log" : "logs/sqr_kara.log", "w"); - for (cnt = 4; cnt <= 10240 / DIGIT_BIT; cnt += 2) { - SLEEP; - mp_rand(&a, cnt); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_sqr(&a, &b)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 100); - printf("Squaring\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(log, "%d %9llu\n", mp_count_bits(&a), tt); - fflush(log); - } - fclose(log); - } - exptmod: +#endif - { - char *primes[] = { - /* 2K large moduli */ - "179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586239334100047359817950870678242457666208137217", - "32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638099733077152121140120031150424541696791951097529546801429027668869927491725169", - "1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085902995208257421855249796721729039744118165938433694823325696642096892124547425283", - /* 2K moduli mersenne primes */ - "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", - "531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127", - "10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087", - "1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007", - "259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071", - "190797007524439073807468042969529173669356994749940177394741882673528979787005053706368049835514900244303495954950709725762186311224148828811920216904542206960744666169364221195289538436845390250168663932838805192055137154390912666527533007309292687539092257043362517857366624699975402375462954490293259233303137330643531556539739921926201438606439020075174723029056838272505051571967594608350063404495977660656269020823960825567012344189908927956646011998057988548630107637380993519826582389781888135705408653045219655801758081251164080554609057468028203308718724654081055323215860189611391296030471108443146745671967766308925858547271507311563765171008318248647110097614890313562856541784154881743146033909602737947385055355960331855614540900081456378659068370317267696980001187750995491090350108417050917991562167972281070161305972518044872048331306383715094854938415738549894606070722584737978176686422134354526989443028353644037187375385397838259511833166416134323695660367676897722287918773420968982326089026150031515424165462111337527431154890666327374921446276833564519776797633875503548665093914556482031482248883127023777039667707976559857333357013727342079099064400455741830654320379350833236245819348824064783585692924881021978332974949906122664421376034687815350484991", + if (should_test("add", argc, argv)) { + log = FOPEN("logs/add.log", "w"); + for (cnt = 8; cnt <= 128; cnt += 8) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_add(&a, &b, &c)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 100000u); + printf("Adding\t\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF(log, "%6d %9" PRIu64 "\n", cnt * MP_DIGIT_BIT, tt); + FFLUSH(log); + } + FCLOSE(log); + } - /* DR moduli */ - "14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368612079", - "101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039", - "736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821797602431", - "38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783", - "542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147", - "1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503", - "1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679", + if (should_test("sub", argc, argv)) { + log = FOPEN("logs/sub.log", "w"); + for (cnt = 8; cnt <= 128; cnt += 8) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_sub(&a, &b, &c)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 100000u); - /* generic unrestricted moduli */ - "17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203", - "2893527720709661239493896562339544088620375736490408468011883030469939904368086092336458298221245707898933583190713188177399401852627749210994595974791782790253946539043962213027074922559572312141181787434278708783207966459019479487", - "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319", - "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887", - "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227", - "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207", - "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979", - NULL + printf("Subtracting\t\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF(log, "%6d %9" PRIu64 "\n", cnt * MP_DIGIT_BIT, tt); + FFLUSH(log); + } + FCLOSE(log); + } + + if (should_test("mulsqr", argc, argv)) { + /* do mult/square twice, first without karatsuba and second with */ + old_kara_m = KARATSUBA_MUL_CUTOFF; + old_kara_s = KARATSUBA_SQR_CUTOFF; + /* currently toom-cook cut-off is too high to kick in, so we just use the karatsuba values */ + old_toom_m = old_kara_m; + old_toom_s = old_kara_s; + for (ix = 0; ix < 3; ix++) { + printf("With%s Karatsuba, With%s Toom\n", (ix == 1) ? "" : "out", (ix == 2) ? "" : "out"); + + KARATSUBA_MUL_CUTOFF = (ix == 1) ? old_kara_m : 9999; + KARATSUBA_SQR_CUTOFF = (ix == 1) ? old_kara_s : 9999; + TOOM_MUL_CUTOFF = (ix == 2) ? old_toom_m : 9999; + TOOM_SQR_CUTOFF = (ix == 2) ? old_toom_s : 9999; + + log = FOPEN((ix == 0) ? "logs/mult.log" : (ix == 1) ? "logs/mult_kara.log" : "logs/mult_toom.log", "w"); + for (cnt = 4; cnt <= (10240 / MP_DIGIT_BIT); cnt += 2) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_mul(&a, &b, &c)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 100u); + printf("Multiplying\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF(log, "%6d %9" PRIu64 "\n", mp_count_bits(&a), tt); + FFLUSH(log); + } + FCLOSE(log); + + log = FOPEN((ix == 0) ? "logs/sqr.log" : (ix == 1) ? "logs/sqr_kara.log" : "logs/sqr_toom.log", "w"); + for (cnt = 4; cnt <= (10240 / MP_DIGIT_BIT); cnt += 2) { + SLEEP; + mp_rand(&a, cnt); + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_sqr(&a, &b)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 100u); + printf("Squaring\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF(log, "%6d %9" PRIu64 "\n", mp_count_bits(&a), tt); + FFLUSH(log); + } + FCLOSE(log); + + } + } + + if (should_test("expt", argc, argv)) { + const char *primes[] = { + /* 2K large moduli */ + "179769313486231590772930519078902473361797697894230657273430081157732675805500963132708477322407536021120113879871393357658789768814416622492847430639474124377767893424865485276302219601246094119453082952085005768838150682342462881473913110540827237163350510684586239334100047359817950870678242457666208137217", + "32317006071311007300714876688669951960444102669715484032130345427524655138867890893197201411522913463688717960921898019494119559150490921095088152386448283120630877367300996091750197750389652106796057638384067568276792218642619756161838094338476170470581645852036305042887575891541065808607552399123930385521914333389668342420684974786564569494856176035326322058077805659331026192708460314150258592864177116725943603718461857357598351152301645904403697613233287231227125684710820209725157101726931323469678542580656697935045997268352998638099733077152121140120031150424541696791951097529546801429027668869927491725169", + "1044388881413152506691752710716624382579964249047383780384233483283953907971557456848826811934997558340890106714439262837987573438185793607263236087851365277945956976543709998340361590134383718314428070011855946226376318839397712745672334684344586617496807908705803704071284048740118609114467977783598029006686938976881787785946905630190260940599579453432823469303026696443059025015972399867714215541693835559885291486318237914434496734087811872639496475100189041349008417061675093668333850551032972088269550769983616369411933015213796825837188091833656751221318492846368125550225998300412344784862595674492194617023806505913245610825731835380087608622102834270197698202313169017678006675195485079921636419370285375124784014907159135459982790513399611551794271106831134090584272884279791554849782954323534517065223269061394905987693002122963395687782878948440616007412945674919823050571642377154816321380631045902916136926708342856440730447899971901781465763473223850267253059899795996090799469201774624817718449867455659250178329070473119433165550807568221846571746373296884912819520317457002440926616910874148385078411929804522981857338977648103126085902995208257421855249796721729039744118165938433694823325696642096892124547425283", + /* 2K moduli mersenne primes */ + "6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", + "531137992816767098689588206552468627329593117727031923199444138200403559860852242739162502265229285668889329486246501015346579337652707239409519978766587351943831270835393219031728127", + "10407932194664399081925240327364085538615262247266704805319112350403608059673360298012239441732324184842421613954281007791383566248323464908139906605677320762924129509389220345773183349661583550472959420547689811211693677147548478866962501384438260291732348885311160828538416585028255604666224831890918801847068222203140521026698435488732958028878050869736186900714720710555703168729087", + "1475979915214180235084898622737381736312066145333169775147771216478570297878078949377407337049389289382748507531496480477281264838760259191814463365330269540496961201113430156902396093989090226259326935025281409614983499388222831448598601834318536230923772641390209490231836446899608210795482963763094236630945410832793769905399982457186322944729636418890623372171723742105636440368218459649632948538696905872650486914434637457507280441823676813517852099348660847172579408422316678097670224011990280170474894487426924742108823536808485072502240519452587542875349976558572670229633962575212637477897785501552646522609988869914013540483809865681250419497686697771007", + "259117086013202627776246767922441530941818887553125427303974923161874019266586362086201209516800483406550695241733194177441689509238807017410377709597512042313066624082916353517952311186154862265604547691127595848775610568757931191017711408826252153849035830401185072116424747461823031471398340229288074545677907941037288235820705892351068433882986888616658650280927692080339605869308790500409503709875902119018371991620994002568935113136548829739112656797303241986517250116412703509705427773477972349821676443446668383119322540099648994051790241624056519054483690809616061625743042361721863339415852426431208737266591962061753535748892894599629195183082621860853400937932839420261866586142503251450773096274235376822938649407127700846077124211823080804139298087057504713825264571448379371125032081826126566649084251699453951887789613650248405739378594599444335231188280123660406262468609212150349937584782292237144339628858485938215738821232393687046160677362909315071", + "190797007524439073807468042969529173669356994749940177394741882673528979787005053706368049835514900244303495954950709725762186311224148828811920216904542206960744666169364221195289538436845390250168663932838805192055137154390912666527533007309292687539092257043362517857366624699975402375462954490293259233303137330643531556539739921926201438606439020075174723029056838272505051571967594608350063404495977660656269020823960825567012344189908927956646011998057988548630107637380993519826582389781888135705408653045219655801758081251164080554609057468028203308718724654081055323215860189611391296030471108443146745671967766308925858547271507311563765171008318248647110097614890313562856541784154881743146033909602737947385055355960331855614540900081456378659068370317267696980001187750995491090350108417050917991562167972281070161305972518044872048331306383715094854938415738549894606070722584737978176686422134354526989443028353644037187375385397838259511833166416134323695660367676897722287918773420968982326089026150031515424165462111337527431154890666327374921446276833564519776797633875503548665093914556482031482248883127023777039667707976559857333357013727342079099064400455741830654320379350833236245819348824064783585692924881021978332974949906122664421376034687815350484991", + + /* DR moduli */ + "14059105607947488696282932836518693308967803494693489478439861164411992439598399594747002144074658928593502845729752797260025831423419686528151609940203368612079", + "101745825697019260773923519755878567461315282017759829107608914364075275235254395622580447400994175578963163918967182013639660669771108475957692810857098847138903161308502419410142185759152435680068435915159402496058513611411688900243039", + "736335108039604595805923406147184530889923370574768772191969612422073040099331944991573923112581267542507986451953227192970402893063850485730703075899286013451337291468249027691733891486704001513279827771740183629161065194874727962517148100775228363421083691764065477590823919364012917984605619526140821797602431", + "38564998830736521417281865696453025806593491967131023221754800625044118265468851210705360385717536794615180260494208076605798671660719333199513807806252394423283413430106003596332513246682903994829528690198205120921557533726473585751382193953592127439965050261476810842071573684505878854588706623484573925925903505747545471088867712185004135201289273405614415899438276535626346098904241020877974002916168099951885406379295536200413493190419727789712076165162175783", + "542189391331696172661670440619180536749994166415993334151601745392193484590296600979602378676624808129613777993466242203025054573692562689251250471628358318743978285860720148446448885701001277560572526947619392551574490839286458454994488665744991822837769918095117129546414124448777033941223565831420390846864429504774477949153794689948747680362212954278693335653935890352619041936727463717926744868338358149568368643403037768649616778526013610493696186055899318268339432671541328195724261329606699831016666359440874843103020666106568222401047720269951530296879490444224546654729111504346660859907296364097126834834235287147", + "1487259134814709264092032648525971038895865645148901180585340454985524155135260217788758027400478312256339496385275012465661575576202252063145698732079880294664220579764848767704076761853197216563262660046602703973050798218246170835962005598561669706844469447435461092542265792444947706769615695252256130901271870341005768912974433684521436211263358097522726462083917939091760026658925757076733484173202927141441492573799914240222628795405623953109131594523623353044898339481494120112723445689647986475279242446083151413667587008191682564376412347964146113898565886683139407005941383669325997475076910488086663256335689181157957571445067490187939553165903773554290260531009121879044170766615232300936675369451260747671432073394867530820527479172464106442450727640226503746586340279816318821395210726268291535648506190714616083163403189943334431056876038286530365757187367147446004855912033137386225053275419626102417236133948503", + "1095121115716677802856811290392395128588168592409109494900178008967955253005183831872715423151551999734857184538199864469605657805519106717529655044054833197687459782636297255219742994736751541815269727940751860670268774903340296040006114013971309257028332849679096824800250742691718610670812374272414086863715763724622797509437062518082383056050144624962776302147890521249477060215148275163688301275847155316042279405557632639366066847442861422164832655874655824221577849928863023018366835675399949740429332468186340518172487073360822220449055340582568461568645259954873303616953776393853174845132081121976327462740354930744487429617202585015510744298530101547706821590188733515880733527449780963163909830077616357506845523215289297624086914545378511082534229620116563260168494523906566709418166011112754529766183554579321224940951177394088465596712620076240067370589036924024728375076210477267488679008016579588696191194060127319035195370137160936882402244399699172017835144537488486396906144217720028992863941288217185353914991583400421682751000603596655790990815525126154394344641336397793791497068253936771017031980867706707490224041075826337383538651825493679503771934836094655802776331664261631740148281763487765852746577808019633679", + + /* generic unrestricted moduli */ + "17933601194860113372237070562165128350027320072176844226673287945873370751245439587792371960615073855669274087805055507977323024886880985062002853331424203", + "2893527720709661239493896562339544088620375736490408468011883030469939904368086092336458298221245707898933583190713188177399401852627749210994595974791782790253946539043962213027074922559572312141181787434278708783207966459019479487", + "347743159439876626079252796797422223177535447388206607607181663903045907591201940478223621722118173270898487582987137708656414344685816179420855160986340457973820182883508387588163122354089264395604796675278966117567294812714812796820596564876450716066283126720010859041484786529056457896367683122960411136319", + "47266428956356393164697365098120418976400602706072312735924071745438532218237979333351774907308168340693326687317443721193266215155735814510792148768576498491199122744351399489453533553203833318691678263241941706256996197460424029012419012634671862283532342656309677173602509498417976091509154360039893165037637034737020327399910409885798185771003505320583967737293415979917317338985837385734747478364242020380416892056650841470869294527543597349250299539682430605173321029026555546832473048600327036845781970289288898317888427517364945316709081173840186150794397479045034008257793436817683392375274635794835245695887", + "436463808505957768574894870394349739623346440601945961161254440072143298152040105676491048248110146278752857839930515766167441407021501229924721335644557342265864606569000117714935185566842453630868849121480179691838399545644365571106757731317371758557990781880691336695584799313313687287468894148823761785582982549586183756806449017542622267874275103877481475534991201849912222670102069951687572917937634467778042874315463238062009202992087620963771759666448266532858079402669920025224220613419441069718482837399612644978839925207109870840278194042158748845445131729137117098529028886770063736487420613144045836803985635654192482395882603511950547826439092832800532152534003936926017612446606135655146445620623395788978726744728503058670046885876251527122350275750995227", + "11424167473351836398078306042624362277956429440521137061889702611766348760692206243140413411077394583180726863277012016602279290144126785129569474909173584789822341986742719230331946072730319555984484911716797058875905400999504305877245849119687509023232790273637466821052576859232452982061831009770786031785669030271542286603956118755585683996118896215213488875253101894663403069677745948305893849505434201763745232895780711972432011344857521691017896316861403206449421332243658855453435784006517202894181640562433575390821384210960117518650374602256601091379644034244332285065935413233557998331562749140202965844219336298970011513882564935538704289446968322281451907487362046511461221329799897350993370560697505809686438782036235372137015731304779072430260986460269894522159103008260495503005267165927542949439526272736586626709581721032189532726389643625590680105784844246152702670169304203783072275089194754889511973916207", + "1214855636816562637502584060163403830270705000634713483015101384881871978446801224798536155406895823305035467591632531067547890948695117172076954220727075688048751022421198712032848890056357845974246560748347918630050853933697792254955890439720297560693579400297062396904306270145886830719309296352765295712183040773146419022875165382778007040109957609739589875590885701126197906063620133954893216612678838507540777138437797705602453719559017633986486649523611975865005712371194067612263330335590526176087004421363598470302731349138773205901447704682181517904064735636518462452242791676541725292378925568296858010151852326316777511935037531017413910506921922450666933202278489024521263798482237150056835746454842662048692127173834433089016107854491097456725016327709663199738238442164843147132789153725513257167915555162094970853584447993125488607696008169807374736711297007473812256272245489405898470297178738029484459690836250560495461579533254473316340608217876781986188705928270735695752830825527963838355419762516246028680280988020401914551825487349990306976304093109384451438813251211051597392127491464898797406789175453067960072008590614886532333015881171367104445044718144312416815712216611576221546455968770801413440778423979", + NULL }; - log = fopen("logs/expt.log", "w"); - logb = fopen("logs/expt_dr.log", "w"); - logc = fopen("logs/expt_2k.log", "w"); - logd = fopen("logs/expt_2kl.log", "w"); - for (n = 0; primes[n]; n++) { - SLEEP; - mp_read_radix(&a, primes[n], 10); - mp_zero(&b); - for (rr = 0; rr < (unsigned) mp_count_bits(&a); rr++) { - mp_mul_2(&b, &b); - b.dp[0] |= lbit(); - b.used += 1; - } - mp_sub_d(&a, 1, &c); - mp_mod(&b, &c, &b); - mp_set(&c, 3); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_exptmod(&c, &b, &a, &d)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 10); - mp_sub_d(&a, 1, &e); - mp_sub(&e, &b, &b); - mp_exptmod(&c, &b, &a, &e); /* c^(p-1-b) mod a */ - mp_mulmod(&e, &d, &a, &d); /* c^b * c^(p-1-b) == c^p-1 == 1 */ - if (mp_cmp_d(&d, 1)) { - printf("Different (%d)!!!\n", mp_count_bits(&a)); - draw(&d); - exit(0); - } - printf("Exponentiating\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(n < 4 ? logd : (n < 9) ? logc : (n < 16) ? logb : log, - "%d %9llu\n", mp_count_bits(&a), tt); + log = FOPEN("logs/expt.log", "w"); + logb = FOPEN("logs/expt_dr.log", "w"); + logc = FOPEN("logs/expt_2k.log", "w"); + logd = FOPEN("logs/expt_2kl.log", "w"); + for (n = 0; primes[n] != NULL; n++) { + SLEEP; + mp_read_radix(&a, primes[n], 10); + mp_zero(&b); + for (rr = 0; rr < (unsigned) mp_count_bits(&a); rr++) { + mp_mul_2(&b, &b); + b.dp[0] |= lbit(); + b.used += 1; + } + mp_sub_d(&a, 1uL, &c); + mp_mod(&b, &c, &b); + mp_set(&c, 3uL); + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_exptmod(&c, &b, &a, &d)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 10u); + mp_sub_d(&a, 1uL, &e); + mp_sub(&e, &b, &b); + mp_exptmod(&c, &b, &a, &e); /* c^(p-1-b) mod a */ + mp_mulmod(&e, &d, &a, &d); /* c^b * c^(p-1-b) == c^p-1 == 1 */ + if (mp_cmp_d(&d, 1uL) != MP_EQ) { + printf("Different (%d)!!!\n", mp_count_bits(&a)); + draw(&d); + exit(0); + } + printf("Exponentiating\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF((n < 3) ? logd : (n < 9) ? logc : (n < 16) ? logb : log, + "%6d %9" PRIu64 "\n", mp_count_bits(&a), tt); } + FCLOSE(log); + FCLOSE(logb); + FCLOSE(logc); + FCLOSE(logd); } - fclose(log); - fclose(logb); - fclose(logc); - fclose(logd); - log = fopen("logs/invmod.log", "w"); - for (cnt = 4; cnt <= 128; cnt += 4) { - SLEEP; - mp_rand(&a, cnt); - mp_rand(&b, cnt); + if (should_test("invmod", argc, argv)) { + log = FOPEN("logs/invmod.log", "w"); + for (cnt = 4; cnt <= 32; cnt += 4) { + SLEEP; + mp_rand(&a, cnt); + mp_rand(&b, cnt); - do { - mp_add_d(&b, 1, &b); - mp_gcd(&a, &b, &c); - } while (mp_cmp_d(&c, 1) != MP_EQ); + do { + mp_add_d(&b, 1uL, &b); + mp_gcd(&a, &b, &c); + } while (mp_cmp_d(&c, 1uL) != MP_EQ); - rr = 0; - tt = -1; - do { - gg = TIMFUNC(); - DO(mp_invmod(&b, &a, &c)); - gg = (TIMFUNC() - gg) >> 1; - if (tt > gg) - tt = gg; - } while (++rr < 1000); - mp_mulmod(&b, &c, &a, &d); - if (mp_cmp_d(&d, 1) != MP_EQ) { - printf("Failed to invert\n"); - return 0; + rr = 0u; + tt = UINT64_MAX; + do { + gg = TIMFUNC(); + DO(mp_invmod(&b, &a, &c)); + gg = (TIMFUNC() - gg) >> 1; + if (tt > gg) + tt = gg; + } while (++rr < 1000u); + mp_mulmod(&b, &c, &a, &d); + if (mp_cmp_d(&d, 1uL) != MP_EQ) { + printf("Failed to invert\n"); + return 0; + } + printf("Inverting mod\t%4d-bit => %9" PRIu64 "/sec, %9" PRIu64 " cycles\n", + mp_count_bits(&a), CLK_PER_SEC / tt, tt); + FPRINTF(log, "%6d %9" PRIu64 "\n", cnt * MP_DIGIT_BIT, tt); } - printf("Inverting mod\t%4d-bit => %9llu/sec, %9llu cycles\n", - mp_count_bits(&a), CLK_PER_SEC / tt, tt); - fprintf(log, "%d %9llu\n", cnt * DIGIT_BIT, tt); + FCLOSE(log); } - fclose(log); return 0; } - -/* $Source: /cvs/libtom/libtommath/demo/timing.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ diff --git a/lib/hcrypto/libtommath/dep.pl b/lib/hcrypto/libtommath/dep.pl deleted file mode 100644 index c39e27ea0..000000000 --- a/lib/hcrypto/libtommath/dep.pl +++ /dev/null @@ -1,123 +0,0 @@ -#!/usr/bin/perl -# -# Walk through source, add labels and make classes -# -#use strict; - -my %deplist; - -#open class file and write preamble -open(CLASS, ">tommath_class.h") or die "Couldn't open tommath_class.h for writing\n"; -print CLASS "#if !(defined(LTM1) && defined(LTM2) && defined(LTM3))\n#if defined(LTM2)\n#define LTM3\n#endif\n#if defined(LTM1)\n#define LTM2\n#endif\n#define LTM1\n\n#if defined(LTM_ALL)\n"; - -foreach my $filename (glob "bn*.c") { - my $define = $filename; - -print "Processing $filename\n"; - - # convert filename to upper case so we can use it as a define - $define =~ tr/[a-z]/[A-Z]/; - $define =~ tr/\./_/; - print CLASS "#define $define\n"; - - # now copy text and apply #ifdef as required - my $apply = 0; - open(SRC, "<$filename"); - open(OUT, ">tmp"); - - # first line will be the #ifdef - my $line = ; - if ($line =~ /include/) { - print OUT $line; - } else { - print OUT "#include \n#ifdef $define\n$line"; - $apply = 1; - } - while () { - if (!($_ =~ /tommath\.h/)) { - print OUT $_; - } - } - if ($apply == 1) { - print OUT "#endif\n"; - } - close SRC; - close OUT; - - unlink($filename); - rename("tmp", $filename); -} -print CLASS "#endif\n\n"; - -# now do classes - -foreach my $filename (glob "bn*.c") { - open(SRC, "<$filename") or die "Can't open source file!\n"; - - # convert filename to upper case so we can use it as a define - $filename =~ tr/[a-z]/[A-Z]/; - $filename =~ tr/\./_/; - - print CLASS "#if defined($filename)\n"; - my $list = $filename; - - # scan for mp_* and make classes - while () { - my $line = $_; - while ($line =~ m/(fast_)*(s_)*mp\_[a-z_0-9]*/) { - $line = $'; - # now $& is the match, we want to skip over LTM keywords like - # mp_int, mp_word, mp_digit - if (!($& eq "mp_digit") && !($& eq "mp_word") && !($& eq "mp_int")) { - my $a = $&; - $a =~ tr/[a-z]/[A-Z]/; - $a = "BN_" . $a . "_C"; - if (!($list =~ /$a/)) { - print CLASS " #define $a\n"; - } - $list = $list . "," . $a; - } - } - } - @deplist{$filename} = $list; - - print CLASS "#endif\n\n"; - close SRC; -} - -print CLASS "#ifdef LTM3\n#define LTM_LAST\n#endif\n#include \n#include \n#else\n#define LTM_LAST\n#endif\n"; -close CLASS; - -#now let's make a cool call graph... - -open(OUT,">callgraph.txt"); -$indent = 0; -foreach (keys %deplist) { - $list = ""; - draw_func(@deplist{$_}); - print OUT "\n\n"; -} -close(OUT); - -sub draw_func() -{ - my @funcs = split(",", $_[0]); - if ($list =~ /@funcs[0]/) { - return; - } else { - $list = $list . @funcs[0]; - } - if ($indent == 0) { } - elsif ($indent >= 1) { print OUT "| " x ($indent - 1) . "+--->"; } - print OUT @funcs[0] . "\n"; - shift @funcs; - my $temp = $list; - foreach my $i (@funcs) { - ++$indent; - draw_func(@deplist{$i}); - --$indent; - } - $list = $temp; -} - - diff --git a/lib/hcrypto/libtommath/doc/bn.pdf b/lib/hcrypto/libtommath/doc/bn.pdf new file mode 100644 index 000000000..fbf05ead1 Binary files /dev/null and b/lib/hcrypto/libtommath/doc/bn.pdf differ diff --git a/lib/hcrypto/libtommath/bn.tex b/lib/hcrypto/libtommath/doc/bn.tex similarity index 56% rename from lib/hcrypto/libtommath/bn.tex rename to lib/hcrypto/libtommath/doc/bn.tex index 9017860e7..5937feef1 100644 --- a/lib/hcrypto/libtommath/bn.tex +++ b/lib/hcrypto/libtommath/doc/bn.tex @@ -6,6 +6,7 @@ \usepackage{alltt} \usepackage{graphicx} \usepackage{layout} +\usepackage{appendix} \def\union{\cup} \def\intersect{\cap} \def\getsrandom{\stackrel{\rm R}{\gets}} @@ -49,10 +50,10 @@ \begin{document} \frontmatter \pagestyle{empty} -\title{LibTomMath User Manual \\ v0.41} -\author{Tom St Denis \\ tomstdenis@gmail.com} +\title{LibTomMath User Manual \\ v1.2.0} +\author{LibTom Projects \\ www.libtom.net} \maketitle -This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been +This text, the library and the accompanying textbook are all hereby placed in the public domain. This book has been formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package. \vspace{10cm} @@ -60,6 +61,9 @@ formatted for B5 [176x250] paper using the \LaTeX{} {\em book} macro package. \begin{flushright}Open Source. Open Academia. Open Minds. \mbox{ } +LibTom Projects + +\& originally Tom St Denis, @@ -74,12 +78,12 @@ Ontario, Canada \section{What is LibTomMath?} LibTomMath is a library of source code which provides a series of efficient and carefully written functions for manipulating large integer numbers. It was written in portable ISO C source code so that it will build on any platform with a conforming -C compiler. +C compiler. In a nutshell the library was written from scratch with verbose comments to help instruct computer science students how -to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous +to implement ``bignum'' math. However, the resulting code has proven to be very useful. It has been used by numerous universities, commercial and open source software developers. It has been used on a variety of platforms ranging from -Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines. +Linux and Windows based x86 to ARM based Gameboys and PPC based MacOS machines. \section{License} As of the v0.25 the library source code has been placed in the public domain with every new release. As of the v0.28 @@ -87,14 +91,27 @@ release the textbook ``Implementing Multiple Precision Arithmetic'' has been pla release as well. This textbook is meant to compliment the project by providing a more solid walkthrough of the development algorithms used in the library. -Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the +Since both\footnote{Note that the MPI files under mtest/ are copyrighted by Michael Fromberger. They are not required to use LibTomMath.} are in the public domain everyone is entitled to do with them as they see fit. \section{Building LibTomMath} LibTomMath is meant to be very ``GCC friendly'' as it comes with a makefile well suited for GCC. However, the library will also build in MSVC, Borland C out of the box. For any other ISO C compiler a makefile will have to be made by the end -developer. +developer. Please consider to commit such a makefile to the LibTomMath developers, currently residing at +\url{http://github.com/libtom/libtommath}, if successfully done so. + +Intel's C-compiler (ICC) is sufficiently compatible with GCC, at least the newer versions, to replace GCC for building the static and the shared library. Editing the makefiles is not needed, just set the shell variable \texttt{CC} as shown below. +\begin{alltt} +CC=/home/czurnieden/intel/bin/icc make +\end{alltt} + +ICC does not know all options available for GCC and LibTomMath uses two diagnostics \texttt{-Wbad-function-cast} and \texttt{-Wcast-align} that are not supported by ICC resulting in the warnings: +\begin{alltt} +icc: command line warning #10148: option '-Wbad-function-cast' not supported +icc: command line warning #10148: option '-Wcast-align' not supported +\end{alltt} +It is possible to mute this ICC warning with the compiler flag \texttt{-diag-disable=10148}\footnote{It is not recommended to suppress warnings without a very good reason but there is no harm in doing so in this very special case.}. \subsection{Static Libraries} To build as a static library for GCC issue the following @@ -102,28 +119,102 @@ To build as a static library for GCC issue the following make \end{alltt} -command. This will build the library and archive the object files in ``libtommath.a''. Now you link against +command. This will build the library and archive the object files in ``libtommath.a''. Now you link against that and include ``tommath.h'' within your programs. Alternatively to build with MSVC issue the following \begin{alltt} nmake -f makefile.msvc \end{alltt} -This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC -version 6.00 with service pack 5. +This will build the library and archive the object files in ``tommath.lib''. This has been tested with MSVC +version 6.00 with service pack 5. + +To run a program to adapt the Toom-Cook cut-off values to your architecture type +\begin{alltt} +make tune +\end{alltt} +This will take some time. \subsection{Shared Libraries} +\subsubsection{GNU based Operating Systems} To build as a shared library for GCC issue the following \begin{alltt} make -f makefile.shared \end{alltt} This requires the ``libtool'' package (common on most Linux/BSD systems). It will build LibTomMath as both shared -and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared -library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally -you use libtool to link your application against the shared object. +and static then install (by default) into /usr/lib as well as install the header files in /usr/include. The shared +library (resource) will be called ``libtommath.la'' while the static library called ``libtommath.a''. Generally +you use libtool to link your application against the shared object. -There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires -Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library +To run a program to adapt the Toom-Cook cut-off values to your architecture type +\begin{alltt} +make -f makefile.shared tune +\end{alltt} +This will take some time. + +\subsubsection{Microsoft Windows based Operating Systems} +There is limited support for making a ``DLL'' in windows via the ``makefile.cygwin\_dll'' makefile. It requires +Cygwin to work with since it requires the auto-export/import functionality. The resulting DLL and import library ``libtommath.dll.a'' can be used to link LibTomMath dynamically to any Windows program using Cygwin. +\subsubsection{OpenBSD} +OpenBSD replaced some of their GNU-tools, especially \texttt{libtool} with their own, slightly different versions. To ease the workload of LibTomMath's developer team, only a static library can be build with the included \texttt{makefile.unix}. + +The wrong \texttt{make} will result in errors like: +\begin{alltt} +*** Parse error in /home/user/GITHUB/libtommath: Need an operator in 'LIBNAME' ) +*** Parse error: Need an operator in 'endif' (makefile.shared:8) +*** Parse error: Need an operator in 'CROSS_COMPILE' (makefile_include.mk:16) +*** Parse error: Need an operator in 'endif' (makefile_include.mk:18) +*** Parse error: Missing dependency operator (makefile_include.mk:22) +*** Parse error: Missing dependency operator (makefile_include.mk:23) +... +\end{alltt} +The wrong \texttt{libtool} will build it all fine but when it comes to the final linking fails with +\begin{alltt} +... +cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... +cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... +cc -I./ -Wall -Wsign-compare -Wextra -Wshadow -Wsystem-headers -Wdeclaration-afo... +libtool --mode=link --tag=CC cc bn_error.lo bn_s_mp_invmod_fast.lo bn_fast_mp_mo +libtool: link: cc bn_error.lo bn_s_mp_invmod_fast.lo bn_s_mp_montgomery_reduce_fast0 +bn_error.lo: file not recognized: File format not recognized +cc: error: linker command failed with exit code 1 (use -v to see invocation) +Error while executing cc bn_error.lo bn_s_mp_invmod_fast.lo bn_fast_mp_montgomery0 +gmake: *** [makefile.shared:64: libtommath.la] Error 1 +\end{alltt} + +To build a shared library with OpenBSD\footnote{Tested with OpenBSD version 6.4} the GNU versions of \texttt{make} and \texttt{libtool} are needed. +\begin{alltt} +$ sudo pkg_add gmake libtool +\end{alltt} +At this time two versions of \texttt{libtool} are installed and both are named \texttt{libtool}, unfortunately but GNU \texttt{libtool} has been placed in \texttt{/usr/local/bin/} and the native version in \texttt{/usr/bin/}. The path might be different in other versions of OpenBSD but both programms differ in the output of \texttt{libtool --version} +\begin{alltt} +$ /usr/local/bin/libtool --version +libtool (GNU libtool) 2.4.2 +Written by Gordon Matzigkeit , 1996 + +Copyright (C) 2011 Free Software Foundation, Inc. +This is free software; see the source for copying conditions. There is NO +warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. +$ libtool --version +libtool (not (GNU libtool)) 1.5.26 +\end{alltt} + +The shared library should build now with +\begin{alltt} +LIBTOOL="/usr/local/bin/libtool" gmake -f makefile.shared +\end{alltt} +You might need to run a \texttt{gmake -f makefile.shared clean} first. + +\subsubsection{NetBSD} +NetBSD is not as strict as OpenBSD but still needs \texttt{gmake} to build the shared library. \texttt{libtool} may also not exist in a fresh install. +\begin{alltt} +pkg_add gmake libtool +\end{alltt} +Please check with \texttt{libtool --version} that installed libtool is indeed a GNU libtool. +Build the shared library by typing: +\begin{alltt} +gmake -f makefile.shared +\end{alltt} \subsection{Testing} To build the library and the test harness type @@ -140,7 +231,7 @@ is included in the package}. Simply pipe mtest into test using mtest/mtest | test \end{alltt} -If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into +If you do not have a ``/dev/urandom'' style RNG source you will have to write your own PRNG and simply pipe that into mtest. For example, if your PRNG program is called ``myprng'' simply invoke \begin{alltt} @@ -149,20 +240,20 @@ myprng | mtest/mtest | test This will output a row of numbers that are increasing. Each column is a different test (such as addition, multiplication, etc) that is being performed. The numbers represent how many times the test was invoked. If an error is detected the program -will exit with a dump of the relevent numbers it was working with. +will exit with a dump of the relevant numbers it was working with. \section{Build Configuration} -LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''. -Each phase changes how the library is built and they are applied one after another respectively. +LibTomMath can configured at build time in three phases we shall call ``depends'', ``tweaks'' and ``trims''. +Each phase changes how the library is built and they are applied one after another respectively. To make the system more powerful you can tweak the build process. Classes are defined in the file -``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply -instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you +``tommath\_superclass.h''. By default, the symbol ``LTM\_ALL'' shall be defined which simply +instructs the system to build all of the functions. This is how LibTomMath used to be packaged. This will give you access to every function LibTomMath offers. -However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You -don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is -another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional +However, there are cases where such a build is not optional. For instance, you want to perform RSA operations. You +don't need the vast majority of the library to perform these operations. Aside from LTM\_ALL there is +another pre--defined class ``SC\_RSA\_1'' which works in conjunction with the RSA from LibTomCrypt. Additional classes can be defined base on the need of the user. \subsection{Build Depends} @@ -172,8 +263,8 @@ file. For instance, BN\_MP\_ADD\_C represents the file ``bn\_mp\_add.c''. When function in the respective file will be compiled and linked into the library. Accordingly when the define is absent the file will not be compiled and not contribute any size to the library. -You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice). -This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined. +You will also note that the header tommath\_class.h is actually recursively included (it includes itself twice). +This is to help resolve as many dependencies as possible. In the last pass the symbol LTM\_LAST will be defined. This is useful for ``trims''. \subsection{Build Tweaks} @@ -193,7 +284,7 @@ They can be enabled at any pass of the configuration phase. \subsection{Build Trims} A trim is a manner of removing functionality from a function that is not required. For instance, to perform -RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed. +RSA cryptography you only require exponentiation with odd moduli so even moduli support can be safely removed. Build trims are meant to be defined on the last pass of the configuration which means they are to be defined only if LTM\_LAST has been defined. @@ -232,7 +323,7 @@ only if LTM\_LAST has been defined. & BN\_S\_MP\_SQR\_C \\ \hline Polynomial Schmolynomial & BN\_MP\_KARATSUBA\_MUL\_C \\ & BN\_MP\_KARATSUBA\_SQR\_C \\ - & BN\_MP\_TOOM\_MUL\_C \\ + & BN\_MP\_TOOM\_MUL\_C \\ & BN\_MP\_TOOM\_SQR\_C \\ \hline @@ -242,11 +333,11 @@ only if LTM\_LAST has been defined. \section{Purpose of LibTomMath} -Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with -bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the +Unlike GNU MP (GMP) Library, LIP, OpenSSL or various other commercial kits (Miracl), LibTomMath was not written with +bleeding edge performance in mind. First and foremost LibTomMath was written to be entirely open. Not only is the source code public domain (unlike various other GPL/etc licensed code), not only is the code freely downloadable but the source code is also accessible for computer science students attempting to learn ``BigNum'' or multiple precision -arithmetic techniques. +arithmetic techniques. LibTomMath was written to be an instructive collection of source code. This is why there are many comments, only one function per source file and often I use a ``middle-road'' approach where I don't cut corners for an extra 2\% speed @@ -258,7 +349,7 @@ the library (beat that!). So you may be thinking ``should I use LibTomMath?'' and the answer is a definite maybe. Let me tabulate what I think are the pros and cons of LibTomMath by comparing it to the math routines from GnuPG\footnote{GnuPG v1.2.3 versus LibTomMath v0.28}. -\newpage\begin{figure}[here] +\newpage\begin{figure}[h] \begin{small} \begin{center} \begin{tabular}{|l|c|c|l|} @@ -277,9 +368,9 @@ are the pros and cons of LibTomMath by comparing it to the math routines from Gn \caption{LibTomMath Valuation} \end{figure} -It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. +It may seem odd to compare LibTomMath to GnuPG since the math in GnuPG is only a small portion of the entire application. However, LibTomMath was written with cryptography in mind. It provides essentially all of the functions a cryptosystem -would require when working with large integers. +would require when working with large integers. So it may feel tempting to just rip the math code out of GnuPG (or GnuMP where it was taken from originally) in your own application but I think there are reasons not to. While LibTomMath is slower than libraries such as GnuMP it is @@ -288,19 +379,19 @@ exponentiations. It depends largely on the processor, compiler and the moduli b Essentially the only time you wouldn't use LibTomMath is when blazing speed is the primary concern. However, on the other side of the coin LibTomMath offers you a totally free (public domain) well structured math library -that is very flexible, complete and performs well in resource contrained environments. Fast RSA for example can -be performed with as little as 8KB of ram for data (again depending on build options). +that is very flexible, complete and performs well in resource constrained environments. Fast RSA for example can +be performed with as little as 8KB of ram for data (again depending on build options). \chapter{Getting Started with LibTomMath} \section{Building Programs} -In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically +In order to use LibTomMath you must include ``tommath.h'' and link against the appropriate library file (typically libtommath.a). There is no library initialization required and the entire library is thread safe. \section{Return Codes} There are three possible return codes a function may return. \index{MP\_OKAY}\index{MP\_YES}\index{MP\_NO}\index{MP\_VAL}\index{MP\_MEM} -\begin{figure}[here!] +\begin{figure}[h!] \begin{center} \begin{small} \begin{tabular}{|l|l|} @@ -327,8 +418,8 @@ to a string use the following function. char *mp_error_to_string(int code); \end{alltt} -This will return a pointer to a string which describes the given error code. It will not work for the return codes -MP\_YES and MP\_NO. +This will return a pointer to a string which describes the given error code. It will not work for the return codes +MP\_YES and MP\_NO. \section{Data Types} The basic ``multiple precision integer'' type is known as the ``mp\_int'' within LibTomMath. This data type is used to @@ -345,7 +436,7 @@ typedef struct \{ Where ``mp\_digit'' is a data type that represents individual digits of the integer. By default, an mp\_digit is the ISO C ``unsigned long'' data type and each digit is $28-$bits long. The mp\_digit type can be configured to suit other -platforms by defining the appropriate macros. +platforms by defining the appropriate macros. All LTM functions that use the mp\_int type will expect a pointer to mp\_int structure. You must allocate memory to hold the structure itself by yourself (whether off stack or heap it doesn't matter). The very first thing that must be @@ -374,7 +465,7 @@ This allows operands to be re-used which can make programming simpler. \section{Initialization} \subsection{Single Initialization} -A single mp\_int can be initialized with the ``mp\_init'' function. +A single mp\_int can be initialized with the ``mp\_init'' function. \index{mp\_init} \begin{alltt} @@ -392,11 +483,11 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the number */ return EXIT_SUCCESS; @@ -404,7 +495,7 @@ int main(void) \end{alltt} \end{small} \subsection{Single Free} -When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function +When you are finished with an mp\_int it is ideal to return the heap it used back to the system. The following function provides this functionality. \index{mp\_clear} @@ -412,9 +503,9 @@ provides this functionality. void mp_clear (mp_int * a); \end{alltt} -The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the -pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. -Is is legal to call mp\_clear() twice on the same mp\_int in a row. +The function expects a pointer to a previously initialized mp\_int structure and frees the heap it uses. It sets the +pointer\footnote{The ``dp'' member.} within the mp\_int to \textbf{NULL} which is used to prevent double free situations. +Is is legal to call mp\_clear() twice on the same mp\_int in a row. \begin{small} \begin{alltt} int main(void) @@ -423,11 +514,11 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the number */ /* We're done with it. */ @@ -451,8 +542,8 @@ int mp_init_multi(mp_int *mp, ...); It accepts a \textbf{NULL} terminated list of pointers to mp\_int structures. It will attempt to initialize them all at once. If the function returns MP\_OKAY then all of the mp\_int variables are ready to use, otherwise none of them -are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd -from the heap at the same time. +are available for use. A complementary mp\_clear\_multi() function allows multiple mp\_int variables to be free'd +from the heap at the same time. \begin{small} \begin{alltt} int main(void) @@ -460,14 +551,14 @@ int main(void) mp_int num1, num2, num3; int result; - if ((result = mp_init_multi(&num1, + if ((result = mp_init_multi(&num1, &num2, - &num3, NULL)) != MP\_OKAY) \{ - printf("Error initializing the numbers. \%s", + &num3, NULL)) != MP\_OKAY) \{ + printf("Error initializing the numbers. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the numbers */ /* We're done with them. */ @@ -478,7 +569,7 @@ int main(void) \end{alltt} \end{small} \subsection{Other Initializers} -To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided. +To initialized and make a copy of an mp\_int the mp\_init\_copy() function has been provided. \index{mp\_init\_copy} \begin{alltt} @@ -497,11 +588,11 @@ int main(void) /* We want a copy of num1 in num2 now */ if ((result = mp_init_copy(&num2, &num1)) != MP_OKAY) \{ - printf("Error initializing the copy. \%s", + printf("Error initializing the copy. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* now num2 is ready and contains a copy of num1 */ /* We're done with them. */ @@ -521,7 +612,7 @@ int mp_init_size (mp_int * a, int size); \end{alltt} The $size$ parameter must be greater than zero. If the function succeeds the mp\_int $a$ will be initialized -to have $size$ digits (which are all initially zero). +to have $size$ digits (which are all initially zero). \begin{small} \begin{alltt} int main(void) @@ -531,11 +622,11 @@ int main(void) /* we need a 60-digit number */ if ((result = mp_init_size(&number, 60)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the number */ return EXIT_SUCCESS; @@ -543,6 +634,25 @@ int main(void) \end{alltt} \end{small} \section{Maintenance Functions} +\subsection{Clear Leading Zeros} + +This is used to ensure that leading zero digits are trimed and the leading "used" digit will be non-zero. +It also fixes the sign if there are no more leading digits. + +\index{mp\_clamp} +\begin{alltt} +void mp_clamp(mp_int *a); +\end{alltt} + +\subsection{Zero Out} + +This function will set the ``bigint'' to zeros without changing the amount of allocated memory. + +\index{mp\_zero} +\begin{alltt} +void mp_zero(mp_int *a); +\end{alltt} + \subsection{Reducing Memory Usage} When an mp\_int is in a state where it won't be changed again\footnote{A Diffie-Hellman modulus for instance.} excess @@ -556,7 +666,7 @@ int mp_shrink (mp_int * a); This will remove excess digits of the mp\_int $a$. If the operation fails the mp\_int should be intact without the excess digits being removed. Note that you can use a shrunk mp\_int in further computations, however, such operations will require heap operations which can be slow. It is not ideal to shrink mp\_int variables that you will further -modify in the system (unless you are seriously low on memory). +modify in the system (unless you are seriously low on memory). \begin{small} \begin{alltt} int main(void) @@ -565,16 +675,16 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the number [e.g. pre-computation] */ /* We're done with it for now. */ if ((result = mp_shrink(&number)) != MP_OKAY) \{ - printf("Error shrinking the number. \%s", + printf("Error shrinking the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -582,7 +692,7 @@ int main(void) /* use it .... */ - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; @@ -595,7 +705,7 @@ Within the mp\_int structure are two parameters which control the limitations of the integer the mp\_int is meant to equal. The \textit{used} parameter dictates how many digits are significant, that is, contribute to the value of the mp\_int. The \textit{alloc} parameter dictates how many digits are currently available in the array. If you need to perform an operation that requires more digits you will have to mp\_grow() the mp\_int to -your desired size. +your desired size. \index{mp\_grow} \begin{alltt} @@ -612,16 +722,16 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* use the number */ /* We need to add 20 digits to the number */ if ((result = mp_grow(&number, number.alloc + 20)) != MP_OKAY) \{ - printf("Error growing the number. \%s", + printf("Error growing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -629,7 +739,7 @@ int main(void) /* use the number */ - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; @@ -637,11 +747,44 @@ int main(void) \end{alltt} \end{small} \chapter{Basic Operations} +\section{Copying} + +A so called ``deep copy'', where new memory is allocated and all contents of $a$ are copied verbatim into $b$ such that $b = a$ at the end. + +\index{mp\_copy} +\begin{alltt} +int mp_copy (mp_int * a, mp_int *b); +\end{alltt} + +You can also just swap $a$ and $b$. It does the normal pointer changing with a temporary pointer variable, just that you do not have to. + +\index{mp\_exch} +\begin{alltt} +void mp_exch (mp_int * a, mp_int *b); +\end{alltt} + +\section{Bit Counting} + +To get the position of the lowest bit set (LSB, the Lowest Significant Bit; the number of bits which are zero before the first zero bit ) + +\index{mp\_cnt\_lsb} +\begin{alltt} +int mp_cnt_lsb(const mp_int *a); +\end{alltt} + +To get the position of the highest bit set (MSB, the Most Significant Bit; the number of bits in teh ``bignum'') + +\index{mp\_count\_bits} +\begin{alltt} +int mp_count_bits(const mp_int *a); +\end{alltt} + + \section{Small Constants} -Setting mp\_ints to small constants is a relatively common operation. To accomodate these instances there are two -small constant assignment functions. The first function is used to set a single digit constant while the second sets -an ISO C style ``unsigned long'' constant. The reason for both functions is efficiency. Setting a single digit is quick but the -domain of a digit can change (it's always at least $0 \ldots 127$). +Setting mp\_ints to small constants is a relatively common operation. To accommodate these instances there is a +small constant assignment function. This function is used to set a single digit constant. +The reason for this function is efficiency. Setting a single digit is quick but the +domain of a digit can change (it's always at least $0 \ldots 127$). \subsection{Single Digit} @@ -663,43 +806,50 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number to 5 */ mp_set(&number, 5); - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; \} \end{alltt} \end{small} -\subsection{Long Constants} +\subsection{Int32 and Int64 Constants} -To set a constant that is the size of an ISO C ``unsigned long'' and larger than a single digit the following function -can be used. +These functions can be used to set a constant with 32 or 64 bits. -\index{mp\_set\_int} +\index{mp\_set\_i32} \index{mp\_set\_u32} +\index{mp\_set\_i64} \index{mp\_set\_u64} \begin{alltt} -int mp_set_int (mp_int * a, unsigned long b); +void mp_set_i32 (mp_int * a, int32_t b); +void mp_set_u32 (mp_int * a, uint32_t b); +void mp_set_i64 (mp_int * a, int64_t b); +void mp_set_u64 (mp_int * a, uint64_t b); \end{alltt} -This will assign the value of the 32-bit variable $b$ to the mp\_int $a$. Unlike mp\_set() this function will always -accept a 32-bit input regardless of the size of a single digit. However, since the value may span several digits -this function can fail if it runs out of heap memory. +These functions assign the sign and value of the input \texttt{b} to \texttt{mp\_int a}. +The value can be obtained again by calling the following functions. -To get the ``unsigned long'' copy of an mp\_int the following function can be used. - -\index{mp\_get\_int} +\index{mp\_get\_i32} \index{mp\_get\_u32} \index{mp\_get\_mag\_u32} +\index{mp\_get\_i64} \index{mp\_get\_u64} \index{mp\_get\_mag\_u64} \begin{alltt} -unsigned long mp_get_int (mp_int * a); +int32_t mp_get_i32 (mp_int * a); +uint32_t mp_get_u32 (mp_int * a); +uint32_t mp_get_mag_u32 (mp_int * a); +int64_t mp_get_i64 (mp_int * a); +uint64_t mp_get_u64 (mp_int * a); +uint64_t mp_get_mag_u64 (mp_int * a); \end{alltt} -This will return the 32 least significant bits of the mp\_int $a$. +These functions return the 32 or 64 least significant bits of $a$ respectively. The unsigned functions +return negative values in a twos complement representation. The absolute value or magnitude can be obtained using the mp\_get\_mag functions. \begin{small} \begin{alltt} int main(void) @@ -708,21 +858,17 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number to 654321 (note this is bigger than 127) */ - if ((result = mp_set_int(&number, 654321)) != MP_OKAY) \{ - printf("Error setting the value of the number. \%s", - mp_error_to_string(result)); - return EXIT_FAILURE; - \} + mp_set_u32(&number, 654321); - printf("number == \%lu", mp_get_int(&number)); + printf("number == \%" PRIi32, mp_get_i32(&number)); - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; @@ -735,15 +881,66 @@ This should output the following if the program succeeds. number == 654321 \end{alltt} +\subsection{Long Constants - platform dependant} + +\index{mp\_set\_l} \index{mp\_set\_ul} +\begin{alltt} +void mp_set_l (mp_int * a, long b); +void mp_set_ul (mp_int * a, unsigned long b); +\end{alltt} + +This will assign the value of the platform-dependent sized variable $b$ to the mp\_int $a$. + +To retrieve the value, the following functions can be used. + +\index{mp\_get\_l} \index{mp\_get\_ul} \index{mp\_get\_mag\_ul} +\begin{alltt} +long mp_get_l (mp_int * a); +unsigned long mp_get_ul (mp_int * a); +unsigned long mp_get_mag_ul (mp_int * a); +\end{alltt} + +This will return the least significant bits of the mp\_int $a$ that fit into a ``long''. + +\subsection{Long Long Constants - platform dependant} + +\index{mp\_set\_ll} \index{mp\_set\_ull} +\begin{alltt} +void mp_set_ll (mp_int * a, long long b); +void mp_set_ull (mp_int * a, unsigned long long b); +\end{alltt} + +This will assign the value of the platform-dependent sized variable $b$ to the mp\_int $a$. + +To retrieve the value, the following functions can be used. + +\index{mp\_get\_ll} +\index{mp\_get\_ull} +\index{mp\_get\_mag\_ull} +\begin{alltt} +long long mp_get_ll (mp_int * a); +unsigned long long mp_get_ull (mp_int * a); +unsigned long long mp_get_mag_ull (mp_int * a); +\end{alltt} + +This will return the least significant bits of the mp\_int $a$ that fit into a ``long long''. + \subsection{Initialize and Setting Constants} To both initialize and set small constants the following two functions are available. \index{mp\_init\_set} \index{mp\_init\_set\_int} \begin{alltt} int mp_init_set (mp_int * a, mp_digit b); -int mp_init_set_int (mp_int * a, unsigned long b); +int mp_init_i32 (mp_int * a, int32_t b); +int mp_init_u32 (mp_int * a, uint32_t b); +int mp_init_i64 (mp_int * a, int64_t b); +int mp_init_u64 (mp_int * a, uint64_t b); +int mp_init_l (mp_int * a, long b); +int mp_init_ul (mp_int * a, unsigned long b); +int mp_init_ll (mp_int * a, long long b); +int mp_init_ull (mp_int * a, unsigned long long b); \end{alltt} -Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values. +Both functions work like the previous counterparts except they first mp\_init $a$ before setting the values. \begin{alltt} int main(void) @@ -753,21 +950,21 @@ int main(void) /* initialize and set a single digit */ if ((result = mp_init_set(&number1, 100)) != MP_OKAY) \{ - printf("Error setting number1: \%s", + printf("Error setting number1: \%s", mp_error_to_string(result)); return EXIT_FAILURE; - \} + \} /* initialize and set a long */ - if ((result = mp_init_set_int(&number2, 1023)) != MP_OKAY) \{ - printf("Error setting number2: \%s", + if ((result = mp_init_l(&number2, 1023)) != MP_OKAY) \{ + printf("Error setting number2: \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* display */ - printf("Number1, Number2 == \%lu, \%lu", - mp_get_int(&number1), mp_get_int(&number2)); + printf("Number1, Number2 == \%" PRIi32 ", \%" PRIi32, + mp_get_i32(&number1), mp_get_i32(&number2)); /* clear */ mp_clear_multi(&number1, &number2, NULL); @@ -787,7 +984,7 @@ Comparisons in LibTomMath are always performed in a ``left to right'' fashion. for any comparison. \index{MP\_GT} \index{MP\_EQ} \index{MP\_LT} -\begin{figure}[here] +\begin{figure}[h] \begin{center} \begin{tabular}{|c|c|} \hline \textbf{Result Code} & \textbf{Meaning} \\ @@ -801,14 +998,14 @@ for any comparison. \label{fig:CMP} \end{figure} -In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of -$b$. +In figure \ref{fig:CMP} two integers $a$ and $b$ are being compared. In this case $a$ is said to be ``to the left'' of +$b$. \subsection{Unsigned comparison} -An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the +An unsigned comparison considers only the digits themselves and not the associated \textit{sign} flag of the mp\_int structures. This is analogous to an absolute comparison. The function mp\_cmp\_mag() will compare two -mp\_int variables based on their digits only. +mp\_int variables based on their digits only. \index{mp\_cmp\_mag} \begin{alltt} @@ -824,18 +1021,18 @@ int main(void) int result; if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ - printf("Error initializing the numbers. \%s", + printf("Error initializing the numbers. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number1 to 5 */ mp_set(&number1, 5); - + /* set the number2 to -6 */ mp_set(&number2, 6); if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ - printf("Error negating number2. \%s", + printf("Error negating number2. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -846,14 +1043,14 @@ int main(void) case MP_LT: printf("|number1| < |number2|"); break; \} - /* we're done with it. */ + /* we're done with it. */ mp_clear_multi(&number1, &number2, NULL); return EXIT_SUCCESS; \} \end{alltt} \end{small} -If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes +If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes successfully it should print the following. \begin{alltt} @@ -882,18 +1079,18 @@ int main(void) int result; if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ - printf("Error initializing the numbers. \%s", + printf("Error initializing the numbers. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number1 to 5 */ mp_set(&number1, 5); - + /* set the number2 to -6 */ mp_set(&number2, 6); if ((result = mp_neg(&number2, &number2)) != MP_OKAY) \{ - printf("Error negating number2. \%s", + printf("Error negating number2. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -904,14 +1101,14 @@ int main(void) case MP_LT: printf("number1 < number2"); break; \} - /* we're done with it. */ + /* we're done with it. */ mp_clear_multi(&number1, &number2, NULL); return EXIT_SUCCESS; \} \end{alltt} \end{small} -If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes +If this program\footnote{This function uses the mp\_neg() function which is discussed in section \ref{sec:NEG}.} completes successfully it should print the following. \begin{alltt} @@ -927,7 +1124,7 @@ To compare a single digit against an mp\_int the following function has been pro int mp_cmp_d(mp_int * a, mp_digit b); \end{alltt} -This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as +This will compare $a$ to the left of $b$ using a signed comparison. Note that it will always treat $b$ as positive. This function is rather handy when you have to compare against small values such as $1$ (which often comes up in cryptography). The function cannot fail and will return one of the tree compare condition codes listed in figure \ref{fig:CMP}. @@ -940,11 +1137,11 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number to 5 */ mp_set(&number, 5); @@ -954,7 +1151,7 @@ int main(void) case MP_LT: printf("number < 7"); break; \} - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; @@ -975,7 +1172,7 @@ AND, XOR and OR directly. These operations are very quick. \subsection{Multiplication by two} Multiplications and divisions by any power of two can be performed with quick logical shifts either left or -right depending on the operation. +right depending on the operation. When multiplying or dividing by two a special case routine can be used which are as follows. \index{mp\_mul\_2} \index{mp\_div\_2} @@ -994,17 +1191,17 @@ int main(void) int result; if ((result = mp_init(&number)) != MP_OKAY) \{ - printf("Error initializing the number. \%s", + printf("Error initializing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* set the number to 5 */ mp_set(&number, 5); /* multiply by two */ if ((result = mp\_mul\_2(&number, &number)) != MP_OKAY) \{ - printf("Error multiplying the number. \%s", + printf("Error multiplying the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -1016,7 +1213,7 @@ int main(void) /* now divide by two */ if ((result = mp\_div\_2(&number, &number)) != MP_OKAY) \{ - printf("Error dividing the number. \%s", + printf("Error dividing the number. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -1026,7 +1223,7 @@ int main(void) case MP_LT: printf("2*number/2 < 7"); break; \} - /* we're done with it. */ + /* we're done with it. */ mp_clear(&number); return EXIT_SUCCESS; @@ -1040,15 +1237,18 @@ If this program is successful it will print out the following text. 2*number/2 < 7 \end{alltt} -Since $10 > 7$ and $5 < 7$. To multiply by a power of two the following function can be used. +Since $10 > 7$ and $5 < 7$. + +To multiply by a power of two the following function can be used. \index{mp\_mul\_2d} \begin{alltt} int mp_mul_2d(mp_int * a, int b, mp_int * c); \end{alltt} -This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to -zero the function will copy $a$ to ``c'' without performing any further actions. +This will multiply $a$ by $2^b$ and store the result in ``c''. If the value of $b$ is less than or equal to +zero the function will copy $a$ to ``c'' without performing any further actions. The multiplication itself +is implemented as a right-shift operation of $a$ by $b$ bits. To divide by a power of two use the following. @@ -1058,14 +1258,23 @@ int mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d); \end{alltt} Which will divide $a$ by $2^b$, store the quotient in ``c'' and the remainder in ``d'. If $b \le 0$ then the function simply copies $a$ over to ``c'' and zeroes $d$. The variable $d$ may be passed as a \textbf{NULL} -value to signal that the remainder is not desired. +value to signal that the remainder is not desired. The division itself is implemented as a left-shift +operation of $a$ by $b$ bits. + +It is also not very uncommon to need just the power of two $2^b$; for example the startvalue for the Newton method. + +\index{mp\_2expt} +\begin{alltt} +int mp_2expt(mp_int *a, int b); +\end{alltt} +It is faster than doing it by shifting $1$ with \texttt{mp\_mul\_2d}. \subsection{Polynomial Basis Operations} -Strictly speaking the organization of the integers within the mp\_int structures is what is known as a +Strictly speaking the organization of the integers within the mp\_int structures is what is known as a ``polynomial basis''. This simply means a field element is stored by divisions of a radix. For example, if -$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be -the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$. +$f(x) = \sum_{i=0}^{k} y_ix^k$ for any vector $\vec y$ then the array of digits in $\vec y$ are said to be +the polynomial basis representation of $z$ if $f(\beta) = z$ for a given radix $\beta$. To multiply by the polynomial $g(x) = x$ all you have todo is shift the digits of the basis left one place. The following function provides this operation. @@ -1085,19 +1294,31 @@ void mp_rshd (mp_int * a, int b) This will divide $a$ in place by $x^b$ and discard the remainder. This function cannot fail as it performs the operations in place and no new digits are required to complete it. -\subsection{AND, OR and XOR Operations} +\subsection{AND, OR, XOR and COMPLEMENT Operations} -While AND, OR and XOR operations are not typical ``bignum functions'' they can be useful in several instances. The -three functions are prototyped as follows. +While AND, OR and XOR operations compute arbitrary-precision bitwise operations. Negative numbers +are treated as if they are in two-complement representation, while internally they are sign-magnitude however. -\index{mp\_or} \index{mp\_and} \index{mp\_xor} +\index{mp\_or} \index{mp\_and} \index{mp\_xor} \index{mp\_complement} \begin{alltt} int mp_or (mp_int * a, mp_int * b, mp_int * c); int mp_and (mp_int * a, mp_int * b, mp_int * c); int mp_xor (mp_int * a, mp_int * b, mp_int * c); +int mp_complement(const mp_int *a, mp_int *b); +int mp_signed_rsh(mp_int * a, int b, mp_int * c, mp_int * d); \end{alltt} -Which compute $c = a \odot b$ where $\odot$ is one of OR, AND or XOR. +The function \texttt{mp\_complement} computes a two-complement $b = \sim a$. The function \texttt{mp\_signed\_rsh} performs +sign extending right shift. For positive numbers it is equivalent to \texttt{mp\_div\_2d}. + +\subsection{Bit Picking} +\index{mp\_get\_bit} +\begin{alltt} +int mp_get_bit(mp_int *a, int b) +\end{alltt} + +Pick a bit: returns \texttt{MP\_YES} if the bit at position $b$ (0-index) is set, that is if it is 1 (one), \texttt{MP\_NO} +if the bit is 0 (zero) and \texttt{MP\_VAL} if $b < 0$. \section{Addition and Subtraction} @@ -1122,17 +1343,17 @@ Simple integer negation can be performed with the following. int mp_neg (mp_int * a, mp_int * b); \end{alltt} -Which assigns $-a$ to $b$. +Which assigns $-a$ to $b$. \subsection{Absolute} Simple integer absolutes can be performed with the following. -\index{mp\_neg} +\index{mp\_abs} \begin{alltt} int mp_abs (mp_int * a, mp_int * b); \end{alltt} -Which assigns $\vert a \vert$ to $b$. +Which assigns $\vert a \vert$ to $b$. \section{Integer Division and Remainder} To perform a complete and general integer division with remainder use the following function. @@ -1141,10 +1362,10 @@ To perform a complete and general integer division with remainder use the follow \begin{alltt} int mp_div (mp_int * a, mp_int * b, mp_int * c, mp_int * d); \end{alltt} - -This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that -$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If -$b$ is zero the function returns \textbf{MP\_VAL}. + +This divides $a$ by $b$ and stores the quotient in $c$ and $d$. The signed quotient is computed such that +$bc + d = a$. Note that either of $c$ or $d$ can be set to \textbf{NULL} if their value is not required. If +$b$ is zero the function returns \textbf{MP\_VAL}. \chapter{Multiplication and Squaring} @@ -1154,13 +1375,13 @@ A full signed integer multiplication can be performed with the following. \begin{alltt} int mp_mul (mp_int * a, mp_int * b, mp_int * c); \end{alltt} -Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are +Which assigns the full signed product $ab$ to $c$. This function actually breaks into one of four cases which are specific multiplication routines optimized for given parameters. First there are the Toom-Cook multiplications which should only be used with very large inputs. This is followed by the Karatsuba multiplications which are for moderate sized inputs. Then followed by the Comba and baseline multipliers. Fortunately for the developer you don't really need to know this unless you really want to fine tune the system. mp\_mul() -will determine on its own\footnote{Some tweaking may be required.} what routine to use automatically when it is called. +will determine on its own\footnote{Some tweaking may be required but \texttt{make tune} will put some reasonable values in \texttt{bncore.c}} what routine to use automatically when it is called. \begin{alltt} int main(void) @@ -1169,43 +1390,34 @@ int main(void) int result; /* Initialize the numbers */ - if ((result = mp_init_multi(&number1, + if ((result = mp_init_multi(&number1, &number2, NULL)) != MP_OKAY) \{ - printf("Error initializing the numbers. \%s", + printf("Error initializing the numbers. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* set the terms */ - if ((result = mp_set_int(&number, 257)) != MP_OKAY) \{ - printf("Error setting number1. \%s", - mp_error_to_string(result)); - return EXIT_FAILURE; - \} - - if ((result = mp_set_int(&number2, 1023)) != MP_OKAY) \{ - printf("Error setting number2. \%s", - mp_error_to_string(result)); - return EXIT_FAILURE; - \} + mp_set_i32(&number, 257); + mp_set_i32(&number2, 1023); /* multiply them */ if ((result = mp_mul(&number1, &number2, &number1)) != MP_OKAY) \{ - printf("Error multiplying terms. \%s", + printf("Error multiplying terms. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* display */ - printf("number1 * number2 == \%lu", mp_get_int(&number1)); + printf("number1 * number2 == \%" PRIi32, mp_get_i32(&number1)); /* free terms and return */ mp_clear_multi(&number1, &number2, NULL); return EXIT_SUCCESS; \} -\end{alltt} +\end{alltt} If this program succeeds it shall output the following. @@ -1224,68 +1436,51 @@ int mp_sqr (mp_int * a, mp_int * b); Will square $a$ and store it in $b$. Like the case of multiplication there are four different squaring algorithms all which can be called from mp\_sqr(). It is ideal to use mp\_sqr over mp\_mul when squaring terms because -of the speed difference. +of the speed difference. \section{Tuning Polynomial Basis Routines} Both of the Toom-Cook and Karatsuba multiplication algorithms are faster than the traditional $O(n^2)$ approach that -the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require +the Comba and baseline algorithms use. At $O(n^{1.464973})$ and $O(n^{1.584962})$ running times respectively they require considerably less work. For example, a 10000-digit multiplication would take roughly 724,000 single precision multiplications with Toom-Cook or 100,000,000 single precision multiplications with the standard Comba (a factor of 138). So why not always use Karatsuba or Toom-Cook? The simple answer is that they have so much overhead that they're not -actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, -GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at +actually faster than Comba until you hit distinct ``cutoff'' points. For Karatsuba with the default configuration, +GCC 3.3.1 and an Athlon XP processor the cutoff point is roughly 110 digits (about 70 for the Intel P4). That is, at 110 digits Karatsuba and Comba multiplications just about break even and for 110+ digits Karatsuba is faster. -Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points +Toom-Cook has incredible overhead and is probably only useful for very large inputs. So far no known cutoff points exist and for the most part I just set the cutoff points very high to make sure they're not called. -A demo program in the ``etc/'' directory of the project called ``tune.c'' can be used to find the cutoff points. This -can be built with GCC as follows +To get reasonable values for the cut-off points for your architecture, type \begin{alltt} -make XXX +make tune \end{alltt} -Where ``XXX'' is one of the following entries from the table \ref{fig:tuning}. -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|l|l|} -\hline \textbf{Value of XXX} & \textbf{Meaning} \\ -\hline tune & Builds portable tuning application \\ -\hline tune86 & Builds x86 (pentium and up) program for COFF \\ -\hline tune86c & Builds x86 program for Cygwin \\ -\hline tune86l & Builds x86 program for Linux (ELF format) \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Build Names for Tuning Programs} -\label{fig:tuning} -\end{figure} +This will run a benchmark, computes the medians, rewrites \texttt{bncore.c}, and recompiles \texttt{bncore.c} and relinks the library. -When the program is running it will output a series of measurements for different cutoff points. It will first find -good Karatsuba squaring and multiplication points. Then it proceeds to find Toom-Cook points. Note that the Toom-Cook -tuning takes a very long time as the cutoff points are likely to be very high. +The benchmark itself can be fine-tuned in the file \texttt{etc/tune\_it.sh}. + +The program \texttt{etc/tune} is also able to print a list of values for printing curves with e.g.: \texttt{gnuplot}. type \texttt{./etc/tune -h} to get a list of all available options. \chapter{Modular Reduction} -Modular reduction is process of taking the remainder of one quantity divided by another. Expressed -as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$. +Modular reduction is process of taking the remainder of one quantity divided by another. Expressed +as (\ref{eqn:mod}) the modular reduction is equivalent to the remainder of $b$ divided by $c$. \begin{equation} a \equiv b \mbox{ (mod }c\mbox{)} \label{eqn:mod} \end{equation} -Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly -fast reduction algorithms can be written for the limited range. +Of particular interest to cryptography are reductions where $b$ is limited to the range $0 \le b < c^2$ since particularly +fast reduction algorithms can be written for the limited range. Note that one of the four optimized reduction algorithms are automatically chosen in the modular exponentiation -algorithm mp\_exptmod when an appropriate modulus is detected. +algorithm mp\_exptmod when an appropriate modulus is detected. \section{Straight Division} In order to effect an arbitrary modular reduction the following algorithm is provided. @@ -1295,7 +1490,7 @@ In order to effect an arbitrary modular reduction the following algorithm is pro int mp_mod(mp_int *a, mp_int *b, mp_int *c); \end{alltt} -This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign +This reduces $a$ modulo $b$ and stores the result in $c$. The sign of $c$ shall agree with the sign of $b$. This algorithm accepts an input $a$ of any range and is not limited by $0 \le a < b^2$. \section{Barrett Reduction} @@ -1325,52 +1520,52 @@ int main(void) mp_int a, b, c, mu; int result; - /* initialize a,b to desired values, mp_init mu, - * c and set c to 1...we want to compute a^3 mod b + /* initialize a,b to desired values, mp_init mu, + * c and set c to 1...we want to compute a^3 mod b */ /* get mu value */ if ((result = mp_reduce_setup(&mu, b)) != MP_OKAY) \{ - printf("Error getting mu. \%s", + printf("Error getting mu. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* square a to get c = a^2 */ if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ - printf("Error squaring. \%s", + printf("Error squaring. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* now reduce `c' modulo b */ if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* multiply a to get c = a^3 */ if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* now reduce `c' modulo b */ if ((result = mp_reduce(&c, &b, &mu)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* c now equals a^3 mod b */ return EXIT_SUCCESS; \} -\end{alltt} +\end{alltt} -This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed. +This program will calculate $a^3 \mbox{ mod }b$ if all the functions succeed. \section{Montgomery Reduction} @@ -1382,7 +1577,7 @@ step is required. This is accomplished with the following. int mp_montgomery_setup(mp_int *a, mp_digit *mp); \end{alltt} -For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the +For the given odd moduli $a$ the precomputation value is placed in $mp$. The reduction is computed with the following. \index{mp\_montgomery\_reduce} @@ -1394,10 +1589,10 @@ $0 \le a < b^2$. Montgomery reduction is faster than Barrett reduction for moduli smaller than the ``comba'' limit. With the default setup for instance, the limit is $127$ digits ($3556$--bits). Note that this function is not limited to -$127$ digits just that it falls back to a baseline algorithm after that point. +$127$ digits just that it falls back to a baseline algorithm after that point. -An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ -where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$). +An important observation is that this reduction does not return $a \mbox{ mod }m$ but $aR^{-1} \mbox{ mod }m$ +where $R = \beta^n$, $n$ is the n number of digits in $m$ and $\beta$ is radix used (default is $2^{28}$). To quickly calculate $R$ the following function was provided. @@ -1405,7 +1600,7 @@ To quickly calculate $R$ the following function was provided. \begin{alltt} int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); \end{alltt} -Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division. +Which calculates $a = R$ for the odd moduli $b$ without using multiplication or division. The normal modus operandi for Montgomery reductions is to normalize the integers before entering the system. For example, to calculate $a^3 \mbox { mod }b$ using Montgomery reduction the value of $a$ can be normalized by @@ -1418,62 +1613,62 @@ int main(void) mp_digit mp; int result; - /* initialize a,b to desired values, - * mp_init R, c and set c to 1.... + /* initialize a,b to desired values, + * mp_init R, c and set c to 1.... */ /* get normalization */ if ((result = mp_montgomery_calc_normalization(&R, b)) != MP_OKAY) \{ - printf("Error getting norm. \%s", + printf("Error getting norm. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* get mp value */ if ((result = mp_montgomery_setup(&c, &mp)) != MP_OKAY) \{ - printf("Error setting up montgomery. \%s", + printf("Error setting up montgomery. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* normalize `a' so now a is equal to aR */ if ((result = mp_mulmod(&a, &R, &b, &a)) != MP_OKAY) \{ - printf("Error computing aR. \%s", + printf("Error computing aR. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* square a to get c = a^2R^2 */ if ((result = mp_sqr(&a, &c)) != MP_OKAY) \{ - printf("Error squaring. \%s", + printf("Error squaring. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* now reduce `c' back down to c = a^2R^2 * R^-1 == a^2R */ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* multiply a to get c = a^3R^2 */ if ((result = mp_mul(&a, &c, &c)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} /* now reduce `c' back down to c = a^3R^2 * R^-1 == a^3R */ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} - + /* now reduce (again) `c' back down to c = a^3R * R^-1 == a^3 */ if ((result = mp_montgomery_reduce(&c, &b, mp)) != MP_OKAY) \{ - printf("Error reducing. \%s", + printf("Error reducing. \%s", mp_error_to_string(result)); return EXIT_FAILURE; \} @@ -1482,19 +1677,19 @@ int main(void) return EXIT_SUCCESS; \} -\end{alltt} +\end{alltt} -This particular example does not look too efficient but it demonstrates the point of the algorithm. By +This particular example does not look too efficient but it demonstrates the point of the algorithm. By normalizing the inputs the reduced results are always of the form $aR$ for some variable $a$. This allows a single final reduction to correct for the normalization and the fast reduction used within the algorithm. For more details consider examining the file \textit{bn\_mp\_exptmod\_fast.c}. -\section{Restricted Dimminished Radix} +\section{Restricted Diminished Radix} -``Dimminished Radix'' reduction refers to reduction with respect to moduli that are ameniable to simple +``Diminished Radix'' reduction refers to reduction with respect to moduli that are amenable to simple digit shifting and small multiplications. In this case the ``restricted'' variant refers to moduli of the -form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$). +form $\beta^k - p$ for some $k \ge 0$ and $0 < p < \beta$ where $\beta$ is the radix (default to $2^{28}$). As in the case of Montgomery reduction there is a pre--computation phase required for a given modulus. @@ -1513,36 +1708,63 @@ int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); \end{alltt} This reduces $a$ in place modulo $b$ with the pre--computed value $mp$. $b$ must be of a restricted -dimminished radix form and $a$ must be in the range $0 \le a < b^2$. Dimminished radix reductions are -much faster than both Barrett and Montgomery reductions as they have a much lower asymtotic running time. +diminished radix form and $a$ must be in the range $0 \le a < b^2$. Diminished radix reductions are +much faster than both Barrett and Montgomery reductions as they have a much lower asymptotic running time. Since the moduli are restricted this algorithm is not particularly useful for something like Rabin, RSA or BBS cryptographic purposes. This reduction algorithm is useful for Diffie-Hellman and ECC where fixed -primes are acceptable. +primes are acceptable. Note that unlike Montgomery reduction there is no normalization process. The result of this function is equal to the correct residue. -\section{Unrestricted Dimminshed Radix} +\section{Unrestricted Diminished Radix} -Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the -form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they -can be applied to a wider range of numbers. +Unrestricted reductions work much like the restricted counterparts except in this case the moduli is of the +form $2^k - p$ for $0 < p < \beta$. In this sense the unrestricted reductions are more flexible as they +can be applied to a wider range of numbers. \index{mp\_reduce\_2k\_setup} \begin{alltt} int mp_reduce_2k_setup(mp_int *a, mp_digit *d); \end{alltt} -This will compute the required $d$ value for the given moduli $a$. +This will compute the required $d$ value for the given moduli $a$. \index{mp\_reduce\_2k} \begin{alltt} int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); \end{alltt} -This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is -slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction. +This will reduce $a$ in place modulo $n$ with the pre--computed value $d$. From my experience this routine is +slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery reduction. + +\section{Combined Modular Reduction} + +Some of the combinations of an arithmetic operations followed by a modular reduction can be done in a faster way. The ones implemented are: + +Addition $d = (a + b) \mod c$ +\index{mp\_addmod} +\begin{alltt} +int mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); +\end{alltt} + +Subtraction $d = (a - b) \mod c$ +\begin{alltt} +int mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); +\end{alltt} + +Multiplication $d = (ab) \mod c$ +\begin{alltt} +int mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); +\end{alltt} + +Squaring $d = (a^2) \mod c$ +\begin{alltt} +int mp_sqrmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d); +\end{alltt} + + \chapter{Exponentiation} \section{Single Digit Exponentiation} @@ -1550,8 +1772,7 @@ slower than mp\_dr\_reduce but faster for most moduli sizes than the Montgomery \begin{alltt} int mp_expt_d (mp_int * a, mp_digit b, mp_int * c) \end{alltt} -This computes $c = a^b$ using a simple binary left-to-right algorithm. It is faster than repeated multiplications by -$a$ for all values of $b$ greater than three. +This function computes $c = a^b$. \section{Modular Exponentiation} \index{mp\_exptmod} @@ -1559,31 +1780,105 @@ $a$ for all values of $b$ greater than three. int mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y) \end{alltt} This computes $Y \equiv G^X \mbox{ (mod }P\mbox{)}$ using a variable width sliding window algorithm. This function -will automatically detect the fastest modular reduction technique to use during the operation. For negative values of -$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that +will automatically detect the fastest modular reduction technique to use during the operation. For negative values of +$X$ the operation is performed as $Y \equiv (G^{-1} \mbox{ mod }P)^{\vert X \vert} \mbox{ (mod }P\mbox{)}$ provided that $gcd(G, P) = 1$. This function is actually a shell around the two internal exponentiation functions. This routine will automatically -detect when Barrett, Montgomery, Restricted and Unrestricted Dimminished Radix based exponentiation can be used. Generally -moduli of the a ``restricted dimminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery +detect when Barrett, Montgomery, Restricted and Unrestricted Diminished Radix based exponentiation can be used. Generally +moduli of the a ``restricted diminished radix'' form lead to the fastest modular exponentiations. Followed by Montgomery and the other two algorithms. +\section{Modulus a Power of Two} +\index{mp\_mod\_2d} +\begin{alltt} +int mp_mod_2d(const mp_int *a, int b, mp_int *c) +\end{alltt} +It calculates $c = a \mod 2^b$. + \section{Root Finding} \index{mp\_n\_root} \begin{alltt} int mp_n_root (mp_int * a, mp_digit b, mp_int * c) \end{alltt} -This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. The implementation of this function is not -ideal for values of $b$ greater than three. It will work but become very slow. So unless you are working with very small -numbers (less than 1000 bits) I'd avoid $b > 3$ situations. Will return a positive root only for even roots and return -a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ -will return $-2$. +This computes $c = a^{1/b}$ such that $c^b \le a$ and $(c+1)^b > a$. Will return a positive root only for even roots and return +a root with the sign of the input for odd roots. For example, performing $4^{1/2}$ will return $2$ whereas $(-8)^{1/3}$ +will return $-2$. + +This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. + +The square root $c = a^{1/2}$ (with the same conditions $c^2 \le a$ and $(c+1)^2 > a$) is implemented with a faster algorithm. + +\index{mp\_sqrt} +\begin{alltt} +int mp_sqrt (mp_int * a, mp_digit b, mp_int * c) +\end{alltt} + + +\chapter{Logarithm} +\section{Integer Logarithm} +A logarithm function for positive integer input \texttt{a, base} computing $\floor{\log_bx}$ such that $(\log_b x)^b \le x$. +\index{mp\_ilogb} +\begin{alltt} +int mp_ilogb(mp_int *a, mp_digit base, mp_int *c) +\end{alltt} +\subsection{Example} +\begin{alltt} +#include +#include +#include + +#include + +int main(int argc, char **argv) +{ + mp_int x, output; + mp_digit base; + int e; + + if (argc != 3) { + fprintf(stderr,"Usage %s base x\textbackslash{}n", argv[0]); + exit(EXIT_FAILURE); + } + if ((e = mp_init_multi(&x, &output, NULL)) != MP_OKAY) { + fprintf(stderr,"mp_init failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", + mp_error_to_string(e)); + exit(EXIT_FAILURE); + } + errno = 0; +#ifdef MP_64BIT + base = (mp_digit)strtoull(argv[1], NULL, 10); +#else + base = (mp_digit)strtoul(argv[1], NULL, 10); +#endif + if ((errno == ERANGE) || (base > (base & MP_MASK))) { + fprintf(stderr,"strtoul(l) failed: input out of range\textbackslash{}n"); + exit(EXIT_FAILURE); + } + if ((e = mp_read_radix(&x, argv[2], 10)) != MP_OKAY) { + fprintf(stderr,"mp_read_radix failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", + mp_error_to_string(e)); + exit(EXIT_FAILURE); + } + if ((e = mp_ilogb(&x, base, &output)) != MP_OKAY) { + fprintf(stderr,"mp_ilogb failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", + mp_error_to_string(e)); + exit(EXIT_FAILURE); + } + + if ((e = mp_fwrite(&output, 10, stdout)) != MP_OKAY) { + fprintf(stderr,"mp_fwrite failed: \textbackslash{}"%s\textbackslash{}"\textbackslash{}n", + mp_error_to_string(e)); + exit(EXIT_FAILURE); + } + putchar('\textbackslash{}n'); + + mp_clear_multi(&x, &output, NULL); + exit(EXIT_SUCCESS); +} +\end{alltt} + -This algorithm uses the ``Newton Approximation'' method and will converge on the correct root fairly quickly. Since -the algorithm requires raising $a$ to the power of $b$ it is not ideal to attempt to find roots for large -values of $b$. If particularly large roots are required then a factor method could be used instead. For example, -$a^{1/16}$ is equivalent to $\left (a^{1/4} \right)^{1/4}$ or simply -$\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$ \chapter{Prime Numbers} \section{Trial Division} @@ -1591,8 +1886,8 @@ $\left ( \left ( \left ( a^{1/2} \right )^{1/2} \right )^{1/2} \right )^{1/2}$ \begin{alltt} int mp_prime_is_divisible (mp_int * a, int *result) \end{alltt} -This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the -outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that +This will attempt to evenly divide $a$ by a list of primes\footnote{Default is the first 256 primes.} and store the +outcome in ``result''. That is if $result = 0$ then $a$ is not divisible by the primes, otherwise it is. Note that if the function does not return \textbf{MP\_OKAY} the value in ``result'' should be considered undefined\footnote{Currently the default is to set it to zero first.}. @@ -1611,10 +1906,10 @@ is set to zero. int mp_prime_miller_rabin (mp_int * a, mp_int * b, int *result) \end{alltt} Performs a Miller-Rabin test to the base $b$ of $a$. This test is much stronger than the Fermat test and is very hard to -fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one. -Otherwise $result$ is set to zero. +fool (besides with Carmichael numbers). If $a$ passes the test (therefore is probably prime) $result$ is set to one. +Otherwise $result$ is set to zero. -Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of +Note that is suggested that you use the Miller-Rabin test instead of the Fermat test since all of the failures of Miller-Rabin are a subset of the failures of the Fermat test. \subsection{Required Number of Tests} @@ -1626,63 +1921,181 @@ This is why a simple function has been provided to help out. \begin{alltt} int mp_prime_rabin_miller_trials(int size) \end{alltt} -This returns the number of trials required for a $2^{-96}$ (or lower) probability of failure for a given ``size'' expressed -in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would -require ten tests whereas a 1024-bit number would only require four tests. +This returns the number of trials required for a low probability of failure for a given ``size'' expressed in bits. This comes in handy specially since larger numbers are slower to test. For example, a 512-bit number would require 18 tests for a probability of $2^{-160}$ whereas a 1024-bit number would only require 12 tests for a probability of $2^{-192}$. The exact values as implemented are listed in table \ref{table:millerrabinrunsimpl}. + +\begin{table}[h] +\begin{center} +\begin{tabular}{c c c} +\textbf{bits} & \textbf{Rounds} & \textbf{Error}\\ + 80 & -1 & Use deterministic algorithm for size <= 80 bits \\ + 81 & 37 & $2^{-96}$ \\ + 96 & 32 & $2^{-96}$ \\ + 128 & 40 & $2^{-112}$ \\ + 160 & 35 & $2^{-112}$ \\ + 256 & 27 & $2^{-128}$ \\ + 384 & 16 & $2^{-128}$ \\ + 512 & 18 & $2^{-160}$ \\ + 768 & 11 & $2^{-160}$ \\ + 896 & 10 & $2^{-160}$ \\ + 1024 & 12 & $2^{-192}$ \\ + 1536 & 8 & $2^{-192}$ \\ + 2048 & 6 & $2^{-192}$ \\ + 3072 & 4 & $2^{-192}$ \\ + 4096 & 5 & $2^{-256}$ \\ + 5120 & 4 & $2^{-256}$ \\ + 6144 & 4 & $2^{-256}$ \\ + 8192 & 3 & $2^{-256}$ \\ + 9216 & 3 & $2^{-256}$ \\ + 10240 & 2 & $2^{-256}$ +\end{tabular} +\caption{ Number of Miller-Rabin rounds as implemented } \label{table:millerrabinrunsimpl} +\end{center} +\end{table} You should always still perform a trial division before a Miller-Rabin test though. +A small table, broke in two for typographical reasons, with the number of rounds of Miller-Rabin tests is shown below. The numbers have been compute with a PARI/GP script listed in appendix \ref{app:numberofmrcomp}. +The first column is the number of bits $b$ in the prime $p = 2^b$, the numbers in the first row represent the +probability that the number that all of the Miller-Rabin tests deemed a pseudoprime is actually a composite. There is a deterministic test for numbers smaller than $2^{80}$. + +\begin{table}[h] +\begin{center} +\begin{tabular}{c c c c c c c} +\textbf{bits} & $\mathbf{2^{-80}}$ & $\mathbf{2^{-96}}$ & $\mathbf{2^{-112}}$ & $\mathbf{2^{-128}}$ & $\mathbf{2^{-160}}$ & $\mathbf{2^{-192}}$ \\ +80 & 31 & 39 & 47 & 55 & 71 & 87 \\ +96 & 29 & 37 & 45 & 53 & 69 & 85 \\ +128 & 24 & 32 & 40 & 48 & 64 & 80 \\ +160 & 19 & 27 & 35 & 43 & 59 & 75 \\ +192 & 15 & 21 & 29 & 37 & 53 & 69 \\ +256 & 10 & 15 & 20 & 27 & 43 & 59 \\ +384 & 7 & 9 & 12 & 16 & 25 & 38 \\ +512 & 5 & 7 & 9 & 12 & 18 & 26 \\ +768 & 4 & 5 & 6 & 8 & 11 & 16 \\ +1024 & 3 & 4 & 5 & 6 & 9 & 12 \\ +1536 & 2 & 3 & 3 & 4 & 6 & 8 \\ +2048 & 2 & 2 & 3 & 3 & 4 & 6 \\ +3072 & 1 & 2 & 2 & 2 & 3 & 4 \\ +4096 & 1 & 1 & 2 & 2 & 2 & 3 \\ +6144 & 1 & 1 & 1 & 1 & 2 & 2 \\ +8192 & 1 & 1 & 1 & 1 & 2 & 2 \\ +12288 & 1 & 1 & 1 & 1 & 1 & 1 \\ +16384 & 1 & 1 & 1 & 1 & 1 & 1 \\ +24576 & 1 & 1 & 1 & 1 & 1 & 1 \\ +32768 & 1 & 1 & 1 & 1 & 1 & 1 +\end{tabular} +\caption{ Number of Miller-Rabin rounds. Part I } \label{table:millerrabinrunsp1} +\end{center} +\end{table} +\newpage +\begin{table}[h] +\begin{center} +\begin{tabular}{c c c c c c c c} +\textbf{bits} &$\mathbf{2^{-224}}$ & $\mathbf{2^{-256}}$ & $\mathbf{2^{-288}}$ & $\mathbf{2^{-320}}$ & $\mathbf{2^{-352}}$ & $\mathbf{2^{-384}}$ & $\mathbf{2^{-416}}$\\ +80 & 103 & 119 & 135 & 151 & 167 & 183 & 199 \\ +96 & 101 & 117 & 133 & 149 & 165 & 181 & 197 \\ +128 & 96 & 112 & 128 & 144 & 160 & 176 & 192 \\ +160 & 91 & 107 & 123 & 139 & 155 & 171 & 187 \\ +192 & 85 & 101 & 117 & 133 & 149 & 165 & 181 \\ +256 & 75 & 91 & 107 & 123 & 139 & 155 & 171 \\ +384 & 54 & 70 & 86 & 102 & 118 & 134 & 150 \\ +512 & 36 & 49 & 65 & 81 & 97 & 113 & 129 \\ +768 & 22 & 29 & 37 & 47 & 58 & 70 & 86 \\ +1024 & 16 & 21 & 26 & 33 & 40 & 48 & 58 \\ +1536 & 10 & 13 & 17 & 21 & 25 & 30 & 35 \\ +2048 & 8 & 10 & 13 & 15 & 18 & 22 & 26 \\ +3072 & 5 & 7 & 8 & 10 & 12 & 14 & 17 \\ +4096 & 4 & 5 & 6 & 8 & 9 & 11 & 12 \\ +6144 & 3 & 4 & 4 & 5 & 6 & 7 & 8 \\ +8192 & 2 & 3 & 3 & 4 & 5 & 6 & 6 \\ +12288 & 2 & 2 & 2 & 3 & 3 & 4 & 4 \\ +16384 & 1 & 2 & 2 & 2 & 3 & 3 & 3 \\ +24576 & 1 & 1 & 2 & 2 & 2 & 2 & 2 \\ +32768 & 1 & 1 & 1 & 1 & 2 & 2 & 2 +\end{tabular} +\caption{ Number of Miller-Rabin rounds. Part II } \label{table:millerrabinrunsp2} +\end{center} +\end{table} + +Determining the probability needed to pick the right column is a bit harder. Fips 186.4, for example has $2^{-80}$ for $512$ bit large numbers, $2^{-112}$ for $1024$ bits, and $2^{128}$ for $1536$ bits. It can be seen in table \ref{table:millerrabinrunsp1} that those combinations follow the diagonal from $(512,2^{-80})$ downwards and to the right to gain a lower probabilty of getting a composite declared a pseudoprime for the same amount of work or less. + +If this version of the library has the strong Lucas-Selfridge and/or the Frobenius-Underwood test implemented only one or two rounds of the Miller-Rabin test with a random base is necesssary for numbers larger than or equal to $1024$ bits. + +This function is meant for RSA. The number of rounds for DSA is $\lceil -log_2(p)/2\rceil$ with $p$ the probability which is just the half of the absolute value of $p$ if given as a power of two. E.g.: with $p = 2^{-128}$, $\lceil -log_2(p)/2\rceil = 64$. + +This function can be used to test a DSA prime directly if these rounds are followed by a Lucas test. + +See also table C.1 in FIPS 186-4. + +\section{Strong Lucas-Selfridge Test} +\index{mp\_prime\_strong\_lucas\_selfridge} +\begin{alltt} +int mp_prime_strong_lucas_selfridge(const mp_int *a, int *result) +\end{alltt} +Performs a strong Lucas-Selfridge test. The strong Lucas-Selfridge test together with the Rabin-Miler test with bases $2$ and $3$ resemble the BPSW test. The single internal use is a compile-time option in \texttt{mp\_prime\_is\_prime} and can be excluded +from the Libtommath build if not needed. + +\section{Frobenius (Underwood) Test} +\index{mp\_prime\_frobenius\_underwood} +\begin{alltt} +int mp_prime_frobenius_underwood(const mp_int *N, int *result) +\end{alltt} +Performs the variant of the Frobenius test as described by Paul Underwood. The single internal use is in +\texttt{mp\_prime\_is\_prime} for \texttt{MP\_8BIT} only but can be included at build-time for all other sizes +if the preprocessor macro \texttt{LTM\_USE\_FROBENIUS\_TEST} is defined. + +It returns \texttt{MP\_ITER} if the number of iterations is exhausted, assumes a composite as the input and sets \texttt{result} accordingly. This will reduce the set of available pseudoprimes by a very small amount: test with large datasets (more than $10^{10}$ numbers, both randomly chosen and sequences of odd numbers with a random start point) found only 31 (thirty-one) numbers with $a > 120$ and none at all with just an additional simple check for divisors $d < 2^8$. + \section{Primality Testing} +Testing if a number is a square can be done a bit faster than just by calculating the square root. It is used by the primality testing function described below. +\index{mp\_is\_square} +\begin{alltt} +int mp_is_square(const mp_int *arg, int *ret); +\end{alltt} + + \index{mp\_prime\_is\_prime} \begin{alltt} int mp_prime_is_prime (mp_int * a, int t, int *result) \end{alltt} -This will perform a trial division followed by $t$ rounds of Miller-Rabin tests on $a$ and store the result in $result$. -If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. Note that $t$ is bounded by -$1 \le t < PRIME\_SIZE$ where $PRIME\_SIZE$ is the number of primes in the prime number table (by default this is $256$). +This will perform a trial division followed by two rounds of Miller-Rabin with bases 2 and 3 and a Lucas-Selfridge test. The Lucas-Selfridge test is replaced with a Frobenius-Underwood for \texttt{MP\_8BIT}. The Frobenius-Underwood test for all other sizes is available as a compile-time option with the preprocessor macro \texttt{LTM\_USE\_FROBENIUS\_TEST}. See file +\texttt{bn\_mp\_prime\_is\_prime.c} for the necessary details. It shall be noted that both functions are much slower than +the Miller-Rabin test and if speed is an essential issue, the macro \texttt{LTM\_USE\_ONLY\_MR} switches both functions, the Frobenius-Underwood test and the Lucas-Selfridge test off and their code will not even be compiled into the library. + +If $t$ is set to a positive value $t$ additional rounds of the Miller-Rabin test with random bases will be performed to allow for Fips 186.4 (vid.~p.~126ff) compliance. The function \texttt{mp\_prime\_rabin\_miller\_trials} can be used to determine the number of rounds. It is vital that the function \texttt{mp\_rand()} has a cryptographically strong random number generator available. + +One Miller-Rabin tests with a random base will be run automatically, so by setting $t$ to a positive value this function will run $t + 1$ Miller-Rabin tests with random bases. + +If $t$ is set to a negative value the test will run the deterministic Miller-Rabin test for the primes up to $3317044064679887385961981$. That limit has to be checked by the caller. + +If $a$ passes all of the tests $result$ is set to one, otherwise it is set to zero. \section{Next Prime} \index{mp\_prime\_next\_prime} \begin{alltt} int mp_prime_next_prime(mp_int *a, int t, int bbs_style) \end{alltt} -This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests. Set $bbs\_style$ to one if you -want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime. +This finds the next prime after $a$ that passes mp\_prime\_is\_prime() with $t$ tests but see the documentation for +mp\_prime\_is\_prime for details regarding the use of the argument $t$. Set $bbs\_style$ to one if you +want only the next prime congruent to $3 \mbox{ mod } 4$, otherwise set it to zero to find any next prime. \section{Random Primes} -\index{mp\_prime\_random} +\index{mp\_prime\_rand} \begin{alltt} -int mp_prime_random(mp_int *a, int t, int size, int bbs, - ltm_prime_callback cb, void *dat) +int mp_prime_rand(mp_int *a, int t, + int size, int flags); \end{alltt} -This will find a prime greater than $256^{size}$ which can be ``bbs\_style'' or not depending on $bbs$ and must pass -$t$ rounds of tests. The ``ltm\_prime\_callback'' is a typedef for +This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. +See the documentation for mp\_prime\_is\_prime for details regarding the use of the argument $t$. +The variable $size$ specifies the bit length of the prime desired. +The variable $flags$ specifies one of several options available +(see fig. \ref{fig:primeopts}) which can be OR'ed together. -\begin{alltt} -typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); -\end{alltt} - -Which is a function that must read $len$ bytes (and return the amount stored) into $dst$. The $dat$ variable is simply -copied from the original input. It can be used to pass RNG context data to the callback. The function -mp\_prime\_random() is more suitable for generating primes which must be secret (as in the case of RSA) since there +The function mp\_prime\_rand() is suitable for generating primes which must be secret (as in the case of RSA) since there is no skew on the least significant bits. -\textit{Note:} As of v0.30 of the LibTomMath library this function has been deprecated. It is still available -but users are encouraged to use the new mp\_prime\_random\_ex() function instead. +\textit{Note:} This function replaces the deprecated mp\_prime\_random and mp\_prime\_random\_ex functions. -\subsection{Extended Generation} -\index{mp\_prime\_random\_ex} -\begin{alltt} -int mp_prime_random_ex(mp_int *a, int t, - int size, int flags, - ltm_prime_callback cb, void *dat); -\end{alltt} -This will generate a prime in $a$ using $t$ tests of the primality testing algorithms. The variable $size$ -specifies the bit length of the prime desired. The variable $flags$ specifies one of several options available -(see fig. \ref{fig:primeopts}) which can be OR'ed together. The callback parameters are used as in -mp\_prime\_random(). - -\begin{figure}[here] +\begin{figure}[h] \begin{center} \begin{small} \begin{tabular}{|r|l|} @@ -1702,23 +2115,55 @@ mp\_prime\_random(). \label{fig:primeopts} \end{figure} +\chapter{Random Number Generation} +\section{PRNG} +\index{mp\_rand\_digit} +\begin{alltt} +int mp_rand_digit(mp_digit *r) +\end{alltt} +This function generates a random number in \texttt{r} of the size given in \texttt{r} (that is, the variable is used for in- and output) but not more than \texttt{MP\_MASK} bits. + +\index{mp\_rand} +\begin{alltt} +int mp_rand(mp_int *a, int digits) +\end{alltt} +This function generates a random number of \texttt{digits} bits. + +The random number generated with these two functions is cryptographically secure if the source of random numbers the operating systems offers is cryptographically secure. It will use \texttt{arc4random()} if the OS is a BSD flavor, Wincrypt on Windows, or \texttt{/dev/urandom} on all operating systems that have it. + + \chapter{Input and Output} \section{ASCII Conversions} \subsection{To ASCII} -\index{mp\_toradix} +\index{mp\_to\_radix} \begin{alltt} -int mp_toradix (mp_int * a, char *str, int radix); +int mp_to_radix (mp_int *a, char *str, size_t maxlen, size_t *written, int radix); \end{alltt} -This still store $a$ in ``str'' as a base-``radix'' string of ASCII chars. This function appends a NUL character -to terminate the string. Valid values of ``radix'' line in the range $[2, 64]$. To determine the size (exact) required -by the conversion before storing any data use the following function. +This stores $a$ in \texttt{str} of maximum length \texttt{maxlen} as a base-\texttt{radix} string of ASCII chars and appends a \texttt{NUL} character to terminate the string. + +Valid values of \texttt{radix} line in the range $[2, 64]$. + +The exact number of characters in \texttt{str} plus the \texttt{NUL} will be put in \texttt{written} if that variable is not set to \texttt{NULL}. + +If \texttt{str} is not big enough to hold $a$, \texttt{str} will be filled with the least-significant digits +of length \texttt{maxlen-1}, then \texttt{str} will be \texttt{NUL} terminated and the error \texttt{MP\_VAL} is returned. + +Please be aware that this function cannot evaluate the actual size of the buffer, it relies on the correctness of \texttt{maxlen}! + \index{mp\_radix\_size} \begin{alltt} int mp_radix_size (mp_int * a, int radix, int *size) \end{alltt} -This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this -function returns an error code and ``size'' will be zero. +This stores in ``size'' the number of characters (including space for the NUL terminator) required. Upon error this +function returns an error code and ``size'' will be zero. + +If \texttt{LTM\_NO\_FILE} is not defined a function to write to a file is also available. +\index{mp\_fwrite} +\begin{alltt} +int mp_fwrite(const mp_int *a, int radix, FILE *stream); +\end{alltt} + \subsection{From ASCII} \index{mp\_read\_radix} @@ -1729,48 +2174,81 @@ This will read the base-``radix'' NUL terminated string from ``str'' into $a$. character it does not recognize (which happens to include th NUL char... imagine that...). A single leading $-$ sign can be used to denote a negative number. +If \texttt{LTM\_NO\_FILE} is not defined a function to read from a file is also available. +\index{mp\_fread} +\begin{alltt} +int mp_fread(mp_int *a, int radix, FILE *stream); +\end{alltt} + + \section{Binary Conversions} Converting an mp\_int to and from binary is another keen idea. -\index{mp\_unsigned\_bin\_size} +\index{mp\_ubin\_size} \begin{alltt} -int mp_unsigned_bin_size(mp_int *a); +size_t mp_ubin_size(mp_int *a); \end{alltt} This will return the number of bytes (octets) required to store the unsigned copy of the integer $a$. -\index{mp\_to\_unsigned\_bin} +\index{mp\_to\_ubin} \begin{alltt} -int mp_to_unsigned_bin(mp_int *a, unsigned char *b); +int mp_to_unsigned_bin(mp_int *a, unsigned char *b, size_t maxlen, size_t *len); \end{alltt} -This will store $a$ into the buffer $b$ in big--endian format. Fortunately this is exactly what DER (or is it ASN?) -requires. It does not store the sign of the integer. +This will store $a$ into the buffer $b$ of size \texttt{maxlen} in big--endian format storing the number of bytes written in \texttt{len}. Fortunately this is exactly what DER (or is it ASN?) requires. It does not store the sign of the integer. -\index{mp\_read\_unsigned\_bin} +\index{mp\_from\_ubin} \begin{alltt} -int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c); +int mp_from_ubin(mp_int *a, unsigned char *b, size_t size); \end{alltt} -This will read in an unsigned big--endian array of bytes (octets) from $b$ of length $c$ into $a$. The resulting -integer $a$ will always be positive. +This will read in an unsigned big--endian array of bytes (octets) from $b$ of length \texttt{size} into $a$. The resulting big-integer $a$ will always be positive. For those who acknowledge the existence of negative numbers (heretic!) there are ``signed'' versions of the previous functions. - +\index{mp\_signed\_bin\_size} \index{mp\_to\_signed\_bin} \index{mp\_read\_signed\_bin} \begin{alltt} -int mp_signed_bin_size(mp_int *a); -int mp_read_signed_bin(mp_int *a, unsigned char *b, int c); -int mp_to_signed_bin(mp_int *a, unsigned char *b); +int mp_sbin_size(mp_int *a); +int mp_from_sbin(mp_int *a, unsigned char *b, size_t size); +int mp_to_sbin(mp_int *a, unsigned char *b, size_t maxsize, size_t *len); \end{alltt} They operate essentially the same as the unsigned copies except they prefix the data with zero or non--zero byte depending on the sign. If the sign is zpos (e.g. not negative) the prefix is zero, otherwise the prefix -is non--zero. +is non--zero. + +The two functions \texttt{mp\_unpack} (get your gifts out of the box, import binary data) and \texttt{mp\_pack} (put your gifts into the box, export binary data) implement the similarly working GMP functions as described at \url{http://gmplib.org/manual/Integer-Import-and-Export.html} with the exception that \texttt{mp\_pack} will not allocate memory if \texttt{rop} is \texttt{NULL}. +\index{mp\_unpack} \index{mp\_pack} +\begin{alltt} +int mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, + mp_endian endian, size_t nails, const void *op, size_t maxsize); +int mp_pack(void *rop, size_t *countp, mp_order order, size_t size, + mp_endian endian, size_t nails, const mp_int *op); +\end{alltt} +The function \texttt{mp\_pack} has the additional variable \texttt{maxsize} which must hold the size of the buffer \texttt{rop} in bytes. Use +\begin{alltt} +/* Parameters "nails" and "size" are the same as in mp_pack */ +size_t mp_pack_size(mp_int *a, size_t nails, size_t size); +\end{alltt} +To get the size in bytes necessary to be put in \texttt{maxsize}). + +To enhance the readability of your code, the following enums have been wrought for your convenience. +\begin{alltt} +typedef enum { + MP_LSB_FIRST = -1, + MP_MSB_FIRST = 1 +} mp_order; +typedef enum { + MP_LITTLE_ENDIAN = -1, + MP_NATIVE_ENDIAN = 0, + MP_BIG_ENDIAN = 1 +} mp_endian; +\end{alltt} \chapter{Algebraic Functions} \section{Extended Euclidean Algorithm} \index{mp\_exteuclid} \begin{alltt} -int mp_exteuclid(mp_int *a, mp_int *b, +int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); \end{alltt} @@ -1780,7 +2258,7 @@ This finds the triple U1/U2/U3 using the Extended Euclidean algorithm such that a \cdot U1 + b \cdot U2 = U3 \end{equation} -Any of the U1/U2/U3 paramters can be set to \textbf{NULL} if they are not desired. +Any of the U1/U2/U3 parameters can be set to \textbf{NULL} if they are not desired. \section{Greatest Common Divisor} \index{mp\_gcd} @@ -1804,7 +2282,36 @@ int mp_jacobi (mp_int * a, mp_int * p, int *c) This will compute the Jacobi symbol for $a$ with respect to $p$. If $p$ is prime this essentially computes the Legendre symbol. The result is stored in $c$ and can take on one of three values $\lbrace -1, 0, 1 \rbrace$. If $p$ is prime then the result will be $-1$ when $a$ is not a quadratic residue modulo $p$. The result will be $0$ if $a$ divides $p$ -and the result will be $1$ if $a$ is a quadratic residue modulo $p$. +and the result will be $1$ if $a$ is a quadratic residue modulo $p$. + +\section{Kronecker Symbol} +\index{mp\_kronecker} +\begin{alltt} +int mp_kronecker (mp_int * a, mp_int * p, int *c) +\end{alltt} +Extension of the Jacoby symbol to all $\lbrace a, p \rbrace \in \mathbb{Z}$ . + + +\section{Modular square root} +\index{mp\_sqrtmod\_prime} +\begin{alltt} +int mp_sqrtmod_prime(mp_int *n, mp_int *p, mp_int *r) +\end{alltt} + +This will solve the modular equatioon $r^2 = n \mod p$ where $p$ is a prime number greater than 2 (odd prime). +The result is returned in the third argument $r$, the function returns \textbf{MP\_OKAY} on success, +other return values indicate failure. + +The implementation is split for two different cases: + +1. if $p \mod 4 == 3$ we apply \href{http://cacr.uwaterloo.ca/hac/}{Handbook of Applied Cryptography algorithm 3.36} and compute $r$ directly as +$r = n^{(p+1)/4} \mod p$ + +2. otherwise we use \href{https://en.wikipedia.org/wiki/Tonelli-Shanks_algorithm}{Tonelli-Shanks algorithm} + +The function does not check the primality of parameter $p$ thus it is up to the caller to assure that this parameter +is a prime number. When $p$ is a composite the function behaviour is undefined, it may even return a false-positive +\textbf{MP\_OKAY}. \section{Modular Inverse} \index{mp\_invmod} @@ -1830,6 +2337,171 @@ These work like the full mp\_int capable variants except the second parameter $b functions fairly handy if you have to work with relatively small numbers since you will not have to allocate an entire mp\_int to store a number like $1$ or $2$. +The functions \texttt{mp\_incr} and \texttt{mp\_decr} mimic the postfix operators \texttt{++} and \texttt{--} respectively, to increment the input by one. They call the full single-digit functions if the addition would carry. Both functions need to be included in a minimized library because they call each other in case of a negative input, These functions change the inputs! +\begin{alltt} +int mp_incr(mp_int *a); +int mp_decr(mp_int *a); +\end{alltt} + + +The division by three can be made faster by replacing the division with a multiplication by the multiplicative inverse of three. + +\index{mp\_div\_3} +\begin{alltt} +int mp_div_3(const mp_int *a, mp_int *c, mp_digit *d); +\end{alltt} + +\chapter{Little Helpers} +It is never wrong to have some useful little shortcuts at hand. +\section{Function Macros} +To make this overview simpler the macros are given as function prototypes. The return of logic macros is \texttt{MP\_NO} or \texttt{MP\_YES} respectively. + +\index{mp\_iseven} +\begin{alltt} +int mp_iseven(mp_int *a) +\end{alltt} +Checks if $a = 0 mod 2$ + +\index{mp\_isodd} +\begin{alltt} +int mp_isodd(mp_int *a) +\end{alltt} +Checks if $a = 1 mod 2$ + +\index{mp\_isneg} +\begin{alltt} +int mp_isneg(mp_int *a) +\end{alltt} +Checks if $a < 0$ + + +\index{mp\_iszero} +\begin{alltt} +int mp_iszero(mp_int *a) +\end{alltt} +Checks if $a = 0$. It does not check if the amount of memory allocated for $a$ is also minimal. + + +Other macros which are either shortcuts to normal functions or just other names for them do have their place in a programmer's life, too! + +\subsection{Renamings} +\index{mp\_mag\_size} +\begin{alltt} +#define mp_mag_size(mp) mp_unsigned_bin_size(mp) +\end{alltt} + + +\index{mp\_raw\_size} +\begin{alltt} +#define mp_raw_size(mp) mp_signed_bin_size(mp) +\end{alltt} + + +\index{mp\_read\_mag} +\begin{alltt} +#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) +\end{alltt} + + +\index{mp\_read\_raw} +\begin{alltt} + #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) +\end{alltt} + + +\index{mp\_tomag} +\begin{alltt} +#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) +\end{alltt} + + +\index{mp\_toraw} +\begin{alltt} +#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) +\end{alltt} + + + +\subsection{Shortcuts} + +\index{mp\_to\_binary} +\begin{alltt} +#define mp_to_binary(M, S, N) mp_to_radix((M), (S), (N), 2) +\end{alltt} + + +\index{mp\_to\_octal} +\begin{alltt} +#define mp_to_octal(M, S, N) mp_to_radix((M), (S), (N), 8) +\end{alltt} + + +\index{mp\_to\_decimal} +\begin{alltt} +#define mp_to_decimal(M, S, N) mp_to_radix((M), (S), (N), 10) +\end{alltt} + + +\index{mp\_to\_hex} +\begin{alltt} +#define mp_to_hex(M, S, N) mp_to_radix((M), (S), (N), 16) +\end{alltt} + +\begin{appendices} +\appendixpage +%\noappendicestocpagenum +\addappheadtotoc +\chapter{Computing Number of Miller-Rabin Trials}\label{app:numberofmrcomp} +The number of Miller-Rabin rounds in the tables \ref{millerrabinrunsimpl}, \ref{millerrabinrunsp1}, and \ref{millerrabinrunsp2} have been calculated with the formula in FIPS 186-4 appendix F.1 (page 117) implemented as a PARI/GP script. +\begin{alltt} +log2(x) = log(x)/log(2) + +fips_f1_sums(k, M, t) = { + local(s = 0); + s = sum(m=3,M, + 2^(m-t*(m-1)) * + sum(j=2,m, + 1/ ( 2^( j + (k-1)/j ) ) + ) + ); + return(s); +} + +fips_f1_2(k, t, M) = { + local(common_factor, t1, t2, f1, f2, ds, res); + + common_factor = 2.00743 * log(2) * k * 2^(-k); + t1 = 2^(k - 2 - M*t); + f1 = (8 * ((Pi^2) - 6))/3; + f2 = 2^(k - 2); + ds = t1 + f1 * f2 * fips_f1_sums(k, M, t); + res = common_factor * ds; + return(res); +} + +fips_f1_1(prime_length, ptarget)={ + local(t, t_end, M, M_end, pkt); + + t_end = ceil(-log2(ptarget)/2); + M_end = floor(2 * sqrt(prime_length-1) - 1); + + for(t = 1, t_end, + for(M = 3, M_end, + pkt = fips_f1_2(prime_length, t, M); + if(pkt <= ptarget, + return(t); + ); + ); + ); +} +\end{alltt} + +To get the number of rounds for a $1024$ bit large prime with a probability of $2^{-160}$: +\begin{alltt} +? fips_f1_1(1024,2^(-160)) +%1 = 9 +\end{alltt} +\end{appendices} \input{bn.ind} \end{document} diff --git a/lib/hcrypto/libtommath/etc/2kprime.c b/lib/hcrypto/libtommath/etc/2kprime.c index 0d2bf21e7..95ed2de42 100644 --- a/lib/hcrypto/libtommath/etc/2kprime.c +++ b/lib/hcrypto/libtommath/etc/2kprime.c @@ -2,12 +2,13 @@ #include #include -int sizes[] = {256, 512, 768, 1024, 1536, 2048, 3072, 4096}; +static int sizes[] = {256, 512, 768, 1024, 1536, 2048, 3072, 4096}; int main(void) { char buf[2000]; - int x, y; + size_t x; + mp_bool y; mp_int q, p; FILE *out; clock_t t1; @@ -16,69 +17,65 @@ int main(void) mp_init_multi(&q, &p, NULL); out = fopen("2kprime.1", "w"); - for (x = 0; x < (int)(sizeof(sizes) / sizeof(sizes[0])); x++) { - top: - mp_2expt(&q, sizes[x]); - mp_add_d(&q, 3, &q); - z = -3; + if (out != NULL) { + for (x = 0; x < (sizeof(sizes) / sizeof(sizes[0])); x++) { +top: + mp_2expt(&q, sizes[x]); + mp_add_d(&q, 3uL, &q); + z = -3; - t1 = clock(); - for(;;) { - mp_sub_d(&q, 4, &q); - z += 4; + t1 = clock(); + for (;;) { + mp_sub_d(&q, 4uL, &q); + z += 4uL; + + if (z > MP_MASK) { + printf("No primes of size %d found\n", sizes[x]); + break; + } + + if ((clock() - t1) > CLOCKS_PER_SEC) { + printf("."); + fflush(stdout); + /* sleep((clock() - t1 + CLOCKS_PER_SEC/2)/CLOCKS_PER_SEC); */ + t1 = clock(); + } + + /* quick test on q */ + mp_prime_is_prime(&q, 1, &y); + if (y == MP_NO) { + continue; + } + + /* find (q-1)/2 */ + mp_sub_d(&q, 1uL, &p); + mp_div_2(&p, &p); + mp_prime_is_prime(&p, 3, &y); + if (y == MP_NO) { + continue; + } + + /* test on q */ + mp_prime_is_prime(&q, 3, &y); + if (y == MP_NO) { + continue; + } - if (z > MP_MASK) { - printf("No primes of size %d found\n", sizes[x]); break; } - if (clock() - t1 > CLOCKS_PER_SEC) { - printf("."); fflush(stdout); -// sleep((clock() - t1 + CLOCKS_PER_SEC/2)/CLOCKS_PER_SEC); - t1 = clock(); + if (y == MP_NO) { + ++sizes[x]; + goto top; } - /* quick test on q */ - mp_prime_is_prime(&q, 1, &y); - if (y == 0) { - continue; - } - - /* find (q-1)/2 */ - mp_sub_d(&q, 1, &p); - mp_div_2(&p, &p); - mp_prime_is_prime(&p, 3, &y); - if (y == 0) { - continue; - } - - /* test on q */ - mp_prime_is_prime(&q, 3, &y); - if (y == 0) { - continue; - } - - break; - } - - if (y == 0) { - ++sizes[x]; - goto top; - } - - mp_toradix(&q, buf, 10); - printf("\n\n%d-bits (k = %lu) = %s\n", sizes[x], z, buf); - fprintf(out, "%d-bits (k = %lu) = %s\n", sizes[x], z, buf); fflush(out); + mp_to_decimal(&q, buf, sizeof(buf)); + printf("\n\n%d-bits (k = %lu) = %s\n", sizes[x], z, buf); + fprintf(out, "%d-bits (k = %lu) = %s\n", sizes[x], z, buf); + fflush(out); + } + fclose(out); } return 0; } - - - - - - -/* $Source: /cvs/libtom/libtommath/etc/2kprime.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ diff --git a/lib/hcrypto/libtommath/etc/drprime.c b/lib/hcrypto/libtommath/etc/drprime.c index d92059327..64e31ef10 100644 --- a/lib/hcrypto/libtommath/etc/drprime.c +++ b/lib/hcrypto/libtommath/etc/drprime.c @@ -1,10 +1,12 @@ /* Makes safe primes of a DR nature */ #include -int sizes[] = { 1+256/DIGIT_BIT, 1+512/DIGIT_BIT, 1+768/DIGIT_BIT, 1+1024/DIGIT_BIT, 1+2048/DIGIT_BIT, 1+4096/DIGIT_BIT }; +static int sizes[] = { 1+256/MP_DIGIT_BIT, 1+512/MP_DIGIT_BIT, 1+768/MP_DIGIT_BIT, 1+1024/MP_DIGIT_BIT, 1+2048/MP_DIGIT_BIT, 1+4096/MP_DIGIT_BIT }; + int main(void) { - int res, x, y; + mp_bool res; + int x, y; char buf[4096]; FILE *out; mp_int a, b; @@ -13,52 +15,53 @@ int main(void) mp_init(&b); out = fopen("drprimes.txt", "w"); - for (x = 0; x < (int)(sizeof(sizes)/sizeof(sizes[0])); x++) { - top: - printf("Seeking a %d-bit safe prime\n", sizes[x] * DIGIT_BIT); - mp_grow(&a, sizes[x]); - mp_zero(&a); - for (y = 1; y < sizes[x]; y++) { - a.dp[y] = MP_MASK; - } + if (out != NULL) { + for (x = 0; x < (int)(sizeof(sizes)/sizeof(sizes[0])); x++) { +top: + printf("Seeking a %d-bit safe prime\n", sizes[x] * MP_DIGIT_BIT); + mp_grow(&a, sizes[x]); + mp_zero(&a); + for (y = 1; y < sizes[x]; y++) { + a.dp[y] = MP_MASK; + } - /* make a DR modulus */ - a.dp[0] = -1; - a.used = sizes[x]; + /* make a DR modulus */ + a.dp[0] = -1; + a.used = sizes[x]; - /* now loop */ - res = 0; - for (;;) { - a.dp[0] += 4; - if (a.dp[0] >= MP_MASK) break; - mp_prime_is_prime(&a, 1, &res); - if (res == 0) continue; - printf("."); fflush(stdout); - mp_sub_d(&a, 1, &b); - mp_div_2(&b, &b); - mp_prime_is_prime(&b, 3, &res); - if (res == 0) continue; - mp_prime_is_prime(&a, 3, &res); - if (res == 1) break; - } + /* now loop */ + res = MP_NO; + for (;;) { + a.dp[0] += 4uL; + if (a.dp[0] >= MP_MASK) break; + mp_prime_is_prime(&a, 1, &res); + if (res == MP_NO) continue; + printf("."); + fflush(stdout); + mp_sub_d(&a, 1uL, &b); + mp_div_2(&b, &b); + mp_prime_is_prime(&b, 3, &res); + if (res == MP_NO) continue; + mp_prime_is_prime(&a, 3, &res); + if (res == MP_YES) break; + } - if (res != 1) { - printf("Error not DR modulus\n"); sizes[x] += 1; goto top; - } else { - mp_toradix(&a, buf, 10); - printf("\n\np == %s\n\n", buf); - fprintf(out, "%d-bit prime:\np == %s\n\n", mp_count_bits(&a), buf); fflush(out); - } + if (res != MP_YES) { + printf("Error not DR modulus\n"); + sizes[x] += 1; + goto top; + } else { + mp_to_decimal(&a, buf, sizeof(buf)); + printf("\n\np == %s\n\n", buf); + fprintf(out, "%d-bit prime:\np == %s\n\n", mp_count_bits(&a), buf); + fflush(out); + } + } + fclose(out); } - fclose(out); mp_clear(&a); mp_clear(&b); return 0; } - - -/* $Source: /cvs/libtom/libtommath/etc/drprime.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ diff --git a/lib/hcrypto/libtommath/etc/makefile b/lib/hcrypto/libtommath/etc/makefile index 99154d864..85bb09efc 100644 --- a/lib/hcrypto/libtommath/etc/makefile +++ b/lib/hcrypto/libtommath/etc/makefile @@ -1,4 +1,5 @@ -CFLAGS += -Wall -W -Wshadow -O3 -fomit-frame-pointer -funroll-loops -I../ +LTM_CFLAGS += -Wall -W -Wextra -Wshadow -O3 -I../ +LTM_CFLAGS += $(CFLAGS) # default lib name (requires install with root) # LIBNAME=-ltommath @@ -8,43 +9,36 @@ LIBNAME=../libtommath.a #provable primes pprime: pprime.o - $(CC) pprime.o $(LIBNAME) -o pprime + $(CC) $(LTM_CFLAGS) pprime.o $(LIBNAME) -o pprime # portable [well requires clock()] tuning app tune: tune.o - $(CC) tune.o $(LIBNAME) -o tune - -# same app but using RDTSC for higher precision [requires 80586+], coff based gcc installs [e.g. ming, cygwin, djgpp] -tune86: tune.c - nasm -f coff timer.asm - $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 - -# for cygwin -tune86c: tune.c - nasm -f gnuwin32 timer.asm - $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 + $(CC) $(LTM_CFLAGS) tune.o $(LIBNAME) -o tune + ./tune_it.sh + +test_standalone: tune.o + # The benchmark program works as a testtool, too + $(CC) $(LTM_CFLAGS) tune.o $(LIBNAME) -o test -#make tune86 for linux or any ELF format -tune86l: tune.c - nasm -f elf -DUSE_ELF timer.asm - $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86l - # spits out mersenne primes mersenne: mersenne.o - $(CC) mersenne.o $(LIBNAME) -o mersenne + $(CC) $(LTM_CFLAGS) mersenne.o $(LIBNAME) -o mersenne -# fines DR safe primes for the given config +# finds DR safe primes for the given config drprime: drprime.o - $(CC) drprime.o $(LIBNAME) -o drprime - -# fines 2k safe primes for the given config + $(CC) $(LTM_CFLAGS) drprime.o $(LIBNAME) -o drprime + +# finds 2k safe primes for the given config 2kprime: 2kprime.o - $(CC) 2kprime.o $(LIBNAME) -o 2kprime + $(CC) $(LTM_CFLAGS) 2kprime.o $(LIBNAME) -o 2kprime mont: mont.o - $(CC) mont.o $(LIBNAME) -o mont + $(CC) $(LTM_CFLAGS) mont.o $(LIBNAME) -o mont + - clean: - rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat \ - *.da *.dyn *.dpi *~ + rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime mont 2kprime pprime.dat \ + tuning_list multiplying squaring test *.da *.dyn *.dpi *~ + rm -rf .libs + +.PHONY: tune diff --git a/lib/hcrypto/libtommath/etc/makefile.icc b/lib/hcrypto/libtommath/etc/makefile.icc index 8a1ffffd2..9217f7b1d 100644 --- a/lib/hcrypto/libtommath/etc/makefile.icc +++ b/lib/hcrypto/libtommath/etc/makefile.icc @@ -8,7 +8,7 @@ CFLAGS += -I../ # -ax? specifies make code specifically for ? but compatible with IA-32 # -x? specifies compile solely for ? [not specifically IA-32 compatible] # -# where ? is +# where ? is # K - PIII # W - first P4 [Williamette] # N - P4 Northwood @@ -28,15 +28,15 @@ LIBNAME=../libtommath.a pprime: pprime.o $(CC) pprime.o $(LIBNAME) -o pprime -# portable [well requires clock()] tuning app tune: tune.o - $(CC) tune.o $(LIBNAME) -o tune - + $(CC) $(CFLAGS) tune.o $(LIBNAME) -o tune + ./tune_it.sh + # same app but using RDTSC for higher precision [requires 80586+], coff based gcc installs [e.g. ming, cygwin, djgpp] tune86: tune.c nasm -f coff timer.asm $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86 - + # for cygwin tune86c: tune.c nasm -f gnuwin32 timer.asm @@ -46,7 +46,7 @@ tune86c: tune.c tune86l: tune.c nasm -f elf -DUSE_ELF timer.asm $(CC) -DX86_TIMER $(CFLAGS) tune.c timer.o $(LIBNAME) -o tune86l - + # spits out mersenne primes mersenne: mersenne.o $(CC) mersenne.o $(LIBNAME) -o mersenne @@ -54,7 +54,7 @@ mersenne: mersenne.o # fines DR safe primes for the given config drprime: drprime.o $(CC) drprime.o $(LIBNAME) -o drprime - + # fines 2k safe primes for the given config 2kprime: 2kprime.o $(CC) 2kprime.o $(LIBNAME) -o 2kprime @@ -62,6 +62,6 @@ drprime: drprime.o mont: mont.o $(CC) mont.o $(LIBNAME) -o mont - + clean: - rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat *.il + rm -f *.log *.o *.obj *.exe pprime tune mersenne drprime tune86 tune86l mont 2kprime pprime.dat *.il tuning_list diff --git a/lib/hcrypto/libtommath/etc/makefile.msvc b/lib/hcrypto/libtommath/etc/makefile.msvc index 2833372e0..592a43715 100644 --- a/lib/hcrypto/libtommath/etc/makefile.msvc +++ b/lib/hcrypto/libtommath/etc/makefile.msvc @@ -9,10 +9,11 @@ pprime: pprime.obj mersenne: mersenne.obj cl mersenne.obj ../tommath.lib - + tune: tune.obj cl tune.obj ../tommath.lib + mont: mont.obj cl mont.obj ../tommath.lib diff --git a/lib/hcrypto/libtommath/etc/mersenne.c b/lib/hcrypto/libtommath/etc/mersenne.c index 78eb0ea27..0c9f52fcf 100644 --- a/lib/hcrypto/libtommath/etc/mersenne.c +++ b/lib/hcrypto/libtommath/etc/mersenne.c @@ -5,140 +5,134 @@ #include #include -int -is_mersenne (long s, int *pp) +static mp_err is_mersenne(long s, mp_bool *pp) { - mp_int n, u; - int res, k; + mp_int n, u; + mp_err res; + int k; - *pp = 0; + *pp = MP_NO; - if ((res = mp_init (&n)) != MP_OKAY) { - return res; - } + if ((res = mp_init(&n)) != MP_OKAY) { + return res; + } - if ((res = mp_init (&u)) != MP_OKAY) { - goto LBL_N; - } + if ((res = mp_init(&u)) != MP_OKAY) { + goto LBL_N; + } - /* n = 2^s - 1 */ - if ((res = mp_2expt(&n, s)) != MP_OKAY) { - goto LBL_MU; - } - if ((res = mp_sub_d (&n, 1, &n)) != MP_OKAY) { - goto LBL_MU; - } - - /* set u=4 */ - mp_set (&u, 4); - - /* for k=1 to s-2 do */ - for (k = 1; k <= s - 2; k++) { - /* u = u^2 - 2 mod n */ - if ((res = mp_sqr (&u, &u)) != MP_OKAY) { + /* n = 2^s - 1 */ + if ((res = mp_2expt(&n, (int)s)) != MP_OKAY) { goto LBL_MU; - } - if ((res = mp_sub_d (&u, 2, &u)) != MP_OKAY) { + } + if ((res = mp_sub_d(&n, 1uL, &n)) != MP_OKAY) { goto LBL_MU; - } + } - /* make sure u is positive */ - while (u.sign == MP_NEG) { - if ((res = mp_add (&u, &n, &u)) != MP_OKAY) { + /* set u=4 */ + mp_set(&u, 4uL); + + /* for k=1 to s-2 do */ + for (k = 1; k <= (s - 2); k++) { + /* u = u^2 - 2 mod n */ + if ((res = mp_sqr(&u, &u)) != MP_OKAY) { + goto LBL_MU; + } + if ((res = mp_sub_d(&u, 2uL, &u)) != MP_OKAY) { goto LBL_MU; } - } - /* reduce */ - if ((res = mp_reduce_2k (&u, &n, 1)) != MP_OKAY) { - goto LBL_MU; - } - } + /* make sure u is positive */ + while (u.sign == MP_NEG) { + if ((res = mp_add(&u, &n, &u)) != MP_OKAY) { + goto LBL_MU; + } + } - /* if u == 0 then its prime */ - if (mp_iszero (&u) == 1) { - mp_prime_is_prime(&n, 8, pp); - if (*pp != 1) printf("FAILURE\n"); - } + /* reduce */ + if ((res = mp_reduce_2k(&u, &n, 1uL)) != MP_OKAY) { + goto LBL_MU; + } + } - res = MP_OKAY; -LBL_MU:mp_clear (&u); -LBL_N:mp_clear (&n); - return res; + /* if u == 0 then its prime */ + if (mp_iszero(&u) == MP_YES) { + mp_prime_is_prime(&n, 8, pp); + if (*pp != MP_YES) printf("FAILURE\n"); + } + + res = MP_OKAY; +LBL_MU: + mp_clear(&u); +LBL_N: + mp_clear(&n); + return res; } /* square root of a long < 65536 */ -long -i_sqrt (long x) +static long i_sqrt(long x) { - long x1, x2; + long x1, x2; - x2 = 16; - do { - x1 = x2; - x2 = x1 - ((x1 * x1) - x) / (2 * x1); - } while (x1 != x2); + x2 = 16; + do { + x1 = x2; + x2 = x1 - ((x1 * x1) - x) / (2 * x1); + } while (x1 != x2); - if (x1 * x1 > x) { - --x1; - } + if ((x1 * x1) > x) { + --x1; + } - return x1; + return x1; } /* is the long prime by brute force */ -int -isprime (long k) +static int isprime(long k) { - long y, z; + long y, z; - y = i_sqrt (k); - for (z = 2; z <= y; z++) { - if ((k % z) == 0) - return 0; - } - return 1; + y = i_sqrt(k); + for (z = 2; z <= y; z++) { + if ((k % z) == 0) + return 0; + } + return 1; } -int -main (void) +int main(void) { - int pp; - long k; - clock_t tt; + mp_bool pp; + long k; + clock_t tt; - k = 3; + k = 3; - for (;;) { - /* start time */ - tt = clock (); + for (;;) { + /* start time */ + tt = clock(); - /* test if 2^k - 1 is prime */ - if (is_mersenne (k, &pp) != MP_OKAY) { - printf ("Whoa error\n"); - return -1; - } + /* test if 2^k - 1 is prime */ + if (is_mersenne(k, &pp) != MP_OKAY) { + printf("Whoa error\n"); + return -1; + } - if (pp == 1) { - /* count time */ - tt = clock () - tt; + if (pp == MP_YES) { + /* count time */ + tt = clock() - tt; - /* display if prime */ - printf ("2^%-5ld - 1 is prime, test took %ld ticks\n", k, tt); - } + /* display if prime */ + printf("2^%-5ld - 1 is prime, test took %ld ticks\n", k, (long)tt); + } - /* goto next odd exponent */ - k += 2; - - /* but make sure its prime */ - while (isprime (k) == 0) { + /* goto next odd exponent */ k += 2; - } - } - return 0; -} -/* $Source: /cvs/libtom/libtommath/etc/mersenne.c,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2006/03/31 14:18:47 $ */ + /* but make sure its prime */ + while (isprime(k) == 0) { + k += 2; + } + } +} diff --git a/lib/hcrypto/libtommath/etc/mont.c b/lib/hcrypto/libtommath/etc/mont.c index 7c5501fc0..4652410d0 100644 --- a/lib/hcrypto/libtommath/etc/mont.c +++ b/lib/hcrypto/libtommath/etc/mont.c @@ -1,50 +1,44 @@ /* tests the montgomery routines */ #include +#include +#include int main(void) { mp_int modulus, R, p, pp; mp_digit mp; - long x, y; + int x, y; srand(time(NULL)); mp_init_multi(&modulus, &R, &p, &pp, NULL); /* loop through various sizes */ for (x = 4; x < 256; x++) { - printf("DIGITS == %3ld...", x); fflush(stdout); + printf("DIGITS == %3d...", x); + fflush(stdout); - /* make up the odd modulus */ - mp_rand(&modulus, x); - modulus.dp[0] |= 1; + /* make up the odd modulus */ + mp_rand(&modulus, x); + modulus.dp[0] |= 1uL; - /* now find the R value */ - mp_montgomery_calc_normalization(&R, &modulus); - mp_montgomery_setup(&modulus, &mp); + /* now find the R value */ + mp_montgomery_calc_normalization(&R, &modulus); + mp_montgomery_setup(&modulus, &mp); - /* now run through a bunch tests */ - for (y = 0; y < 1000; y++) { - mp_rand(&p, x/2); /* p = random */ - mp_mul(&p, &R, &pp); /* pp = R * p */ - mp_montgomery_reduce(&pp, &modulus, mp); + /* now run through a bunch tests */ + for (y = 0; y < 1000; y++) { + mp_rand(&p, x/2); /* p = random */ + mp_mul(&p, &R, &pp); /* pp = R * p */ + mp_montgomery_reduce(&pp, &modulus, mp); - /* should be equal to p */ - if (mp_cmp(&pp, &p) != MP_EQ) { - printf("FAILURE!\n"); - exit(-1); - } - } - printf("PASSED\n"); - } + /* should be equal to p */ + if (mp_cmp(&pp, &p) != MP_EQ) { + printf("FAILURE!\n"); + exit(-1); + } + } + printf("PASSED\n"); + } - return 0; + return 0; } - - - - - - -/* $Source: /cvs/libtom/libtommath/etc/mont.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ diff --git a/lib/hcrypto/libtommath/etc/pprime.c b/lib/hcrypto/libtommath/etc/pprime.c index 317e2a0fc..009a18cb9 100644 --- a/lib/hcrypto/libtommath/etc/pprime.c +++ b/lib/hcrypto/libtommath/etc/pprime.c @@ -4,397 +4,408 @@ * * Tom St Denis, tomstdenis@gmail.com, http://tom.gmail.com */ +#include #include #include "tommath.h" -int n_prime; -FILE *primes; +static int n_prime; +static FILE *primes; /* fast square root */ -static mp_digit -i_sqrt (mp_word x) +static mp_digit i_sqrt(mp_word x) { - mp_word x1, x2; + mp_word x1, x2; - x2 = x; - do { - x1 = x2; - x2 = x1 - ((x1 * x1) - x) / (2 * x1); - } while (x1 != x2); + x2 = x; + do { + x1 = x2; + x2 = x1 - ((x1 * x1) - x) / (2u * x1); + } while (x1 != x2); - if (x1 * x1 > x) { - --x1; - } + if ((x1 * x1) > x) { + --x1; + } - return x1; + return x1; } /* generates a prime digit */ -static void gen_prime (void) +static void gen_prime(void) { - mp_digit r, x, y, next; - FILE *out; + mp_digit r, x, y, next; + FILE *out; - out = fopen("pprime.dat", "wb"); + out = fopen("pprime.dat", "wb"); + if (out != NULL) { - /* write first set of primes */ - r = 3; fwrite(&r, 1, sizeof(mp_digit), out); - r = 5; fwrite(&r, 1, sizeof(mp_digit), out); - r = 7; fwrite(&r, 1, sizeof(mp_digit), out); - r = 11; fwrite(&r, 1, sizeof(mp_digit), out); - r = 13; fwrite(&r, 1, sizeof(mp_digit), out); - r = 17; fwrite(&r, 1, sizeof(mp_digit), out); - r = 19; fwrite(&r, 1, sizeof(mp_digit), out); - r = 23; fwrite(&r, 1, sizeof(mp_digit), out); - r = 29; fwrite(&r, 1, sizeof(mp_digit), out); - r = 31; fwrite(&r, 1, sizeof(mp_digit), out); + /* write first set of primes */ + /* *INDENT-OFF* */ + r = 3uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 5uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 7uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 11uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 13uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 17uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 19uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 23uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 29uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + r = 31uL; fwrite(&r, 1uL, sizeof(mp_digit), out); + /* *INDENT-ON* */ - /* get square root, since if 'r' is composite its factors must be < than this */ - y = i_sqrt (r); - next = (y + 1) * (y + 1); + /* get square root, since if 'r' is composite its factors must be < than this */ + y = i_sqrt(r); + next = (y + 1uL) * (y + 1uL); - for (;;) { - do { - r += 2; /* next candidate */ - r &= MP_MASK; - if (r < 31) break; + for (;;) { + do { + r += 2uL; /* next candidate */ + r &= MP_MASK; + if (r < 31uL) break; - /* update sqrt ? */ - if (next <= r) { - ++y; - next = (y + 1) * (y + 1); - } + /* update sqrt ? */ + if (next <= r) { + ++y; + next = (y + 1uL) * (y + 1uL); + } - /* loop if divisible by 3,5,7,11,13,17,19,23,29 */ - if ((r % 3) == 0) { - x = 0; - continue; - } - if ((r % 5) == 0) { - x = 0; - continue; - } - if ((r % 7) == 0) { - x = 0; - continue; - } - if ((r % 11) == 0) { - x = 0; - continue; - } - if ((r % 13) == 0) { - x = 0; - continue; - } - if ((r % 17) == 0) { - x = 0; - continue; - } - if ((r % 19) == 0) { - x = 0; - continue; - } - if ((r % 23) == 0) { - x = 0; - continue; - } - if ((r % 29) == 0) { - x = 0; - continue; - } + /* loop if divisible by 3,5,7,11,13,17,19,23,29 */ + if ((r % 3uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 5uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 7uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 11uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 13uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 17uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 19uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 23uL) == 0uL) { + x = 0uL; + continue; + } + if ((r % 29uL) == 0uL) { + x = 0uL; + continue; + } - /* now check if r is divisible by x + k={1,7,11,13,17,19,23,29} */ - for (x = 30; x <= y; x += 30) { - if ((r % (x + 1)) == 0) { - x = 0; - break; + /* now check if r is divisible by x + k={1,7,11,13,17,19,23,29} */ + for (x = 30uL; x <= y; x += 30uL) { + if ((r % (x + 1uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 7uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 11uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 13uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 17uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 19uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 23uL)) == 0uL) { + x = 0uL; + break; + } + if ((r % (x + 29uL)) == 0uL) { + x = 0uL; + break; + } + } + } while (x == 0uL); + if (r > 31uL) { + fwrite(&r, 1uL, sizeof(mp_digit), out); + printf("%9lu\r", r); + fflush(stdout); + } + if (r < 31uL) break; } - if ((r % (x + 7)) == 0) { - x = 0; - break; - } - if ((r % (x + 11)) == 0) { - x = 0; - break; - } - if ((r % (x + 13)) == 0) { - x = 0; - break; - } - if ((r % (x + 17)) == 0) { - x = 0; - break; - } - if ((r % (x + 19)) == 0) { - x = 0; - break; - } - if ((r % (x + 23)) == 0) { - x = 0; - break; - } - if ((r % (x + 29)) == 0) { - x = 0; - break; - } - } - } while (x == 0); - if (r > 31) { fwrite(&r, 1, sizeof(mp_digit), out); printf("%9d\r", r); fflush(stdout); } - if (r < 31) break; - } - fclose(out); + fclose(out); + } } -void load_tab(void) +static void load_tab(void) { primes = fopen("pprime.dat", "rb"); if (primes == NULL) { gen_prime(); primes = fopen("pprime.dat", "rb"); } - fseek(primes, 0, SEEK_END); + fseek(primes, 0L, SEEK_END); n_prime = ftell(primes) / sizeof(mp_digit); } -mp_digit prime_digit(void) +static mp_digit prime_digit(void) { int n; mp_digit d; n = abs(rand()) % n_prime; fseek(primes, n * sizeof(mp_digit), SEEK_SET); - fread(&d, 1, sizeof(mp_digit), primes); + fread(&d, 1uL, sizeof(mp_digit), primes); return d; } /* makes a prime of at least k bits */ -int -pprime (int k, int li, mp_int * p, mp_int * q) +static mp_err pprime(int k, int li, mp_int *p, mp_int *q) { - mp_int a, b, c, n, x, y, z, v; - int res, ii; - static const mp_digit bases[] = { 2, 3, 5, 7, 11, 13, 17, 19 }; + mp_int a, b, c, n, x, y, z, v; + mp_err res; + int ii; + static const mp_digit bases[] = { 2, 3, 5, 7, 11, 13, 17, 19 }; - /* single digit ? */ - if (k <= (int) DIGIT_BIT) { - mp_set (p, prime_digit ()); - return MP_OKAY; - } + /* single digit ? */ + if (k <= (int) MP_DIGIT_BIT) { + mp_set(p, prime_digit()); + return MP_OKAY; + } - if ((res = mp_init (&c)) != MP_OKAY) { - return res; - } + if ((res = mp_init(&c)) != MP_OKAY) { + return res; + } - if ((res = mp_init (&v)) != MP_OKAY) { - goto LBL_C; - } + if ((res = mp_init(&v)) != MP_OKAY) { + goto LBL_C; + } - /* product of first 50 primes */ - if ((res = - mp_read_radix (&v, - "19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190", - 10)) != MP_OKAY) { - goto LBL_V; - } + /* product of first 50 primes */ + if ((res = + mp_read_radix(&v, + "19078266889580195013601891820992757757219839668357012055907516904309700014933909014729740190", + 10)) != MP_OKAY) { + goto LBL_V; + } - if ((res = mp_init (&a)) != MP_OKAY) { - goto LBL_V; - } + if ((res = mp_init(&a)) != MP_OKAY) { + goto LBL_V; + } - /* set the prime */ - mp_set (&a, prime_digit ()); + /* set the prime */ + mp_set(&a, prime_digit()); - if ((res = mp_init (&b)) != MP_OKAY) { - goto LBL_A; - } + if ((res = mp_init(&b)) != MP_OKAY) { + goto LBL_A; + } - if ((res = mp_init (&n)) != MP_OKAY) { - goto LBL_B; - } + if ((res = mp_init(&n)) != MP_OKAY) { + goto LBL_B; + } - if ((res = mp_init (&x)) != MP_OKAY) { - goto LBL_N; - } + if ((res = mp_init(&x)) != MP_OKAY) { + goto LBL_N; + } - if ((res = mp_init (&y)) != MP_OKAY) { - goto LBL_X; - } + if ((res = mp_init(&y)) != MP_OKAY) { + goto LBL_X; + } - if ((res = mp_init (&z)) != MP_OKAY) { - goto LBL_Y; - } + if ((res = mp_init(&z)) != MP_OKAY) { + goto LBL_Y; + } - /* now loop making the single digit */ - while (mp_count_bits (&a) < k) { - fprintf (stderr, "prime has %4d bits left\r", k - mp_count_bits (&a)); - fflush (stderr); - top: - mp_set (&b, prime_digit ()); + /* now loop making the single digit */ + while (mp_count_bits(&a) < k) { + fprintf(stderr, "prime has %4d bits left\r", k - mp_count_bits(&a)); + fflush(stderr); +top: + mp_set(&b, prime_digit()); - /* now compute z = a * b * 2 */ - if ((res = mp_mul (&a, &b, &z)) != MP_OKAY) { /* z = a * b */ - goto LBL_Z; - } - - if ((res = mp_copy (&z, &c)) != MP_OKAY) { /* c = a * b */ - goto LBL_Z; - } - - if ((res = mp_mul_2 (&z, &z)) != MP_OKAY) { /* z = 2 * a * b */ - goto LBL_Z; - } - - /* n = z + 1 */ - if ((res = mp_add_d (&z, 1, &n)) != MP_OKAY) { /* n = z + 1 */ - goto LBL_Z; - } - - /* check (n, v) == 1 */ - if ((res = mp_gcd (&n, &v, &y)) != MP_OKAY) { /* y = (n, v) */ - goto LBL_Z; - } - - if (mp_cmp_d (&y, 1) != MP_EQ) - goto top; - - /* now try base x=bases[ii] */ - for (ii = 0; ii < li; ii++) { - mp_set (&x, bases[ii]); - - /* compute x^a mod n */ - if ((res = mp_exptmod (&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */ - goto LBL_Z; + /* now compute z = a * b * 2 */ + if ((res = mp_mul(&a, &b, &z)) != MP_OKAY) { /* z = a * b */ + goto LBL_Z; } - /* if y == 1 loop */ - if (mp_cmp_d (&y, 1) == MP_EQ) - continue; - - /* now x^2a mod n */ - if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */ - goto LBL_Z; + if ((res = mp_copy(&z, &c)) != MP_OKAY) { /* c = a * b */ + goto LBL_Z; } - if (mp_cmp_d (&y, 1) == MP_EQ) - continue; - - /* compute x^b mod n */ - if ((res = mp_exptmod (&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */ - goto LBL_Z; + if ((res = mp_mul_2(&z, &z)) != MP_OKAY) { /* z = 2 * a * b */ + goto LBL_Z; } - /* if y == 1 loop */ - if (mp_cmp_d (&y, 1) == MP_EQ) - continue; - - /* now x^2b mod n */ - if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */ - goto LBL_Z; + /* n = z + 1 */ + if ((res = mp_add_d(&z, 1uL, &n)) != MP_OKAY) { /* n = z + 1 */ + goto LBL_Z; } - if (mp_cmp_d (&y, 1) == MP_EQ) - continue; - - /* compute x^c mod n == x^ab mod n */ - if ((res = mp_exptmod (&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */ - goto LBL_Z; + /* check (n, v) == 1 */ + if ((res = mp_gcd(&n, &v, &y)) != MP_OKAY) { /* y = (n, v) */ + goto LBL_Z; } - /* if y == 1 loop */ - if (mp_cmp_d (&y, 1) == MP_EQ) - continue; + if (mp_cmp_d(&y, 1uL) != MP_EQ) + goto top; - /* now compute (x^c mod n)^2 */ - if ((res = mp_sqrmod (&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */ - goto LBL_Z; + /* now try base x=bases[ii] */ + for (ii = 0; ii < li; ii++) { + mp_set(&x, bases[ii]); + + /* compute x^a mod n */ + if ((res = mp_exptmod(&x, &a, &n, &y)) != MP_OKAY) { /* y = x^a mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d(&y, 1uL) == MP_EQ) + continue; + + /* now x^2a mod n */ + if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2a mod n */ + goto LBL_Z; + } + + if (mp_cmp_d(&y, 1uL) == MP_EQ) + continue; + + /* compute x^b mod n */ + if ((res = mp_exptmod(&x, &b, &n, &y)) != MP_OKAY) { /* y = x^b mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d(&y, 1uL) == MP_EQ) + continue; + + /* now x^2b mod n */ + if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2b mod n */ + goto LBL_Z; + } + + if (mp_cmp_d(&y, 1uL) == MP_EQ) + continue; + + /* compute x^c mod n == x^ab mod n */ + if ((res = mp_exptmod(&x, &c, &n, &y)) != MP_OKAY) { /* y = x^ab mod n */ + goto LBL_Z; + } + + /* if y == 1 loop */ + if (mp_cmp_d(&y, 1uL) == MP_EQ) + continue; + + /* now compute (x^c mod n)^2 */ + if ((res = mp_sqrmod(&y, &n, &y)) != MP_OKAY) { /* y = x^2ab mod n */ + goto LBL_Z; + } + + /* y should be 1 */ + if (mp_cmp_d(&y, 1uL) != MP_EQ) + continue; + break; } - /* y should be 1 */ - if (mp_cmp_d (&y, 1) != MP_EQ) - continue; - break; - } + /* no bases worked? */ + if (ii == li) + goto top; - /* no bases worked? */ - if (ii == li) - goto top; + { + char buf[4096]; -{ - char buf[4096]; + mp_to_decimal(&n, buf, sizeof(buf)); + printf("Certificate of primality for:\n%s\n\n", buf); + mp_to_decimal(&a, buf, sizeof(buf)); + printf("A == \n%s\n\n", buf); + mp_to_decimal(&b, buf, sizeof(buf)); + printf("B == \n%s\n\nG == %lu\n", buf, bases[ii]); + printf("----------------------------------------------------------------\n"); + } - mp_toradix(&n, buf, 10); - printf("Certificate of primality for:\n%s\n\n", buf); - mp_toradix(&a, buf, 10); - printf("A == \n%s\n\n", buf); - mp_toradix(&b, buf, 10); - printf("B == \n%s\n\nG == %d\n", buf, bases[ii]); - printf("----------------------------------------------------------------\n"); -} + /* a = n */ + mp_copy(&n, &a); + } - /* a = n */ - mp_copy (&n, &a); - } + /* get q to be the order of the large prime subgroup */ + mp_sub_d(&n, 1uL, q); + mp_div_2(q, q); + mp_div(q, &b, q, NULL); - /* get q to be the order of the large prime subgroup */ - mp_sub_d (&n, 1, q); - mp_div_2 (q, q); - mp_div (q, &b, q, NULL); + mp_exch(&n, p); - mp_exch (&n, p); - - res = MP_OKAY; -LBL_Z:mp_clear (&z); -LBL_Y:mp_clear (&y); -LBL_X:mp_clear (&x); -LBL_N:mp_clear (&n); -LBL_B:mp_clear (&b); -LBL_A:mp_clear (&a); -LBL_V:mp_clear (&v); -LBL_C:mp_clear (&c); - return res; + res = MP_OKAY; +LBL_Z: + mp_clear(&z); +LBL_Y: + mp_clear(&y); +LBL_X: + mp_clear(&x); +LBL_N: + mp_clear(&n); +LBL_B: + mp_clear(&b); +LBL_A: + mp_clear(&a); +LBL_V: + mp_clear(&v); +LBL_C: + mp_clear(&c); + return res; } -int -main (void) +int main(void) { - mp_int p, q; - char buf[4096]; - int k, li; - clock_t t1; + mp_int p, q; + char buf[4096]; + int k, li; + clock_t t1; - srand (time (NULL)); - load_tab(); + srand(time(NULL)); + load_tab(); - printf ("Enter # of bits: \n"); - fgets (buf, sizeof (buf), stdin); - sscanf (buf, "%d", &k); + printf("Enter # of bits: \n"); + fgets(buf, sizeof(buf), stdin); + sscanf(buf, "%d", &k); - printf ("Enter number of bases to try (1 to 8):\n"); - fgets (buf, sizeof (buf), stdin); - sscanf (buf, "%d", &li); + printf("Enter number of bases to try (1 to 8):\n"); + fgets(buf, sizeof(buf), stdin); + sscanf(buf, "%d", &li); - mp_init (&p); - mp_init (&q); + mp_init(&p); + mp_init(&q); - t1 = clock (); - pprime (k, li, &p, &q); - t1 = clock () - t1; + t1 = clock(); + pprime(k, li, &p, &q); + t1 = clock() - t1; - printf ("\n\nTook %ld ticks, %d bits\n", t1, mp_count_bits (&p)); + printf("\n\nTook %d ticks, %d bits\n", t1, mp_count_bits(&p)); - mp_toradix (&p, buf, 10); - printf ("P == %s\n", buf); - mp_toradix (&q, buf, 10); - printf ("Q == %s\n", buf); + mp_to_decimal(&p, buf, sizeof(buf)); + printf("P == %s\n", buf); + mp_to_decimal(&q, buf, sizeof(buf)); + printf("Q == %s\n", buf); - return 0; + return 0; } - -/* $Source: /cvs/libtom/libtommath/etc/pprime.c,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2006/03/31 14:18:47 $ */ diff --git a/lib/hcrypto/libtommath/etc/timer.asm b/lib/hcrypto/libtommath/etc/timer.asm index f1f84bdb4..35890d985 100644 --- a/lib/hcrypto/libtommath/etc/timer.asm +++ b/lib/hcrypto/libtommath/etc/timer.asm @@ -34,4 +34,4 @@ _t_read: sub eax,[time+4] sbb edx,[time+0] ret - + \ No newline at end of file diff --git a/lib/hcrypto/libtommath/etc/tune.c b/lib/hcrypto/libtommath/etc/tune.c index e09e8ef77..bc2cdfe6e 100644 --- a/lib/hcrypto/libtommath/etc/tune.c +++ b/lib/hcrypto/libtommath/etc/tune.c @@ -1,142 +1,542 @@ /* Tune the Karatsuba parameters * - * Tom St Denis, tomstdenis@gmail.com + * Tom St Denis, tstdenis82@gmail.com */ -#include +#include "../tommath.h" +#include "../tommath_private.h" #include +#include +#include -/* how many times todo each size mult. Depends on your computer. For slow computers - * this can be low like 5 or 10. For fast [re: Athlon] should be 25 - 50 or so - */ -#define TIMES (1UL<<14UL) +/* + Please take in mind that both multiplicands are of the same size. The balancing + mechanism in mp_balance works well but has some overhead itself. You can test + the behaviour of it with the option "-o" followed by a (small) positive number 'x' + to generate ratios of the form 1:x. +*/ -/* RDTSC from Scott Duplichan */ -static ulong64 TIMFUNC (void) - { - #if defined __GNUC__ - #if defined(__i386__) || defined(__x86_64__) - unsigned long long a; - __asm__ __volatile__ ("rdtsc\nmovl %%eax,%0\nmovl %%edx,4+%0\n"::"m"(a):"%eax","%edx"); - return a; - #else /* gcc-IA64 version */ - unsigned long result; - __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory"); - while (__builtin_expect ((int) result == -1, 0)) - __asm__ __volatile__("mov %0=ar.itc" : "=r"(result) :: "memory"); - return result; - #endif +static uint64_t s_timer_function(void); +static void s_timer_start(void); +static uint64_t s_timer_stop(void); +static uint64_t s_time_mul(int size); +static uint64_t s_time_sqr(int size); +static void s_usage(char *s); - // Microsoft and Intel Windows compilers - #elif defined _M_IX86 - __asm rdtsc - #elif defined _M_AMD64 - return __rdtsc (); - #elif defined _M_IA64 - #if defined __INTEL_COMPILER - #include - #endif - return __getReg (3116); - #else - #error need rdtsc function for this build - #endif +static uint64_t s_timer_function(void) +{ +#if _POSIX_C_SOURCE >= 199309L +#define LTM_BILLION 1000000000 + struct timespec ts; + + /* TODO: Sets errno in case of error. Use? */ + clock_gettime(CLOCK_MONOTONIC, &ts); + return (((uint64_t)ts.tv_sec) * LTM_BILLION + (uint64_t)ts.tv_nsec); +#else + clock_t t; + t = clock(); + if (t < (clock_t)(0)) { + return (uint64_t)(0); } - - -#ifndef X86_TIMER + return (uint64_t)(t); +#endif +} /* generic ISO C timer */ -ulong64 LBL_T; -void t_start(void) { LBL_T = TIMFUNC(); } -ulong64 t_read(void) { return TIMFUNC() - LBL_T; } - -#else -extern void t_start(void); -extern ulong64 t_read(void); -#endif - -ulong64 time_mult(int size, int s) +static uint64_t s_timer_tmp; +static void s_timer_start(void) { - unsigned long x; - mp_int a, b, c; - ulong64 t1; - - mp_init (&a); - mp_init (&b); - mp_init (&c); - - mp_rand (&a, size); - mp_rand (&b, size); - - if (s == 1) { - KARATSUBA_MUL_CUTOFF = size; - } else { - KARATSUBA_MUL_CUTOFF = 100000; - } - - t_start(); - for (x = 0; x < TIMES; x++) { - mp_mul(&a,&b,&c); - } - t1 = t_read(); - mp_clear (&a); - mp_clear (&b); - mp_clear (&c); - return t1; + s_timer_tmp = s_timer_function(); +} +static uint64_t s_timer_stop(void) +{ + return s_timer_function() - s_timer_tmp; } -ulong64 time_sqr(int size, int s) + +static int s_check_result; +static int s_number_of_test_loops; +static int s_stabilization_extra; +static int s_offset = 1; + +#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) +static uint64_t s_time_mul(int size) { - unsigned long x; - mp_int a, b; - ulong64 t1; + int x; + mp_err e; + mp_int a, b, c, d; + uint64_t t1; - mp_init (&a); - mp_init (&b); + if ((e = mp_init_multi(&a, &b, &c, &d, NULL)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } - mp_rand (&a, size); + if ((e = mp_rand(&a, size * s_offset)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } + if ((e = mp_rand(&b, size)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } - if (s == 1) { - KARATSUBA_SQR_CUTOFF = size; - } else { - KARATSUBA_SQR_CUTOFF = 100000; - } + s_timer_start(); + for (x = 0; x < s_number_of_test_loops; x++) { + if ((e = mp_mul(&a,&b,&c)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } + if (s_check_result == 1) { + if ((e = s_mp_mul(&a,&b,&d)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } + if (mp_cmp(&c, &d) != MP_EQ) { + /* Time of 0 cannot happen (famous last words?) */ + t1 = 0uLL; + goto LTM_ERR; + } + } + } - t_start(); - for (x = 0; x < TIMES; x++) { - mp_sqr(&a,&b); - } - t1 = t_read(); - mp_clear (&a); - mp_clear (&b); - return t1; + t1 = s_timer_stop(); +LTM_ERR: + mp_clear_multi(&a, &b, &c, &d, NULL); + return t1; } -int -main (void) +static uint64_t s_time_sqr(int size) { - ulong64 t1, t2; - int x, y; + int x; + mp_err e; + mp_int a, b, c; + uint64_t t1; - for (x = 8; ; x += 2) { - t1 = time_mult(x, 0); - t2 = time_mult(x, 1); - printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1); - if (t2 < t1) break; - } - y = x; + if ((e = mp_init_multi(&a, &b, &c, NULL)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } - for (x = 8; ; x += 2) { - t1 = time_sqr(x, 0); - t2 = time_sqr(x, 1); - printf("%d: %9llu %9llu, %9llu\n", x, t1, t2, t2 - t1); - if (t2 < t1) break; - } - printf("KARATSUBA_MUL_CUTOFF = %d\n", y); - printf("KARATSUBA_SQR_CUTOFF = %d\n", x); + if ((e = mp_rand(&a, size)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } - return 0; + s_timer_start(); + for (x = 0; x < s_number_of_test_loops; x++) { + if ((e = mp_sqr(&a,&b)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } + if (s_check_result == 1) { + if ((e = s_mp_sqr(&a,&c)) != MP_OKAY) { + t1 = UINT64_MAX; + goto LTM_ERR; + } + if (mp_cmp(&c, &b) != MP_EQ) { + t1 = 0uLL; + goto LTM_ERR; + } + } + } + + t1 = s_timer_stop(); +LTM_ERR: + mp_clear_multi(&a, &b, &c, NULL); + return t1; } -/* $Source: /cvs/libtom/libtommath/etc/tune.c,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2006/03/31 14:18:47 $ */ +struct tune_args { + int testmode; + int verbose; + int print; + int bncore; + int terse; + int upper_limit_print; + int increment_print; +} args; + +static void s_run(const char *name, uint64_t (*op)(int), int *cutoff) +{ + int x, count = 0; + uint64_t t1, t2; + if ((args.verbose == 1) || (args.testmode == 1)) { + printf("# %s.\n", name); + } + for (x = 8; x < args.upper_limit_print; x += args.increment_print) { + *cutoff = INT_MAX; + t1 = op(x); + if ((t1 == 0uLL) || (t1 == UINT64_MAX)) { + fprintf(stderr,"%s failed at x = INT_MAX (%s)\n", name, + (t1 == 0uLL)?"wrong result":"internal error"); + exit(EXIT_FAILURE); + } + *cutoff = x; + t2 = op(x); + if ((t2 == 0uLL) || (t2 == UINT64_MAX)) { + fprintf(stderr,"%s failed (%s)\n", name, + (t2 == 0uLL)?"wrong result":"internal error"); + exit(EXIT_FAILURE); + } + if (args.verbose == 1) { + printf("%d: %9"PRIu64" %9"PRIu64", %9"PRIi64"\n", x, t1, t2, (int64_t)t2 - (int64_t)t1); + } + if (t2 < t1) { + if (count == s_stabilization_extra) { + count = 0; + break; + } else if (count < s_stabilization_extra) { + count++; + } + } else if (count > 0) { + count--; + } + } + *cutoff = x - s_stabilization_extra * args.increment_print; +} + +static long s_strtol(const char *str, char **endptr, const char *err) +{ + const int base = 10; + char *_endptr; + long val; + errno = 0; + val = strtol(str, &_endptr, base); + if ((val > INT_MAX || val < 0) || (errno != 0)) { + fprintf(stderr, "Value %s not usable\n", str); + exit(EXIT_FAILURE); + } + if (_endptr == str) { + fprintf(stderr, "%s\n", err); + exit(EXIT_FAILURE); + } + if (endptr) *endptr = _endptr; + return val; +} + +static int s_exit_code = EXIT_FAILURE; +static void s_usage(char *s) +{ + fprintf(stderr,"Usage: %s [TvcpGbtrSLFfMmosh]\n",s); + fprintf(stderr," -T testmode, for use with testme.sh\n"); + fprintf(stderr," -v verbose, print all timings\n"); + fprintf(stderr," -c check results\n"); + fprintf(stderr," -p print benchmark of final cutoffs in files \"multiplying\"\n"); + fprintf(stderr," and \"squaring\"\n"); + fprintf(stderr," -G [string] suffix for the filenames listed above\n"); + fprintf(stderr," Implies '-p'\n"); + fprintf(stderr," -b print benchmark of bncore.c\n"); + fprintf(stderr," -t prints space (0x20) separated results\n"); + fprintf(stderr," -r [64] number of rounds\n"); + fprintf(stderr," -S [0xdeadbeef] seed for PRNG\n"); + fprintf(stderr," -L [3] number of negative values accumulated until the result is accepted\n"); + fprintf(stderr," -M [3000] upper limit of T-C tests/prints\n"); + fprintf(stderr," -m [1] increment of T-C tests/prints\n"); + fprintf(stderr," -o [1] multiplier for the second multiplicand\n"); + fprintf(stderr," (Not for computing the cut-offs!)\n"); + fprintf(stderr," -s 'preset' use values in 'preset' for printing.\n"); + fprintf(stderr," 'preset' is a comma separated string with cut-offs for\n"); + fprintf(stderr," ksm, kss, tc3m, tc3s in that order\n"); + fprintf(stderr," ksm = karatsuba multiplication\n"); + fprintf(stderr," kss = karatsuba squaring\n"); + fprintf(stderr," tc3m = Toom-Cook 3-way multiplication\n"); + fprintf(stderr," tc3s = Toom-Cook 3-way squaring\n"); + fprintf(stderr," Implies '-p'\n"); + fprintf(stderr," -h this message\n"); + exit(s_exit_code); +} + +struct cutoffs { + int KARATSUBA_MUL, KARATSUBA_SQR; + int TOOM_MUL, TOOM_SQR; +}; + +const struct cutoffs max_cutoffs = +{ INT_MAX, INT_MAX, INT_MAX, INT_MAX }; + +static void set_cutoffs(const struct cutoffs *c) +{ + KARATSUBA_MUL_CUTOFF = c->KARATSUBA_MUL; + KARATSUBA_SQR_CUTOFF = c->KARATSUBA_SQR; + TOOM_MUL_CUTOFF = c->TOOM_MUL; + TOOM_SQR_CUTOFF = c->TOOM_SQR; +} + +static void get_cutoffs(struct cutoffs *c) +{ + c->KARATSUBA_MUL = KARATSUBA_MUL_CUTOFF; + c->KARATSUBA_SQR = KARATSUBA_SQR_CUTOFF; + c->TOOM_MUL = TOOM_MUL_CUTOFF; + c->TOOM_SQR = TOOM_SQR_CUTOFF; + +} + +int main(int argc, char **argv) +{ + uint64_t t1, t2; + int x, i, j; + size_t n; + + int printpreset = 0; + /*int preset[8];*/ + char *endptr, *str; + + uint64_t seed = 0xdeadbeef; + + int opt; + struct cutoffs orig, updated; + + FILE *squaring, *multiplying; + char mullog[256] = "multiplying"; + char sqrlog[256] = "squaring"; + s_number_of_test_loops = 64; + s_stabilization_extra = 3; + + MP_ZERO_BUFFER(&args, sizeof(args)); + + args.testmode = 0; + args.verbose = 0; + args.print = 0; + args.bncore = 0; + args.terse = 0; + + args.upper_limit_print = 3000; + args.increment_print = 1; + + /* Very simple option parser, please treat it nicely. */ + if (argc != 1) { + for (opt = 1; (opt < argc) && (argv[opt][0] == '-'); opt++) { + switch (argv[opt][1]) { + case 'T': + args.testmode = 1; + s_check_result = 1; + args.upper_limit_print = 1000; + args.increment_print = 11; + s_number_of_test_loops = 1; + s_stabilization_extra = 1; + s_offset = 1; + break; + case 'v': + args.verbose = 1; + break; + case 'c': + s_check_result = 1; + break; + case 'p': + args.print = 1; + break; + case 'G': + args.print = 1; + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + /* manual strcat() */ + for (i = 0; i < 255; i++) { + if (mullog[i] == '\0') { + break; + } + } + for (j = 0; i < 255; j++, i++) { + mullog[i] = argv[opt][j]; + if (argv[opt][j] == '\0') { + break; + } + } + for (i = 0; i < 255; i++) { + if (sqrlog[i] == '\0') { + break; + } + } + for (j = 0; i < 255; j++, i++) { + sqrlog[i] = argv[opt][j]; + if (argv[opt][j] == '\0') { + break; + } + } + break; + case 'b': + args.bncore = 1; + break; + case 't': + args.terse = 1; + break; + case 'S': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + str = argv[opt]; + errno = 0; + seed = (uint64_t)s_strtol(argv[opt], NULL, "No seed given?\n"); + break; + case 'L': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + s_stabilization_extra = (int)s_strtol(argv[opt], NULL, "No value for option \"-L\"given"); + break; + case 'o': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + s_offset = (int)s_strtol(argv[opt], NULL, "No value for the offset given"); + break; + case 'r': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + s_number_of_test_loops = (int)s_strtol(argv[opt], NULL, "No value for the number of rounds given"); + break; + + case 'M': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + args.upper_limit_print = (int)s_strtol(argv[opt], NULL, "No value for the upper limit of T-C tests given"); + break; + case 'm': + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + args.increment_print = (int)s_strtol(argv[opt], NULL, "No value for the increment for the T-C tests given"); + break; + case 's': + printpreset = 1; + args.print = 1; + opt++; + if (opt >= argc) { + s_usage(argv[0]); + } + str = argv[opt]; + KARATSUBA_MUL_CUTOFF = (int)s_strtol(str, &endptr, "[1/4] No value for KARATSUBA_MUL_CUTOFF given"); + str = endptr + 1; + KARATSUBA_SQR_CUTOFF = (int)s_strtol(str, &endptr, "[2/4] No value for KARATSUBA_SQR_CUTOFF given"); + str = endptr + 1; + TOOM_MUL_CUTOFF = (int)s_strtol(str, &endptr, "[3/4] No value for TOOM_MUL_CUTOFF given"); + str = endptr + 1; + TOOM_SQR_CUTOFF = (int)s_strtol(str, &endptr, "[4/4] No value for TOOM_SQR_CUTOFF given"); + break; + case 'h': + s_exit_code = EXIT_SUCCESS; + /* FALLTHROUGH */ + default: + s_usage(argv[0]); + } + } + } + + /* + mp_rand uses the cryptographically secure + source of the OS by default. That is too expensive, too slow and + most important for a benchmark: it is not repeatable. + */ + s_mp_rand_jenkins_init(seed); + mp_rand_source(s_mp_rand_jenkins); + + get_cutoffs(&orig); + + updated = max_cutoffs; + if ((args.bncore == 0) && (printpreset == 0)) { + struct { + const char *name; + int *cutoff, *update; + uint64_t (*fn)(int); + } test[] = { +#define T_MUL_SQR(n, o, f) { #n, &o##_CUTOFF, &(updated.o), MP_HAS(S_MP_##o) ? f : NULL } + /* + The influence of the Comba multiplication cannot be + eradicated programmatically. It depends on the size + of the macro MP_WPARRAY in tommath.h which needs to + be changed manually (to 0 (zero)). + */ + T_MUL_SQR("Karatsuba multiplication", KARATSUBA_MUL, s_time_mul), + T_MUL_SQR("Karatsuba squaring", KARATSUBA_SQR, s_time_sqr), + T_MUL_SQR("Toom-Cook 3-way multiplying", TOOM_MUL, s_time_mul), + T_MUL_SQR("Toom-Cook 3-way squaring", TOOM_SQR, s_time_sqr), +#undef T_MUL_SQR + }; + /* Turn all limits from bncore.c to the max */ + set_cutoffs(&max_cutoffs); + for (n = 0; n < sizeof(test)/sizeof(test[0]); ++n) { + if (test[n].fn) { + s_run(test[n].name, test[n].fn, test[n].cutoff); + *test[n].update = *test[n].cutoff; + *test[n].cutoff = INT_MAX; + } + } + } + if (args.terse == 1) { + printf("%d %d %d %d\n", + updated.KARATSUBA_MUL, + updated.KARATSUBA_SQR, + updated.TOOM_MUL, + updated.TOOM_SQR); + } else { + printf("KARATSUBA_MUL_CUTOFF = %d\n", updated.KARATSUBA_MUL); + printf("KARATSUBA_SQR_CUTOFF = %d\n", updated.KARATSUBA_SQR); + printf("TOOM_MUL_CUTOFF = %d\n", updated.TOOM_MUL); + printf("TOOM_SQR_CUTOFF = %d\n", updated.TOOM_SQR); + } + + if (args.print == 1) { + printf("Printing data for graphing to \"%s\" and \"%s\"\n",mullog, sqrlog); + + multiplying = fopen(mullog, "w+"); + if (multiplying == NULL) { + fprintf(stderr, "Opening file \"%s\" failed\n", mullog); + exit(EXIT_FAILURE); + } + + squaring = fopen(sqrlog, "w+"); + if (squaring == NULL) { + fprintf(stderr, "Opening file \"%s\" failed\n",sqrlog); + exit(EXIT_FAILURE); + } + + for (x = 8; x < args.upper_limit_print; x += args.increment_print) { + set_cutoffs(&max_cutoffs); + t1 = s_time_mul(x); + set_cutoffs(&orig); + t2 = s_time_mul(x); + fprintf(multiplying, "%d: %9"PRIu64" %9"PRIu64", %9"PRIi64"\n", x, t1, t2, (int64_t)t2 - (int64_t)t1); + fflush(multiplying); + if (args.verbose == 1) { + printf("MUL %d: %9"PRIu64" %9"PRIu64", %9"PRIi64"\n", x, t1, t2, (int64_t)t2 - (int64_t)t1); + fflush(stdout); + } + set_cutoffs(&max_cutoffs); + t1 = s_time_sqr(x); + set_cutoffs(&orig); + t2 = s_time_sqr(x); + fprintf(squaring,"%d: %9"PRIu64" %9"PRIu64", %9"PRIi64"\n", x, t1, t2, (int64_t)t2 - (int64_t)t1); + fflush(squaring); + if (args.verbose == 1) { + printf("SQR %d: %9"PRIu64" %9"PRIu64", %9"PRIi64"\n", x, t1, t2, (int64_t)t2 - (int64_t)t1); + fflush(stdout); + } + } + printf("Finished. Data for graphing in \"%s\" and \"%s\"\n",mullog, sqrlog); + if (args.verbose == 1) { + set_cutoffs(&orig); + if (args.terse == 1) { + printf("%d %d %d %d\n", + KARATSUBA_MUL_CUTOFF, + KARATSUBA_SQR_CUTOFF, + TOOM_MUL_CUTOFF, + TOOM_SQR_CUTOFF); + } else { + printf("KARATSUBA_MUL_CUTOFF = %d\n", KARATSUBA_MUL_CUTOFF); + printf("KARATSUBA_SQR_CUTOFF = %d\n", KARATSUBA_SQR_CUTOFF); + printf("TOOM_MUL_CUTOFF = %d\n", TOOM_MUL_CUTOFF); + printf("TOOM_SQR_CUTOFF = %d\n", TOOM_SQR_CUTOFF); + } + } + } + exit(EXIT_SUCCESS); +} diff --git a/lib/hcrypto/libtommath/etc/tune_it.sh b/lib/hcrypto/libtommath/etc/tune_it.sh new file mode 100755 index 000000000..5e0fe7c3e --- /dev/null +++ b/lib/hcrypto/libtommath/etc/tune_it.sh @@ -0,0 +1,107 @@ +#!/bin/sh + +die() { + echo "$1 failed" + echo "Exiting" + exit $2 +} +# A linear congruential generator is sufficient for the purpose. +SEED=3735928559 +LCG() { + SEED=$(((1103515245 * $SEED + 12345) % 2147483648)) + echo $SEED +} +median() { +# read everything besides the header from file $1 +# | cut-out the required column $2 +# | sort all the entries numerically +# | show only the first $3 entries +# | show only the last entry + tail -n +2 $1 | cut -d' ' -f$2 | sort -n | head -n $3 | tail -n 1 +} + +MPWD=$(dirname $(readlink -f "$0")) +FILE_NAME="tuning_list" +TOMMATH_CUTOFFS_H="$MPWD/../tommath_cutoffs.h" +BACKUP_SUFFIX=".orig" +RNUM=0 + +############################################################################# +# It would be a good idea to isolate these processes (with e.g.: cpuset) # +# # +# It is not a good idea to e.g: watch high resolution videos while this # +# test are running if you do not have enough memory to avoid page faults. # +############################################################################# + +# Number of rounds overall. +LIMIT=100 +# Number of loops for each input. +RLOOPS=10 +# Offset ( > 0 ) . Runs tests with asymmetric input of the form 1:OFFSET +# Please use another destination for TOMMATH_CUTOFFS_H if you change OFFSET, because the numbers +# with an offset different from 1 (one) are not usable as the general cut-off values +# in "tommath_cutoffs.h". +OFFSET=1 +# Number ( >= 3 ) of positive results (TC-is-faster) accumulated until it is accepted. +# Due to the algorithm used to compute the median in this Posix compliant shell script +# the value needs to be 3 (three), not less, to keep the variation small. +LAG=3 +# Keep the temporary file $FILE_NAME. Set to 0 (zero) to remove it at the end. +# The file is in a format fit to feed into R directly. If you do it and find the median +# of this program to be off by more than a couple: please contact the authors and report +# the numbers from this program and R and the standard deviation. This program is known +# to get larger errors if the standard deviation is larger than ~50. +KEEP_TEMP=1 + +echo "You might like to watch the numbers go up to $LIMIT but it will take a long time!" + +# Might not have sufficient rights or disc full. +echo "km ks tc3m tc3s" > $FILE_NAME || die "Writing header to $FILE_NAME" $? +i=1 +while [ $i -le $LIMIT ]; do + RNUM=$(LCG) + printf "\r%d" $i + "$MPWD"/tune -t -r $RLOOPS -L $LAG -S "$RNUM" -o $OFFSET >> $FILE_NAME || die "tune" $? + i=$((i + 1)) +done + +if [ $KEEP_TEMP -eq 0 ]; then + rm -v $FILE_NAME || die "Removing $KEEP_TEMP" $? +fi + +echo "Writing cut-off values to \"$TOMMATH_CUTOFFS_H\"." +echo "In case of failure: a copy of \"$TOMMATH_CUTOFFS_H\" is in \"$TOMMATH_CUTOFFS_H$BACKUP_SUFFIX\"" + +cp -v $TOMMATH_CUTOFFS_H $TOMMATH_CUTOFFS_H$BACKUP_SUFFIX || die "Making backup copy of $TOMMATH_CUTOFFS_H" $? + +cat << END_OF_INPUT > $TOMMATH_CUTOFFS_H || die "Writing header to $TOMMATH_CUTOFFS_H" $? +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ +/* + Current values evaluated on an AMD A8-6600K (64-bit). + Type "make tune" to optimize them for your machine but + be aware that it may take a long time. It took 2:30 minutes + on the aforementioned machine for example. + */ +END_OF_INPUT + +# The Posix shell does not offer an array data type so we create +# the median with 'standard tools'^TM + +# read the file (without the first line) and count the lines +i=$(tail -n +2 $FILE_NAME | wc -l) +# our median point will be at $i entries +i=$(( (i / 2) + 1 )) +TMP=$(median $FILE_NAME 1 $i) +echo "#define MP_DEFAULT_KARATSUBA_MUL_CUTOFF $TMP" +echo "#define MP_DEFAULT_KARATSUBA_MUL_CUTOFF $TMP" >> $TOMMATH_CUTOFFS_H || die "(km) Appending to $TOMMATH_CUTOFFS_H" $? +TMP=$(median $FILE_NAME 2 $i) +echo "#define MP_DEFAULT_KARATSUBA_SQR_CUTOFF $TMP" +echo "#define MP_DEFAULT_KARATSUBA_SQR_CUTOFF $TMP" >> $TOMMATH_CUTOFFS_H || die "(ks) Appending to $TOMMATH_CUTOFFS_H" $? +TMP=$(median $FILE_NAME 3 $i) +echo "#define MP_DEFAULT_TOOM_MUL_CUTOFF $TMP" +echo "#define MP_DEFAULT_TOOM_MUL_CUTOFF $TMP" >> $TOMMATH_CUTOFFS_H || die "(tc3m) Appending to $TOMMATH_CUTOFFS_H" $? +TMP=$(median $FILE_NAME 4 $i) +echo "#define MP_DEFAULT_TOOM_SQR_CUTOFF $TMP" +echo "#define MP_DEFAULT_TOOM_SQR_CUTOFF $TMP" >> $TOMMATH_CUTOFFS_H || die "(tc3s) Appending to $TOMMATH_CUTOFFS_H" $? + diff --git a/lib/hcrypto/libtommath/gen.pl b/lib/hcrypto/libtommath/gen.pl index 28c78bc29..332994d5c 100644 --- a/lib/hcrypto/libtommath/gen.pl +++ b/lib/hcrypto/libtommath/gen.pl @@ -4,14 +4,17 @@ # add the whole source without any makefile troubles # use strict; +use warnings; -open( OUT, ">mpi.c" ) or die "Couldn't open mpi.c for writing: $!"; -foreach my $filename (glob "bn*.c") { - open( SRC, "<$filename" ) or die "Couldn't open $filename for reading: $!"; - print OUT "/* Start: $filename */\n"; - print OUT while ; - print OUT "\n/* End: $filename */\n\n"; - close SRC or die "Error closing $filename after reading: $!"; +open(my $out, '>', 'mpi.c') or die "Couldn't open mpi.c for writing: $!"; +foreach my $filename (glob 'bn*.c') { + open(my $src, '<', $filename) or die "Couldn't open $filename for reading: $!"; + print {$out} "/* Start: $filename */\n"; + print {$out} $_ while <$src>; + print {$out} "\n/* End: $filename */\n\n"; + close $src or die "Error closing $filename after reading: $!"; } -print OUT "\n/* EOF */\n"; -close OUT or die "Error closing mpi.c after writing: $!"; +print {$out} "\n/* EOF */\n"; +close $out or die "Error closing mpi.c after writing: $!"; + +system('perl -pli -e "s/\s*$//" mpi.c'); diff --git a/lib/hcrypto/libtommath/helper.pl b/lib/hcrypto/libtommath/helper.pl new file mode 100755 index 000000000..e60c1a775 --- /dev/null +++ b/lib/hcrypto/libtommath/helper.pl @@ -0,0 +1,482 @@ +#!/usr/bin/env perl + +use strict; +use warnings; + +use Getopt::Long; +use File::Find 'find'; +use File::Basename 'basename'; +use File::Glob 'bsd_glob'; + +sub read_file { + my $f = shift; + open my $fh, "<", $f or die "FATAL: read_rawfile() cannot open file '$f': $!"; + binmode $fh; + return do { local $/; <$fh> }; +} + +sub write_file { + my ($f, $data) = @_; + die "FATAL: write_file() no data" unless defined $data; + open my $fh, ">", $f or die "FATAL: write_file() cannot open file '$f': $!"; + binmode $fh; + print $fh $data or die "FATAL: write_file() cannot write to '$f': $!"; + close $fh or die "FATAL: write_file() cannot close '$f': $!"; + return; +} + +sub sanitize_comments { + my($content) = @_; + $content =~ s{/\*(.*?)\*/}{my $x=$1; $x =~ s/\w/x/g; "/*$x*/";}egs; + return $content; +} + +sub check_source { + my @all_files = ( + bsd_glob("makefile*"), + bsd_glob("*.{h,c,sh,pl}"), + bsd_glob("*/*.{h,c,sh,pl}"), + ); + + my $fails = 0; + for my $file (sort @all_files) { + my $troubles = {}; + my $lineno = 1; + my $content = read_file($file); + $content = sanitize_comments $content; + push @{$troubles->{crlf_line_end}}, '?' if $content =~ /\r/; + for my $l (split /\n/, $content) { + push @{$troubles->{merge_conflict}}, $lineno if $l =~ /^(<<<<<<<|=======|>>>>>>>)([^<=>]|$)/; + push @{$troubles->{trailing_space}}, $lineno if $l =~ / $/; + push @{$troubles->{tab}}, $lineno if $l =~ /\t/ && basename($file) !~ /^makefile/i; + push @{$troubles->{non_ascii_char}}, $lineno if $l =~ /[^[:ascii:]]/; + push @{$troubles->{cpp_comment}}, $lineno if $file =~ /\.(c|h)$/ && ($l =~ /\s\/\// || $l =~ /\/\/\s/); + # we prefer using XMALLOC, XFREE, XREALLOC, XCALLOC ... + push @{$troubles->{unwanted_malloc}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmalloc\s*\(/; + push @{$troubles->{unwanted_realloc}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\brealloc\s*\(/; + push @{$troubles->{unwanted_calloc}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bcalloc\s*\(/; + push @{$troubles->{unwanted_free}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bfree\s*\(/; + # and we probably want to also avoid the following + push @{$troubles->{unwanted_memcpy}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmemcpy\s*\(/; + push @{$troubles->{unwanted_memset}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmemset\s*\(/; + push @{$troubles->{unwanted_memcpy}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmemcpy\s*\(/; + push @{$troubles->{unwanted_memmove}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmemmove\s*\(/; + push @{$troubles->{unwanted_memcmp}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bmemcmp\s*\(/; + push @{$troubles->{unwanted_strcmp}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bstrcmp\s*\(/; + push @{$troubles->{unwanted_strcpy}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bstrcpy\s*\(/; + push @{$troubles->{unwanted_strncpy}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bstrncpy\s*\(/; + push @{$troubles->{unwanted_clock}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bclock\s*\(/; + push @{$troubles->{unwanted_qsort}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bqsort\s*\(/; + push @{$troubles->{sizeof_no_brackets}}, $lineno if $file =~ /^[^\/]+\.c$/ && $l =~ /\bsizeof\s*[^\(]/; + if ($file =~ m|^[^\/]+\.c$| && $l =~ /^static(\s+[a-zA-Z0-9_]+)+\s+([a-zA-Z0-9_]+)\s*\(/) { + my $funcname = $2; + # static functions should start with s_ + push @{$troubles->{staticfunc_name}}, "$lineno($funcname)" if $funcname !~ /^s_/; + } + $lineno++; + } + for my $k (sort keys %$troubles) { + warn "[$k] $file line:" . join(",", @{$troubles->{$k}}) . "\n"; + $fails++; + } + } + + warn( $fails > 0 ? "check-source: FAIL $fails\n" : "check-source: PASS\n" ); + return $fails; +} + +sub check_comments { + my $fails = 0; + my $first_comment = <<'MARKER'; +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ +MARKER + #my @all_files = (bsd_glob("*.{h,c}"), bsd_glob("*/*.{h,c}")); + my @all_files = (bsd_glob("*.{h,c}")); + for my $f (@all_files) { + my $txt = read_file($f); + if ($txt !~ /\Q$first_comment\E/s) { + warn "[first_comment] $f\n"; + $fails++; + } + } + warn( $fails > 0 ? "check-comments: FAIL $fails\n" : "check-comments: PASS\n" ); + return $fails; +} + +sub check_doc { + my $fails = 0; + my $tex = read_file('doc/bn.tex'); + my $tmh = read_file('tommath.h'); + my @functions = $tmh =~ /\n\s*[a-zA-Z0-9_* ]+?(mp_[a-z0-9_]+)\s*\([^\)]+\)\s*;/sg; + my @macros = $tmh =~ /\n\s*#define\s+([a-z0-9_]+)\s*\([^\)]+\)/sg; + for my $n (sort @functions) { + (my $nn = $n) =~ s/_/\\_/g; # mp_sub_d >> mp\_sub\_d + if ($tex !~ /index\Q{$nn}\E/) { + warn "[missing_doc_for_function] $n\n"; + $fails++ + } + } + for my $n (sort @macros) { + (my $nn = $n) =~ s/_/\\_/g; # mp_iszero >> mp\_iszero + if ($tex !~ /index\Q{$nn}\E/) { + warn "[missing_doc_for_macro] $n\n"; + $fails++ + } + } + warn( $fails > 0 ? "check_doc: FAIL $fails\n" : "check-doc: PASS\n" ); + return $fails; +} + +sub prepare_variable { + my ($varname, @list) = @_; + my $output = "$varname="; + my $len = length($output); + foreach my $obj (sort @list) { + $len = $len + length $obj; + $obj =~ s/\*/\$/; + if ($len > 100) { + $output .= "\\\n"; + $len = length $obj; + } + $output .= $obj . ' '; + } + $output =~ s/ $//; + return $output; +} + +sub prepare_msvc_files_xml { + my ($all, $exclude_re, $targets) = @_; + my $last = []; + my $depth = 2; + + # sort files in the same order as visual studio (ugly, I know) + my @parts = (); + for my $orig (@$all) { + my $p = $orig; + $p =~ s|/|/~|g; + $p =~ s|/~([^/]+)$|/$1|g; + my @l = map { sprintf "% -99s", $_ } split /\//, $p; + push @parts, [ $orig, join(':', @l) ]; + } + my @sorted = map { $_->[0] } sort { $a->[1] cmp $b->[1] } @parts; + + my $files = "\r\n"; + for my $full (@sorted) { + my @items = split /\//, $full; # split by '/' + $full =~ s|/|\\|g; # replace '/' bt '\' + shift @items; # drop first one (src) + pop @items; # drop last one (filename.ext) + my $current = \@items; + if (join(':', @$current) ne join(':', @$last)) { + my $common = 0; + $common++ while ($last->[$common] && $current->[$common] && $last->[$common] eq $current->[$common]); + my $back = @$last - $common; + if ($back > 0) { + $files .= ("\t" x --$depth) . "\r\n" for (1..$back); + } + my $fwd = [ @$current ]; splice(@$fwd, 0, $common); + for my $i (0..scalar(@$fwd) - 1) { + $files .= ("\t" x $depth) . "[$i]\"\r\n"; + $files .= ("\t" x $depth) . "\t>\r\n"; + $depth++; + } + $last = $current; + } + $files .= ("\t" x $depth) . "\r\n"; + if ($full =~ $exclude_re) { + for (@$targets) { + $files .= ("\t" x $depth) . "\t\r\n"; + $files .= ("\t" x $depth) . "\t\t\r\n"; + $files .= ("\t" x $depth) . "\t\r\n"; + } + } + $files .= ("\t" x $depth) . "\r\n"; + } + $files .= ("\t" x --$depth) . "\r\n" for (@$last); + $files .= "\t"; + return $files; +} + +sub patch_file { + my ($content, @variables) = @_; + for my $v (@variables) { + if ($v =~ /^([A-Z0-9_]+)\s*=.*$/si) { + my $name = $1; + $content =~ s/\n\Q$name\E\b.*?[^\\]\n/\n$v\n/s; + } + else { + die "patch_file failed: " . substr($v, 0, 30) . ".."; + } + } + return $content; +} + +sub process_makefiles { + my $write = shift; + my $changed_count = 0; + my @o = map { my $x = $_; $x =~ s/\.c$/.o/; $x } bsd_glob("*.c"); + my @all = bsd_glob("*.{c,h}"); + + my $var_o = prepare_variable("OBJECTS", @o); + (my $var_obj = $var_o) =~ s/\.o\b/.obj/sg; + + # update MSVC project files + my $msvc_files = prepare_msvc_files_xml(\@all, qr/NOT_USED_HERE/, ['Debug|Win32', 'Release|Win32', 'Debug|x64', 'Release|x64']); + for my $m (qw/libtommath_VS2008.vcproj/) { + my $old = read_file($m); + my $new = $old; + $new =~ s|.*|$msvc_files|s; + if ($old ne $new) { + write_file($m, $new) if $write; + warn "changed: $m\n"; + $changed_count++; + } + } + + # update OBJECTS + HEADERS in makefile* + for my $m (qw/ makefile makefile.shared makefile_include.mk makefile.msvc makefile.unix makefile.mingw /) { + my $old = read_file($m); + my $new = $m eq 'makefile.msvc' ? patch_file($old, $var_obj) + : patch_file($old, $var_o); + if ($old ne $new) { + write_file($m, $new) if $write; + warn "changed: $m\n"; + $changed_count++; + } + } + + if ($write) { + return 0; # no failures + } + else { + warn( $changed_count > 0 ? "check-makefiles: FAIL $changed_count\n" : "check-makefiles: PASS\n" ); + return $changed_count; + } +} + +sub draw_func +{ + my ($deplist, $depmap, $out, $indent, $funcslist) = @_; + my @funcs = split ',', $funcslist; + # try this if you want to have a look at a minimized version of the callgraph without all the trivial functions + #if ($deplist =~ /$funcs[0]/ || $funcs[0] =~ /BN_MP_(ADD|SUB|CLEAR|CLEAR_\S+|DIV|MUL|COPY|ZERO|GROW|CLAMP|INIT|INIT_\S+|SET|ABS|CMP|CMP_D|EXCH)_C/) { + if ($deplist =~ /$funcs[0]/) { + return $deplist; + } else { + $deplist = $deplist . $funcs[0]; + } + if ($indent == 0) { + } elsif ($indent >= 1) { + print {$out} '| ' x ($indent - 1) . '+--->'; + } + print {$out} $funcs[0] . "\n"; + shift @funcs; + my $olddeplist = $deplist; + foreach my $i (@funcs) { + $deplist = draw_func($deplist, $depmap, $out, $indent + 1, ${$depmap}{$i}) if exists ${$depmap}{$i}; + } + return $olddeplist; +} + +sub update_dep +{ + #open class file and write preamble + open(my $class, '>', 'tommath_class.h') or die "Couldn't open tommath_class.h for writing\n"; + print {$class} << 'EOS'; +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#if !(defined(LTM1) && defined(LTM2) && defined(LTM3)) +#define LTM_INSIDE +#if defined(LTM2) +# define LTM3 +#endif +#if defined(LTM1) +# define LTM2 +#endif +#define LTM1 +#if defined(LTM_ALL) +EOS + + foreach my $filename (glob 'bn*.c') { + my $define = $filename; + + print "Processing $filename\n"; + + # convert filename to upper case so we can use it as a define + $define =~ tr/[a-z]/[A-Z]/; + $define =~ tr/\./_/; + print {$class} "# define $define\n"; + + # now copy text and apply #ifdef as required + my $apply = 0; + open(my $src, '<', $filename); + open(my $out, '>', 'tmp'); + + # first line will be the #ifdef + my $line = <$src>; + if ($line =~ /include/) { + print {$out} $line; + } else { + print {$out} << "EOS"; +#include "tommath_private.h" +#ifdef $define +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ +$line +EOS + $apply = 1; + } + while (<$src>) { + if ($_ !~ /tommath\.h/) { + print {$out} $_; + } + } + if ($apply == 1) { + print {$out} "#endif\n"; + } + close $src; + close $out; + + unlink $filename; + rename 'tmp', $filename; + } + print {$class} "#endif\n#endif\n"; + + # now do classes + my %depmap; + foreach my $filename (glob 'bn*.c') { + my $content; + if ($filename =~ "bn_deprecated.c") { + open(my $src, '<', $filename) or die "Can't open source file!\n"; + read $src, $content, -s $src; + close $src; + } else { + my $cc = $ENV{'CC'} || 'gcc'; + $content = `$cc -E -x c -DLTM_ALL $filename`; + $content =~ s/^# 1 "$filename".*?^# 2 "$filename"//ms; + } + + # convert filename to upper case so we can use it as a define + $filename =~ tr/[a-z]/[A-Z]/; + $filename =~ tr/\./_/; + + print {$class} "#if defined($filename)\n"; + my $list = $filename; + + # strip comments + $content =~ s{/\*.*?\*/}{}gs; + + # scan for mp_* and make classes + my @deps = (); + foreach my $line (split /\n/, $content) { + while ($line =~ /(fast_)?(s_)?mp\_[a-z_0-9]*((?=\;)|(?=\())|(?<=\()mp\_[a-z_0-9]*(?=\()/g) { + my $a = $&; + next if $a eq "mp_err"; + $a =~ tr/[a-z]/[A-Z]/; + $a = 'BN_' . $a . '_C'; + push @deps, $a; + } + } + @deps = sort(@deps); + foreach my $a (@deps) { + if ($list !~ /$a/) { + print {$class} "# define $a\n"; + } + $list = $list . ',' . $a; + } + $depmap{$filename} = $list; + + print {$class} "#endif\n\n"; + } + + print {$class} << 'EOS'; +#ifdef LTM_INSIDE +#undef LTM_INSIDE +#ifdef LTM3 +# define LTM_LAST +#endif + +#include "tommath_superclass.h" +#include "tommath_class.h" +#else +# define LTM_LAST +#endif +EOS + close $class; + + #now let's make a cool call graph... + + open(my $out, '>', 'callgraph.txt'); + foreach (sort keys %depmap) { + draw_func("", \%depmap, $out, 0, $depmap{$_}); + print {$out} "\n\n"; + } + close $out; + + return 0; +} + +sub generate_def { + my @files = split /\n/, `git ls-files`; + @files = grep(/\.c/, @files); + @files = map { my $x = $_; $x =~ s/^bn_|\.c$//g; $x; } @files; + @files = grep(!/mp_radix_smap/, @files); + + push(@files, qw(mp_set_int mp_set_long mp_set_long_long mp_get_int mp_get_long mp_get_long_long mp_init_set_int)); + + my $files = join("\n ", sort(grep(/^mp_/, @files))); + write_file "tommath.def", "; libtommath +; +; Use this command to produce a 32-bit .lib file, for use in any MSVC version +; lib -machine:X86 -name:libtommath.dll -def:tommath.def -out:tommath.lib +; Use this command to produce a 64-bit .lib file, for use in any MSVC version +; lib -machine:X64 -name:libtommath.dll -def:tommath.def -out:tommath.lib +; +EXPORTS + $files +"; + return 0; +} + +sub die_usage { + die <<"MARKER"; +usage: $0 -s OR $0 --check-source + $0 -o OR $0 --check-comments + $0 -m OR $0 --check-makefiles + $0 -a OR $0 --check-all + $0 -u OR $0 --update-files +MARKER +} + +GetOptions( "s|check-source" => \my $check_source, + "o|check-comments" => \my $check_comments, + "m|check-makefiles" => \my $check_makefiles, + "d|check-doc" => \my $check_doc, + "a|check-all" => \my $check_all, + "u|update-files" => \my $update_files, + "h|help" => \my $help + ) or die_usage; + +my $failure; +$failure ||= check_source() if $check_all || $check_source; +$failure ||= check_comments() if $check_all || $check_comments; +$failure ||= check_doc() if $check_doc; # temporarily excluded from --check-all +$failure ||= process_makefiles(0) if $check_all || $check_makefiles; +$failure ||= process_makefiles(1) if $update_files; +$failure ||= update_dep() if $update_files; +$failure ||= generate_def() if $update_files; + +die_usage unless defined $failure; +exit $failure ? 1 : 0; diff --git a/lib/hcrypto/libtommath/libtommath.dsp b/lib/hcrypto/libtommath/libtommath.dsp deleted file mode 100644 index 6b8908f07..000000000 --- a/lib/hcrypto/libtommath/libtommath.dsp +++ /dev/null @@ -1,572 +0,0 @@ -# Microsoft Developer Studio Project File - Name="libtommath" - Package Owner=<4> -# Microsoft Developer Studio Generated Build File, Format Version 6.00 -# ** DO NOT EDIT ** - -# TARGTYPE "Win32 (x86) Static Library" 0x0104 - -CFG=libtommath - Win32 Debug -!MESSAGE This is not a valid makefile. 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-SOURCE=.\tommath_class.h -# End Source File -# Begin Source File - -SOURCE=.\tommath_superclass.h -# End Source File -# End Target -# End Project diff --git a/lib/hcrypto/libtommath/libtommath.pc.in b/lib/hcrypto/libtommath/libtommath.pc.in new file mode 100644 index 000000000..099b1cd74 --- /dev/null +++ b/lib/hcrypto/libtommath/libtommath.pc.in @@ -0,0 +1,10 @@ +prefix=@to-be-replaced@ +exec_prefix=${prefix} +libdir=${exec_prefix}/lib +includedir=${prefix}/include + +Name: LibTomMath +Description: public domain library for manipulating large integer numbers +Version: @to-be-replaced@ +Libs: -L${libdir} -ltommath +Cflags: -I${includedir} diff --git a/lib/hcrypto/libtommath/libtommath_VS2008.sln b/lib/hcrypto/libtommath/libtommath_VS2008.sln new file mode 100644 index 000000000..6bfc159b7 --- /dev/null +++ b/lib/hcrypto/libtommath/libtommath_VS2008.sln @@ -0,0 +1,29 @@ + +Microsoft Visual Studio Solution File, Format Version 10.00 +# Visual Studio 2008 +Project("{8BC9CEB8-8B4A-11D0-8D11-00A0C91BC942}") = "tommath", "libtommath_VS2008.vcproj", "{42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}" +EndProject +Global + GlobalSection(SolutionConfigurationPlatforms) = preSolution + Debug|Win32 = Debug|Win32 + Debug|x64 = Debug|x64 + Release|Win32 = Release|Win32 + Release|x64 = Release|x64 + EndGlobalSection + GlobalSection(ProjectConfigurationPlatforms) = postSolution + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Debug|Win32.ActiveCfg = Debug|Win32 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Debug|Win32.Build.0 = Debug|Win32 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Debug|x64.ActiveCfg = Debug|x64 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Debug|x64.Build.0 = Debug|x64 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Release|Win32.ActiveCfg = Release|Win32 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Release|Win32.Build.0 = Release|Win32 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Release|x64.ActiveCfg = Release|x64 + {42109FEE-B0B9-4FCD-9E56-2863BF8C55D2}.Release|x64.Build.0 = Release|x64 + EndGlobalSection + GlobalSection(SolutionProperties) = preSolution + HideSolutionNode = FALSE + EndGlobalSection + GlobalSection(ExtensibilityGlobals) = postSolution + SolutionGuid = {83B84178-7B4F-4B78-9C5D-17B8201D5B61} + EndGlobalSection +EndGlobal diff --git a/lib/hcrypto/libtommath/libtommath_VS2008.vcproj b/lib/hcrypto/libtommath/libtommath_VS2008.vcproj new file mode 100644 index 000000000..67cc89bf7 --- /dev/null +++ b/lib/hcrypto/libtommath/libtommath_VS2008.vcproj @@ -0,0 +1,966 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/lib/hcrypto/libtommath/logs/README b/lib/hcrypto/libtommath/logs/README index 4c4e5f3aa..ea20c8137 100644 --- a/lib/hcrypto/libtommath/logs/README +++ b/lib/hcrypto/libtommath/logs/README @@ -1,13 +1,13 @@ -To use the pretty graphs you have to first build/run the ltmtest from the root directory of the package. -Todo this type - -make timing ; ltmtest - -in the root. It will run for a while [about ten minutes on most PCs] and produce a series of .log files in logs/. - -After doing that run "gnuplot graphs.dem" to make the PNGs. If you managed todo that all so far just open index.html to view -them all :-) - -Have fun - -Tom +To use the pretty graphs you have to first build/run the ltmtest from the root directory of the package. +Todo this type + +make timing ; ltmtest + +in the root. It will run for a while [about ten minutes on most PCs] and produce a series of .log files in logs/. + +After doing that run "gnuplot graphs.dem" to make the PNGs. If you managed todo that all so far just open index.html to view +them all :-) + +Have fun + +Tom \ No newline at end of file diff --git a/lib/hcrypto/libtommath/logs/add.log b/lib/hcrypto/libtommath/logs/add.log index 43503ac13..0ed7b70d9 100644 --- a/lib/hcrypto/libtommath/logs/add.log +++ b/lib/hcrypto/libtommath/logs/add.log @@ -1,16 +1,16 @@ -480 87 -960 111 -1440 135 -1920 159 -2400 200 -2880 224 -3360 248 -3840 272 -4320 296 -4800 320 -5280 344 -5760 368 -6240 392 -6720 416 -7200 440 -7680 464 + 480 48 + 960 61 + 1440 82 + 1920 97 + 2400 106 + 2880 112 + 3360 127 + 3840 130 + 4320 146 + 4800 157 + 5280 174 + 5760 185 + 6240 200 + 6720 214 + 7200 230 + 7680 244 diff --git a/lib/hcrypto/libtommath/logs/addsub.png b/lib/hcrypto/libtommath/logs/addsub.png index a5679ac56..b8ffef74b 100644 Binary files a/lib/hcrypto/libtommath/logs/addsub.png and b/lib/hcrypto/libtommath/logs/addsub.png differ diff --git a/lib/hcrypto/libtommath/logs/expt.log b/lib/hcrypto/libtommath/logs/expt.log index 70932abc4..2e5ee308f 100644 --- a/lib/hcrypto/libtommath/logs/expt.log +++ b/lib/hcrypto/libtommath/logs/expt.log @@ -1,7 +1,7 @@ -513 1435869 -769 3544970 -1025 7791638 -2049 46902238 -2561 85334899 -3073 141451412 -4097 308770310 + 513 446633 + 769 1110301 + 1025 2414927 + 2049 14870787 + 2561 26299761 + 3073 44323310 + 4097 98934292 diff --git a/lib/hcrypto/libtommath/logs/expt.png b/lib/hcrypto/libtommath/logs/expt.png index 9ee8bb769..27c53eecf 100644 Binary files a/lib/hcrypto/libtommath/logs/expt.png and b/lib/hcrypto/libtommath/logs/expt.png differ diff --git a/lib/hcrypto/libtommath/logs/expt_2k.log b/lib/hcrypto/libtommath/logs/expt_2k.log index 97d325f3f..140b92f2c 100644 --- a/lib/hcrypto/libtommath/logs/expt_2k.log +++ b/lib/hcrypto/libtommath/logs/expt_2k.log @@ -1,5 +1,6 @@ -607 2109225 -1279 10148314 -2203 34126877 -3217 82716424 -4253 161569606 + 521 533515 + 607 675230 + 1279 2560713 + 2203 7468422 + 3217 17314246 + 4253 33899969 diff --git a/lib/hcrypto/libtommath/logs/expt_2kl.log b/lib/hcrypto/libtommath/logs/expt_2kl.log index d9ad4be24..1dc495f9e 100644 --- a/lib/hcrypto/libtommath/logs/expt_2kl.log +++ b/lib/hcrypto/libtommath/logs/expt_2kl.log @@ -1,4 +1,3 @@ -1024 7705271 -2048 34286851 -4096 165207491 -521 1618631 + 1024 2210287 + 2048 7940364 + 4096 35903891 diff --git a/lib/hcrypto/libtommath/logs/expt_dr.log b/lib/hcrypto/libtommath/logs/expt_dr.log index c6bbe0777..3752ea81a 100644 --- a/lib/hcrypto/libtommath/logs/expt_dr.log +++ b/lib/hcrypto/libtommath/logs/expt_dr.log @@ -1,7 +1,7 @@ -532 1928550 -784 3763908 -1036 7564221 -1540 16566059 -2072 32283784 -3080 79851565 -4116 157843530 + 532 642330 + 784 1138699 + 1036 1972796 + 1540 3912241 + 2072 7075836 + 3080 16420867 + 4116 32477173 diff --git a/lib/hcrypto/libtommath/logs/graphs.dem b/lib/hcrypto/libtommath/logs/graphs.dem index dfaf6138a..538e5c075 100644 --- a/lib/hcrypto/libtommath/logs/graphs.dem +++ b/lib/hcrypto/libtommath/logs/graphs.dem @@ -1,5 +1,4 @@ set terminal png -set size 1.75 set ylabel "Cycles per Operation" set xlabel "Operand size (bits)" diff --git a/lib/hcrypto/libtommath/logs/invmod.log b/lib/hcrypto/libtommath/logs/invmod.log index e69de29bb..7d22449ef 100644 --- a/lib/hcrypto/libtommath/logs/invmod.log +++ b/lib/hcrypto/libtommath/logs/invmod.log @@ -0,0 +1,8 @@ + 240 58197 + 480 86617 + 720 255279 + 960 399626 + 1200 533330 + 1440 470046 + 1680 906754 + 1920 1132009 diff --git a/lib/hcrypto/libtommath/logs/invmod.png b/lib/hcrypto/libtommath/logs/invmod.png index 0a8a4ad77..5c09e9012 100644 Binary files a/lib/hcrypto/libtommath/logs/invmod.png and b/lib/hcrypto/libtommath/logs/invmod.png differ diff --git a/lib/hcrypto/libtommath/logs/mult.log b/lib/hcrypto/libtommath/logs/mult.log index 33563fc67..841b40b96 100644 --- a/lib/hcrypto/libtommath/logs/mult.log +++ b/lib/hcrypto/libtommath/logs/mult.log @@ -1,84 +1,84 @@ -271 555 -390 855 -508 1161 -631 1605 -749 2117 -871 2687 -991 3329 -1108 4084 -1231 4786 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a/lib/hcrypto/libtommath/logs/mult.png b/lib/hcrypto/libtommath/logs/mult.png index 4f7a4eed7..9681183a2 100644 Binary files a/lib/hcrypto/libtommath/logs/mult.png and b/lib/hcrypto/libtommath/logs/mult.png differ diff --git a/lib/hcrypto/libtommath/logs/mult_kara.log b/lib/hcrypto/libtommath/logs/mult_kara.log index 7136c7931..91b59cbfd 100644 --- a/lib/hcrypto/libtommath/logs/mult_kara.log +++ b/lib/hcrypto/libtommath/logs/mult_kara.log @@ -1,84 +1,84 @@ -271 560 -391 870 -511 1159 -631 1605 -750 2111 -871 2737 -991 3361 -1111 4054 -1231 4778 -1351 5600 -1471 6404 -1591 7323 -1710 8255 -1831 9239 -1948 10257 -2070 11397 -2190 12531 -2308 13665 -2429 14870 -2550 16175 -2671 17539 -2787 18879 -2911 20350 -3031 21807 -3150 23415 -3270 24897 -3388 26567 -3511 28205 -3627 30076 -3751 31744 -3869 33657 -3991 35425 -4111 37522 -4229 39363 -4351 41503 -4470 43491 -4590 45827 -4711 47795 -4828 50166 -4951 52318 -5070 54911 -5191 57036 -5308 58237 -5431 60248 -5551 62678 -5671 64786 -5791 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15825 + 5520 16512 + 5639 17379 + 5757 17596 + 5879 18350 + 6000 18976 + 6115 19601 + 6240 20076 + 6354 20515 + 6480 21670 + 6600 22312 + 6716 22647 + 6839 23437 + 6960 24164 + 7080 24723 + 7199 25454 + 7320 26092 + 7440 26912 + 7557 27521 + 7677 28015 + 7800 28885 + 7919 29483 + 8040 30115 + 8160 31236 + 8280 31975 + 8400 30835 + 8520 31565 + 8639 32380 + 8760 32760 + 8879 33590 + 8996 34553 + 9119 35185 + 9239 36146 + 9358 36815 + 9480 39630 + 9596 43022 + 9720 41219 + 9840 41596 + 9960 42354 + 10080 43352 + 10200 43915 diff --git a/lib/hcrypto/libtommath/logs/sqr.log b/lib/hcrypto/libtommath/logs/sqr.log index cd29fc5c1..93234a145 100644 --- a/lib/hcrypto/libtommath/logs/sqr.log +++ b/lib/hcrypto/libtommath/logs/sqr.log @@ -1,84 +1,84 @@ -265 562 -389 882 -509 1207 -631 1572 -750 1990 -859 2433 -991 2894 -1109 3555 -1230 4228 -1350 5018 -1471 5805 -1591 6579 -1709 7415 -1829 8329 -1949 9225 -2071 10139 -2188 11239 -2309 12178 -2431 13212 -2551 14294 -2671 15551 -2791 16512 -2911 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b/lib/hcrypto/libtommath/logs/sqr_kara.log @@ -1,84 +1,84 @@ -271 560 -388 878 -511 1179 -629 1625 -751 1988 -871 2423 -989 2896 -1111 3561 -1231 4209 -1350 5015 -1470 5804 -1591 6556 -1709 7420 -1831 8263 -1951 9173 -2070 10153 -2191 11229 -2310 12167 -2431 13211 -2550 14309 -2671 15524 -2788 16525 -2910 17712 -3028 18822 -3148 20220 -3271 21343 -3391 22652 -3511 23944 -3630 25485 -3750 26778 -3868 28201 -3990 29653 -4111 31393 -4225 32841 -4350 34328 -4471 35786 -4590 37652 -4711 39245 -4830 40876 -4951 42433 -5068 44547 -5191 46321 -5311 48140 -5430 49727 -5550 52034 -5671 53954 -5791 55921 -5908 57597 -6031 60084 -6148 62226 -6270 64295 -6390 66045 -6511 68779 -6629 71003 -6751 73169 -6871 74992 -6991 77895 -7110 80376 -7231 82628 -7351 84468 -7470 87664 -7591 90284 -7711 91352 -7828 93995 -7950 96276 -8071 98691 -8190 101256 -8308 103631 -8431 105222 -8550 108343 -8671 110281 -8787 112764 -8911 115397 -9031 117690 -9151 120266 -9271 122715 -9391 124624 -9510 127937 -9630 130313 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22394 + 9480 23105 + 9598 23334 + 9718 25301 + 9840 26053 + 9960 26565 + 10079 26812 + 10200 27300 diff --git a/lib/hcrypto/libtommath/logs/sub.log b/lib/hcrypto/libtommath/logs/sub.log index 9f84fa2ef..87c0160f8 100644 --- a/lib/hcrypto/libtommath/logs/sub.log +++ b/lib/hcrypto/libtommath/logs/sub.log @@ -1,16 +1,16 @@ -480 94 -960 116 -1440 140 -1920 164 -2400 205 -2880 229 -3360 253 -3840 277 -4320 299 -4800 321 -5280 345 -5760 371 -6240 395 -6720 419 -7200 441 -7680 465 + 480 36 + 960 51 + 1440 64 + 1920 78 + 2400 90 + 2880 105 + 3360 118 + 3840 133 + 4320 146 + 4800 161 + 5280 182 + 5760 201 + 6240 201 + 6720 214 + 7200 228 + 7680 243 diff --git a/lib/hcrypto/libtommath/makefile b/lib/hcrypto/libtommath/makefile index 3e254d475..be9fac657 100644 --- a/lib/hcrypto/libtommath/makefile +++ b/lib/hcrypto/libtommath/makefile @@ -2,42 +2,10 @@ # #Tom St Denis -#version of library -VERSION=0.41 - -CFLAGS += -I./ -Wall -W -Wshadow -Wsign-compare - -ifndef MAKE - MAKE=make -endif - -ifndef IGNORE_SPEED - -#for speed -CFLAGS += -O3 -funroll-loops - -#for size -#CFLAGS += -Os - -#x86 optimizations [should be valid for any GCC install though] -CFLAGS += -fomit-frame-pointer - -#debug -#CFLAGS += -g3 - -endif - -#install as this user -ifndef INSTALL_GROUP - GROUP=wheel +ifeq ($V,1) +silent= else - GROUP=$(INSTALL_GROUP) -endif - -ifndef INSTALL_USER - USER=root -else - USER=$(INSTALL_USER) +silent=@ endif #default files to install @@ -45,142 +13,153 @@ ifndef LIBNAME LIBNAME=libtommath.a endif -default: ${LIBNAME} +coverage: LIBNAME:=-Wl,--whole-archive $(LIBNAME) -Wl,--no-whole-archive -HEADERS=tommath.h tommath_class.h tommath_superclass.h +include makefile_include.mk -#LIBPATH-The directory for libtommath to be installed to. -#INCPATH-The directory to install the header files for libtommath. -#DATAPATH-The directory to install the pdf docs. -DESTDIR= -LIBPATH=/usr/lib -INCPATH=/usr/include -DATAPATH=/usr/share/doc/libtommath/pdf +%.o: %.c $(HEADERS) +ifneq ($V,1) + @echo " * ${CC} $@" +endif + ${silent} ${CC} -c ${LTM_CFLAGS} $< -o $@ -OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ -bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ -bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ -bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ -bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ -bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ -bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ -bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ -bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ -bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ -bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ -bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ -bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ -bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ -bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ -bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ -bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ -bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ -bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ -bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ -bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \ -bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ -bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ -bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ -bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ -bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \ -bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o +LCOV_ARGS=--directory . + +#START_INS +OBJECTS=bn_cutoffs.o bn_deprecated.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o bn_mp_addmod.o \ +bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \ +bn_mp_cnt_lsb.o bn_mp_complement.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_decr.o bn_mp_div.o bn_mp_div_2.o \ +bn_mp_div_2d.o bn_mp_div_3.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o \ +bn_mp_error_to_string.o bn_mp_exch.o bn_mp_expt_u32.o bn_mp_exptmod.o bn_mp_exteuclid.o bn_mp_fread.o \ +bn_mp_from_sbin.o bn_mp_from_ubin.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_double.o bn_mp_get_i32.o \ +bn_mp_get_i64.o bn_mp_get_l.o bn_mp_get_ll.o bn_mp_get_mag_u32.o bn_mp_get_mag_u64.o bn_mp_get_mag_ul.o \ +bn_mp_get_mag_ull.o bn_mp_grow.o bn_mp_incr.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_i32.o \ +bn_mp_init_i64.o bn_mp_init_l.o bn_mp_init_ll.o bn_mp_init_multi.o bn_mp_init_set.o bn_mp_init_size.o \ +bn_mp_init_u32.o bn_mp_init_u64.o bn_mp_init_ul.o bn_mp_init_ull.o bn_mp_invmod.o bn_mp_is_square.o \ +bn_mp_iseven.o bn_mp_isodd.o bn_mp_kronecker.o bn_mp_lcm.o bn_mp_log_u32.o bn_mp_lshd.o bn_mp_mod.o \ +bn_mp_mod_2d.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \ +bn_mp_montgomery_setup.o bn_mp_mul.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \ +bn_mp_or.o bn_mp_pack.o bn_mp_pack_count.o bn_mp_prime_fermat.o bn_mp_prime_frobenius_underwood.o \ +bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \ +bn_mp_prime_rabin_miller_trials.o bn_mp_prime_rand.o bn_mp_prime_strong_lucas_selfridge.o \ +bn_mp_radix_size.o bn_mp_radix_smap.o bn_mp_rand.o bn_mp_read_radix.o bn_mp_reduce.o bn_mp_reduce_2k.o \ +bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce_is_2k.o \ +bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_root_u32.o bn_mp_rshd.o bn_mp_sbin_size.o bn_mp_set.o \ +bn_mp_set_double.o bn_mp_set_i32.o bn_mp_set_i64.o bn_mp_set_l.o bn_mp_set_ll.o bn_mp_set_u32.o \ +bn_mp_set_u64.o bn_mp_set_ul.o bn_mp_set_ull.o bn_mp_shrink.o bn_mp_signed_rsh.o bn_mp_sqr.o \ +bn_mp_sqrmod.o bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o \ +bn_mp_to_radix.o bn_mp_to_sbin.o bn_mp_to_ubin.o bn_mp_ubin_size.o bn_mp_unpack.o bn_mp_xor.o bn_mp_zero.o \ +bn_prime_tab.o bn_s_mp_add.o bn_s_mp_balance_mul.o bn_s_mp_exptmod.o bn_s_mp_exptmod_fast.o \ +bn_s_mp_get_bit.o bn_s_mp_invmod_fast.o bn_s_mp_invmod_slow.o bn_s_mp_karatsuba_mul.o \ +bn_s_mp_karatsuba_sqr.o bn_s_mp_montgomery_reduce_fast.o bn_s_mp_mul_digs.o bn_s_mp_mul_digs_fast.o \ +bn_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs_fast.o bn_s_mp_prime_is_divisible.o \ +bn_s_mp_rand_jenkins.o bn_s_mp_rand_platform.o bn_s_mp_reverse.o bn_s_mp_sqr.o bn_s_mp_sqr_fast.o \ +bn_s_mp_sub.o bn_s_mp_toom_mul.o bn_s_mp_toom_sqr.o + +#END_INS $(LIBNAME): $(OBJECTS) $(AR) $(ARFLAGS) $@ $(OBJECTS) - ranlib $@ + $(RANLIB) $@ #make a profiled library (takes a while!!!) # # This will build the library with profile generation # then run the test demo and rebuild the library. -# +# # So far I've seen improvements in the MP math profiled: make CFLAGS="$(CFLAGS) -fprofile-arcs -DTESTING" timing - ./ltmtest - rm -f *.a *.o ltmtest + ./timing + rm -f *.a *.o timing make CFLAGS="$(CFLAGS) -fbranch-probabilities" -#make a single object profiled library +#make a single object profiled library profiled_single: perl gen.pl - $(CC) $(CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o - $(CC) $(CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -o ltmtest - ./ltmtest - rm -f *.o ltmtest - $(CC) $(CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o + $(CC) $(LTM_CFLAGS) -fprofile-arcs -DTESTING -c mpi.c -o mpi.o + $(CC) $(LTM_CFLAGS) -DTESTING -DTIMER demo/timing.c mpi.o -lgcov -o timing + ./timing + rm -f *.o timing + $(CC) $(LTM_CFLAGS) -fbranch-probabilities -DTESTING -c mpi.c -o mpi.o $(AR) $(ARFLAGS) $(LIBNAME) mpi.o - ranlib $(LIBNAME) + ranlib $(LIBNAME) install: $(LIBNAME) - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH) - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) - install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH) - install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) + install -d $(DESTDIR)$(LIBPATH) + install -d $(DESTDIR)$(INCPATH) + install -m 644 $(LIBNAME) $(DESTDIR)$(LIBPATH) + install -m 644 $(HEADERS_PUB) $(DESTDIR)$(INCPATH) -test: $(LIBNAME) demo/demo.o - $(CC) $(CFLAGS) demo/demo.o $(LIBNAME) -o test - -mtest: test - cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest - -timing: $(LIBNAME) - $(CC) $(CFLAGS) -DTIMER demo/timing.c $(LIBNAME) -o ltmtest +uninstall: + rm $(DESTDIR)$(LIBPATH)/$(LIBNAME) + rm $(HEADERS_PUB:%=$(DESTDIR)$(INCPATH)/%) -# makes the LTM book DVI file, requires tetex, perl and makeindex [part of tetex I think] -docdvi: tommath.src - cd pics ; MAKE=${MAKE} ${MAKE} - echo "hello" > tommath.ind - perl booker.pl - latex tommath > /dev/null - latex tommath > /dev/null - makeindex tommath - latex tommath > /dev/null +test_standalone: test + @echo "test_standalone is deprecated, please use make-target 'test'" -# poster, makes the single page PDF poster -poster: poster.tex - pdflatex poster - rm -f poster.aux poster.log +DEMOS=test mtest_opponent -# makes the LTM book PDF file, requires tetex, cleans up the LaTeX temp files -docs: docdvi - dvipdf tommath - rm -f tommath.log tommath.aux tommath.dvi tommath.idx tommath.toc tommath.lof tommath.ind tommath.ilg - cd pics ; MAKE=${MAKE} ${MAKE} clean - -#LTM user manual -mandvi: bn.tex - echo "hello" > bn.ind - latex bn > /dev/null - latex bn > /dev/null - makeindex bn - latex bn > /dev/null +define DEMO_template +$(1): demo/$(1).o demo/shared.o $$(LIBNAME) + $$(CC) $$(LTM_CFLAGS) $$(LTM_LFLAGS) $$^ -o $$@ +endef -#LTM user manual [pdf] -manual: mandvi - pdflatex bn >/dev/null - rm -f bn.aux bn.dvi bn.log bn.idx bn.lof bn.out bn.toc +$(foreach demo, $(strip $(DEMOS)), $(eval $(call DEMO_template,$(demo)))) -pretty: - perl pretty.build +.PHONY: mtest +mtest: + cd mtest ; $(CC) $(LTM_CFLAGS) -O0 mtest.c $(LTM_LFLAGS) -o mtest -clean: - rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \ - *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.da *.dyn *.dpi tommath.tex `find . -type f | grep [~] | xargs` *.lo *.la - rm -rf .libs - cd etc ; MAKE=${MAKE} ${MAKE} clean - cd pics ; MAKE=${MAKE} ${MAKE} clean +timing: $(LIBNAME) demo/timing.c + $(CC) $(LTM_CFLAGS) -DTIMER demo/timing.c $(LIBNAME) $(LTM_LFLAGS) -o timing -#zipup the project (take that!) -no_oops: clean - cd .. ; cvs commit - echo Scanning for scratch/dirty files - find . -type f | grep -v CVS | xargs -n 1 bash mess.sh +tune: $(LIBNAME) + $(MAKE) -C etc tune CFLAGS="$(LTM_CFLAGS)" + $(MAKE) -zipup: clean manual poster docs - perl gen.pl ; mv mpi.c pre_gen/ ; \ - cd .. ; rm -rf ltm* libtommath-$(VERSION) ; mkdir libtommath-$(VERSION) ; \ - cp -R ./libtommath/* ./libtommath-$(VERSION)/ ; \ - tar -c libtommath-$(VERSION)/* | bzip2 -9vvc > ltm-$(VERSION).tar.bz2 ; \ - zip -9 -r ltm-$(VERSION).zip libtommath-$(VERSION)/* ; \ - mv -f ltm* ~ ; rm -rf libtommath-$(VERSION) +# You have to create a file .coveralls.yml with the content "repo_token: " +# in the base folder to be able to submit to coveralls +coveralls: lcov + coveralls-lcov + +docs manual: + $(MAKE) -C doc/ $@ V=$(V) + +.PHONY: pre_gen +pre_gen: + mkdir -p pre_gen + perl gen.pl + sed -e 's/[[:blank:]]*$$//' mpi.c > pre_gen/mpi.c + rm mpi.c + +zipup: clean astyle new_file docs + @# Update the index, so diff-index won't fail in case the pdf has been created. + @# As the pdf creation modifies the tex files, git sometimes detects the + @# modified files, but misses that it's put back to its original version. + @git update-index --refresh + @git diff-index --quiet HEAD -- || ( echo "FAILURE: uncommited changes or not a git" && exit 1 ) + rm -rf libtommath-$(VERSION) ltm-$(VERSION).* + @# files/dirs excluded from "git archive" are defined in .gitattributes + git archive --format=tar --prefix=libtommath-$(VERSION)/ HEAD | tar x + @echo 'fixme check' + -@(find libtommath-$(VERSION)/ -type f | xargs grep 'FIXM[E]') && echo '############## BEWARE: the "fixme" marker was found !!! ##############' || true + mkdir -p libtommath-$(VERSION)/doc + cp doc/bn.pdf libtommath-$(VERSION)/doc/ + $(MAKE) -C libtommath-$(VERSION)/ pre_gen + tar -c libtommath-$(VERSION)/ | xz -6e -c - > ltm-$(VERSION).tar.xz + zip -9rq ltm-$(VERSION).zip libtommath-$(VERSION) + cp doc/bn.pdf bn-$(VERSION).pdf + rm -rf libtommath-$(VERSION) + gpg -b -a ltm-$(VERSION).tar.xz + gpg -b -a ltm-$(VERSION).zip + +new_file: + perl helper.pl --update-files + +perlcritic: + perlcritic *.pl doc/*.pl + +astyle: + @echo " * run astyle on all sources" + @astyle --options=astylerc --formatted $(OBJECTS:.o=.c) tommath*.h demo/*.c etc/*.c mtest/mtest.c diff --git a/lib/hcrypto/libtommath/makefile.bcc b/lib/hcrypto/libtommath/makefile.bcc deleted file mode 100644 index 67743d962..000000000 --- a/lib/hcrypto/libtommath/makefile.bcc +++ /dev/null @@ -1,44 +0,0 @@ -# -# Borland C++Builder Makefile (makefile.bcc) -# - - -LIB = tlib -CC = bcc32 -CFLAGS = -c -O2 -I. - -OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \ -bn_mp_clamp.obj bn_mp_zero.obj bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \ -bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \ -bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \ -bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \ -bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \ -bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \ -bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \ -bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \ -bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \ -bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \ -bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \ -bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \ -bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj \ -bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \ -bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \ -bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \ -bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \ -bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ -bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \ -bn_mp_reduce_2k_l.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_2k_setup_l.obj \ -bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \ -bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \ -bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \ -bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \ -bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj \ -bn_mp_to_signed_bin_n.obj bn_mp_to_unsigned_bin_n.obj - -TARGET = libtommath.lib - -$(TARGET): $(OBJECTS) - -.c.obj: - $(CC) $(CFLAGS) $< - $(LIB) $(TARGET) -+$@ diff --git a/lib/hcrypto/libtommath/makefile.cygwin_dll b/lib/hcrypto/libtommath/makefile.cygwin_dll deleted file mode 100644 index 85a9b2053..000000000 --- a/lib/hcrypto/libtommath/makefile.cygwin_dll +++ /dev/null @@ -1,55 +0,0 @@ -#Makefile for Cygwin-GCC -# -#This makefile will build a Windows DLL [doesn't require cygwin to run] in the file -#libtommath.dll. The import library is in libtommath.dll.a. Remember to add -#"-Wl,--enable-auto-import" to your client build to avoid the auto-import warnings -# -#Tom St Denis -CFLAGS += -I./ -Wall -W -Wshadow -O3 -funroll-loops -mno-cygwin - -#x86 optimizations [should be valid for any GCC install though] -CFLAGS += -fomit-frame-pointer - -default: windll - -OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ -bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ -bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ -bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ -bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ -bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ -bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ -bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ -bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ -bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ -bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ -bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ -bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ -bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ -bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ -bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ -bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ -bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ -bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ -bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ -bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \ -bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ -bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ -bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ -bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ -bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \ -bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o - -# make a Windows DLL via Cygwin -windll: $(OBJECTS) - gcc -mno-cygwin -mdll -o libtommath.dll -Wl,--out-implib=libtommath.dll.a -Wl,--export-all-symbols *.o - ranlib libtommath.dll.a - -# build the test program using the windows DLL -test: $(OBJECTS) windll - gcc $(CFLAGS) demo/demo.c libtommath.dll.a -Wl,--enable-auto-import -o test -s - cd mtest ; $(CC) -O3 -fomit-frame-pointer -funroll-loops mtest.c -o mtest -s - -/* $Source: /cvs/libtom/libtommath/makefile.cygwin_dll,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:45 $ */ diff --git a/lib/hcrypto/libtommath/makefile.icc b/lib/hcrypto/libtommath/makefile.icc deleted file mode 100644 index cf70ab0c2..000000000 --- a/lib/hcrypto/libtommath/makefile.icc +++ /dev/null @@ -1,116 +0,0 @@ -#Makefile for ICC -# -#Tom St Denis -CC=icc - -CFLAGS += -I./ - -# optimize for SPEED -# -# -mcpu= can be pentium, pentiumpro (covers PII through PIII) or pentium4 -# -ax? specifies make code specifically for ? but compatible with IA-32 -# -x? specifies compile solely for ? [not specifically IA-32 compatible] -# -# where ? is -# K - PIII -# W - first P4 [Williamette] -# N - P4 Northwood -# P - P4 Prescott -# B - Blend of P4 and PM [mobile] -# -# Default to just generic max opts -CFLAGS += -O3 -xP -ip - -#install as this user -USER=root -GROUP=root - -default: libtommath.a - -#default files to install -LIBNAME=libtommath.a -HEADERS=tommath.h - -#LIBPATH-The directory for libtomcrypt to be installed to. -#INCPATH-The directory to install the header files for libtommath. -#DATAPATH-The directory to install the pdf docs. -DESTDIR= -LIBPATH=/usr/lib -INCPATH=/usr/include -DATAPATH=/usr/share/doc/libtommath/pdf - -OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ -bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ -bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ -bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ -bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ -bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ -bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ -bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ -bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ -bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ -bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ -bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ -bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ -bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ -bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ -bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ -bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ -bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ -bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ -bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ -bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \ -bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ -bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ -bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ -bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ -bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \ -bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o - -libtommath.a: $(OBJECTS) - $(AR) $(ARFLAGS) libtommath.a $(OBJECTS) - ranlib libtommath.a - -#make a profiled library (takes a while!!!) -# -# This will build the library with profile generation -# then run the test demo and rebuild the library. -# -# So far I've seen improvements in the MP math -profiled: - make -f makefile.icc CFLAGS="$(CFLAGS) -prof_gen -DTESTING" timing - ./ltmtest - rm -f *.a *.o ltmtest - make -f makefile.icc CFLAGS="$(CFLAGS) -prof_use" - -#make a single object profiled library -profiled_single: - perl gen.pl - $(CC) $(CFLAGS) -prof_gen -DTESTING -c mpi.c -o mpi.o - $(CC) $(CFLAGS) -DTESTING -DTIMER demo/demo.c mpi.o -o ltmtest - ./ltmtest - rm -f *.o ltmtest - $(CC) $(CFLAGS) -prof_use -ip -DTESTING -c mpi.c -o mpi.o - $(AR) $(ARFLAGS) libtommath.a mpi.o - ranlib libtommath.a - -install: libtommath.a - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH) - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) - install -g $(GROUP) -o $(USER) $(LIBNAME) $(DESTDIR)$(LIBPATH) - install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) - -test: libtommath.a demo/demo.o - $(CC) demo/demo.o libtommath.a -o test - -mtest: test - cd mtest ; $(CC) $(CFLAGS) mtest.c -o mtest - -timing: libtommath.a - $(CC) $(CFLAGS) -DTIMER demo/timing.c libtommath.a -o ltmtest - -clean: - rm -f *.bat *.pdf *.o *.a *.obj *.lib *.exe *.dll etclib/*.o demo/demo.o test ltmtest mpitest mtest/mtest mtest/mtest.exe \ - *.idx *.toc *.log *.aux *.dvi *.lof *.ind *.ilg *.ps *.log *.s mpi.c *.il etc/*.il *.dyn - cd etc ; make clean - cd pics ; make clean diff --git a/lib/hcrypto/libtommath/makefile.mingw b/lib/hcrypto/libtommath/makefile.mingw new file mode 100644 index 000000000..7eee57dc4 --- /dev/null +++ b/lib/hcrypto/libtommath/makefile.mingw @@ -0,0 +1,109 @@ +# MAKEFILE for MS Windows (mingw + gcc + gmake) +# +# BEWARE: variable OBJECTS is updated via helper.pl + +### USAGE: +# Open a command prompt with gcc + gmake in PATH and start: +# +# gmake -f makefile.mingw all +# test.exe +# gmake -f makefile.mingw PREFIX=c:\devel\libtom install + +#The following can be overridden from command line e.g. make -f makefile.mingw CC=gcc ARFLAGS=rcs +PREFIX = c:\mingw +CC = gcc +AR = ar +ARFLAGS = r +RANLIB = ranlib +STRIP = strip +CFLAGS = -O2 +LDFLAGS = + +#Compilation flags +LTM_CFLAGS = -I. $(CFLAGS) +LTM_LDFLAGS = $(LDFLAGS) -static-libgcc + +#Libraries to be created +LIBMAIN_S =libtommath.a +LIBMAIN_I =libtommath.dll.a +LIBMAIN_D =libtommath.dll + +#List of objects to compile (all goes to libtommath.a) +OBJECTS=bn_cutoffs.o bn_deprecated.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o bn_mp_addmod.o \ +bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \ +bn_mp_cnt_lsb.o bn_mp_complement.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_decr.o bn_mp_div.o bn_mp_div_2.o \ +bn_mp_div_2d.o bn_mp_div_3.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o \ +bn_mp_error_to_string.o bn_mp_exch.o bn_mp_expt_u32.o bn_mp_exptmod.o bn_mp_exteuclid.o bn_mp_fread.o \ +bn_mp_from_sbin.o bn_mp_from_ubin.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_double.o bn_mp_get_i32.o \ +bn_mp_get_i64.o bn_mp_get_l.o bn_mp_get_ll.o bn_mp_get_mag_u32.o bn_mp_get_mag_u64.o bn_mp_get_mag_ul.o \ +bn_mp_get_mag_ull.o bn_mp_grow.o bn_mp_incr.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_i32.o \ +bn_mp_init_i64.o bn_mp_init_l.o bn_mp_init_ll.o bn_mp_init_multi.o bn_mp_init_set.o bn_mp_init_size.o \ +bn_mp_init_u32.o bn_mp_init_u64.o bn_mp_init_ul.o bn_mp_init_ull.o bn_mp_invmod.o bn_mp_is_square.o \ +bn_mp_iseven.o bn_mp_isodd.o bn_mp_kronecker.o bn_mp_lcm.o bn_mp_log_u32.o bn_mp_lshd.o bn_mp_mod.o \ +bn_mp_mod_2d.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \ +bn_mp_montgomery_setup.o bn_mp_mul.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \ +bn_mp_or.o bn_mp_pack.o bn_mp_pack_count.o bn_mp_prime_fermat.o bn_mp_prime_frobenius_underwood.o \ +bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \ +bn_mp_prime_rabin_miller_trials.o bn_mp_prime_rand.o bn_mp_prime_strong_lucas_selfridge.o \ +bn_mp_radix_size.o bn_mp_radix_smap.o bn_mp_rand.o bn_mp_read_radix.o bn_mp_reduce.o bn_mp_reduce_2k.o \ +bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce_is_2k.o \ +bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_root_u32.o bn_mp_rshd.o bn_mp_sbin_size.o bn_mp_set.o \ +bn_mp_set_double.o bn_mp_set_i32.o bn_mp_set_i64.o bn_mp_set_l.o bn_mp_set_ll.o bn_mp_set_u32.o \ +bn_mp_set_u64.o bn_mp_set_ul.o bn_mp_set_ull.o bn_mp_shrink.o bn_mp_signed_rsh.o bn_mp_sqr.o \ +bn_mp_sqrmod.o bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o \ +bn_mp_to_radix.o bn_mp_to_sbin.o bn_mp_to_ubin.o bn_mp_ubin_size.o bn_mp_unpack.o bn_mp_xor.o bn_mp_zero.o \ +bn_prime_tab.o bn_s_mp_add.o bn_s_mp_balance_mul.o bn_s_mp_exptmod.o bn_s_mp_exptmod_fast.o \ +bn_s_mp_get_bit.o bn_s_mp_invmod_fast.o bn_s_mp_invmod_slow.o bn_s_mp_karatsuba_mul.o \ +bn_s_mp_karatsuba_sqr.o bn_s_mp_montgomery_reduce_fast.o bn_s_mp_mul_digs.o bn_s_mp_mul_digs_fast.o \ +bn_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs_fast.o bn_s_mp_prime_is_divisible.o \ +bn_s_mp_rand_jenkins.o bn_s_mp_rand_platform.o bn_s_mp_reverse.o bn_s_mp_sqr.o bn_s_mp_sqr_fast.o \ +bn_s_mp_sub.o bn_s_mp_toom_mul.o bn_s_mp_toom_sqr.o + +HEADERS_PUB=tommath.h +HEADERS=tommath_private.h tommath_class.h tommath_superclass.h tommath_cutoffs.h $(HEADERS_PUB) + +#The default rule for make builds the libtommath.a library (static) +default: $(LIBMAIN_S) + +#Dependencies on *.h +$(OBJECTS): $(HEADERS) + +.c.o: + $(CC) $(LTM_CFLAGS) -c $< -o $@ + +#Create libtommath.a +$(LIBMAIN_S): $(OBJECTS) + $(AR) $(ARFLAGS) $@ $(OBJECTS) + $(RANLIB) $@ + +#Create DLL + import library libtommath.dll.a +$(LIBMAIN_D) $(LIBMAIN_I): $(OBJECTS) + $(CC) -s -shared -o $(LIBMAIN_D) $^ -Wl,--enable-auto-import,--export-all -Wl,--out-implib=$(LIBMAIN_I) $(LTM_LDFLAGS) + $(STRIP) -S $(LIBMAIN_D) + +#Build test suite +test.exe: demo/shared.o demo/test.o $(LIBMAIN_S) + $(CC) $(LTM_CFLAGS) $(LTM_LDFLAGS) $^ -o $@ + @echo NOTICE: start the tests by launching test.exe + +test_standalone: test.exe + @echo test_standalone is deprecated, please use make-target 'test.exe' + +all: $(LIBMAIN_S) test.exe + +tune: $(LIBNAME_S) + $(MAKE) -C etc tune + $(MAKE) + +clean: + @-cmd /c del /Q /S *.o *.a *.exe *.dll 2>nul + +#Install the library + headers +install: $(LIBMAIN_S) $(LIBMAIN_I) $(LIBMAIN_D) + cmd /c if not exist "$(PREFIX)\bin" mkdir "$(PREFIX)\bin" + cmd /c if not exist "$(PREFIX)\lib" mkdir "$(PREFIX)\lib" + cmd /c if not exist "$(PREFIX)\include" mkdir "$(PREFIX)\include" + copy /Y $(LIBMAIN_S) "$(PREFIX)\lib" + copy /Y $(LIBMAIN_I) "$(PREFIX)\lib" + copy /Y $(LIBMAIN_D) "$(PREFIX)\bin" + copy /Y tommath*.h "$(PREFIX)\include" diff --git a/lib/hcrypto/libtommath/makefile.msvc b/lib/hcrypto/libtommath/makefile.msvc index 5edebec92..aa8d8bec1 100644 --- a/lib/hcrypto/libtommath/makefile.msvc +++ b/lib/hcrypto/libtommath/makefile.msvc @@ -1,40 +1,93 @@ -#MSVC Makefile +# MAKEFILE for MS Windows (nmake + Windows SDK) # -#Tom St Denis +# BEWARE: variable OBJECTS is updated via helper.pl -CFLAGS = /I. /Ox /DWIN32 /W3 /Fo$@ +### USAGE: +# Open a command prompt with WinSDK variables set and start: +# +# nmake -f makefile.msvc all +# test.exe +# nmake -f makefile.msvc PREFIX=c:\devel\libtom install -default: library +#The following can be overridden from command line e.g. make -f makefile.msvc CC=gcc ARFLAGS=rcs +PREFIX = c:\devel +CFLAGS = /Ox -OBJECTS=bncore.obj bn_mp_init.obj bn_mp_clear.obj bn_mp_exch.obj bn_mp_grow.obj bn_mp_shrink.obj \ -bn_mp_clamp.obj bn_mp_zero.obj bn_mp_set.obj bn_mp_set_int.obj bn_mp_init_size.obj bn_mp_copy.obj \ -bn_mp_init_copy.obj bn_mp_abs.obj bn_mp_neg.obj bn_mp_cmp_mag.obj bn_mp_cmp.obj bn_mp_cmp_d.obj \ -bn_mp_rshd.obj bn_mp_lshd.obj bn_mp_mod_2d.obj bn_mp_div_2d.obj bn_mp_mul_2d.obj bn_mp_div_2.obj \ -bn_mp_mul_2.obj bn_s_mp_add.obj bn_s_mp_sub.obj bn_fast_s_mp_mul_digs.obj bn_s_mp_mul_digs.obj \ -bn_fast_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs.obj bn_fast_s_mp_sqr.obj bn_s_mp_sqr.obj \ -bn_mp_add.obj bn_mp_sub.obj bn_mp_karatsuba_mul.obj bn_mp_mul.obj bn_mp_karatsuba_sqr.obj \ -bn_mp_sqr.obj bn_mp_div.obj bn_mp_mod.obj bn_mp_add_d.obj bn_mp_sub_d.obj bn_mp_mul_d.obj \ -bn_mp_div_d.obj bn_mp_mod_d.obj bn_mp_expt_d.obj bn_mp_addmod.obj bn_mp_submod.obj \ -bn_mp_mulmod.obj bn_mp_sqrmod.obj bn_mp_gcd.obj bn_mp_lcm.obj bn_fast_mp_invmod.obj bn_mp_invmod.obj \ -bn_mp_reduce.obj bn_mp_montgomery_setup.obj bn_fast_mp_montgomery_reduce.obj bn_mp_montgomery_reduce.obj \ -bn_mp_exptmod_fast.obj bn_mp_exptmod.obj bn_mp_2expt.obj bn_mp_n_root.obj bn_mp_jacobi.obj bn_reverse.obj \ -bn_mp_count_bits.obj bn_mp_read_unsigned_bin.obj bn_mp_read_signed_bin.obj bn_mp_to_unsigned_bin.obj \ -bn_mp_to_signed_bin.obj bn_mp_unsigned_bin_size.obj bn_mp_signed_bin_size.obj \ -bn_mp_xor.obj bn_mp_and.obj bn_mp_or.obj bn_mp_rand.obj bn_mp_montgomery_calc_normalization.obj \ -bn_mp_prime_is_divisible.obj bn_prime_tab.obj bn_mp_prime_fermat.obj bn_mp_prime_miller_rabin.obj \ -bn_mp_prime_is_prime.obj bn_mp_prime_next_prime.obj bn_mp_dr_reduce.obj \ -bn_mp_dr_is_modulus.obj bn_mp_dr_setup.obj bn_mp_reduce_setup.obj \ -bn_mp_toom_mul.obj bn_mp_toom_sqr.obj bn_mp_div_3.obj bn_s_mp_exptmod.obj \ -bn_mp_reduce_2k.obj bn_mp_reduce_is_2k.obj bn_mp_reduce_2k_setup.obj \ -bn_mp_reduce_2k_l.obj bn_mp_reduce_is_2k_l.obj bn_mp_reduce_2k_setup_l.obj \ -bn_mp_radix_smap.obj bn_mp_read_radix.obj bn_mp_toradix.obj bn_mp_radix_size.obj \ -bn_mp_fread.obj bn_mp_fwrite.obj bn_mp_cnt_lsb.obj bn_error.obj \ -bn_mp_init_multi.obj bn_mp_clear_multi.obj bn_mp_exteuclid.obj bn_mp_toradix_n.obj \ -bn_mp_prime_random_ex.obj bn_mp_get_int.obj bn_mp_sqrt.obj bn_mp_is_square.obj \ -bn_mp_init_set.obj bn_mp_init_set_int.obj bn_mp_invmod_slow.obj bn_mp_prime_rabin_miller_trials.obj \ -bn_mp_to_signed_bin_n.obj bn_mp_to_unsigned_bin_n.obj +#Compilation flags +LTM_CFLAGS = /nologo /I./ /D_CRT_SECURE_NO_WARNINGS /D_CRT_NONSTDC_NO_DEPRECATE /D__STDC_WANT_SECURE_LIB__=1 /D_CRT_HAS_CXX17=0 /Wall /wd4146 /wd4127 /wd4668 /wd4710 /wd4711 /wd4820 /wd5045 /WX $(CFLAGS) +LTM_LDFLAGS = advapi32.lib -HEADERS=tommath.h tommath_class.h tommath_superclass.h +#Libraries to be created (this makefile builds only static libraries) +LIBMAIN_S =tommath.lib -library: $(OBJECTS) - lib /out:tommath.lib $(OBJECTS) +#List of objects to compile (all goes to tommath.lib) +OBJECTS=bn_cutoffs.obj bn_deprecated.obj bn_mp_2expt.obj bn_mp_abs.obj bn_mp_add.obj bn_mp_add_d.obj bn_mp_addmod.obj \ +bn_mp_and.obj bn_mp_clamp.obj bn_mp_clear.obj bn_mp_clear_multi.obj bn_mp_cmp.obj bn_mp_cmp_d.obj bn_mp_cmp_mag.obj \ +bn_mp_cnt_lsb.obj bn_mp_complement.obj bn_mp_copy.obj bn_mp_count_bits.obj bn_mp_decr.obj bn_mp_div.obj bn_mp_div_2.obj \ +bn_mp_div_2d.obj bn_mp_div_3.obj bn_mp_div_d.obj bn_mp_dr_is_modulus.obj bn_mp_dr_reduce.obj bn_mp_dr_setup.obj \ +bn_mp_error_to_string.obj bn_mp_exch.obj bn_mp_expt_u32.obj bn_mp_exptmod.obj bn_mp_exteuclid.obj bn_mp_fread.obj \ +bn_mp_from_sbin.obj bn_mp_from_ubin.obj bn_mp_fwrite.obj bn_mp_gcd.obj bn_mp_get_double.obj bn_mp_get_i32.obj \ +bn_mp_get_i64.obj bn_mp_get_l.obj bn_mp_get_ll.obj bn_mp_get_mag_u32.obj bn_mp_get_mag_u64.obj bn_mp_get_mag_ul.obj \ +bn_mp_get_mag_ull.obj bn_mp_grow.obj bn_mp_incr.obj bn_mp_init.obj bn_mp_init_copy.obj bn_mp_init_i32.obj \ +bn_mp_init_i64.obj bn_mp_init_l.obj bn_mp_init_ll.obj bn_mp_init_multi.obj bn_mp_init_set.obj bn_mp_init_size.obj \ +bn_mp_init_u32.obj bn_mp_init_u64.obj bn_mp_init_ul.obj bn_mp_init_ull.obj bn_mp_invmod.obj bn_mp_is_square.obj \ +bn_mp_iseven.obj bn_mp_isodd.obj bn_mp_kronecker.obj bn_mp_lcm.obj bn_mp_log_u32.obj bn_mp_lshd.obj bn_mp_mod.obj \ +bn_mp_mod_2d.obj bn_mp_mod_d.obj bn_mp_montgomery_calc_normalization.obj bn_mp_montgomery_reduce.obj \ +bn_mp_montgomery_setup.obj bn_mp_mul.obj bn_mp_mul_2.obj bn_mp_mul_2d.obj bn_mp_mul_d.obj bn_mp_mulmod.obj bn_mp_neg.obj \ +bn_mp_or.obj bn_mp_pack.obj bn_mp_pack_count.obj bn_mp_prime_fermat.obj bn_mp_prime_frobenius_underwood.obj \ +bn_mp_prime_is_prime.obj bn_mp_prime_miller_rabin.obj bn_mp_prime_next_prime.obj \ +bn_mp_prime_rabin_miller_trials.obj bn_mp_prime_rand.obj bn_mp_prime_strong_lucas_selfridge.obj \ +bn_mp_radix_size.obj bn_mp_radix_smap.obj bn_mp_rand.obj bn_mp_read_radix.obj bn_mp_reduce.obj bn_mp_reduce_2k.obj \ +bn_mp_reduce_2k_l.obj bn_mp_reduce_2k_setup.obj bn_mp_reduce_2k_setup_l.obj bn_mp_reduce_is_2k.obj \ +bn_mp_reduce_is_2k_l.obj bn_mp_reduce_setup.obj bn_mp_root_u32.obj bn_mp_rshd.obj bn_mp_sbin_size.obj bn_mp_set.obj \ +bn_mp_set_double.obj bn_mp_set_i32.obj bn_mp_set_i64.obj bn_mp_set_l.obj bn_mp_set_ll.obj bn_mp_set_u32.obj \ +bn_mp_set_u64.obj bn_mp_set_ul.obj bn_mp_set_ull.obj bn_mp_shrink.obj bn_mp_signed_rsh.obj bn_mp_sqr.obj \ +bn_mp_sqrmod.obj bn_mp_sqrt.obj bn_mp_sqrtmod_prime.obj bn_mp_sub.obj bn_mp_sub_d.obj bn_mp_submod.obj \ +bn_mp_to_radix.obj bn_mp_to_sbin.obj bn_mp_to_ubin.obj bn_mp_ubin_size.obj bn_mp_unpack.obj bn_mp_xor.obj bn_mp_zero.obj \ +bn_prime_tab.obj bn_s_mp_add.obj bn_s_mp_balance_mul.obj bn_s_mp_exptmod.obj bn_s_mp_exptmod_fast.obj \ +bn_s_mp_get_bit.obj bn_s_mp_invmod_fast.obj bn_s_mp_invmod_slow.obj bn_s_mp_karatsuba_mul.obj \ +bn_s_mp_karatsuba_sqr.obj bn_s_mp_montgomery_reduce_fast.obj bn_s_mp_mul_digs.obj bn_s_mp_mul_digs_fast.obj \ +bn_s_mp_mul_high_digs.obj bn_s_mp_mul_high_digs_fast.obj bn_s_mp_prime_is_divisible.obj \ +bn_s_mp_rand_jenkins.obj bn_s_mp_rand_platform.obj bn_s_mp_reverse.obj bn_s_mp_sqr.obj bn_s_mp_sqr_fast.obj \ +bn_s_mp_sub.obj bn_s_mp_toom_mul.obj bn_s_mp_toom_sqr.obj + +HEADERS_PUB=tommath.h +HEADERS=tommath_private.h tommath_class.h tommath_superclass.h tommath_cutoffs.h $(HEADERS_PUB) + +#The default rule for make builds the tommath.lib library (static) +default: $(LIBMAIN_S) + +#Dependencies on *.h +$(OBJECTS): $(HEADERS) + +.c.obj: + $(CC) $(LTM_CFLAGS) /c $< /Fo$@ + +#Create tommath.lib +$(LIBMAIN_S): $(OBJECTS) + lib /out:$(LIBMAIN_S) $(OBJECTS) + +#Build test suite +test.exe: $(LIBMAIN_S) demo/shared.obj demo/test.obj + cl $(LTM_CFLAGS) $(TOBJECTS) $(LIBMAIN_S) $(LTM_LDFLAGS) demo/shared.c demo/test.c /Fe$@ + @echo NOTICE: start the tests by launching test.exe + +test_standalone: test.exe + @echo test_standalone is deprecated, please use make-target 'test.exe' + +all: $(LIBMAIN_S) test.exe + +tune: $(LIBMAIN_S) + $(MAKE) -C etc tune + $(MAKE) + +clean: + @-cmd /c del /Q /S *.OBJ *.LIB *.EXE *.DLL 2>nul + +#Install the library + headers +install: $(LIBMAIN_S) + cmd /c if not exist "$(PREFIX)\bin" mkdir "$(PREFIX)\bin" + cmd /c if not exist "$(PREFIX)\lib" mkdir "$(PREFIX)\lib" + cmd /c if not exist "$(PREFIX)\include" mkdir "$(PREFIX)\include" + copy /Y $(LIBMAIN_S) "$(PREFIX)\lib" + copy /Y tommath*.h "$(PREFIX)\include" diff --git a/lib/hcrypto/libtommath/makefile.shared b/lib/hcrypto/libtommath/makefile.shared index f17bbbd48..680210736 100644 --- a/lib/hcrypto/libtommath/makefile.shared +++ b/lib/hcrypto/libtommath/makefile.shared @@ -1,102 +1,99 @@ #Makefile for GCC # #Tom St Denis -VERSION=0:41 - -CC = libtool --mode=compile --tag=CC gcc - -CFLAGS += -I./ -Wall -W -Wshadow -Wsign-compare - -ifndef IGNORE_SPEED - -#for speed -CFLAGS += -O3 -funroll-loops - -#for size -#CFLAGS += -Os - -#x86 optimizations [should be valid for any GCC install though] -CFLAGS += -fomit-frame-pointer - -endif - -#install as this user -ifndef INSTALL_GROUP - GROUP=wheel -else - GROUP=$(INSTALL_GROUP) -endif - -ifndef INSTALL_USER - USER=root -else - USER=$(INSTALL_USER) -endif - -default: libtommath.la #default files to install ifndef LIBNAME LIBNAME=libtommath.la endif -ifndef LIBNAME_S - LIBNAME_S=libtommath.a + +include makefile_include.mk + + +ifndef LIBTOOL + ifeq ($(PLATFORM), Darwin) + LIBTOOL:=glibtool + else + LIBTOOL:=libtool + endif endif -HEADERS=tommath.h tommath_class.h tommath_superclass.h +LTCOMPILE = $(LIBTOOL) --mode=compile --tag=CC $(CC) +LTLINK = $(LIBTOOL) --mode=link --tag=CC $(CC) -#LIBPATH-The directory for libtommath to be installed to. -#INCPATH-The directory to install the header files for libtommath. -#DATAPATH-The directory to install the pdf docs. -DESTDIR= -LIBPATH=/usr/lib -INCPATH=/usr/include -DATAPATH=/usr/share/doc/libtommath/pdf +LCOV_ARGS=--directory .libs --directory . -OBJECTS=bncore.o bn_mp_init.o bn_mp_clear.o bn_mp_exch.o bn_mp_grow.o bn_mp_shrink.o \ -bn_mp_clamp.o bn_mp_zero.o bn_mp_set.o bn_mp_set_int.o bn_mp_init_size.o bn_mp_copy.o \ -bn_mp_init_copy.o bn_mp_abs.o bn_mp_neg.o bn_mp_cmp_mag.o bn_mp_cmp.o bn_mp_cmp_d.o \ -bn_mp_rshd.o bn_mp_lshd.o bn_mp_mod_2d.o bn_mp_div_2d.o bn_mp_mul_2d.o bn_mp_div_2.o \ -bn_mp_mul_2.o bn_s_mp_add.o bn_s_mp_sub.o bn_fast_s_mp_mul_digs.o bn_s_mp_mul_digs.o \ -bn_fast_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs.o bn_fast_s_mp_sqr.o bn_s_mp_sqr.o \ -bn_mp_add.o bn_mp_sub.o bn_mp_karatsuba_mul.o bn_mp_mul.o bn_mp_karatsuba_sqr.o \ -bn_mp_sqr.o bn_mp_div.o bn_mp_mod.o bn_mp_add_d.o bn_mp_sub_d.o bn_mp_mul_d.o \ -bn_mp_div_d.o bn_mp_mod_d.o bn_mp_expt_d.o bn_mp_addmod.o bn_mp_submod.o \ -bn_mp_mulmod.o bn_mp_sqrmod.o bn_mp_gcd.o bn_mp_lcm.o bn_fast_mp_invmod.o bn_mp_invmod.o \ -bn_mp_reduce.o bn_mp_montgomery_setup.o bn_fast_mp_montgomery_reduce.o bn_mp_montgomery_reduce.o \ -bn_mp_exptmod_fast.o bn_mp_exptmod.o bn_mp_2expt.o bn_mp_n_root.o bn_mp_jacobi.o bn_reverse.o \ -bn_mp_count_bits.o bn_mp_read_unsigned_bin.o bn_mp_read_signed_bin.o bn_mp_to_unsigned_bin.o \ -bn_mp_to_signed_bin.o bn_mp_unsigned_bin_size.o bn_mp_signed_bin_size.o \ -bn_mp_xor.o bn_mp_and.o bn_mp_or.o bn_mp_rand.o bn_mp_montgomery_calc_normalization.o \ -bn_mp_prime_is_divisible.o bn_prime_tab.o bn_mp_prime_fermat.o bn_mp_prime_miller_rabin.o \ -bn_mp_prime_is_prime.o bn_mp_prime_next_prime.o bn_mp_dr_reduce.o \ -bn_mp_dr_is_modulus.o bn_mp_dr_setup.o bn_mp_reduce_setup.o \ -bn_mp_toom_mul.o bn_mp_toom_sqr.o bn_mp_div_3.o bn_s_mp_exptmod.o \ -bn_mp_reduce_2k.o bn_mp_reduce_is_2k.o bn_mp_reduce_2k_setup.o \ -bn_mp_reduce_2k_l.o bn_mp_reduce_is_2k_l.o bn_mp_reduce_2k_setup_l.o \ -bn_mp_radix_smap.o bn_mp_read_radix.o bn_mp_toradix.o bn_mp_radix_size.o \ -bn_mp_fread.o bn_mp_fwrite.o bn_mp_cnt_lsb.o bn_error.o \ -bn_mp_init_multi.o bn_mp_clear_multi.o bn_mp_exteuclid.o bn_mp_toradix_n.o \ -bn_mp_prime_random_ex.o bn_mp_get_int.o bn_mp_sqrt.o bn_mp_is_square.o bn_mp_init_set.o \ -bn_mp_init_set_int.o bn_mp_invmod_slow.o bn_mp_prime_rabin_miller_trials.o \ -bn_mp_to_signed_bin_n.o bn_mp_to_unsigned_bin_n.o +#START_INS +OBJECTS=bn_cutoffs.o bn_deprecated.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o bn_mp_addmod.o \ +bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \ +bn_mp_cnt_lsb.o bn_mp_complement.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_decr.o bn_mp_div.o bn_mp_div_2.o \ +bn_mp_div_2d.o bn_mp_div_3.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o \ +bn_mp_error_to_string.o bn_mp_exch.o bn_mp_expt_u32.o bn_mp_exptmod.o bn_mp_exteuclid.o bn_mp_fread.o \ +bn_mp_from_sbin.o bn_mp_from_ubin.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_double.o bn_mp_get_i32.o \ +bn_mp_get_i64.o bn_mp_get_l.o bn_mp_get_ll.o bn_mp_get_mag_u32.o bn_mp_get_mag_u64.o bn_mp_get_mag_ul.o \ +bn_mp_get_mag_ull.o bn_mp_grow.o bn_mp_incr.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_i32.o \ +bn_mp_init_i64.o bn_mp_init_l.o bn_mp_init_ll.o bn_mp_init_multi.o bn_mp_init_set.o bn_mp_init_size.o \ +bn_mp_init_u32.o bn_mp_init_u64.o bn_mp_init_ul.o bn_mp_init_ull.o bn_mp_invmod.o bn_mp_is_square.o \ +bn_mp_iseven.o bn_mp_isodd.o bn_mp_kronecker.o bn_mp_lcm.o bn_mp_log_u32.o bn_mp_lshd.o bn_mp_mod.o \ +bn_mp_mod_2d.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \ +bn_mp_montgomery_setup.o bn_mp_mul.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \ +bn_mp_or.o bn_mp_pack.o bn_mp_pack_count.o bn_mp_prime_fermat.o bn_mp_prime_frobenius_underwood.o \ +bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \ +bn_mp_prime_rabin_miller_trials.o bn_mp_prime_rand.o bn_mp_prime_strong_lucas_selfridge.o \ +bn_mp_radix_size.o bn_mp_radix_smap.o bn_mp_rand.o bn_mp_read_radix.o bn_mp_reduce.o bn_mp_reduce_2k.o \ +bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce_is_2k.o \ +bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_root_u32.o bn_mp_rshd.o bn_mp_sbin_size.o bn_mp_set.o \ +bn_mp_set_double.o bn_mp_set_i32.o bn_mp_set_i64.o bn_mp_set_l.o bn_mp_set_ll.o bn_mp_set_u32.o \ +bn_mp_set_u64.o bn_mp_set_ul.o bn_mp_set_ull.o bn_mp_shrink.o bn_mp_signed_rsh.o bn_mp_sqr.o \ +bn_mp_sqrmod.o bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o \ +bn_mp_to_radix.o bn_mp_to_sbin.o bn_mp_to_ubin.o bn_mp_ubin_size.o bn_mp_unpack.o bn_mp_xor.o bn_mp_zero.o \ +bn_prime_tab.o bn_s_mp_add.o bn_s_mp_balance_mul.o bn_s_mp_exptmod.o bn_s_mp_exptmod_fast.o \ +bn_s_mp_get_bit.o bn_s_mp_invmod_fast.o bn_s_mp_invmod_slow.o bn_s_mp_karatsuba_mul.o \ +bn_s_mp_karatsuba_sqr.o bn_s_mp_montgomery_reduce_fast.o bn_s_mp_mul_digs.o bn_s_mp_mul_digs_fast.o \ +bn_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs_fast.o bn_s_mp_prime_is_divisible.o \ +bn_s_mp_rand_jenkins.o bn_s_mp_rand_platform.o bn_s_mp_reverse.o bn_s_mp_sqr.o bn_s_mp_sqr_fast.o \ +bn_s_mp_sub.o bn_s_mp_toom_mul.o bn_s_mp_toom_sqr.o + +#END_INS objs: $(OBJECTS) +.c.o: $(HEADERS) + $(LTCOMPILE) $(LTM_CFLAGS) $(LTM_LDFLAGS) -o $@ -c $< + +LOBJECTS = $(OBJECTS:.o=.lo) + $(LIBNAME): $(OBJECTS) - libtool --mode=link gcc *.lo -o $(LIBNAME) -rpath $(LIBPATH) -version-info $(VERSION) + $(LTLINK) $(LTM_LDFLAGS) $(LOBJECTS) -o $(LIBNAME) -rpath $(LIBPATH) -version-info $(VERSION_SO) $(LTM_LIBTOOLFLAGS) install: $(LIBNAME) - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(LIBPATH) - libtool --mode=install install -c $(LIBNAME) $(DESTDIR)$(LIBPATH)/$(LIBNAME) - install -d -g $(GROUP) -o $(USER) $(DESTDIR)$(INCPATH) - install -g $(GROUP) -o $(USER) $(HEADERS) $(DESTDIR)$(INCPATH) + install -d $(DESTDIR)$(LIBPATH) + install -d $(DESTDIR)$(INCPATH) + $(LIBTOOL) --mode=install install -m 644 $(LIBNAME) $(DESTDIR)$(LIBPATH)/$(LIBNAME) + install -m 644 $(HEADERS_PUB) $(DESTDIR)$(INCPATH) + sed -e 's,^prefix=.*,prefix=$(PREFIX),' -e 's,^Version:.*,Version: $(VERSION_PC),' libtommath.pc.in > libtommath.pc + install -d $(DESTDIR)$(LIBPATH)/pkgconfig + install -m 644 libtommath.pc $(DESTDIR)$(LIBPATH)/pkgconfig/ -test: $(LIBNAME) demo/demo.o - gcc $(CFLAGS) -c demo/demo.c -o demo/demo.o - libtool --mode=link gcc -o test demo/demo.o $(LIBNAME_S) - -mtest: test - cd mtest ; gcc $(CFLAGS) mtest.c -o mtest - -timing: $(LIBNAME) - gcc $(CFLAGS) -DTIMER demo/timing.c $(LIBNAME_S) -o ltmtest +uninstall: + $(LIBTOOL) --mode=uninstall rm $(DESTDIR)$(LIBPATH)/$(LIBNAME) + rm $(HEADERS_PUB:%=$(DESTDIR)$(INCPATH)/%) + rm $(DESTDIR)$(LIBPATH)/pkgconfig/libtommath.pc + +test_standalone: test + @echo "test_standalone is deprecated, please use make-target 'test'" + +test mtest_opponent: demo/shared.o $(LIBNAME) | demo/test.o demo/mtest_opponent.o + $(LTLINK) $(LTM_LDFLAGS) demo/$@.o $^ -o $@ + +.PHONY: mtest +mtest: + cd mtest ; $(CC) $(LTM_CFLAGS) -O0 mtest.c $(LTM_LDFLAGS) -o mtest + +timing: $(LIBNAME) demo/timing.c + $(LTLINK) $(LTM_CFLAGS) $(LTM_LDFLAGS) -DTIMER demo/timing.c $(LIBNAME) -o timing + +tune: $(LIBNAME) + $(LTCOMPILE) $(LTM_CFLAGS) -c etc/tune.c -o etc/tune.o + $(LTLINK) $(LTM_LDFLAGS) -o etc/tune etc/tune.o $(LIBNAME) + cd etc/; /bin/sh tune_it.sh; cd .. + $(MAKE) -f makefile.shared diff --git a/lib/hcrypto/libtommath/makefile.unix b/lib/hcrypto/libtommath/makefile.unix new file mode 100644 index 000000000..4cefc7e17 --- /dev/null +++ b/lib/hcrypto/libtommath/makefile.unix @@ -0,0 +1,106 @@ +# MAKEFILE that is intended to be compatible with any kind of make (GNU make, BSD make, ...) +# works on: Linux, *BSD, Cygwin, AIX, HP-UX and hopefully other UNIX systems +# +# Please do not use here neither any special make syntax nor any unusual tools/utilities! + +# using ICC compiler: +# make -f makefile.unix CC=icc CFLAGS="-O3 -xP -ip" + +# using Borland C++Builder: +# make -f makefile.unix CC=bcc32 + +#The following can be overridden from command line e.g. "make -f makefile.unix CC=gcc ARFLAGS=rcs" +DESTDIR = +PREFIX = /usr/local +LIBPATH = $(PREFIX)/lib +INCPATH = $(PREFIX)/include +CC = cc +AR = ar +ARFLAGS = r +RANLIB = ranlib +CFLAGS = -O2 +LDFLAGS = + +VERSION = 1.2.0 + +#Compilation flags +LTM_CFLAGS = -I. $(CFLAGS) +LTM_LDFLAGS = $(LDFLAGS) + +#Library to be created (this makefile builds only static library) +LIBMAIN_S = libtommath.a + +OBJECTS=bn_cutoffs.o bn_deprecated.o bn_mp_2expt.o bn_mp_abs.o bn_mp_add.o bn_mp_add_d.o bn_mp_addmod.o \ +bn_mp_and.o bn_mp_clamp.o bn_mp_clear.o bn_mp_clear_multi.o bn_mp_cmp.o bn_mp_cmp_d.o bn_mp_cmp_mag.o \ +bn_mp_cnt_lsb.o bn_mp_complement.o bn_mp_copy.o bn_mp_count_bits.o bn_mp_decr.o bn_mp_div.o bn_mp_div_2.o \ +bn_mp_div_2d.o bn_mp_div_3.o bn_mp_div_d.o bn_mp_dr_is_modulus.o bn_mp_dr_reduce.o bn_mp_dr_setup.o \ +bn_mp_error_to_string.o bn_mp_exch.o bn_mp_expt_u32.o bn_mp_exptmod.o bn_mp_exteuclid.o bn_mp_fread.o \ +bn_mp_from_sbin.o bn_mp_from_ubin.o bn_mp_fwrite.o bn_mp_gcd.o bn_mp_get_double.o bn_mp_get_i32.o \ +bn_mp_get_i64.o bn_mp_get_l.o bn_mp_get_ll.o bn_mp_get_mag_u32.o bn_mp_get_mag_u64.o bn_mp_get_mag_ul.o \ +bn_mp_get_mag_ull.o bn_mp_grow.o bn_mp_incr.o bn_mp_init.o bn_mp_init_copy.o bn_mp_init_i32.o \ +bn_mp_init_i64.o bn_mp_init_l.o bn_mp_init_ll.o bn_mp_init_multi.o bn_mp_init_set.o bn_mp_init_size.o \ +bn_mp_init_u32.o bn_mp_init_u64.o bn_mp_init_ul.o bn_mp_init_ull.o bn_mp_invmod.o bn_mp_is_square.o \ +bn_mp_iseven.o bn_mp_isodd.o bn_mp_kronecker.o bn_mp_lcm.o bn_mp_log_u32.o bn_mp_lshd.o bn_mp_mod.o \ +bn_mp_mod_2d.o bn_mp_mod_d.o bn_mp_montgomery_calc_normalization.o bn_mp_montgomery_reduce.o \ +bn_mp_montgomery_setup.o bn_mp_mul.o bn_mp_mul_2.o bn_mp_mul_2d.o bn_mp_mul_d.o bn_mp_mulmod.o bn_mp_neg.o \ +bn_mp_or.o bn_mp_pack.o bn_mp_pack_count.o bn_mp_prime_fermat.o bn_mp_prime_frobenius_underwood.o \ +bn_mp_prime_is_prime.o bn_mp_prime_miller_rabin.o bn_mp_prime_next_prime.o \ +bn_mp_prime_rabin_miller_trials.o bn_mp_prime_rand.o bn_mp_prime_strong_lucas_selfridge.o \ +bn_mp_radix_size.o bn_mp_radix_smap.o bn_mp_rand.o bn_mp_read_radix.o bn_mp_reduce.o bn_mp_reduce_2k.o \ +bn_mp_reduce_2k_l.o bn_mp_reduce_2k_setup.o bn_mp_reduce_2k_setup_l.o bn_mp_reduce_is_2k.o \ +bn_mp_reduce_is_2k_l.o bn_mp_reduce_setup.o bn_mp_root_u32.o bn_mp_rshd.o bn_mp_sbin_size.o bn_mp_set.o \ +bn_mp_set_double.o bn_mp_set_i32.o bn_mp_set_i64.o bn_mp_set_l.o bn_mp_set_ll.o bn_mp_set_u32.o \ +bn_mp_set_u64.o bn_mp_set_ul.o bn_mp_set_ull.o bn_mp_shrink.o bn_mp_signed_rsh.o bn_mp_sqr.o \ +bn_mp_sqrmod.o bn_mp_sqrt.o bn_mp_sqrtmod_prime.o bn_mp_sub.o bn_mp_sub_d.o bn_mp_submod.o \ +bn_mp_to_radix.o bn_mp_to_sbin.o bn_mp_to_ubin.o bn_mp_ubin_size.o bn_mp_unpack.o bn_mp_xor.o bn_mp_zero.o \ +bn_prime_tab.o bn_s_mp_add.o bn_s_mp_balance_mul.o bn_s_mp_exptmod.o bn_s_mp_exptmod_fast.o \ +bn_s_mp_get_bit.o bn_s_mp_invmod_fast.o bn_s_mp_invmod_slow.o bn_s_mp_karatsuba_mul.o \ +bn_s_mp_karatsuba_sqr.o bn_s_mp_montgomery_reduce_fast.o bn_s_mp_mul_digs.o bn_s_mp_mul_digs_fast.o \ +bn_s_mp_mul_high_digs.o bn_s_mp_mul_high_digs_fast.o bn_s_mp_prime_is_divisible.o \ +bn_s_mp_rand_jenkins.o bn_s_mp_rand_platform.o bn_s_mp_reverse.o bn_s_mp_sqr.o bn_s_mp_sqr_fast.o \ +bn_s_mp_sub.o bn_s_mp_toom_mul.o bn_s_mp_toom_sqr.o + +HEADERS_PUB=tommath.h +HEADERS=tommath_private.h tommath_class.h tommath_superclass.h tommath_cutoffs.h $(HEADERS_PUB) + +#The default rule for make builds the libtommath.a library (static) +default: $(LIBMAIN_S) + +#Dependencies on *.h +$(OBJECTS): $(HEADERS) + +#This is necessary for compatibility with BSD make (namely on OpenBSD) +.SUFFIXES: .o .c +.c.o: + $(CC) $(LTM_CFLAGS) -c $< -o $@ + +#Create libtommath.a +$(LIBMAIN_S): $(OBJECTS) + $(AR) $(ARFLAGS) $@ $(OBJECTS) + $(RANLIB) $@ + +#Build test_standalone suite +test: demo/shared.o demo/test.o $(LIBMAIN_S) + $(CC) $(LTM_CFLAGS) $(LTM_LDFLAGS) $^ -o $@ + @echo "NOTICE: start the tests by: ./test" + +test_standalone: test + @echo "test_standalone is deprecated, please use make-target 'test'" + +all: $(LIBMAIN_S) test + +tune: $(LIBMAIN_S) + $(MAKE) -C etc tune + $(MAKE) + +#NOTE: this makefile works also on cygwin, thus we need to delete *.exe +clean: + -@rm -f $(OBJECTS) $(LIBMAIN_S) + -@rm -f demo/main.o demo/opponent.o demo/test.o test test.exe + +#Install the library + headers +install: $(LIBMAIN_S) + @mkdir -p $(DESTDIR)$(INCPATH) $(DESTDIR)$(LIBPATH)/pkgconfig + @cp $(LIBMAIN_S) $(DESTDIR)$(LIBPATH)/ + @cp $(HEADERS_PUB) $(DESTDIR)$(INCPATH)/ + @sed -e 's,^prefix=.*,prefix=$(PREFIX),' -e 's,^Version:.*,Version: $(VERSION),' libtommath.pc.in > $(DESTDIR)$(LIBPATH)/pkgconfig/libtommath.pc diff --git a/lib/hcrypto/libtommath/makefile_include.mk b/lib/hcrypto/libtommath/makefile_include.mk new file mode 100644 index 000000000..7b025e8e9 --- /dev/null +++ b/lib/hcrypto/libtommath/makefile_include.mk @@ -0,0 +1,166 @@ +# +# Include makefile for libtommath +# + +#version of library +VERSION=1.2.0 +VERSION_PC=1.2.0 +VERSION_SO=3:0:2 + +PLATFORM := $(shell uname | sed -e 's/_.*//') + +# default make target +default: ${LIBNAME} + +# Compiler and Linker Names +ifndef CROSS_COMPILE + CROSS_COMPILE= +endif + +# We only need to go through this dance of determining the right compiler if we're using +# cross compilation, otherwise $(CC) is fine as-is. +ifneq (,$(CROSS_COMPILE)) +ifeq ($(origin CC),default) +CSTR := "\#ifdef __clang__\nCLANG\n\#endif\n" +ifeq ($(PLATFORM),FreeBSD) + # XXX: FreeBSD needs extra escaping for some reason + CSTR := $$$(CSTR) +endif +ifneq (,$(shell echo $(CSTR) | $(CC) -E - | grep CLANG)) + CC := $(CROSS_COMPILE)clang +else + CC := $(CROSS_COMPILE)gcc +endif # Clang +endif # cc is Make's default +endif # CROSS_COMPILE non-empty + +LD=$(CROSS_COMPILE)ld +AR=$(CROSS_COMPILE)ar +RANLIB=$(CROSS_COMPILE)ranlib + +ifndef MAKE +# BSDs refer to GNU Make as gmake +ifneq (,$(findstring $(PLATFORM),FreeBSD OpenBSD DragonFly NetBSD)) + MAKE=gmake +else + MAKE=make +endif +endif + +LTM_CFLAGS += -I./ -Wall -Wsign-compare -Wextra -Wshadow + +ifdef SANITIZER +LTM_CFLAGS += -fsanitize=undefined -fno-sanitize-recover=all -fno-sanitize=float-divide-by-zero +endif + +ifndef NO_ADDTL_WARNINGS +# additional warnings +LTM_CFLAGS += -Wdeclaration-after-statement -Wbad-function-cast -Wcast-align +LTM_CFLAGS += -Wstrict-prototypes -Wpointer-arith +endif + +ifdef CONV_WARNINGS +LTM_CFLAGS += -std=c89 -Wconversion -Wsign-conversion +ifeq ($(CONV_WARNINGS), strict) +LTM_CFLAGS += -DMP_USE_ENUMS -Wc++-compat +endif +else +LTM_CFLAGS += -Wsystem-headers +endif + +ifdef COMPILE_DEBUG +#debug +LTM_CFLAGS += -g3 +endif + +ifdef COMPILE_SIZE +#for size +LTM_CFLAGS += -Os +else + +ifndef IGNORE_SPEED +#for speed +LTM_CFLAGS += -O3 -funroll-loops + +#x86 optimizations [should be valid for any GCC install though] +LTM_CFLAGS += -fomit-frame-pointer +endif + +endif # COMPILE_SIZE + +ifneq ($(findstring clang,$(CC)),) +LTM_CFLAGS += -Wno-typedef-redefinition -Wno-tautological-compare -Wno-builtin-requires-header +endif +ifneq ($(findstring mingw,$(CC)),) +LTM_CFLAGS += -Wno-shadow +endif +ifeq ($(PLATFORM), Darwin) +LTM_CFLAGS += -Wno-nullability-completeness +endif +ifeq ($(PLATFORM), CYGWIN) +LIBTOOLFLAGS += -no-undefined +endif + +# add in the standard FLAGS +LTM_CFLAGS += $(CFLAGS) +LTM_LFLAGS += $(LFLAGS) +LTM_LDFLAGS += $(LDFLAGS) +LTM_LIBTOOLFLAGS += $(LIBTOOLFLAGS) + + +ifeq ($(PLATFORM),FreeBSD) + _ARCH := $(shell sysctl -b hw.machine_arch) +else + _ARCH := $(shell uname -m) +endif + +# adjust coverage set +ifneq ($(filter $(_ARCH), i386 i686 x86_64 amd64 ia64),) + COVERAGE = test_standalone timing + COVERAGE_APP = ./test && ./timing +else + COVERAGE = test_standalone + COVERAGE_APP = ./test +endif + +HEADERS_PUB=tommath.h +HEADERS=tommath_private.h tommath_class.h tommath_superclass.h tommath_cutoffs.h $(HEADERS_PUB) + +#LIBPATH The directory for libtommath to be installed to. +#INCPATH The directory to install the header files for libtommath. +#DATAPATH The directory to install the pdf docs. +DESTDIR ?= +PREFIX ?= /usr/local +LIBPATH ?= $(PREFIX)/lib +INCPATH ?= $(PREFIX)/include +DATAPATH ?= $(PREFIX)/share/doc/libtommath/pdf + +#make the code coverage of the library +# +coverage: LTM_CFLAGS += -fprofile-arcs -ftest-coverage -DTIMING_NO_LOGS +coverage: LTM_LFLAGS += -lgcov +coverage: LTM_LDFLAGS += -lgcov + +coverage: $(COVERAGE) + $(COVERAGE_APP) + +lcov: coverage + rm -f coverage.info + lcov --capture --no-external --no-recursion $(LCOV_ARGS) --output-file coverage.info -q + genhtml coverage.info --output-directory coverage -q + +# target that removes all coverage output +cleancov-clean: + rm -f `find . -type f -name "*.info" | xargs` + rm -rf coverage/ + +# cleans everything - coverage output and standard 'clean' +cleancov: cleancov-clean clean + +clean: + rm -f *.gcda *.gcno *.gcov *.bat *.o *.a *.obj *.lib *.exe *.dll etclib/*.o \ + demo/*.o test timing mtest_opponent mtest/mtest mtest/mtest.exe tuning_list \ + *.s mpi.c *.da *.dyn *.dpi tommath.tex `find . -type f | grep [~] | xargs` *.lo *.la + rm -rf .libs/ demo/.libs + ${MAKE} -C etc/ clean MAKE=${MAKE} + ${MAKE} -C doc/ clean MAKE=${MAKE} diff --git a/lib/hcrypto/libtommath/mess.sh b/lib/hcrypto/libtommath/mess.sh deleted file mode 100644 index bf639ce3b..000000000 --- a/lib/hcrypto/libtommath/mess.sh +++ /dev/null @@ -1,4 +0,0 @@ -#!/bin/bash -if cvs log $1 >/dev/null 2>/dev/null; then exit 0; else echo "$1 shouldn't be here" ; exit 1; fi - - diff --git a/lib/hcrypto/libtommath/mtest/logtab.h b/lib/hcrypto/libtommath/mtest/logtab.h index bbefaefc0..dae3344d6 100644 --- a/lib/hcrypto/libtommath/mtest/logtab.h +++ b/lib/hcrypto/libtommath/mtest/logtab.h @@ -1,24 +1,24 @@ const float s_logv_2[] = { - 0.000000000, 0.000000000, 1.000000000, 0.630929754, /* 0 1 2 3 */ - 0.500000000, 0.430676558, 0.386852807, 0.356207187, /* 4 5 6 7 */ - 0.333333333, 0.315464877, 0.301029996, 0.289064826, /* 8 9 10 11 */ - 0.278942946, 0.270238154, 0.262649535, 0.255958025, /* 12 13 14 15 */ - 0.250000000, 0.244650542, 0.239812467, 0.235408913, /* 16 17 18 19 */ - 0.231378213, 0.227670249, 0.224243824, 0.221064729, /* 20 21 22 23 */ - 0.218104292, 0.215338279, 0.212746054, 0.210309918, /* 24 25 26 27 */ - 0.208014598, 0.205846832, 0.203795047, 0.201849087, /* 28 29 30 31 */ - 0.200000000, 0.198239863, 0.196561632, 0.194959022, /* 32 33 34 35 */ - 0.193426404, 0.191958720, 0.190551412, 0.189200360, /* 36 37 38 39 */ - 0.187901825, 0.186652411, 0.185449023, 0.184288833, /* 40 41 42 43 */ - 0.183169251, 0.182087900, 0.181042597, 0.180031327, /* 44 45 46 47 */ - 0.179052232, 0.178103594, 0.177183820, 0.176291434, /* 48 49 50 51 */ - 0.175425064, 0.174583430, 0.173765343, 0.172969690, /* 52 53 54 55 */ - 0.172195434, 0.171441601, 0.170707280, 0.169991616, /* 56 57 58 59 */ - 0.169293808, 0.168613099, 0.167948779, 0.167300179, /* 60 61 62 63 */ + 0.000000000, 0.000000000, 1.000000000, 0.630929754, /* 0 1 2 3 */ + 0.500000000, 0.430676558, 0.386852807, 0.356207187, /* 4 5 6 7 */ + 0.333333333, 0.315464877, 0.301029996, 0.289064826, /* 8 9 10 11 */ + 0.278942946, 0.270238154, 0.262649535, 0.255958025, /* 12 13 14 15 */ + 0.250000000, 0.244650542, 0.239812467, 0.235408913, /* 16 17 18 19 */ + 0.231378213, 0.227670249, 0.224243824, 0.221064729, /* 20 21 22 23 */ + 0.218104292, 0.215338279, 0.212746054, 0.210309918, /* 24 25 26 27 */ + 0.208014598, 0.205846832, 0.203795047, 0.201849087, /* 28 29 30 31 */ + 0.200000000, 0.198239863, 0.196561632, 0.194959022, /* 32 33 34 35 */ + 0.193426404, 0.191958720, 0.190551412, 0.189200360, /* 36 37 38 39 */ + 0.187901825, 0.186652411, 0.185449023, 0.184288833, /* 40 41 42 43 */ + 0.183169251, 0.182087900, 0.181042597, 0.180031327, /* 44 45 46 47 */ + 0.179052232, 0.178103594, 0.177183820, 0.176291434, /* 48 49 50 51 */ + 0.175425064, 0.174583430, 0.173765343, 0.172969690, /* 52 53 54 55 */ + 0.172195434, 0.171441601, 0.170707280, 0.169991616, /* 56 57 58 59 */ + 0.169293808, 0.168613099, 0.167948779, 0.167300179, /* 60 61 62 63 */ 0.166666667 }; -/* $Source: /cvs/libtom/libtommath/mtest/logtab.h,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/mtest/mpi-config.h b/lib/hcrypto/libtommath/mtest/mpi-config.h index f69e36d65..ea576e563 100644 --- a/lib/hcrypto/libtommath/mtest/mpi-config.h +++ b/lib/hcrypto/libtommath/mtest/mpi-config.h @@ -1,5 +1,5 @@ /* Default configuration for MPI library */ -/* $Id: mpi-config.h,v 1.2 2005/05/05 14:38:47 tom Exp $ */ +/* $Id$ */ #ifndef MPI_CONFIG_H_ #define MPI_CONFIG_H_ @@ -85,6 +85,6 @@ /* crc==3287762869, version==2, Sat Feb 02 06:43:53 2002 */ -/* $Source: /cvs/libtom/libtommath/mtest/mpi-config.h,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/mtest/mpi-types.h b/lib/hcrypto/libtommath/mtest/mpi-types.h index 026de589d..f99d7eeae 100644 --- a/lib/hcrypto/libtommath/mtest/mpi-types.h +++ b/lib/hcrypto/libtommath/mtest/mpi-types.h @@ -15,6 +15,6 @@ typedef int mp_err; #define RADIX (MP_DIGIT_MAX+1) -/* $Source: /cvs/libtom/libtommath/mtest/mpi-types.h,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/mtest/mpi.c b/lib/hcrypto/libtommath/mtest/mpi.c index d8d05b00b..7e71ad609 100644 --- a/lib/hcrypto/libtommath/mtest/mpi.c +++ b/lib/hcrypto/libtommath/mtest/mpi.c @@ -2,11 +2,13 @@ mpi.c by Michael J. Fromberger - Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved + Copyright (C) 1998 Michael J. Fromberger Arbitrary precision integer arithmetic library - $Id: mpi.c,v 1.2 2005/05/05 14:38:47 tom Exp $ + SPDX-License-Identifier: Unlicense + + $Id$ */ #include "mpi.h" @@ -176,7 +178,7 @@ mp_err s_mp_add_d(mp_int *mp, mp_digit d); /* unsigned digit addition */ mp_err s_mp_sub_d(mp_int *mp, mp_digit d); /* unsigned digit subtract */ mp_err s_mp_mul_d(mp_int *mp, mp_digit d); /* unsigned digit multiply */ mp_err s_mp_div_d(mp_int *mp, mp_digit d, mp_digit *r); - /* unsigned digit divide */ + /* unsigned digit divide */ mp_err s_mp_reduce(mp_int *x, mp_int *m, mp_int *mu); /* Barrett reduction */ mp_err s_mp_add(mp_int *a, mp_int *b); /* magnitude addition */ @@ -358,15 +360,15 @@ mp_err mp_copy(mp_int *from, mp_int *to) } else { if((tmp = s_mp_alloc(USED(from), sizeof(mp_digit))) == NULL) - return MP_MEM; + return MP_MEM; s_mp_copy(DIGITS(from), tmp, USED(from)); if(DIGITS(to) != NULL) { #if MP_CRYPTO - s_mp_setz(DIGITS(to), ALLOC(to)); + s_mp_setz(DIGITS(to), ALLOC(to)); #endif - s_mp_free(DIGITS(to)); + s_mp_free(DIGITS(to)); } DIGITS(to) = tmp; @@ -507,7 +509,7 @@ mp_err mp_set_int(mp_int *mp, long z) return res; res = s_mp_add_d(mp, - (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX)); + (mp_digit)((v >> (ix * CHAR_BIT)) & UCHAR_MAX)); if(res != MP_OKAY) return res; @@ -772,7 +774,7 @@ mp_err mp_expt_d(mp_int *a, mp_digit d, mp_int *c) while(d != 0) { if(d & 1) { if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } d >>= 1; @@ -875,13 +877,13 @@ mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) */ if(c == b) { if((res = s_mp_add(c, a)) != MP_OKAY) - return res; + return res; } else { if(c != a && (res = mp_copy(a, c)) != MP_OKAY) - return res; + return res; if((res = s_mp_add(c, b)) != MP_OKAY) - return res; + return res; } } else if((cmp = s_mp_cmp(a, b)) > 0) { /* different sign: a > b */ @@ -894,10 +896,10 @@ mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) mp_int tmp; if((res = mp_init_copy(&tmp, a)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { - mp_clear(&tmp); - return res; + mp_clear(&tmp); + return res; } s_mp_exch(&tmp, c); @@ -906,9 +908,9 @@ mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) } else { if(c != a && (res = mp_copy(a, c)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(c, b)) != MP_OKAY) - return res; + return res; } @@ -924,10 +926,10 @@ mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) mp_int tmp; if((res = mp_init_copy(&tmp, b)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { - mp_clear(&tmp); - return res; + mp_clear(&tmp); + return res; } s_mp_exch(&tmp, c); @@ -936,9 +938,9 @@ mp_err mp_add(mp_int *a, mp_int *b, mp_int *c) } else { if(c != b && (res = mp_copy(b, c)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(c, a)) != MP_OKAY) - return res; + return res; } } @@ -970,12 +972,12 @@ mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c) if(SIGN(a) != SIGN(b)) { if(c == a) { if((res = s_mp_add(c, b)) != MP_OKAY) - return res; + return res; } else { if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) - return res; + return res; if((res = s_mp_add(c, a)) != MP_OKAY) - return res; + return res; SIGN(c) = SIGN(a); } @@ -984,20 +986,20 @@ mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c) mp_int tmp; if((res = mp_init_copy(&tmp, a)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(&tmp, b)) != MP_OKAY) { - mp_clear(&tmp); - return res; + mp_clear(&tmp); + return res; } s_mp_exch(&tmp, c); mp_clear(&tmp); } else { if(c != a && ((res = mp_copy(a, c)) != MP_OKAY)) - return res; + return res; if((res = s_mp_sub(c, b)) != MP_OKAY) - return res; + return res; } } else if(cmp == 0) { /* Same sign, equal magnitude */ @@ -1009,21 +1011,21 @@ mp_err mp_sub(mp_int *a, mp_int *b, mp_int *c) mp_int tmp; if((res = mp_init_copy(&tmp, b)) != MP_OKAY) - return res; + return res; if((res = s_mp_sub(&tmp, a)) != MP_OKAY) { - mp_clear(&tmp); - return res; + mp_clear(&tmp); + return res; } s_mp_exch(&tmp, c); mp_clear(&tmp); } else { if(c != b && ((res = mp_copy(b, c)) != MP_OKAY)) - return res; + return res; if((res = s_mp_sub(c, a)) != MP_OKAY) - return res; + return res; } SIGN(c) = !SIGN(b); @@ -1157,7 +1159,7 @@ mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r) if((cmp = s_mp_cmp(a, b)) < 0) { if(r) { if((res = mp_copy(a, r)) != MP_OKAY) - return res; + return res; } if(q) @@ -1173,7 +1175,7 @@ mp_err mp_div(mp_int *a, mp_int *b, mp_int *q, mp_int *r) mp_set(q, 1); if(qneg) - SIGN(q) = MP_NEG; + SIGN(q) = MP_NEG; } if(r) @@ -1264,7 +1266,7 @@ mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) mp_int s, x; mp_err res; mp_digit d; - int dig, bit; + unsigned int bit, dig; ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); @@ -1286,14 +1288,14 @@ mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) /* Loop over bits of each non-maximal digit */ for(bit = 0; bit < DIGIT_BIT; bit++) { if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; } d >>= 1; if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } } @@ -1303,7 +1305,7 @@ mp_err mp_expt(mp_int *a, mp_int *b, mp_int *c) while(d) { if(d & 1) { if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } d >>= 1; @@ -1379,7 +1381,7 @@ mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c) if(SIGN(c) == MP_NEG) { if((res = mp_add(c, m, c)) != MP_OKAY) - return res; + return res; } } else if(mag < 0) { @@ -1388,7 +1390,7 @@ mp_err mp_mod(mp_int *a, mp_int *m, mp_int *c) if(mp_cmp_z(a) < 0) { if((res = mp_add(c, m, c)) != MP_OKAY) - return res; + return res; } @@ -1637,7 +1639,7 @@ mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) mp_err res; mp_digit d, *db = DIGITS(b); mp_size ub = USED(b); - int dig, bit; + unsigned int bit, dig; ARGCHK(a != NULL && b != NULL && c != NULL, MP_BADARG); @@ -1667,18 +1669,18 @@ mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) /* Loop over the bits of the lower-order digits */ for(bit = 0; bit < DIGIT_BIT; bit++) { if(d & 1) { - if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; - if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) - goto CLEANUP; + if((res = s_mp_mul(&s, &x)) != MP_OKAY) + goto CLEANUP; + if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) + goto CLEANUP; } d >>= 1; if((res = s_mp_sqr(&x)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; if((res = s_mp_reduce(&x, m, &mu)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } } @@ -1688,9 +1690,9 @@ mp_err mp_exptmod(mp_int *a, mp_int *b, mp_int *m, mp_int *c) while(d) { if(d & 1) { if((res = s_mp_mul(&s, &x)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; if((res = s_mp_reduce(&s, m, &mu)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } d >>= 1; @@ -1735,8 +1737,8 @@ mp_err mp_exptmod_d(mp_int *a, mp_digit d, mp_int *m, mp_int *c) while(d != 0) { if(d & 1) { if((res = s_mp_mul(&s, &x)) != MP_OKAY || - (res = mp_mod(&s, m, &s)) != MP_OKAY) - goto CLEANUP; + (res = mp_mod(&s, m, &s)) != MP_OKAY) + goto CLEANUP; } d /= 2; @@ -1973,17 +1975,17 @@ mp_err mp_gcd(mp_int *a, mp_int *b, mp_int *c) if(mp_cmp_z(&t) == MP_GT) { if((res = mp_copy(&t, &u)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; } else { if((res = mp_copy(&t, &v)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; /* v = -t */ if(SIGN(&t) == MP_ZPOS) - SIGN(&v) = MP_NEG; + SIGN(&v) = MP_NEG; else - SIGN(&v) = MP_ZPOS; + SIGN(&v) = MP_ZPOS; } if((res = mp_sub(&u, &v, &t)) != MP_OKAY) @@ -2111,12 +2113,12 @@ mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y) s_mp_div_2(&u); if(mp_iseven(&A) && mp_iseven(&B)) { - s_mp_div_2(&A); s_mp_div_2(&B); + s_mp_div_2(&A); s_mp_div_2(&B); } else { - if((res = mp_add(&A, &yc, &A)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&A); - if((res = mp_sub(&B, &xc, &B)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&B); + if((res = mp_add(&A, &yc, &A)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&A); + if((res = mp_sub(&B, &xc, &B)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&B); } } @@ -2124,12 +2126,12 @@ mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y) s_mp_div_2(&v); if(mp_iseven(&C) && mp_iseven(&D)) { - s_mp_div_2(&C); s_mp_div_2(&D); + s_mp_div_2(&C); s_mp_div_2(&D); } else { - if((res = mp_add(&C, &yc, &C)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&C); - if((res = mp_sub(&D, &xc, &D)) != MP_OKAY) goto CLEANUP; - s_mp_div_2(&D); + if((res = mp_add(&C, &yc, &C)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&C); + if((res = mp_sub(&D, &xc, &D)) != MP_OKAY) goto CLEANUP; + s_mp_div_2(&D); } } @@ -2148,13 +2150,13 @@ mp_err mp_xgcd(mp_int *a, mp_int *b, mp_int *g, mp_int *x, mp_int *y) /* If we're done, copy results to output */ if(mp_cmp_z(&u) == 0) { if(x) - if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP; + if((res = mp_copy(&C, x)) != MP_OKAY) goto CLEANUP; if(y) - if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP; + if((res = mp_copy(&D, y)) != MP_OKAY) goto CLEANUP; if(g) - if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP; + if((res = mp_mul(&gx, &v, g)) != MP_OKAY) goto CLEANUP; break; } @@ -2387,7 +2389,7 @@ mp_err mp_to_unsigned_bin(mp_int *mp, unsigned char *str) /* Generate digits in reverse order */ while(dp < end) { - int ix; + unsigned int ix; d = *dp; for(ix = 0; ix < sizeof(mp_digit); ++ix) { @@ -2463,15 +2465,15 @@ mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix) mp_sign sig = MP_ZPOS; ARGCHK(mp != NULL && str != NULL && radix >= 2 && radix <= MAX_RADIX, - MP_BADARG); + MP_BADARG); mp_zero(mp); /* Skip leading non-digit characters until a digit or '-' or '+' */ while(str[ix] && - (s_mp_tovalue(str[ix], radix) < 0) && - str[ix] != '-' && - str[ix] != '+') { + (s_mp_tovalue(str[ix], radix) < 0) && + str[ix] != '-' && + str[ix] != '+') { ++ix; } @@ -2541,7 +2543,7 @@ int mp_value_radix_size(int num, int qty, int radix) /* {{{ mp_toradix(mp, str, radix) */ -mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix) +mp_err mp_toradix(mp_int *mp, char *str, int radix) { int ix, pos = 0; @@ -2567,8 +2569,8 @@ mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix) /* Generate output digits in reverse order */ while(mp_cmp_z(&tmp) != 0) { if((res = s_mp_div_d(&tmp, rdx, &rem)) != MP_OKAY) { - mp_clear(&tmp); - return res; + mp_clear(&tmp); + return res; } /* Generate digits, use capital letters */ @@ -2587,10 +2589,10 @@ mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix) /* Reverse the digits and sign indicator */ ix = 0; while(ix < pos) { - char tmp = str[ix]; + char _tmp = str[ix]; str[ix] = str[pos]; - str[pos] = tmp; + str[pos] = _tmp; ++ix; --pos; } @@ -2817,7 +2819,7 @@ mp_err s_mp_lshd(mp_int *mp, mp_size p) mp_err res; mp_size pos; mp_digit *dp; - int ix; + int ix; if(p == 0) return MP_OKAY; @@ -2833,7 +2835,7 @@ mp_err s_mp_lshd(mp_int *mp, mp_size p) dp[ix + p] = dp[ix]; /* Fill the bottom digits with zeroes */ - for(ix = 0; ix < p; ix++) + for(ix = 0; (unsigned)ix < p; ix++) dp[ix] = 0; return MP_OKAY; @@ -2898,7 +2900,7 @@ void s_mp_div_2(mp_int *mp) mp_err s_mp_mul_2(mp_int *mp) { - int ix; + unsigned int ix; mp_digit kin = 0, kout, *dp = DIGITS(mp); mp_err res; @@ -2914,7 +2916,7 @@ mp_err s_mp_mul_2(mp_int *mp) if(kin) { if(ix >= ALLOC(mp)) { if((res = s_mp_grow(mp, ALLOC(mp) + 1)) != MP_OKAY) - return res; + return res; dp = DIGITS(mp); } @@ -2970,7 +2972,7 @@ mp_err s_mp_mul_2d(mp_int *mp, mp_digit d) mp_err res; mp_digit save, next, mask, *dp; mp_size used; - int ix; + unsigned int ix; if((res = s_mp_lshd(mp, d / DIGIT_BIT)) != MP_OKAY) return res; @@ -3555,12 +3557,12 @@ mp_err s_mp_sqr(mp_int *a) w = *pa1 * *pa2; /* If w is more than half MP_WORD_MAX, the doubling will - overflow, and we need to record a carry out into the next - word */ + overflow, and we need to record a carry out into the next + word */ u = (w >> (MP_WORD_BIT - 1)) & 1; /* Double what we've got, overflow will be ignored as defined - for C arithmetic (we've already noted if it is to occur) + for C arithmetic (we've already noted if it is to occur) */ w *= 2; @@ -3568,7 +3570,7 @@ mp_err s_mp_sqr(mp_int *a) v = *pt + k; /* If we do not already have an overflow carry, check to see - if the addition will cause one, and set the carry out if so + if the addition will cause one, and set the carry out if so */ u |= ((MP_WORD_MAX - v) < w); @@ -3579,7 +3581,7 @@ mp_err s_mp_sqr(mp_int *a) *pt = ACCUM(w); /* Save carry information for the next iteration of the loop. - This is why k must be an mp_word, instead of an mp_digit */ + This is why k must be an mp_word, instead of an mp_digit */ k = CARRYOUT(w) | (u << DIGIT_BIT); } /* for(jx ...) */ @@ -3665,10 +3667,10 @@ mp_err s_mp_div(mp_int *a, mp_int *b) /* Find a partial substring of a which is at least b */ while(s_mp_cmp(&rem, b) < 0 && ix >= 0) { if((res = s_mp_lshd(&rem, 1)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; if((res = s_mp_lshd(", 1)) != MP_OKAY) - goto CLEANUP; + goto CLEANUP; DIGIT(&rem, 0) = DIGIT(a, ix); s_mp_clamp(&rem); @@ -3790,9 +3792,9 @@ int s_mp_cmp(mp_int *a, mp_int *b) while(ix >= 0) { if(*ap > *bp) - return MP_GT; + return MP_GT; else if(*ap < *bp) - return MP_LT; + return MP_LT; --ap; --bp; --ix; } @@ -3851,7 +3853,7 @@ int s_mp_ispow2(mp_int *v) while(ix >= 0) { if(*dp) - return -1; /* not a power of two */ + return -1; /* not a power of two */ --dp; --ix; } @@ -3980,6 +3982,6 @@ int s_mp_outlen(int bits, int r) /* HERE THERE BE DRAGONS */ /* crc==4242132123, version==2, Sat Feb 02 06:43:52 2002 */ -/* $Source: /cvs/libtom/libtommath/mtest/mpi.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/mtest/mpi.h b/lib/hcrypto/libtommath/mtest/mpi.h index 66ae87311..9a9cc4101 100644 --- a/lib/hcrypto/libtommath/mtest/mpi.h +++ b/lib/hcrypto/libtommath/mtest/mpi.h @@ -2,11 +2,13 @@ mpi.h by Michael J. Fromberger - Copyright (C) 1998 Michael J. Fromberger, All Rights Reserved + Copyright (C) 1998 Michael J. Fromberger Arbitrary precision integer arithmetic library - $Id: mpi.h,v 1.2 2005/05/05 14:38:47 tom Exp $ + SPDX-License-Identifier: Unlicense + + $Id$ */ #ifndef _H_MPI_ @@ -210,7 +212,7 @@ int mp_count_bits(mp_int *mp); mp_err mp_read_radix(mp_int *mp, unsigned char *str, int radix); int mp_radix_size(mp_int *mp, int radix); int mp_value_radix_size(int num, int qty, int radix); -mp_err mp_toradix(mp_int *mp, unsigned char *str, int radix); +mp_err mp_toradix(mp_int *mp, char *str, int radix); int mp_char2value(char ch, int r); @@ -226,6 +228,6 @@ const char *mp_strerror(mp_err ec); #endif /* end _H_MPI_ */ -/* $Source: /cvs/libtom/libtommath/mtest/mpi.h,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/mtest/mtest.c b/lib/hcrypto/libtommath/mtest/mtest.c index bdfe6127a..06c9afb1f 100644 --- a/lib/hcrypto/libtommath/mtest/mtest.c +++ b/lib/hcrypto/libtommath/mtest/mtest.c @@ -39,39 +39,71 @@ mulmod #include #include "mpi.c" +#ifdef LTM_MTEST_REAL_RAND +#define getRandChar() fgetc(rng) FILE *rng; +#else +#define getRandChar() (rand()&0xFF) +#endif void rand_num(mp_int *a) { - int n, size; + int size; unsigned char buf[2048]; + size_t sz; - size = 1 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; - buf[0] = (fgetc(rng)&1)?1:0; - fread(buf+1, 1, size, rng); - while (buf[1] == 0) buf[1] = fgetc(rng); + size = 1 + ((getRandChar()<<8) + getRandChar()) % 101; + buf[0] = (getRandChar()&1)?1:0; +#ifdef LTM_MTEST_REAL_RAND + sz = fread(buf+1, 1, size, rng); +#else + sz = 1; + while (sz < (unsigned)size) { + buf[sz] = getRandChar(); + ++sz; + } +#endif + if (sz != (unsigned)size) { + fprintf(stderr, "\nWarning: fread failed\n\n"); + } + while (buf[1] == 0) buf[1] = getRandChar(); mp_read_raw(a, buf, 1+size); } void rand_num2(mp_int *a) { - int n, size; + int size; unsigned char buf[2048]; + size_t sz; - size = 10 + ((fgetc(rng)<<8) + fgetc(rng)) % 101; - buf[0] = (fgetc(rng)&1)?1:0; - fread(buf+1, 1, size, rng); - while (buf[1] == 0) buf[1] = fgetc(rng); + size = 10 + ((getRandChar()<<8) + getRandChar()) % 101; + buf[0] = (getRandChar()&1)?1:0; +#ifdef LTM_MTEST_REAL_RAND + sz = fread(buf+1, 1, size, rng); +#else + sz = 1; + while (sz < (unsigned)size) { + buf[sz] = getRandChar(); + ++sz; + } +#endif + if (sz != (unsigned)size) { + fprintf(stderr, "\nWarning: fread failed\n\n"); + } + while (buf[1] == 0) buf[1] = getRandChar(); mp_read_raw(a, buf, 1+size); } #define mp_to64(a, b) mp_toradix(a, b, 64) -int main(void) +int main(int argc, char *argv[]) { int n, tmp; + long long max; mp_int a, b, c, d, e; +#ifdef MTEST_NO_FULLSPEED clock_t t1; +#endif char buf[4096]; mp_init(&a); @@ -80,229 +112,263 @@ int main(void) mp_init(&d); mp_init(&e); + if (argc > 1) { + max = strtol(argv[1], NULL, 0); + if (max < 0) { + if (max > -64) { + max = (1 << -(max)) + 1; + } else { + max = 1; + } + } else if (max == 0) { + max = 1; + } + } else { + max = 0; + } + /* initial (2^n - 1)^2 testing, makes sure the comba multiplier works [it has the new carry code] */ -/* - mp_set(&a, 1); - for (n = 1; n < 8192; n++) { - mp_mul(&a, &a, &c); - printf("mul\n"); - mp_to64(&a, buf); - printf("%s\n%s\n", buf, buf); - mp_to64(&c, buf); - printf("%s\n", buf); + /* + mp_set(&a, 1); + for (n = 1; n < 8192; n++) { + mp_mul(&a, &a, &c); + printf("mul\n"); + mp_to64(&a, buf); + printf("%s\n%s\n", buf, buf); + mp_to64(&c, buf); + printf("%s\n", buf); - mp_add_d(&a, 1, &a); - mp_mul_2(&a, &a); - mp_sub_d(&a, 1, &a); - } -*/ + mp_add_d(&a, 1, &a); + mp_mul_2(&a, &a); + mp_sub_d(&a, 1, &a); + } + */ +#ifdef LTM_MTEST_REAL_RAND rng = fopen("/dev/urandom", "rb"); if (rng == NULL) { rng = fopen("/dev/random", "rb"); if (rng == NULL) { - fprintf(stderr, "\nWarning: stdin used as random source\n\n"); - rng = stdin; + fprintf(stderr, "\nWarning: no /dev/[u]random available\n\n"); + printf("exit\n"); + return 1; } } +#else + srand(23); +#endif +#ifdef MTEST_NO_FULLSPEED t1 = clock(); +#endif for (;;) { -#if 0 +#ifdef MTEST_NO_FULLSPEED if (clock() - t1 > CLOCKS_PER_SEC) { sleep(2); t1 = clock(); } #endif - n = fgetc(rng) % 15; + n = getRandChar() % 15; + + if (max != 0) { + --max; + if (max == 0) + n = 255; + } + + if (n == 0) { + /* add tests */ + rand_num(&a); + rand_num(&b); + mp_add(&a, &b, &c); + printf("add\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 1) { + /* sub tests */ + rand_num(&a); + rand_num(&b); + mp_sub(&a, &b, &c); + printf("sub\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 2) { + /* mul tests */ + rand_num(&a); + rand_num(&b); + mp_mul(&a, &b, &c); + printf("mul\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 3) { + /* div tests */ + rand_num(&a); + rand_num(&b); + mp_div(&a, &b, &c, &d); + printf("div\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + mp_to64(&d, buf); + printf("%s\n", buf); + } else if (n == 4) { + /* sqr tests */ + rand_num(&a); + mp_sqr(&a, &b); + printf("sqr\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 5) { + /* mul_2d test */ + rand_num(&a); + mp_copy(&a, &b); + n = getRandChar() & 63; + mp_mul_2d(&b, n, &b); + mp_to64(&a, buf); + printf("mul2d\n"); + printf("%s\n", buf); + printf("%d\n", n); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 6) { + /* div_2d test */ + rand_num(&a); + mp_copy(&a, &b); + n = getRandChar() & 63; + mp_div_2d(&b, n, &b, NULL); + mp_to64(&a, buf); + printf("div2d\n"); + printf("%s\n", buf); + printf("%d\n", n); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 7) { + /* gcd test */ + rand_num(&a); + rand_num(&b); + a.sign = MP_ZPOS; + b.sign = MP_ZPOS; + mp_gcd(&a, &b, &c); + printf("gcd\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 8) { + /* lcm test */ + rand_num(&a); + rand_num(&b); + a.sign = MP_ZPOS; + b.sign = MP_ZPOS; + mp_lcm(&a, &b, &c); + printf("lcm\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 9) { + /* exptmod test */ + rand_num2(&a); + rand_num2(&b); + rand_num2(&c); + /* if (c.dp[0]&1) mp_add_d(&c, 1, &c); */ + a.sign = b.sign = c.sign = 0; + mp_exptmod(&a, &b, &c, &d); + printf("expt\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + mp_to64(&d, buf); + printf("%s\n", buf); + } else if (n == 10) { + /* invmod test */ + do { + rand_num2(&a); + rand_num2(&b); + b.sign = MP_ZPOS; + a.sign = MP_ZPOS; + mp_gcd(&a, &b, &c); + } while (mp_cmp_d(&c, 1) != 0 || mp_cmp_d(&b, 1) == 0); + mp_invmod(&a, &b, &c); + printf("invmod\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + mp_to64(&c, buf); + printf("%s\n", buf); + } else if (n == 11) { + rand_num(&a); + mp_mul_2(&a, &a); + mp_div_2(&a, &b); + printf("div2\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 12) { + rand_num2(&a); + mp_mul_2(&a, &b); + printf("mul2\n"); + mp_to64(&a, buf); + printf("%s\n", buf); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 13) { + rand_num2(&a); + tmp = abs(rand()) & THE_MASK; + mp_add_d(&a, tmp, &b); + printf("add_d\n"); + mp_to64(&a, buf); + printf("%s\n%d\n", buf, tmp); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 14) { + rand_num2(&a); + tmp = abs(rand()) & THE_MASK; + mp_sub_d(&a, tmp, &b); + printf("sub_d\n"); + mp_to64(&a, buf); + printf("%s\n%d\n", buf, tmp); + mp_to64(&b, buf); + printf("%s\n", buf); + } else if (n == 255) { + printf("exit\n"); + break; + } - if (n == 0) { - /* add tests */ - rand_num(&a); - rand_num(&b); - mp_add(&a, &b, &c); - printf("add\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 1) { - /* sub tests */ - rand_num(&a); - rand_num(&b); - mp_sub(&a, &b, &c); - printf("sub\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 2) { - /* mul tests */ - rand_num(&a); - rand_num(&b); - mp_mul(&a, &b, &c); - printf("mul\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 3) { - /* div tests */ - rand_num(&a); - rand_num(&b); - mp_div(&a, &b, &c, &d); - printf("div\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - mp_to64(&d, buf); - printf("%s\n", buf); - } else if (n == 4) { - /* sqr tests */ - rand_num(&a); - mp_sqr(&a, &b); - printf("sqr\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 5) { - /* mul_2d test */ - rand_num(&a); - mp_copy(&a, &b); - n = fgetc(rng) & 63; - mp_mul_2d(&b, n, &b); - mp_to64(&a, buf); - printf("mul2d\n"); - printf("%s\n", buf); - printf("%d\n", n); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 6) { - /* div_2d test */ - rand_num(&a); - mp_copy(&a, &b); - n = fgetc(rng) & 63; - mp_div_2d(&b, n, &b, NULL); - mp_to64(&a, buf); - printf("div2d\n"); - printf("%s\n", buf); - printf("%d\n", n); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 7) { - /* gcd test */ - rand_num(&a); - rand_num(&b); - a.sign = MP_ZPOS; - b.sign = MP_ZPOS; - mp_gcd(&a, &b, &c); - printf("gcd\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 8) { - /* lcm test */ - rand_num(&a); - rand_num(&b); - a.sign = MP_ZPOS; - b.sign = MP_ZPOS; - mp_lcm(&a, &b, &c); - printf("lcm\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 9) { - /* exptmod test */ - rand_num2(&a); - rand_num2(&b); - rand_num2(&c); -// if (c.dp[0]&1) mp_add_d(&c, 1, &c); - a.sign = b.sign = c.sign = 0; - mp_exptmod(&a, &b, &c, &d); - printf("expt\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - mp_to64(&d, buf); - printf("%s\n", buf); - } else if (n == 10) { - /* invmod test */ - rand_num2(&a); - rand_num2(&b); - b.sign = MP_ZPOS; - a.sign = MP_ZPOS; - mp_gcd(&a, &b, &c); - if (mp_cmp_d(&c, 1) != 0) continue; - if (mp_cmp_d(&b, 1) == 0) continue; - mp_invmod(&a, &b, &c); - printf("invmod\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - mp_to64(&c, buf); - printf("%s\n", buf); - } else if (n == 11) { - rand_num(&a); - mp_mul_2(&a, &a); - mp_div_2(&a, &b); - printf("div2\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 12) { - rand_num2(&a); - mp_mul_2(&a, &b); - printf("mul2\n"); - mp_to64(&a, buf); - printf("%s\n", buf); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 13) { - rand_num2(&a); - tmp = abs(rand()) & THE_MASK; - mp_add_d(&a, tmp, &b); - printf("add_d\n"); - mp_to64(&a, buf); - printf("%s\n%d\n", buf, tmp); - mp_to64(&b, buf); - printf("%s\n", buf); - } else if (n == 14) { - rand_num2(&a); - tmp = abs(rand()) & THE_MASK; - mp_sub_d(&a, tmp, &b); - printf("sub_d\n"); - mp_to64(&a, buf); - printf("%s\n%d\n", buf, tmp); - mp_to64(&b, buf); - printf("%s\n", buf); - } } +#ifdef LTM_MTEST_REAL_RAND fclose(rng); +#endif return 0; } -/* $Source: /cvs/libtom/libtommath/mtest/mtest.c,v $ */ -/* $Revision: 1.2 $ */ -/* $Date: 2005/05/05 14:38:47 $ */ +/* $Source$ */ +/* $Revision$ */ +/* $Date$ */ diff --git a/lib/hcrypto/libtommath/pics/design_process.sxd b/lib/hcrypto/libtommath/pics/design_process.sxd deleted file mode 100644 index 7414dbb27..000000000 Binary files a/lib/hcrypto/libtommath/pics/design_process.sxd and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/design_process.tif b/lib/hcrypto/libtommath/pics/design_process.tif deleted file mode 100644 index 4a0c012d5..000000000 Binary files a/lib/hcrypto/libtommath/pics/design_process.tif and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/expt_state.sxd b/lib/hcrypto/libtommath/pics/expt_state.sxd deleted file mode 100644 index 6518404d5..000000000 Binary files a/lib/hcrypto/libtommath/pics/expt_state.sxd and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/expt_state.tif b/lib/hcrypto/libtommath/pics/expt_state.tif deleted file mode 100644 index cb06e8edf..000000000 Binary files a/lib/hcrypto/libtommath/pics/expt_state.tif and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/makefile b/lib/hcrypto/libtommath/pics/makefile deleted file mode 100644 index f55a600b6..000000000 --- a/lib/hcrypto/libtommath/pics/makefile +++ /dev/null @@ -1,35 +0,0 @@ -# makes the images... yeah - -default: pses - -design_process.ps: design_process.tif - tiff2ps -s -e design_process.tif > design_process.ps - -sliding_window.ps: sliding_window.tif - tiff2ps -s -e sliding_window.tif > sliding_window.ps - -expt_state.ps: expt_state.tif - tiff2ps -s -e expt_state.tif > expt_state.ps - -primality.ps: primality.tif - tiff2ps -s -e primality.tif > primality.ps - -design_process.pdf: design_process.ps - epstopdf design_process.ps - -sliding_window.pdf: sliding_window.ps - epstopdf sliding_window.ps - -expt_state.pdf: expt_state.ps - epstopdf expt_state.ps - -primality.pdf: primality.ps - epstopdf primality.ps - - -pses: sliding_window.ps expt_state.ps primality.ps design_process.ps -pdfes: sliding_window.pdf expt_state.pdf primality.pdf design_process.pdf - -clean: - rm -rf *.ps *.pdf .xvpics - diff --git a/lib/hcrypto/libtommath/pics/primality.tif b/lib/hcrypto/libtommath/pics/primality.tif deleted file mode 100644 index 76d6be3fa..000000000 Binary files a/lib/hcrypto/libtommath/pics/primality.tif and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/radix.sxd b/lib/hcrypto/libtommath/pics/radix.sxd deleted file mode 100644 index b9eb9a032..000000000 Binary files a/lib/hcrypto/libtommath/pics/radix.sxd and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/sliding_window.sxd b/lib/hcrypto/libtommath/pics/sliding_window.sxd deleted file mode 100644 index 91e7c0d9f..000000000 Binary files a/lib/hcrypto/libtommath/pics/sliding_window.sxd and /dev/null differ diff --git a/lib/hcrypto/libtommath/pics/sliding_window.tif b/lib/hcrypto/libtommath/pics/sliding_window.tif deleted file mode 100644 index bb4cb96eb..000000000 Binary files a/lib/hcrypto/libtommath/pics/sliding_window.tif and /dev/null differ diff --git a/lib/hcrypto/libtommath/poster.out b/lib/hcrypto/libtommath/poster.out deleted file mode 100644 index e69de29bb..000000000 diff --git a/lib/hcrypto/libtommath/poster.pdf b/lib/hcrypto/libtommath/poster.pdf deleted file mode 100644 index f3768d769..000000000 Binary files a/lib/hcrypto/libtommath/poster.pdf and /dev/null differ diff --git a/lib/hcrypto/libtommath/poster.tex b/lib/hcrypto/libtommath/poster.tex deleted file mode 100644 index e7388f44d..000000000 --- a/lib/hcrypto/libtommath/poster.tex +++ /dev/null @@ -1,35 +0,0 @@ -\documentclass[landscape,11pt]{article} -\usepackage{amsmath, amssymb} -\usepackage{hyperref} -\begin{document} -\hspace*{-3in} -\begin{tabular}{llllll} -$c = a + b$ & {\tt mp\_add(\&a, \&b, \&c)} & $b = 2a$ & {\tt mp\_mul\_2(\&a, \&b)} & \\ -$c = a - b$ & {\tt mp\_sub(\&a, \&b, \&c)} & $b = a/2$ & {\tt mp\_div\_2(\&a, \&b)} & \\ -$c = ab $ & {\tt mp\_mul(\&a, \&b, \&c)} & $c = 2^ba$ & {\tt mp\_mul\_2d(\&a, b, \&c)} \\ -$b = a^2 $ & {\tt mp\_sqr(\&a, \&b)} & $c = a/2^b, d = a \mod 2^b$ & {\tt mp\_div\_2d(\&a, b, \&c, \&d)} \\ -$c = \lfloor a/b \rfloor, d = a \mod b$ & {\tt mp\_div(\&a, \&b, \&c, \&d)} & $c = a \mod 2^b $ & {\tt mp\_mod\_2d(\&a, b, \&c)} \\ - && \\ -$a = b $ & {\tt mp\_set\_int(\&a, b)} & $c = a \vee b$ & {\tt mp\_or(\&a, \&b, \&c)} \\ -$b = a $ & {\tt mp\_copy(\&a, \&b)} & $c = a \wedge b$ & {\tt mp\_and(\&a, \&b, \&c)} \\ - && $c = a \oplus b$ & {\tt mp\_xor(\&a, \&b, \&c)} \\ - & \\ -$b = -a $ & {\tt mp\_neg(\&a, \&b)} & $d = a + b \mod c$ & {\tt mp\_addmod(\&a, \&b, \&c, \&d)} \\ -$b = |a| $ & {\tt mp\_abs(\&a, \&b)} & $d = a - b \mod c$ & {\tt mp\_submod(\&a, \&b, \&c, \&d)} \\ - && $d = ab \mod c$ & {\tt mp\_mulmod(\&a, \&b, \&c, \&d)} \\ -Compare $a$ and $b$ & {\tt mp\_cmp(\&a, \&b)} & $c = a^2 \mod b$ & {\tt mp\_sqrmod(\&a, \&b, \&c)} \\ -Is Zero? & {\tt mp\_iszero(\&a)} & $c = a^{-1} \mod b$ & {\tt mp\_invmod(\&a, \&b, \&c)} \\ -Is Even? & {\tt mp\_iseven(\&a)} & $d = a^b \mod c$ & {\tt mp\_exptmod(\&a, \&b, \&c, \&d)} \\ -Is Odd ? & {\tt mp\_isodd(\&a)} \\ -&\\ -$\vert \vert a \vert \vert$ & {\tt mp\_unsigned\_bin\_size(\&a)} & $res$ = 1 if $a$ prime to $t$ rounds? & {\tt mp\_prime\_is\_prime(\&a, t, \&res)} \\ -$buf \leftarrow a$ & {\tt mp\_to\_unsigned\_bin(\&a, buf)} & Next prime after $a$ to $t$ rounds. & {\tt mp\_prime\_next\_prime(\&a, t, bbs\_style)} \\ -$a \leftarrow buf[0..len-1]$ & {\tt mp\_read\_unsigned\_bin(\&a, buf, len)} \\ -&\\ -$b = \sqrt{a}$ & {\tt mp\_sqrt(\&a, \&b)} & $c = \mbox{gcd}(a, b)$ & {\tt mp\_gcd(\&a, \&b, \&c)} \\ -$c = a^{1/b}$ & {\tt mp\_n\_root(\&a, b, \&c)} & $c = \mbox{lcm}(a, b)$ & {\tt mp\_lcm(\&a, \&b, \&c)} \\ -&\\ -Greater Than & MP\_GT & Equal To & MP\_EQ \\ -Less Than & MP\_LT & Bits per digit & DIGIT\_BIT \\ -\end{tabular} -\end{document} diff --git a/lib/hcrypto/libtommath/pre_gen/mpi.c b/lib/hcrypto/libtommath/pre_gen/mpi.c new file mode 100644 index 000000000..96f001d1f --- /dev/null +++ b/lib/hcrypto/libtommath/pre_gen/mpi.c @@ -0,0 +1,9541 @@ +/* Start: bn_cutoffs.c */ +#include "tommath_private.h" +#ifdef BN_CUTOFFS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifndef MP_FIXED_CUTOFFS +#include "tommath_cutoffs.h" +int KARATSUBA_MUL_CUTOFF = MP_DEFAULT_KARATSUBA_MUL_CUTOFF, + KARATSUBA_SQR_CUTOFF = MP_DEFAULT_KARATSUBA_SQR_CUTOFF, + TOOM_MUL_CUTOFF = MP_DEFAULT_TOOM_MUL_CUTOFF, + TOOM_SQR_CUTOFF = MP_DEFAULT_TOOM_SQR_CUTOFF; +#endif + +#endif + +/* End: bn_cutoffs.c */ + +/* Start: bn_deprecated.c */ +#include "tommath_private.h" +#ifdef BN_DEPRECATED_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifdef BN_MP_GET_BIT_C +int mp_get_bit(const mp_int *a, int b) +{ + if (b < 0) { + return MP_VAL; + } + return (s_mp_get_bit(a, (unsigned int)b) == MP_YES) ? MP_YES : MP_NO; +} +#endif +#ifdef BN_MP_JACOBI_C +mp_err mp_jacobi(const mp_int *a, const mp_int *n, int *c) +{ + if (a->sign == MP_NEG) { + return MP_VAL; + } + if (mp_cmp_d(n, 0uL) != MP_GT) { + return MP_VAL; + } + return mp_kronecker(a, n, c); +} +#endif +#ifdef BN_MP_PRIME_RANDOM_EX_C +mp_err mp_prime_random_ex(mp_int *a, int t, int size, int flags, private_mp_prime_callback cb, void *dat) +{ + return s_mp_prime_random_ex(a, t, size, flags, cb, dat); +} +#endif +#ifdef BN_MP_RAND_DIGIT_C +mp_err mp_rand_digit(mp_digit *r) +{ + mp_err err = s_mp_rand_source(r, sizeof(mp_digit)); + *r &= MP_MASK; + return err; +} +#endif +#ifdef BN_FAST_MP_INVMOD_C +mp_err fast_mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_invmod_fast(a, b, c); +} +#endif +#ifdef BN_FAST_MP_MONTGOMERY_REDUCE_C +mp_err fast_mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) +{ + return s_mp_montgomery_reduce_fast(x, n, rho); +} +#endif +#ifdef BN_FAST_S_MP_MUL_DIGS_C +mp_err fast_s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + return s_mp_mul_digs_fast(a, b, c, digs); +} +#endif +#ifdef BN_FAST_S_MP_MUL_HIGH_DIGS_C +mp_err fast_s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + return s_mp_mul_high_digs_fast(a, b, c, digs); +} +#endif +#ifdef BN_FAST_S_MP_SQR_C +mp_err fast_s_mp_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_sqr_fast(a, b); +} +#endif +#ifdef BN_MP_BALANCE_MUL_C +mp_err mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_balance_mul(a, b, c); +} +#endif +#ifdef BN_MP_EXPTMOD_FAST_C +mp_err mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) +{ + return s_mp_exptmod_fast(G, X, P, Y, redmode); +} +#endif +#ifdef BN_MP_INVMOD_SLOW_C +mp_err mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_invmod_slow(a, b, c); +} +#endif +#ifdef BN_MP_KARATSUBA_MUL_C +mp_err mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_karatsuba_mul(a, b, c); +} +#endif +#ifdef BN_MP_KARATSUBA_SQR_C +mp_err mp_karatsuba_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_karatsuba_sqr(a, b); +} +#endif +#ifdef BN_MP_TOOM_MUL_C +mp_err mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + return s_mp_toom_mul(a, b, c); +} +#endif +#ifdef BN_MP_TOOM_SQR_C +mp_err mp_toom_sqr(const mp_int *a, mp_int *b) +{ + return s_mp_toom_sqr(a, b); +} +#endif +#ifdef S_MP_REVERSE_C +void bn_reverse(unsigned char *s, int len) +{ + if (len > 0) { + s_mp_reverse(s, (size_t)len); + } +} +#endif +#ifdef BN_MP_TC_AND_C +mp_err mp_tc_and(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_and(a, b, c); +} +#endif +#ifdef BN_MP_TC_OR_C +mp_err mp_tc_or(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_or(a, b, c); +} +#endif +#ifdef BN_MP_TC_XOR_C +mp_err mp_tc_xor(const mp_int *a, const mp_int *b, mp_int *c) +{ + return mp_xor(a, b, c); +} +#endif +#ifdef BN_MP_TC_DIV_2D_C +mp_err mp_tc_div_2d(const mp_int *a, int b, mp_int *c) +{ + return mp_signed_rsh(a, b, c); +} +#endif +#ifdef BN_MP_INIT_SET_INT_C +mp_err mp_init_set_int(mp_int *a, unsigned long b) +{ + return mp_init_u32(a, (uint32_t)b); +} +#endif +#ifdef BN_MP_SET_INT_C +mp_err mp_set_int(mp_int *a, unsigned long b) +{ + mp_set_u32(a, (uint32_t)b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_SET_LONG_C +mp_err mp_set_long(mp_int *a, unsigned long b) +{ + mp_set_u64(a, b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_SET_LONG_LONG_C +mp_err mp_set_long_long(mp_int *a, unsigned long long b) +{ + mp_set_u64(a, b); + return MP_OKAY; +} +#endif +#ifdef BN_MP_GET_INT_C +unsigned long mp_get_int(const mp_int *a) +{ + return (unsigned long)mp_get_mag_u32(a); +} +#endif +#ifdef BN_MP_GET_LONG_C +unsigned long mp_get_long(const mp_int *a) +{ + return (unsigned long)mp_get_mag_ul(a); +} +#endif +#ifdef BN_MP_GET_LONG_LONG_C +unsigned long long mp_get_long_long(const mp_int *a) +{ + return mp_get_mag_ull(a); +} +#endif +#ifdef BN_MP_PRIME_IS_DIVISIBLE_C +mp_err mp_prime_is_divisible(const mp_int *a, mp_bool *result) +{ + return s_mp_prime_is_divisible(a, result); +} +#endif +#ifdef BN_MP_EXPT_D_EX_C +mp_err mp_expt_d_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) +{ + (void)fast; + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_expt_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_EXPT_D_C +mp_err mp_expt_d(const mp_int *a, mp_digit b, mp_int *c) +{ + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_expt_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_N_ROOT_EX_C +mp_err mp_n_root_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) +{ + (void)fast; + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_root_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_N_ROOT_C +mp_err mp_n_root(const mp_int *a, mp_digit b, mp_int *c) +{ + if (b > MP_MIN(MP_DIGIT_MAX, UINT32_MAX)) { + return MP_VAL; + } + return mp_root_u32(a, (uint32_t)b, c); +} +#endif +#ifdef BN_MP_UNSIGNED_BIN_SIZE_C +int mp_unsigned_bin_size(const mp_int *a) +{ + return (int)mp_ubin_size(a); +} +#endif +#ifdef BN_MP_READ_UNSIGNED_BIN_C +mp_err mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c) +{ + return mp_from_ubin(a, b, (size_t) c); +} +#endif +#ifdef BN_MP_TO_UNSIGNED_BIN_C +mp_err mp_to_unsigned_bin(const mp_int *a, unsigned char *b) +{ + return mp_to_ubin(a, b, SIZE_MAX, NULL); +} +#endif +#ifdef BN_MP_TO_UNSIGNED_BIN_N_C +mp_err mp_to_unsigned_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) +{ + size_t n = mp_ubin_size(a); + if (*outlen < (unsigned long)n) { + return MP_VAL; + } + *outlen = (unsigned long)n; + return mp_to_ubin(a, b, n, NULL); +} +#endif +#ifdef BN_MP_SIGNED_BIN_SIZE_C +int mp_signed_bin_size(const mp_int *a) +{ + return (int)mp_sbin_size(a); +} +#endif +#ifdef BN_MP_READ_SIGNED_BIN_C +mp_err mp_read_signed_bin(mp_int *a, const unsigned char *b, int c) +{ + return mp_from_sbin(a, b, (size_t) c); +} +#endif +#ifdef BN_MP_TO_SIGNED_BIN_C +mp_err mp_to_signed_bin(const mp_int *a, unsigned char *b) +{ + return mp_to_sbin(a, b, SIZE_MAX, NULL); +} +#endif +#ifdef BN_MP_TO_SIGNED_BIN_N_C +mp_err mp_to_signed_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) +{ + size_t n = mp_sbin_size(a); + if (*outlen < (unsigned long)n) { + return MP_VAL; + } + *outlen = (unsigned long)n; + return mp_to_sbin(a, b, n, NULL); +} +#endif +#ifdef BN_MP_TORADIX_N_C +mp_err mp_toradix_n(const mp_int *a, char *str, int radix, int maxlen) +{ + if (maxlen < 0) { + return MP_VAL; + } + return mp_to_radix(a, str, (size_t)maxlen, NULL, radix); +} +#endif +#ifdef BN_MP_TORADIX_C +mp_err mp_toradix(const mp_int *a, char *str, int radix) +{ + return mp_to_radix(a, str, SIZE_MAX, NULL, radix); +} +#endif +#ifdef BN_MP_IMPORT_C +mp_err mp_import(mp_int *rop, size_t count, int order, size_t size, int endian, size_t nails, + const void *op) +{ + return mp_unpack(rop, count, order, size, endian, nails, op); +} +#endif +#ifdef BN_MP_EXPORT_C +mp_err mp_export(void *rop, size_t *countp, int order, size_t size, + int endian, size_t nails, const mp_int *op) +{ + return mp_pack(rop, SIZE_MAX, countp, order, size, endian, nails, op); +} +#endif +#endif + +/* End: bn_deprecated.c */ + +/* Start: bn_mp_2expt.c */ +#include "tommath_private.h" +#ifdef BN_MP_2EXPT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes a = 2**b + * + * Simple algorithm which zeroes the int, grows it then just sets one bit + * as required. + */ +mp_err mp_2expt(mp_int *a, int b) +{ + mp_err err; + + /* zero a as per default */ + mp_zero(a); + + /* grow a to accomodate the single bit */ + if ((err = mp_grow(a, (b / MP_DIGIT_BIT) + 1)) != MP_OKAY) { + return err; + } + + /* set the used count of where the bit will go */ + a->used = (b / MP_DIGIT_BIT) + 1; + + /* put the single bit in its place */ + a->dp[b / MP_DIGIT_BIT] = (mp_digit)1 << (mp_digit)(b % MP_DIGIT_BIT); + + return MP_OKAY; +} +#endif + +/* End: bn_mp_2expt.c */ + +/* Start: bn_mp_abs.c */ +#include "tommath_private.h" +#ifdef BN_MP_ABS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = |a| + * + * Simple function copies the input and fixes the sign to positive + */ +mp_err mp_abs(const mp_int *a, mp_int *b) +{ + mp_err err; + + /* copy a to b */ + if (a != b) { + if ((err = mp_copy(a, b)) != MP_OKAY) { + return err; + } + } + + /* force the sign of b to positive */ + b->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_abs.c */ + +/* Start: bn_mp_add.c */ +#include "tommath_private.h" +#ifdef BN_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* high level addition (handles signs) */ +mp_err mp_add(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_sign sa, sb; + mp_err err; + + /* get sign of both inputs */ + sa = a->sign; + sb = b->sign; + + /* handle two cases, not four */ + if (sa == sb) { + /* both positive or both negative */ + /* add their magnitudes, copy the sign */ + c->sign = sa; + err = s_mp_add(a, b, c); + } else { + /* one positive, the other negative */ + /* subtract the one with the greater magnitude from */ + /* the one of the lesser magnitude. The result gets */ + /* the sign of the one with the greater magnitude. */ + if (mp_cmp_mag(a, b) == MP_LT) { + c->sign = sb; + err = s_mp_sub(b, a, c); + } else { + c->sign = sa; + err = s_mp_sub(a, b, c); + } + } + return err; +} + +#endif + +/* End: bn_mp_add.c */ + +/* Start: bn_mp_add_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_ADD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* single digit addition */ +mp_err mp_add_d(const mp_int *a, mp_digit b, mp_int *c) +{ + mp_err err; + int ix, oldused; + mp_digit *tmpa, *tmpc; + + /* grow c as required */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } + + /* if a is negative and |a| >= b, call c = |a| - b */ + if ((a->sign == MP_NEG) && ((a->used > 1) || (a->dp[0] >= b))) { + mp_int a_ = *a; + /* temporarily fix sign of a */ + a_.sign = MP_ZPOS; + + /* c = |a| - b */ + err = mp_sub_d(&a_, b, c); + + /* fix sign */ + c->sign = MP_NEG; + + /* clamp */ + mp_clamp(c); + + return err; + } + + /* old number of used digits in c */ + oldused = c->used; + + /* source alias */ + tmpa = a->dp; + + /* destination alias */ + tmpc = c->dp; + + /* if a is positive */ + if (a->sign == MP_ZPOS) { + /* add digits, mu is carry */ + mp_digit mu = b; + for (ix = 0; ix < a->used; ix++) { + *tmpc = *tmpa++ + mu; + mu = *tmpc >> MP_DIGIT_BIT; + *tmpc++ &= MP_MASK; + } + /* set final carry */ + ix++; + *tmpc++ = mu; + + /* setup size */ + c->used = a->used + 1; + } else { + /* a was negative and |a| < b */ + c->used = 1; + + /* the result is a single digit */ + if (a->used == 1) { + *tmpc++ = b - a->dp[0]; + } else { + *tmpc++ = b; + } + + /* setup count so the clearing of oldused + * can fall through correctly + */ + ix = 1; + } + + /* sign always positive */ + c->sign = MP_ZPOS; + + /* now zero to oldused */ + MP_ZERO_DIGITS(tmpc, oldused - ix); + mp_clamp(c); + + return MP_OKAY; +} + +#endif + +/* End: bn_mp_add_d.c */ + +/* Start: bn_mp_addmod.c */ +#include "tommath_private.h" +#ifdef BN_MP_ADDMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* d = a + b (mod c) */ +mp_err mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) +{ + mp_err err; + mp_int t; + + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + if ((err = mp_add(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_addmod.c */ + +/* Start: bn_mp_and.c */ +#include "tommath_private.h" +#ifdef BN_MP_AND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* two complement and */ +mp_err mp_and(const mp_int *a, const mp_int *b, mp_int *c) +{ + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = ((a->sign == MP_NEG) && (b->sign == MP_NEG)) ? MP_NEG : MP_ZPOS; + + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } + + for (i = 0; i < used; i++) { + mp_digit x, y; + + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } + + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x & y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_and.c */ + +/* Start: bn_mp_clamp.c */ +#include "tommath_private.h" +#ifdef BN_MP_CLAMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* trim unused digits + * + * This is used to ensure that leading zero digits are + * trimed and the leading "used" digit will be non-zero + * Typically very fast. Also fixes the sign if there + * are no more leading digits + */ +void mp_clamp(mp_int *a) +{ + /* decrease used while the most significant digit is + * zero. + */ + while ((a->used > 0) && (a->dp[a->used - 1] == 0u)) { + --(a->used); + } + + /* reset the sign flag if used == 0 */ + if (a->used == 0) { + a->sign = MP_ZPOS; + } +} +#endif + +/* End: bn_mp_clamp.c */ + +/* Start: bn_mp_clear.c */ +#include "tommath_private.h" +#ifdef BN_MP_CLEAR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* clear one (frees) */ +void mp_clear(mp_int *a) +{ + /* only do anything if a hasn't been freed previously */ + if (a->dp != NULL) { + /* free ram */ + MP_FREE_DIGITS(a->dp, a->alloc); + + /* reset members to make debugging easier */ + a->dp = NULL; + a->alloc = a->used = 0; + a->sign = MP_ZPOS; + } +} +#endif + +/* End: bn_mp_clear.c */ + +/* Start: bn_mp_clear_multi.c */ +#include "tommath_private.h" +#ifdef BN_MP_CLEAR_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#include + +void mp_clear_multi(mp_int *mp, ...) +{ + mp_int *next_mp = mp; + va_list args; + va_start(args, mp); + while (next_mp != NULL) { + mp_clear(next_mp); + next_mp = va_arg(args, mp_int *); + } + va_end(args); +} +#endif + +/* End: bn_mp_clear_multi.c */ + +/* Start: bn_mp_cmp.c */ +#include "tommath_private.h" +#ifdef BN_MP_CMP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* compare two ints (signed)*/ +mp_ord mp_cmp(const mp_int *a, const mp_int *b) +{ + /* compare based on sign */ + if (a->sign != b->sign) { + if (a->sign == MP_NEG) { + return MP_LT; + } else { + return MP_GT; + } + } + + /* compare digits */ + if (a->sign == MP_NEG) { + /* if negative compare opposite direction */ + return mp_cmp_mag(b, a); + } else { + return mp_cmp_mag(a, b); + } +} +#endif + +/* End: bn_mp_cmp.c */ + +/* Start: bn_mp_cmp_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_CMP_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* compare a digit */ +mp_ord mp_cmp_d(const mp_int *a, mp_digit b) +{ + /* compare based on sign */ + if (a->sign == MP_NEG) { + return MP_LT; + } + + /* compare based on magnitude */ + if (a->used > 1) { + return MP_GT; + } + + /* compare the only digit of a to b */ + if (a->dp[0] > b) { + return MP_GT; + } else if (a->dp[0] < b) { + return MP_LT; + } else { + return MP_EQ; + } +} +#endif + +/* End: bn_mp_cmp_d.c */ + +/* Start: bn_mp_cmp_mag.c */ +#include "tommath_private.h" +#ifdef BN_MP_CMP_MAG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* compare maginitude of two ints (unsigned) */ +mp_ord mp_cmp_mag(const mp_int *a, const mp_int *b) +{ + int n; + const mp_digit *tmpa, *tmpb; + + /* compare based on # of non-zero digits */ + if (a->used > b->used) { + return MP_GT; + } + + if (a->used < b->used) { + return MP_LT; + } + + /* alias for a */ + tmpa = a->dp + (a->used - 1); + + /* alias for b */ + tmpb = b->dp + (a->used - 1); + + /* compare based on digits */ + for (n = 0; n < a->used; ++n, --tmpa, --tmpb) { + if (*tmpa > *tmpb) { + return MP_GT; + } + + if (*tmpa < *tmpb) { + return MP_LT; + } + } + return MP_EQ; +} +#endif + +/* End: bn_mp_cmp_mag.c */ + +/* Start: bn_mp_cnt_lsb.c */ +#include "tommath_private.h" +#ifdef BN_MP_CNT_LSB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +static const int lnz[16] = { + 4, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0 +}; + +/* Counts the number of lsbs which are zero before the first zero bit */ +int mp_cnt_lsb(const mp_int *a) +{ + int x; + mp_digit q, qq; + + /* easy out */ + if (MP_IS_ZERO(a)) { + return 0; + } + + /* scan lower digits until non-zero */ + for (x = 0; (x < a->used) && (a->dp[x] == 0u); x++) {} + q = a->dp[x]; + x *= MP_DIGIT_BIT; + + /* now scan this digit until a 1 is found */ + if ((q & 1u) == 0u) { + do { + qq = q & 15u; + x += lnz[qq]; + q >>= 4; + } while (qq == 0u); + } + return x; +} + +#endif + +/* End: bn_mp_cnt_lsb.c */ + +/* Start: bn_mp_complement.c */ +#include "tommath_private.h" +#ifdef BN_MP_COMPLEMENT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = ~a */ +mp_err mp_complement(const mp_int *a, mp_int *b) +{ + mp_err err = mp_neg(a, b); + return (err == MP_OKAY) ? mp_sub_d(b, 1uL, b) : err; +} +#endif + +/* End: bn_mp_complement.c */ + +/* Start: bn_mp_copy.c */ +#include "tommath_private.h" +#ifdef BN_MP_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* copy, b = a */ +mp_err mp_copy(const mp_int *a, mp_int *b) +{ + int n; + mp_digit *tmpa, *tmpb; + mp_err err; + + /* if dst == src do nothing */ + if (a == b) { + return MP_OKAY; + } + + /* grow dest */ + if (b->alloc < a->used) { + if ((err = mp_grow(b, a->used)) != MP_OKAY) { + return err; + } + } + + /* zero b and copy the parameters over */ + /* pointer aliases */ + + /* source */ + tmpa = a->dp; + + /* destination */ + tmpb = b->dp; + + /* copy all the digits */ + for (n = 0; n < a->used; n++) { + *tmpb++ = *tmpa++; + } + + /* clear high digits */ + MP_ZERO_DIGITS(tmpb, b->used - n); + + /* copy used count and sign */ + b->used = a->used; + b->sign = a->sign; + return MP_OKAY; +} +#endif + +/* End: bn_mp_copy.c */ + +/* Start: bn_mp_count_bits.c */ +#include "tommath_private.h" +#ifdef BN_MP_COUNT_BITS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* returns the number of bits in an int */ +int mp_count_bits(const mp_int *a) +{ + int r; + mp_digit q; + + /* shortcut */ + if (MP_IS_ZERO(a)) { + return 0; + } + + /* get number of digits and add that */ + r = (a->used - 1) * MP_DIGIT_BIT; + + /* take the last digit and count the bits in it */ + q = a->dp[a->used - 1]; + while (q > 0u) { + ++r; + q >>= 1u; + } + return r; +} +#endif + +/* End: bn_mp_count_bits.c */ + +/* Start: bn_mp_decr.c */ +#include "tommath_private.h" +#ifdef BN_MP_DECR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Decrement "a" by one like "a--". Changes input! */ +mp_err mp_decr(mp_int *a) +{ + if (MP_IS_ZERO(a)) { + mp_set(a,1uL); + a->sign = MP_NEG; + return MP_OKAY; + } else if (a->sign == MP_NEG) { + mp_err err; + a->sign = MP_ZPOS; + if ((err = mp_incr(a)) != MP_OKAY) { + return err; + } + /* There is no -0 in LTM */ + if (!MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + return MP_OKAY; + } else if (a->dp[0] > 1uL) { + a->dp[0]--; + if (a->dp[0] == 0u) { + mp_zero(a); + } + return MP_OKAY; + } else { + return mp_sub_d(a, 1uL,a); + } +} +#endif + +/* End: bn_mp_decr.c */ + +/* Start: bn_mp_div.c */ +#include "tommath_private.h" +#ifdef BN_MP_DIV_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifdef BN_MP_DIV_SMALL + +/* slower bit-bang division... also smaller */ +mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) +{ + mp_int ta, tb, tq, q; + int n, n2; + mp_err err; + + /* is divisor zero ? */ + if (MP_IS_ZERO(b)) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag(a, b) == MP_LT) { + if (d != NULL) { + err = mp_copy(a, d); + } else { + err = MP_OKAY; + } + if (c != NULL) { + mp_zero(c); + } + return err; + } + + /* init our temps */ + if ((err = mp_init_multi(&ta, &tb, &tq, &q, NULL)) != MP_OKAY) { + return err; + } + + + mp_set(&tq, 1uL); + n = mp_count_bits(a) - mp_count_bits(b); + if ((err = mp_abs(a, &ta)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_abs(b, &tb)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul_2d(&tb, n, &tb)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul_2d(&tq, n, &tq)) != MP_OKAY) goto LBL_ERR; + + while (n-- >= 0) { + if (mp_cmp(&tb, &ta) != MP_GT) { + if ((err = mp_sub(&ta, &tb, &ta)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&q, &tq, &q)) != MP_OKAY) goto LBL_ERR; + } + if ((err = mp_div_2d(&tb, 1, &tb, NULL)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2d(&tq, 1, &tq, NULL)) != MP_OKAY) goto LBL_ERR; + } + + /* now q == quotient and ta == remainder */ + n = a->sign; + n2 = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + if (c != NULL) { + mp_exch(c, &q); + c->sign = MP_IS_ZERO(c) ? MP_ZPOS : n2; + } + if (d != NULL) { + mp_exch(d, &ta); + d->sign = MP_IS_ZERO(d) ? MP_ZPOS : n; + } +LBL_ERR: + mp_clear_multi(&ta, &tb, &tq, &q, NULL); + return err; +} + +#else + +/* integer signed division. + * c*b + d == a [e.g. a/b, c=quotient, d=remainder] + * HAC pp.598 Algorithm 14.20 + * + * Note that the description in HAC is horribly + * incomplete. For example, it doesn't consider + * the case where digits are removed from 'x' in + * the inner loop. It also doesn't consider the + * case that y has fewer than three digits, etc.. + * + * The overall algorithm is as described as + * 14.20 from HAC but fixed to treat these cases. +*/ +mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) +{ + mp_int q, x, y, t1, t2; + int n, t, i, norm; + mp_sign neg; + mp_err err; + + /* is divisor zero ? */ + if (MP_IS_ZERO(b)) { + return MP_VAL; + } + + /* if a < b then q=0, r = a */ + if (mp_cmp_mag(a, b) == MP_LT) { + if (d != NULL) { + err = mp_copy(a, d); + } else { + err = MP_OKAY; + } + if (c != NULL) { + mp_zero(c); + } + return err; + } + + if ((err = mp_init_size(&q, a->used + 2)) != MP_OKAY) { + return err; + } + q.used = a->used + 2; + + if ((err = mp_init(&t1)) != MP_OKAY) goto LBL_Q; + + if ((err = mp_init(&t2)) != MP_OKAY) goto LBL_T1; + + if ((err = mp_init_copy(&x, a)) != MP_OKAY) goto LBL_T2; + + if ((err = mp_init_copy(&y, b)) != MP_OKAY) goto LBL_X; + + /* fix the sign */ + neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + x.sign = y.sign = MP_ZPOS; + + /* normalize both x and y, ensure that y >= b/2, [b == 2**MP_DIGIT_BIT] */ + norm = mp_count_bits(&y) % MP_DIGIT_BIT; + if (norm < (MP_DIGIT_BIT - 1)) { + norm = (MP_DIGIT_BIT - 1) - norm; + if ((err = mp_mul_2d(&x, norm, &x)) != MP_OKAY) goto LBL_Y; + if ((err = mp_mul_2d(&y, norm, &y)) != MP_OKAY) goto LBL_Y; + } else { + norm = 0; + } + + /* note hac does 0 based, so if used==5 then its 0,1,2,3,4, e.g. use 4 */ + n = x.used - 1; + t = y.used - 1; + + /* while (x >= y*b**n-t) do { q[n-t] += 1; x -= y*b**{n-t} } */ + /* y = y*b**{n-t} */ + if ((err = mp_lshd(&y, n - t)) != MP_OKAY) goto LBL_Y; + + while (mp_cmp(&x, &y) != MP_LT) { + ++(q.dp[n - t]); + if ((err = mp_sub(&x, &y, &x)) != MP_OKAY) goto LBL_Y; + } + + /* reset y by shifting it back down */ + mp_rshd(&y, n - t); + + /* step 3. for i from n down to (t + 1) */ + for (i = n; i >= (t + 1); i--) { + if (i > x.used) { + continue; + } + + /* step 3.1 if xi == yt then set q{i-t-1} to b-1, + * otherwise set q{i-t-1} to (xi*b + x{i-1})/yt */ + if (x.dp[i] == y.dp[t]) { + q.dp[(i - t) - 1] = ((mp_digit)1 << (mp_digit)MP_DIGIT_BIT) - (mp_digit)1; + } else { + mp_word tmp; + tmp = (mp_word)x.dp[i] << (mp_word)MP_DIGIT_BIT; + tmp |= (mp_word)x.dp[i - 1]; + tmp /= (mp_word)y.dp[t]; + if (tmp > (mp_word)MP_MASK) { + tmp = MP_MASK; + } + q.dp[(i - t) - 1] = (mp_digit)(tmp & (mp_word)MP_MASK); + } + + /* while (q{i-t-1} * (yt * b + y{t-1})) > + xi * b**2 + xi-1 * b + xi-2 + + do q{i-t-1} -= 1; + */ + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] + 1uL) & (mp_digit)MP_MASK; + do { + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & (mp_digit)MP_MASK; + + /* find left hand */ + mp_zero(&t1); + t1.dp[0] = ((t - 1) < 0) ? 0u : y.dp[t - 1]; + t1.dp[1] = y.dp[t]; + t1.used = 2; + if ((err = mp_mul_d(&t1, q.dp[(i - t) - 1], &t1)) != MP_OKAY) goto LBL_Y; + + /* find right hand */ + t2.dp[0] = ((i - 2) < 0) ? 0u : x.dp[i - 2]; + t2.dp[1] = x.dp[i - 1]; /* i >= 1 always holds */ + t2.dp[2] = x.dp[i]; + t2.used = 3; + } while (mp_cmp_mag(&t1, &t2) == MP_GT); + + /* step 3.3 x = x - q{i-t-1} * y * b**{i-t-1} */ + if ((err = mp_mul_d(&y, q.dp[(i - t) - 1], &t1)) != MP_OKAY) goto LBL_Y; + + if ((err = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) goto LBL_Y; + + if ((err = mp_sub(&x, &t1, &x)) != MP_OKAY) goto LBL_Y; + + /* if x < 0 then { x = x + y*b**{i-t-1}; q{i-t-1} -= 1; } */ + if (x.sign == MP_NEG) { + if ((err = mp_copy(&y, &t1)) != MP_OKAY) goto LBL_Y; + if ((err = mp_lshd(&t1, (i - t) - 1)) != MP_OKAY) goto LBL_Y; + if ((err = mp_add(&x, &t1, &x)) != MP_OKAY) goto LBL_Y; + + q.dp[(i - t) - 1] = (q.dp[(i - t) - 1] - 1uL) & MP_MASK; + } + } + + /* now q is the quotient and x is the remainder + * [which we have to normalize] + */ + + /* get sign before writing to c */ + x.sign = (x.used == 0) ? MP_ZPOS : a->sign; + + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + c->sign = neg; + } + + if (d != NULL) { + if ((err = mp_div_2d(&x, norm, &x, NULL)) != MP_OKAY) goto LBL_Y; + mp_exch(&x, d); + } + + err = MP_OKAY; + +LBL_Y: + mp_clear(&y); +LBL_X: + mp_clear(&x); +LBL_T2: + mp_clear(&t2); +LBL_T1: + mp_clear(&t1); +LBL_Q: + mp_clear(&q); + return err; +} + +#endif + +#endif + +/* End: bn_mp_div.c */ + +/* Start: bn_mp_div_2.c */ +#include "tommath_private.h" +#ifdef BN_MP_DIV_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = a/2 */ +mp_err mp_div_2(const mp_int *a, mp_int *b) +{ + int x, oldused; + mp_digit r, rr, *tmpa, *tmpb; + mp_err err; + + /* copy */ + if (b->alloc < a->used) { + if ((err = mp_grow(b, a->used)) != MP_OKAY) { + return err; + } + } + + oldused = b->used; + b->used = a->used; + + /* source alias */ + tmpa = a->dp + b->used - 1; + + /* dest alias */ + tmpb = b->dp + b->used - 1; + + /* carry */ + r = 0; + for (x = b->used - 1; x >= 0; x--) { + /* get the carry for the next iteration */ + rr = *tmpa & 1u; + + /* shift the current digit, add in carry and store */ + *tmpb-- = (*tmpa-- >> 1) | (r << (MP_DIGIT_BIT - 1)); + + /* forward carry to next iteration */ + r = rr; + } + + /* zero excess digits */ + MP_ZERO_DIGITS(b->dp + b->used, oldused - b->used); + + b->sign = a->sign; + mp_clamp(b); + return MP_OKAY; +} +#endif + +/* End: bn_mp_div_2.c */ + +/* Start: bn_mp_div_2d.c */ +#include "tommath_private.h" +#ifdef BN_MP_DIV_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift right by a certain bit count (store quotient in c, optional remainder in d) */ +mp_err mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d) +{ + mp_digit D, r, rr; + int x; + mp_err err; + + /* if the shift count is <= 0 then we do no work */ + if (b <= 0) { + err = mp_copy(a, c); + if (d != NULL) { + mp_zero(d); + } + return err; + } + + /* copy */ + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } + /* 'a' should not be used after here - it might be the same as d */ + + /* get the remainder */ + if (d != NULL) { + if ((err = mp_mod_2d(a, b, d)) != MP_OKAY) { + return err; + } + } + + /* shift by as many digits in the bit count */ + if (b >= MP_DIGIT_BIT) { + mp_rshd(c, b / MP_DIGIT_BIT); + } + + /* shift any bit count < MP_DIGIT_BIT */ + D = (mp_digit)(b % MP_DIGIT_BIT); + if (D != 0u) { + mp_digit *tmpc, mask, shift; + + /* mask */ + mask = ((mp_digit)1 << D) - 1uL; + + /* shift for lsb */ + shift = (mp_digit)MP_DIGIT_BIT - D; + + /* alias */ + tmpc = c->dp + (c->used - 1); + + /* carry */ + r = 0; + for (x = c->used - 1; x >= 0; x--) { + /* get the lower bits of this word in a temp */ + rr = *tmpc & mask; + + /* shift the current word and mix in the carry bits from the previous word */ + *tmpc = (*tmpc >> D) | (r << shift); + --tmpc; + + /* set the carry to the carry bits of the current word found above */ + r = rr; + } + } + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_div_2d.c */ + +/* Start: bn_mp_div_3.c */ +#include "tommath_private.h" +#ifdef BN_MP_DIV_3_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* divide by three (based on routine from MPI and the GMP manual) */ +mp_err mp_div_3(const mp_int *a, mp_int *c, mp_digit *d) +{ + mp_int q; + mp_word w, t; + mp_digit b; + mp_err err; + int ix; + + /* b = 2**MP_DIGIT_BIT / 3 */ + b = ((mp_word)1 << (mp_word)MP_DIGIT_BIT) / (mp_word)3; + + if ((err = mp_init_size(&q, a->used)) != MP_OKAY) { + return err; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << (mp_word)MP_DIGIT_BIT) | (mp_word)a->dp[ix]; + + if (w >= 3u) { + /* multiply w by [1/3] */ + t = (w * (mp_word)b) >> (mp_word)MP_DIGIT_BIT; + + /* now subtract 3 * [w/3] from w, to get the remainder */ + w -= t+t+t; + + /* fixup the remainder as required since + * the optimization is not exact. + */ + while (w >= 3u) { + t += 1u; + w -= 3u; + } + } else { + t = 0; + } + q.dp[ix] = (mp_digit)t; + } + + /* [optional] store the remainder */ + if (d != NULL) { + *d = (mp_digit)w; + } + + /* [optional] store the quotient */ + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return err; +} + +#endif + +/* End: bn_mp_div_3.c */ + +/* Start: bn_mp_div_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_DIV_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* single digit division (based on routine from MPI) */ +mp_err mp_div_d(const mp_int *a, mp_digit b, mp_int *c, mp_digit *d) +{ + mp_int q; + mp_word w; + mp_digit t; + mp_err err; + int ix; + + /* cannot divide by zero */ + if (b == 0u) { + return MP_VAL; + } + + /* quick outs */ + if ((b == 1u) || MP_IS_ZERO(a)) { + if (d != NULL) { + *d = 0; + } + if (c != NULL) { + return mp_copy(a, c); + } + return MP_OKAY; + } + + /* power of two ? */ + if ((b & (b - 1u)) == 0u) { + ix = 1; + while ((ix < MP_DIGIT_BIT) && (b != (((mp_digit)1)<dp[0] & (((mp_digit)1<<(mp_digit)ix) - 1uL); + } + if (c != NULL) { + return mp_div_2d(a, ix, c, NULL); + } + return MP_OKAY; + } + + /* three? */ + if (MP_HAS(MP_DIV_3) && (b == 3u)) { + return mp_div_3(a, c, d); + } + + /* no easy answer [c'est la vie]. Just division */ + if ((err = mp_init_size(&q, a->used)) != MP_OKAY) { + return err; + } + + q.used = a->used; + q.sign = a->sign; + w = 0; + for (ix = a->used - 1; ix >= 0; ix--) { + w = (w << (mp_word)MP_DIGIT_BIT) | (mp_word)a->dp[ix]; + + if (w >= b) { + t = (mp_digit)(w / b); + w -= (mp_word)t * (mp_word)b; + } else { + t = 0; + } + q.dp[ix] = t; + } + + if (d != NULL) { + *d = (mp_digit)w; + } + + if (c != NULL) { + mp_clamp(&q); + mp_exch(&q, c); + } + mp_clear(&q); + + return err; +} + +#endif + +/* End: bn_mp_div_d.c */ + +/* Start: bn_mp_dr_is_modulus.c */ +#include "tommath_private.h" +#ifdef BN_MP_DR_IS_MODULUS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines if a number is a valid DR modulus */ +mp_bool mp_dr_is_modulus(const mp_int *a) +{ + int ix; + + /* must be at least two digits */ + if (a->used < 2) { + return MP_NO; + } + + /* must be of the form b**k - a [a <= b] so all + * but the first digit must be equal to -1 (mod b). + */ + for (ix = 1; ix < a->used; ix++) { + if (a->dp[ix] != MP_MASK) { + return MP_NO; + } + } + return MP_YES; +} + +#endif + +/* End: bn_mp_dr_is_modulus.c */ + +/* Start: bn_mp_dr_reduce.c */ +#include "tommath_private.h" +#ifdef BN_MP_DR_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reduce "x" in place modulo "n" using the Diminished Radix algorithm. + * + * Based on algorithm from the paper + * + * "Generating Efficient Primes for Discrete Log Cryptosystems" + * Chae Hoon Lim, Pil Joong Lee, + * POSTECH Information Research Laboratories + * + * The modulus must be of a special format [see manual] + * + * Has been modified to use algorithm 7.10 from the LTM book instead + * + * Input x must be in the range 0 <= x <= (n-1)**2 + */ +mp_err mp_dr_reduce(mp_int *x, const mp_int *n, mp_digit k) +{ + mp_err err; + int i, m; + mp_word r; + mp_digit mu, *tmpx1, *tmpx2; + + /* m = digits in modulus */ + m = n->used; + + /* ensure that "x" has at least 2m digits */ + if (x->alloc < (m + m)) { + if ((err = mp_grow(x, m + m)) != MP_OKAY) { + return err; + } + } + + /* top of loop, this is where the code resumes if + * another reduction pass is required. + */ +top: + /* aliases for digits */ + /* alias for lower half of x */ + tmpx1 = x->dp; + + /* alias for upper half of x, or x/B**m */ + tmpx2 = x->dp + m; + + /* set carry to zero */ + mu = 0; + + /* compute (x mod B**m) + k * [x/B**m] inline and inplace */ + for (i = 0; i < m; i++) { + r = ((mp_word)*tmpx2++ * (mp_word)k) + *tmpx1 + mu; + *tmpx1++ = (mp_digit)(r & MP_MASK); + mu = (mp_digit)(r >> ((mp_word)MP_DIGIT_BIT)); + } + + /* set final carry */ + *tmpx1++ = mu; + + /* zero words above m */ + MP_ZERO_DIGITS(tmpx1, (x->used - m) - 1); + + /* clamp, sub and return */ + mp_clamp(x); + + /* if x >= n then subtract and reduce again + * Each successive "recursion" makes the input smaller and smaller. + */ + if (mp_cmp_mag(x, n) != MP_LT) { + if ((err = s_mp_sub(x, n, x)) != MP_OKAY) { + return err; + } + goto top; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_dr_reduce.c */ + +/* Start: bn_mp_dr_setup.c */ +#include "tommath_private.h" +#ifdef BN_MP_DR_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines the setup value */ +void mp_dr_setup(const mp_int *a, mp_digit *d) +{ + /* the casts are required if MP_DIGIT_BIT is one less than + * the number of bits in a mp_digit [e.g. MP_DIGIT_BIT==31] + */ + *d = (mp_digit)(((mp_word)1 << (mp_word)MP_DIGIT_BIT) - (mp_word)a->dp[0]); +} + +#endif + +/* End: bn_mp_dr_setup.c */ + +/* Start: bn_mp_error_to_string.c */ +#include "tommath_private.h" +#ifdef BN_MP_ERROR_TO_STRING_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* return a char * string for a given code */ +const char *mp_error_to_string(mp_err code) +{ + switch (code) { + case MP_OKAY: + return "Successful"; + case MP_ERR: + return "Unknown error"; + case MP_MEM: + return "Out of heap"; + case MP_VAL: + return "Value out of range"; + case MP_ITER: + return "Max. iterations reached"; + case MP_BUF: + return "Buffer overflow"; + default: + return "Invalid error code"; + } +} + +#endif + +/* End: bn_mp_error_to_string.c */ + +/* Start: bn_mp_exch.c */ +#include "tommath_private.h" +#ifdef BN_MP_EXCH_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* swap the elements of two integers, for cases where you can't simply swap the + * mp_int pointers around + */ +void mp_exch(mp_int *a, mp_int *b) +{ + mp_int t; + + t = *a; + *a = *b; + *b = t; +} +#endif + +/* End: bn_mp_exch.c */ + +/* Start: bn_mp_expt_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_EXPT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* calculate c = a**b using a square-multiply algorithm */ +mp_err mp_expt_u32(const mp_int *a, uint32_t b, mp_int *c) +{ + mp_err err; + + mp_int g; + + if ((err = mp_init_copy(&g, a)) != MP_OKAY) { + return err; + } + + /* set initial result */ + mp_set(c, 1uL); + + while (b > 0u) { + /* if the bit is set multiply */ + if ((b & 1u) != 0u) { + if ((err = mp_mul(c, &g, c)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* square */ + if (b > 1u) { + if ((err = mp_sqr(&g, &g)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* shift to next bit */ + b >>= 1; + } + + err = MP_OKAY; + +LBL_ERR: + mp_clear(&g); + return err; +} + +#endif + +/* End: bn_mp_expt_u32.c */ + +/* Start: bn_mp_exptmod.c */ +#include "tommath_private.h" +#ifdef BN_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* this is a shell function that calls either the normal or Montgomery + * exptmod functions. Originally the call to the montgomery code was + * embedded in the normal function but that wasted alot of stack space + * for nothing (since 99% of the time the Montgomery code would be called) + */ +mp_err mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y) +{ + int dr; + + /* modulus P must be positive */ + if (P->sign == MP_NEG) { + return MP_VAL; + } + + /* if exponent X is negative we have to recurse */ + if (X->sign == MP_NEG) { + mp_int tmpG, tmpX; + mp_err err; + + if (!MP_HAS(MP_INVMOD)) { + return MP_VAL; + } + + if ((err = mp_init_multi(&tmpG, &tmpX, NULL)) != MP_OKAY) { + return err; + } + + /* first compute 1/G mod P */ + if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) { + goto LBL_ERR; + } + + /* now get |X| */ + if ((err = mp_abs(X, &tmpX)) != MP_OKAY) { + goto LBL_ERR; + } + + /* and now compute (1/G)**|X| instead of G**X [X < 0] */ + err = mp_exptmod(&tmpG, &tmpX, P, Y); +LBL_ERR: + mp_clear_multi(&tmpG, &tmpX, NULL); + return err; + } + + /* modified diminished radix reduction */ + if (MP_HAS(MP_REDUCE_IS_2K_L) && MP_HAS(MP_REDUCE_2K_L) && MP_HAS(S_MP_EXPTMOD) && + (mp_reduce_is_2k_l(P) == MP_YES)) { + return s_mp_exptmod(G, X, P, Y, 1); + } + + /* is it a DR modulus? default to no */ + dr = (MP_HAS(MP_DR_IS_MODULUS) && (mp_dr_is_modulus(P) == MP_YES)) ? 1 : 0; + + /* if not, is it a unrestricted DR modulus? */ + if (MP_HAS(MP_REDUCE_IS_2K) && (dr == 0)) { + dr = (mp_reduce_is_2k(P) == MP_YES) ? 2 : 0; + } + + /* if the modulus is odd or dr != 0 use the montgomery method */ + if (MP_HAS(S_MP_EXPTMOD_FAST) && (MP_IS_ODD(P) || (dr != 0))) { + return s_mp_exptmod_fast(G, X, P, Y, dr); + } else if (MP_HAS(S_MP_EXPTMOD)) { + /* otherwise use the generic Barrett reduction technique */ + return s_mp_exptmod(G, X, P, Y, 0); + } else { + /* no exptmod for evens */ + return MP_VAL; + } +} + +#endif + +/* End: bn_mp_exptmod.c */ + +/* Start: bn_mp_exteuclid.c */ +#include "tommath_private.h" +#ifdef BN_MP_EXTEUCLID_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Extended euclidean algorithm of (a, b) produces + a*u1 + b*u2 = u3 + */ +mp_err mp_exteuclid(const mp_int *a, const mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) +{ + mp_int u1, u2, u3, v1, v2, v3, t1, t2, t3, q, tmp; + mp_err err; + + if ((err = mp_init_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL)) != MP_OKAY) { + return err; + } + + /* initialize, (u1,u2,u3) = (1,0,a) */ + mp_set(&u1, 1uL); + if ((err = mp_copy(a, &u3)) != MP_OKAY) goto LBL_ERR; + + /* initialize, (v1,v2,v3) = (0,1,b) */ + mp_set(&v2, 1uL); + if ((err = mp_copy(b, &v3)) != MP_OKAY) goto LBL_ERR; + + /* loop while v3 != 0 */ + while (!MP_IS_ZERO(&v3)) { + /* q = u3/v3 */ + if ((err = mp_div(&u3, &v3, &q, NULL)) != MP_OKAY) goto LBL_ERR; + + /* (t1,t2,t3) = (u1,u2,u3) - (v1,v2,v3)q */ + if ((err = mp_mul(&v1, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u1, &tmp, &t1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul(&v2, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u2, &tmp, &t2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_mul(&v3, &q, &tmp)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&u3, &tmp, &t3)) != MP_OKAY) goto LBL_ERR; + + /* (u1,u2,u3) = (v1,v2,v3) */ + if ((err = mp_copy(&v1, &u1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&v2, &u2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&v3, &u3)) != MP_OKAY) goto LBL_ERR; + + /* (v1,v2,v3) = (t1,t2,t3) */ + if ((err = mp_copy(&t1, &v1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&t2, &v2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&t3, &v3)) != MP_OKAY) goto LBL_ERR; + } + + /* make sure U3 >= 0 */ + if (u3.sign == MP_NEG) { + if ((err = mp_neg(&u1, &u1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_neg(&u2, &u2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_neg(&u3, &u3)) != MP_OKAY) goto LBL_ERR; + } + + /* copy result out */ + if (U1 != NULL) { + mp_exch(U1, &u1); + } + if (U2 != NULL) { + mp_exch(U2, &u2); + } + if (U3 != NULL) { + mp_exch(U3, &u3); + } + + err = MP_OKAY; +LBL_ERR: + mp_clear_multi(&u1, &u2, &u3, &v1, &v2, &v3, &t1, &t2, &t3, &q, &tmp, NULL); + return err; +} +#endif + +/* End: bn_mp_exteuclid.c */ + +/* Start: bn_mp_fread.c */ +#include "tommath_private.h" +#ifdef BN_MP_FREAD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifndef MP_NO_FILE +/* read a bigint from a file stream in ASCII */ +mp_err mp_fread(mp_int *a, int radix, FILE *stream) +{ + mp_err err; + mp_sign neg; + + /* if first digit is - then set negative */ + int ch = fgetc(stream); + if (ch == (int)'-') { + neg = MP_NEG; + ch = fgetc(stream); + } else { + neg = MP_ZPOS; + } + + /* no digits, return error */ + if (ch == EOF) { + return MP_ERR; + } + + /* clear a */ + mp_zero(a); + + do { + int y; + unsigned pos = (unsigned)(ch - (int)'('); + if (mp_s_rmap_reverse_sz < pos) { + break; + } + + y = (int)mp_s_rmap_reverse[pos]; + + if ((y == 0xff) || (y >= radix)) { + break; + } + + /* shift up and add */ + if ((err = mp_mul_d(a, (mp_digit)radix, a)) != MP_OKAY) { + return err; + } + if ((err = mp_add_d(a, (mp_digit)y, a)) != MP_OKAY) { + return err; + } + } while ((ch = fgetc(stream)) != EOF); + + if (a->used != 0) { + a->sign = neg; + } + + return MP_OKAY; +} +#endif + +#endif + +/* End: bn_mp_fread.c */ + +/* Start: bn_mp_from_sbin.c */ +#include "tommath_private.h" +#ifdef BN_MP_FROM_SBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* read signed bin, big endian, first byte is 0==positive or 1==negative */ +mp_err mp_from_sbin(mp_int *a, const unsigned char *buf, size_t size) +{ + mp_err err; + + /* read magnitude */ + if ((err = mp_from_ubin(a, buf + 1, size - 1u)) != MP_OKAY) { + return err; + } + + /* first byte is 0 for positive, non-zero for negative */ + if (buf[0] == (unsigned char)0) { + a->sign = MP_ZPOS; + } else { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_from_sbin.c */ + +/* Start: bn_mp_from_ubin.c */ +#include "tommath_private.h" +#ifdef BN_MP_FROM_UBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reads a unsigned char array, assumes the msb is stored first [big endian] */ +mp_err mp_from_ubin(mp_int *a, const unsigned char *buf, size_t size) +{ + mp_err err; + + /* make sure there are at least two digits */ + if (a->alloc < 2) { + if ((err = mp_grow(a, 2)) != MP_OKAY) { + return err; + } + } + + /* zero the int */ + mp_zero(a); + + /* read the bytes in */ + while (size-- > 0u) { + if ((err = mp_mul_2d(a, 8, a)) != MP_OKAY) { + return err; + } + +#ifndef MP_8BIT + a->dp[0] |= *buf++; + a->used += 1; +#else + a->dp[0] = (*buf & MP_MASK); + a->dp[1] |= ((*buf++ >> 7) & 1u); + a->used += 2; +#endif + } + mp_clamp(a); + return MP_OKAY; +} +#endif + +/* End: bn_mp_from_ubin.c */ + +/* Start: bn_mp_fwrite.c */ +#include "tommath_private.h" +#ifdef BN_MP_FWRITE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifndef MP_NO_FILE +mp_err mp_fwrite(const mp_int *a, int radix, FILE *stream) +{ + char *buf; + mp_err err; + int len; + size_t written; + + /* TODO: this function is not in this PR */ + if (MP_HAS(MP_RADIX_SIZE_OVERESTIMATE)) { + /* if ((err = mp_radix_size_overestimate(&t, base, &len)) != MP_OKAY) goto LBL_ERR; */ + } else { + if ((err = mp_radix_size(a, radix, &len)) != MP_OKAY) { + return err; + } + } + + buf = (char *) MP_MALLOC((size_t)len); + if (buf == NULL) { + return MP_MEM; + } + + if ((err = mp_to_radix(a, buf, (size_t)len, &written, radix)) != MP_OKAY) { + goto LBL_ERR; + } + + if (fwrite(buf, written, 1uL, stream) != 1uL) { + err = MP_ERR; + goto LBL_ERR; + } + err = MP_OKAY; + + +LBL_ERR: + MP_FREE_BUFFER(buf, (size_t)len); + return err; +} +#endif + +#endif + +/* End: bn_mp_fwrite.c */ + +/* Start: bn_mp_gcd.c */ +#include "tommath_private.h" +#ifdef BN_MP_GCD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Greatest Common Divisor using the binary method */ +mp_err mp_gcd(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int u, v; + int k, u_lsb, v_lsb; + mp_err err; + + /* either zero than gcd is the largest */ + if (MP_IS_ZERO(a)) { + return mp_abs(b, c); + } + if (MP_IS_ZERO(b)) { + return mp_abs(a, c); + } + + /* get copies of a and b we can modify */ + if ((err = mp_init_copy(&u, a)) != MP_OKAY) { + return err; + } + + if ((err = mp_init_copy(&v, b)) != MP_OKAY) { + goto LBL_U; + } + + /* must be positive for the remainder of the algorithm */ + u.sign = v.sign = MP_ZPOS; + + /* B1. Find the common power of two for u and v */ + u_lsb = mp_cnt_lsb(&u); + v_lsb = mp_cnt_lsb(&v); + k = MP_MIN(u_lsb, v_lsb); + + if (k > 0) { + /* divide the power of two out */ + if ((err = mp_div_2d(&u, k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + + if ((err = mp_div_2d(&v, k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* divide any remaining factors of two out */ + if (u_lsb != k) { + if ((err = mp_div_2d(&u, u_lsb - k, &u, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + if (v_lsb != k) { + if ((err = mp_div_2d(&v, v_lsb - k, &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + while (!MP_IS_ZERO(&v)) { + /* make sure v is the largest */ + if (mp_cmp_mag(&u, &v) == MP_GT) { + /* swap u and v to make sure v is >= u */ + mp_exch(&u, &v); + } + + /* subtract smallest from largest */ + if ((err = s_mp_sub(&v, &u, &v)) != MP_OKAY) { + goto LBL_V; + } + + /* Divide out all factors of two */ + if ((err = mp_div_2d(&v, mp_cnt_lsb(&v), &v, NULL)) != MP_OKAY) { + goto LBL_V; + } + } + + /* multiply by 2**k which we divided out at the beginning */ + if ((err = mp_mul_2d(&u, k, c)) != MP_OKAY) { + goto LBL_V; + } + c->sign = MP_ZPOS; + err = MP_OKAY; +LBL_V: + mp_clear(&u); +LBL_U: + mp_clear(&v); + return err; +} +#endif + +/* End: bn_mp_gcd.c */ + +/* Start: bn_mp_get_double.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_DOUBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +double mp_get_double(const mp_int *a) +{ + int i; + double d = 0.0, fac = 1.0; + for (i = 0; i < MP_DIGIT_BIT; ++i) { + fac *= 2.0; + } + for (i = a->used; i --> 0;) { + d = (d * fac) + (double)a->dp[i]; + } + return (a->sign == MP_NEG) ? -d : d; +} +#endif + +/* End: bn_mp_get_double.c */ + +/* Start: bn_mp_get_i32.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_i32, mp_get_mag_u32, int32_t, uint32_t) +#endif + +/* End: bn_mp_get_i32.c */ + +/* Start: bn_mp_get_i64.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_i64, mp_get_mag_u64, int64_t, uint64_t) +#endif + +/* End: bn_mp_get_i64.c */ + +/* Start: bn_mp_get_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_l, mp_get_mag_ul, long, unsigned long) +#endif + +/* End: bn_mp_get_l.c */ + +/* Start: bn_mp_get_ll.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_SIGNED(mp_get_ll, mp_get_mag_ull, long long, unsigned long long) +#endif + +/* End: bn_mp_get_ll.c */ + +/* Start: bn_mp_get_mag_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_u32, uint32_t) +#endif + +/* End: bn_mp_get_mag_u32.c */ + +/* Start: bn_mp_get_mag_u64.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_u64, uint64_t) +#endif + +/* End: bn_mp_get_mag_u64.c */ + +/* Start: bn_mp_get_mag_ul.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_ul, unsigned long) +#endif + +/* End: bn_mp_get_mag_ul.c */ + +/* Start: bn_mp_get_mag_ull.c */ +#include "tommath_private.h" +#ifdef BN_MP_GET_MAG_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_GET_MAG(mp_get_mag_ull, unsigned long long) +#endif + +/* End: bn_mp_get_mag_ull.c */ + +/* Start: bn_mp_grow.c */ +#include "tommath_private.h" +#ifdef BN_MP_GROW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* grow as required */ +mp_err mp_grow(mp_int *a, int size) +{ + int i; + mp_digit *tmp; + + /* if the alloc size is smaller alloc more ram */ + if (a->alloc < size) { + /* reallocate the array a->dp + * + * We store the return in a temporary variable + * in case the operation failed we don't want + * to overwrite the dp member of a. + */ + tmp = (mp_digit *) MP_REALLOC(a->dp, + (size_t)a->alloc * sizeof(mp_digit), + (size_t)size * sizeof(mp_digit)); + if (tmp == NULL) { + /* reallocation failed but "a" is still valid [can be freed] */ + return MP_MEM; + } + + /* reallocation succeeded so set a->dp */ + a->dp = tmp; + + /* zero excess digits */ + i = a->alloc; + a->alloc = size; + MP_ZERO_DIGITS(a->dp + i, a->alloc - i); + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_grow.c */ + +/* Start: bn_mp_incr.c */ +#include "tommath_private.h" +#ifdef BN_MP_INCR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Increment "a" by one like "a++". Changes input! */ +mp_err mp_incr(mp_int *a) +{ + if (MP_IS_ZERO(a)) { + mp_set(a,1uL); + return MP_OKAY; + } else if (a->sign == MP_NEG) { + mp_err err; + a->sign = MP_ZPOS; + if ((err = mp_decr(a)) != MP_OKAY) { + return err; + } + /* There is no -0 in LTM */ + if (!MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + return MP_OKAY; + } else if (a->dp[0] < MP_DIGIT_MAX) { + a->dp[0]++; + return MP_OKAY; + } else { + return mp_add_d(a, 1uL,a); + } +} +#endif + +/* End: bn_mp_incr.c */ + +/* Start: bn_mp_init.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* init a new mp_int */ +mp_err mp_init(mp_int *a) +{ + /* allocate memory required and clear it */ + a->dp = (mp_digit *) MP_CALLOC((size_t)MP_PREC, sizeof(mp_digit)); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the used to zero, allocated digits to the default precision + * and sign to positive */ + a->used = 0; + a->alloc = MP_PREC; + a->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_init.c */ + +/* Start: bn_mp_init_copy.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_COPY_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* creates "a" then copies b into it */ +mp_err mp_init_copy(mp_int *a, const mp_int *b) +{ + mp_err err; + + if ((err = mp_init_size(a, b->used)) != MP_OKAY) { + return err; + } + + if ((err = mp_copy(b, a)) != MP_OKAY) { + mp_clear(a); + } + + return err; +} +#endif + +/* End: bn_mp_init_copy.c */ + +/* Start: bn_mp_init_i32.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_i32, mp_set_i32, int32_t) +#endif + +/* End: bn_mp_init_i32.c */ + +/* Start: bn_mp_init_i64.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_i64, mp_set_i64, int64_t) +#endif + +/* End: bn_mp_init_i64.c */ + +/* Start: bn_mp_init_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_l, mp_set_l, long) +#endif + +/* End: bn_mp_init_l.c */ + +/* Start: bn_mp_init_ll.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ll, mp_set_ll, long long) +#endif + +/* End: bn_mp_init_ll.c */ + +/* Start: bn_mp_init_multi.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_MULTI_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#include + +mp_err mp_init_multi(mp_int *mp, ...) +{ + mp_err err = MP_OKAY; /* Assume ok until proven otherwise */ + int n = 0; /* Number of ok inits */ + mp_int *cur_arg = mp; + va_list args; + + va_start(args, mp); /* init args to next argument from caller */ + while (cur_arg != NULL) { + if (mp_init(cur_arg) != MP_OKAY) { + /* Oops - error! Back-track and mp_clear what we already + succeeded in init-ing, then return error. + */ + va_list clean_args; + + /* now start cleaning up */ + cur_arg = mp; + va_start(clean_args, mp); + while (n-- != 0) { + mp_clear(cur_arg); + cur_arg = va_arg(clean_args, mp_int *); + } + va_end(clean_args); + err = MP_MEM; + break; + } + n++; + cur_arg = va_arg(args, mp_int *); + } + va_end(args); + return err; /* Assumed ok, if error flagged above. */ +} + +#endif + +/* End: bn_mp_init_multi.c */ + +/* Start: bn_mp_init_set.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* initialize and set a digit */ +mp_err mp_init_set(mp_int *a, mp_digit b) +{ + mp_err err; + if ((err = mp_init(a)) != MP_OKAY) { + return err; + } + mp_set(a, b); + return err; +} +#endif + +/* End: bn_mp_init_set.c */ + +/* Start: bn_mp_init_size.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* init an mp_init for a given size */ +mp_err mp_init_size(mp_int *a, int size) +{ + size = MP_MAX(MP_MIN_PREC, size); + + /* alloc mem */ + a->dp = (mp_digit *) MP_CALLOC((size_t)size, sizeof(mp_digit)); + if (a->dp == NULL) { + return MP_MEM; + } + + /* set the members */ + a->used = 0; + a->alloc = size; + a->sign = MP_ZPOS; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_init_size.c */ + +/* Start: bn_mp_init_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_u32, mp_set_u32, uint32_t) +#endif + +/* End: bn_mp_init_u32.c */ + +/* Start: bn_mp_init_u64.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_u64, mp_set_u64, uint64_t) +#endif + +/* End: bn_mp_init_u64.c */ + +/* Start: bn_mp_init_ul.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ul, mp_set_ul, unsigned long) +#endif + +/* End: bn_mp_init_ul.c */ + +/* Start: bn_mp_init_ull.c */ +#include "tommath_private.h" +#ifdef BN_MP_INIT_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_INIT_INT(mp_init_ull, mp_set_ull, unsigned long long) +#endif + +/* End: bn_mp_init_ull.c */ + +/* Start: bn_mp_invmod.c */ +#include "tommath_private.h" +#ifdef BN_MP_INVMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* hac 14.61, pp608 */ +mp_err mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) +{ + /* b cannot be negative and has to be >1 */ + if ((b->sign == MP_NEG) || (mp_cmp_d(b, 1uL) != MP_GT)) { + return MP_VAL; + } + + /* if the modulus is odd we can use a faster routine instead */ + if (MP_HAS(S_MP_INVMOD_FAST) && MP_IS_ODD(b)) { + return s_mp_invmod_fast(a, b, c); + } + + return MP_HAS(S_MP_INVMOD_SLOW) + ? s_mp_invmod_slow(a, b, c) + : MP_VAL; +} +#endif + +/* End: bn_mp_invmod.c */ + +/* Start: bn_mp_is_square.c */ +#include "tommath_private.h" +#ifdef BN_MP_IS_SQUARE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Check if remainders are possible squares - fast exclude non-squares */ +static const char rem_128[128] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1 +}; + +static const char rem_105[105] = { + 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, + 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, + 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, + 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, + 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, + 1, 1, 1, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, + 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1 +}; + +/* Store non-zero to ret if arg is square, and zero if not */ +mp_err mp_is_square(const mp_int *arg, mp_bool *ret) +{ + mp_err err; + mp_digit c; + mp_int t; + unsigned long r; + + /* Default to Non-square :) */ + *ret = MP_NO; + + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + if (MP_IS_ZERO(arg)) { + return MP_OKAY; + } + + /* First check mod 128 (suppose that MP_DIGIT_BIT is at least 7) */ + if (rem_128[127u & arg->dp[0]] == (char)1) { + return MP_OKAY; + } + + /* Next check mod 105 (3*5*7) */ + if ((err = mp_mod_d(arg, 105uL, &c)) != MP_OKAY) { + return err; + } + if (rem_105[c] == (char)1) { + return MP_OKAY; + } + + + if ((err = mp_init_u32(&t, 11u*13u*17u*19u*23u*29u*31u)) != MP_OKAY) { + return err; + } + if ((err = mp_mod(arg, &t, &t)) != MP_OKAY) { + goto LBL_ERR; + } + r = mp_get_u32(&t); + /* Check for other prime modules, note it's not an ERROR but we must + * free "t" so the easiest way is to goto LBL_ERR. We know that err + * is already equal to MP_OKAY from the mp_mod call + */ + if (((1uL<<(r%11uL)) & 0x5C4uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%13uL)) & 0x9E4uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%17uL)) & 0x5CE8uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%19uL)) & 0x4F50CuL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%23uL)) & 0x7ACCA0uL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%29uL)) & 0xC2EDD0CuL) != 0uL) goto LBL_ERR; + if (((1uL<<(r%31uL)) & 0x6DE2B848uL) != 0uL) goto LBL_ERR; + + /* Final check - is sqr(sqrt(arg)) == arg ? */ + if ((err = mp_sqrt(arg, &t)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_sqr(&t, &t)) != MP_OKAY) { + goto LBL_ERR; + } + + *ret = (mp_cmp_mag(&t, arg) == MP_EQ) ? MP_YES : MP_NO; +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_is_square.c */ + +/* Start: bn_mp_iseven.c */ +#include "tommath_private.h" +#ifdef BN_MP_ISEVEN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_bool mp_iseven(const mp_int *a) +{ + return MP_IS_EVEN(a) ? MP_YES : MP_NO; +} +#endif + +/* End: bn_mp_iseven.c */ + +/* Start: bn_mp_isodd.c */ +#include "tommath_private.h" +#ifdef BN_MP_ISODD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_bool mp_isodd(const mp_int *a) +{ + return MP_IS_ODD(a) ? MP_YES : MP_NO; +} +#endif + +/* End: bn_mp_isodd.c */ + +/* Start: bn_mp_kronecker.c */ +#include "tommath_private.h" +#ifdef BN_MP_KRONECKER_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + Kronecker symbol (a|p) + Straightforward implementation of algorithm 1.4.10 in + Henri Cohen: "A Course in Computational Algebraic Number Theory" + + @book{cohen2013course, + title={A course in computational algebraic number theory}, + author={Cohen, Henri}, + volume={138}, + year={2013}, + publisher={Springer Science \& Business Media} + } + */ +mp_err mp_kronecker(const mp_int *a, const mp_int *p, int *c) +{ + mp_int a1, p1, r; + mp_err err; + int v, k; + + static const int table[8] = {0, 1, 0, -1, 0, -1, 0, 1}; + + if (MP_IS_ZERO(p)) { + if ((a->used == 1) && (a->dp[0] == 1u)) { + *c = 1; + } else { + *c = 0; + } + return MP_OKAY; + } + + if (MP_IS_EVEN(a) && MP_IS_EVEN(p)) { + *c = 0; + return MP_OKAY; + } + + if ((err = mp_init_copy(&a1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_init_copy(&p1, p)) != MP_OKAY) { + goto LBL_KRON_0; + } + + v = mp_cnt_lsb(&p1); + if ((err = mp_div_2d(&p1, v, &p1, NULL)) != MP_OKAY) { + goto LBL_KRON_1; + } + + if ((v & 1) == 0) { + k = 1; + } else { + k = table[a->dp[0] & 7u]; + } + + if (p1.sign == MP_NEG) { + p1.sign = MP_ZPOS; + if (a1.sign == MP_NEG) { + k = -k; + } + } + + if ((err = mp_init(&r)) != MP_OKAY) { + goto LBL_KRON_1; + } + + for (;;) { + if (MP_IS_ZERO(&a1)) { + if (mp_cmp_d(&p1, 1uL) == MP_EQ) { + *c = k; + goto LBL_KRON; + } else { + *c = 0; + goto LBL_KRON; + } + } + + v = mp_cnt_lsb(&a1); + if ((err = mp_div_2d(&a1, v, &a1, NULL)) != MP_OKAY) { + goto LBL_KRON; + } + + if ((v & 1) == 1) { + k = k * table[p1.dp[0] & 7u]; + } + + if (a1.sign == MP_NEG) { + /* + * Compute k = (-1)^((a1)*(p1-1)/4) * k + * a1.dp[0] + 1 cannot overflow because the MSB + * of the type mp_digit is not set by definition + */ + if (((a1.dp[0] + 1u) & p1.dp[0] & 2u) != 0u) { + k = -k; + } + } else { + /* compute k = (-1)^((a1-1)*(p1-1)/4) * k */ + if ((a1.dp[0] & p1.dp[0] & 2u) != 0u) { + k = -k; + } + } + + if ((err = mp_copy(&a1, &r)) != MP_OKAY) { + goto LBL_KRON; + } + r.sign = MP_ZPOS; + if ((err = mp_mod(&p1, &r, &a1)) != MP_OKAY) { + goto LBL_KRON; + } + if ((err = mp_copy(&r, &p1)) != MP_OKAY) { + goto LBL_KRON; + } + } + +LBL_KRON: + mp_clear(&r); +LBL_KRON_1: + mp_clear(&p1); +LBL_KRON_0: + mp_clear(&a1); + + return err; +} + +#endif + +/* End: bn_mp_kronecker.c */ + +/* Start: bn_mp_lcm.c */ +#include "tommath_private.h" +#ifdef BN_MP_LCM_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes least common multiple as |a*b|/(a, b) */ +mp_err mp_lcm(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_err err; + mp_int t1, t2; + + + if ((err = mp_init_multi(&t1, &t2, NULL)) != MP_OKAY) { + return err; + } + + /* t1 = get the GCD of the two inputs */ + if ((err = mp_gcd(a, b, &t1)) != MP_OKAY) { + goto LBL_T; + } + + /* divide the smallest by the GCD */ + if (mp_cmp_mag(a, b) == MP_LT) { + /* store quotient in t2 such that t2 * b is the LCM */ + if ((err = mp_div(a, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + err = mp_mul(b, &t2, c); + } else { + /* store quotient in t2 such that t2 * a is the LCM */ + if ((err = mp_div(b, &t1, &t2, NULL)) != MP_OKAY) { + goto LBL_T; + } + err = mp_mul(a, &t2, c); + } + + /* fix the sign to positive */ + c->sign = MP_ZPOS; + +LBL_T: + mp_clear_multi(&t1, &t2, NULL); + return err; +} +#endif + +/* End: bn_mp_lcm.c */ + +/* Start: bn_mp_log_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_LOG_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Compute log_{base}(a) */ +static mp_word s_pow(mp_word base, mp_word exponent) +{ + mp_word result = 1uLL; + while (exponent != 0u) { + if ((exponent & 1u) == 1u) { + result *= base; + } + exponent >>= 1; + base *= base; + } + + return result; +} + +static mp_digit s_digit_ilogb(mp_digit base, mp_digit n) +{ + mp_word bracket_low = 1uLL, bracket_mid, bracket_high, N; + mp_digit ret, high = 1uL, low = 0uL, mid; + + if (n < base) { + return 0uL; + } + if (n == base) { + return 1uL; + } + + bracket_high = (mp_word) base ; + N = (mp_word) n; + + while (bracket_high < N) { + low = high; + bracket_low = bracket_high; + high <<= 1; + bracket_high *= bracket_high; + } + + while (((mp_digit)(high - low)) > 1uL) { + mid = (low + high) >> 1; + bracket_mid = bracket_low * s_pow(base, (mp_word)(mid - low)); + + if (N < bracket_mid) { + high = mid ; + bracket_high = bracket_mid ; + } + if (N > bracket_mid) { + low = mid ; + bracket_low = bracket_mid ; + } + if (N == bracket_mid) { + return (mp_digit) mid; + } + } + + if (bracket_high == N) { + ret = high; + } else { + ret = low; + } + + return ret; +} + +/* TODO: output could be "int" because the output of mp_radix_size is int, too, + as is the output of mp_bitcount. + With the same problem: max size is INT_MAX * MP_DIGIT not INT_MAX only! +*/ +mp_err mp_log_u32(const mp_int *a, uint32_t base, uint32_t *c) +{ + mp_err err; + mp_ord cmp; + uint32_t high, low, mid; + mp_int bracket_low, bracket_high, bracket_mid, t, bi_base; + + err = MP_OKAY; + + if (a->sign == MP_NEG) { + return MP_VAL; + } + + if (MP_IS_ZERO(a)) { + return MP_VAL; + } + + if (base < 2u) { + return MP_VAL; + } + + /* A small shortcut for bases that are powers of two. */ + if ((base & (base - 1u)) == 0u) { + int y, bit_count; + for (y=0; (y < 7) && ((base & 1u) == 0u); y++) { + base >>= 1; + } + bit_count = mp_count_bits(a) - 1; + *c = (uint32_t)(bit_count/y); + return MP_OKAY; + } + + if (a->used == 1) { + *c = (uint32_t)s_digit_ilogb(base, a->dp[0]); + return err; + } + + cmp = mp_cmp_d(a, base); + if ((cmp == MP_LT) || (cmp == MP_EQ)) { + *c = cmp == MP_EQ; + return err; + } + + if ((err = + mp_init_multi(&bracket_low, &bracket_high, + &bracket_mid, &t, &bi_base, NULL)) != MP_OKAY) { + return err; + } + + low = 0u; + mp_set(&bracket_low, 1uL); + high = 1u; + + mp_set(&bracket_high, base); + + /* + A kind of Giant-step/baby-step algorithm. + Idea shamelessly stolen from https://programmingpraxis.com/2010/05/07/integer-logarithms/2/ + The effect is asymptotic, hence needs benchmarks to test if the Giant-step should be skipped + for small n. + */ + while (mp_cmp(&bracket_high, a) == MP_LT) { + low = high; + if ((err = mp_copy(&bracket_high, &bracket_low)) != MP_OKAY) { + goto LBL_ERR; + } + high <<= 1; + if ((err = mp_sqr(&bracket_high, &bracket_high)) != MP_OKAY) { + goto LBL_ERR; + } + } + mp_set(&bi_base, base); + + while ((high - low) > 1u) { + mid = (high + low) >> 1; + + if ((err = mp_expt_u32(&bi_base, (uint32_t)(mid - low), &t)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_mul(&bracket_low, &t, &bracket_mid)) != MP_OKAY) { + goto LBL_ERR; + } + cmp = mp_cmp(a, &bracket_mid); + if (cmp == MP_LT) { + high = mid; + mp_exch(&bracket_mid, &bracket_high); + } + if (cmp == MP_GT) { + low = mid; + mp_exch(&bracket_mid, &bracket_low); + } + if (cmp == MP_EQ) { + *c = mid; + goto LBL_END; + } + } + + *c = (mp_cmp(&bracket_high, a) == MP_EQ) ? high : low; + +LBL_END: +LBL_ERR: + mp_clear_multi(&bracket_low, &bracket_high, &bracket_mid, + &t, &bi_base, NULL); + return err; +} + + +#endif + +/* End: bn_mp_log_u32.c */ + +/* Start: bn_mp_lshd.c */ +#include "tommath_private.h" +#ifdef BN_MP_LSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift left a certain amount of digits */ +mp_err mp_lshd(mp_int *a, int b) +{ + int x; + mp_err err; + mp_digit *top, *bottom; + + /* if its less than zero return */ + if (b <= 0) { + return MP_OKAY; + } + /* no need to shift 0 around */ + if (MP_IS_ZERO(a)) { + return MP_OKAY; + } + + /* grow to fit the new digits */ + if (a->alloc < (a->used + b)) { + if ((err = mp_grow(a, a->used + b)) != MP_OKAY) { + return err; + } + } + + /* increment the used by the shift amount then copy upwards */ + a->used += b; + + /* top */ + top = a->dp + a->used - 1; + + /* base */ + bottom = (a->dp + a->used - 1) - b; + + /* much like mp_rshd this is implemented using a sliding window + * except the window goes the otherway around. Copying from + * the bottom to the top. see bn_mp_rshd.c for more info. + */ + for (x = a->used - 1; x >= b; x--) { + *top-- = *bottom--; + } + + /* zero the lower digits */ + MP_ZERO_DIGITS(a->dp, b); + + return MP_OKAY; +} +#endif + +/* End: bn_mp_lshd.c */ + +/* Start: bn_mp_mod.c */ +#include "tommath_private.h" +#ifdef BN_MP_MOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* c = a mod b, 0 <= c < b if b > 0, b < c <= 0 if b < 0 */ +mp_err mp_mod(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int t; + mp_err err; + + if ((err = mp_init_size(&t, b->used)) != MP_OKAY) { + return err; + } + + if ((err = mp_div(a, b, NULL, &t)) != MP_OKAY) { + goto LBL_ERR; + } + + if (MP_IS_ZERO(&t) || (t.sign == b->sign)) { + err = MP_OKAY; + mp_exch(&t, c); + } else { + err = mp_add(b, &t, c); + } + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_mod.c */ + +/* Start: bn_mp_mod_2d.c */ +#include "tommath_private.h" +#ifdef BN_MP_MOD_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* calc a value mod 2**b */ +mp_err mp_mod_2d(const mp_int *a, int b, mp_int *c) +{ + int x; + mp_err err; + + /* if b is <= 0 then zero the int */ + if (b <= 0) { + mp_zero(c); + return MP_OKAY; + } + + /* if the modulus is larger than the value than return */ + if (b >= (a->used * MP_DIGIT_BIT)) { + return mp_copy(a, c); + } + + /* copy */ + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } + + /* zero digits above the last digit of the modulus */ + x = (b / MP_DIGIT_BIT) + (((b % MP_DIGIT_BIT) == 0) ? 0 : 1); + MP_ZERO_DIGITS(c->dp + x, c->used - x); + + /* clear the digit that is not completely outside/inside the modulus */ + c->dp[b / MP_DIGIT_BIT] &= + ((mp_digit)1 << (mp_digit)(b % MP_DIGIT_BIT)) - (mp_digit)1; + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_mod_2d.c */ + +/* Start: bn_mp_mod_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_MOD_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_err mp_mod_d(const mp_int *a, mp_digit b, mp_digit *c) +{ + return mp_div_d(a, b, NULL, c); +} +#endif + +/* End: bn_mp_mod_d.c */ + +/* Start: bn_mp_montgomery_calc_normalization.c */ +#include "tommath_private.h" +#ifdef BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + * shifts with subtractions when the result is greater than b. + * + * The method is slightly modified to shift B unconditionally upto just under + * the leading bit of b. This saves alot of multiple precision shifting. + */ +mp_err mp_montgomery_calc_normalization(mp_int *a, const mp_int *b) +{ + int x, bits; + mp_err err; + + /* how many bits of last digit does b use */ + bits = mp_count_bits(b) % MP_DIGIT_BIT; + + if (b->used > 1) { + if ((err = mp_2expt(a, ((b->used - 1) * MP_DIGIT_BIT) + bits - 1)) != MP_OKAY) { + return err; + } + } else { + mp_set(a, 1uL); + bits = 1; + } + + + /* now compute C = A * B mod b */ + for (x = bits - 1; x < (int)MP_DIGIT_BIT; x++) { + if ((err = mp_mul_2(a, a)) != MP_OKAY) { + return err; + } + if (mp_cmp_mag(a, b) != MP_LT) { + if ((err = s_mp_sub(a, b, a)) != MP_OKAY) { + return err; + } + } + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_calc_normalization.c */ + +/* Start: bn_mp_montgomery_reduce.c */ +#include "tommath_private.h" +#ifdef BN_MP_MONTGOMERY_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes xR**-1 == x (mod N) via Montgomery Reduction */ +mp_err mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) +{ + int ix, digs; + mp_err err; + mp_digit mu; + + /* can the fast reduction [comba] method be used? + * + * Note that unlike in mul you're safely allowed *less* + * than the available columns [255 per default] since carries + * are fixed up in the inner loop. + */ + digs = (n->used * 2) + 1; + if ((digs < MP_WARRAY) && + (x->used <= MP_WARRAY) && + (n->used < MP_MAXFAST)) { + return s_mp_montgomery_reduce_fast(x, n, rho); + } + + /* grow the input as required */ + if (x->alloc < digs) { + if ((err = mp_grow(x, digs)) != MP_OKAY) { + return err; + } + } + x->used = digs; + + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * rho mod b + * + * The value of rho must be precalculated via + * montgomery_setup() such that + * it equals -1/n0 mod b this allows the + * following inner loop to reduce the + * input one digit at a time + */ + mu = (mp_digit)(((mp_word)x->dp[ix] * (mp_word)rho) & MP_MASK); + + /* a = a + mu * m * b**i */ + { + int iy; + mp_digit *tmpn, *tmpx, u; + mp_word r; + + /* alias for digits of the modulus */ + tmpn = n->dp; + + /* alias for the digits of x [the input] */ + tmpx = x->dp + ix; + + /* set the carry to zero */ + u = 0; + + /* Multiply and add in place */ + for (iy = 0; iy < n->used; iy++) { + /* compute product and sum */ + r = ((mp_word)mu * (mp_word)*tmpn++) + + (mp_word)u + (mp_word)*tmpx; + + /* get carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + + /* fix digit */ + *tmpx++ = (mp_digit)(r & (mp_word)MP_MASK); + } + /* At this point the ix'th digit of x should be zero */ + + + /* propagate carries upwards as required*/ + while (u != 0u) { + *tmpx += u; + u = *tmpx >> MP_DIGIT_BIT; + *tmpx++ &= MP_MASK; + } + } + } + + /* at this point the n.used'th least + * significant digits of x are all zero + * which means we can shift x to the + * right by n.used digits and the + * residue is unchanged. + */ + + /* x = x/b**n.used */ + mp_clamp(x); + mp_rshd(x, n->used); + + /* if x >= n then x = x - n */ + if (mp_cmp_mag(x, n) != MP_LT) { + return s_mp_sub(x, n, x); + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_reduce.c */ + +/* Start: bn_mp_montgomery_setup.c */ +#include "tommath_private.h" +#ifdef BN_MP_MONTGOMERY_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* setups the montgomery reduction stuff */ +mp_err mp_montgomery_setup(const mp_int *n, mp_digit *rho) +{ + mp_digit x, b; + + /* fast inversion mod 2**k + * + * Based on the fact that + * + * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n) + * => 2*X*A - X*X*A*A = 1 + * => 2*(1) - (1) = 1 + */ + b = n->dp[0]; + + if ((b & 1u) == 0u) { + return MP_VAL; + } + + x = (((b + 2u) & 4u) << 1) + b; /* here x*a==1 mod 2**4 */ + x *= 2u - (b * x); /* here x*a==1 mod 2**8 */ +#if !defined(MP_8BIT) + x *= 2u - (b * x); /* here x*a==1 mod 2**16 */ +#endif +#if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT)) + x *= 2u - (b * x); /* here x*a==1 mod 2**32 */ +#endif +#ifdef MP_64BIT + x *= 2u - (b * x); /* here x*a==1 mod 2**64 */ +#endif + + /* rho = -1/m mod b */ + *rho = (mp_digit)(((mp_word)1 << (mp_word)MP_DIGIT_BIT) - x) & MP_MASK; + + return MP_OKAY; +} +#endif + +/* End: bn_mp_montgomery_setup.c */ + +/* Start: bn_mp_mul.c */ +#include "tommath_private.h" +#ifdef BN_MP_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* high level multiplication (handles sign) */ +mp_err mp_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_err err; + int min_len = MP_MIN(a->used, b->used), + max_len = MP_MAX(a->used, b->used), + digs = a->used + b->used + 1; + mp_sign neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG; + + if (MP_HAS(S_MP_BALANCE_MUL) && + /* Check sizes. The smaller one needs to be larger than the Karatsuba cut-off. + * The bigger one needs to be at least about one MP_KARATSUBA_MUL_CUTOFF bigger + * to make some sense, but it depends on architecture, OS, position of the + * stars... so YMMV. + * Using it to cut the input into slices small enough for fast_s_mp_mul_digs + * was actually slower on the author's machine, but YMMV. + */ + (min_len >= MP_KARATSUBA_MUL_CUTOFF) && + ((max_len / 2) >= MP_KARATSUBA_MUL_CUTOFF) && + /* Not much effect was observed below a ratio of 1:2, but again: YMMV. */ + (max_len >= (2 * min_len))) { + err = s_mp_balance_mul(a,b,c); + } else if (MP_HAS(S_MP_TOOM_MUL) && + (min_len >= MP_TOOM_MUL_CUTOFF)) { + err = s_mp_toom_mul(a, b, c); + } else if (MP_HAS(S_MP_KARATSUBA_MUL) && + (min_len >= MP_KARATSUBA_MUL_CUTOFF)) { + err = s_mp_karatsuba_mul(a, b, c); + } else if (MP_HAS(S_MP_MUL_DIGS_FAST) && + /* can we use the fast multiplier? + * + * The fast multiplier can be used if the output will + * have less than MP_WARRAY digits and the number of + * digits won't affect carry propagation + */ + (digs < MP_WARRAY) && + (min_len <= MP_MAXFAST)) { + err = s_mp_mul_digs_fast(a, b, c, digs); + } else if (MP_HAS(S_MP_MUL_DIGS)) { + err = s_mp_mul_digs(a, b, c, digs); + } else { + err = MP_VAL; + } + c->sign = (c->used > 0) ? neg : MP_ZPOS; + return err; +} +#endif + +/* End: bn_mp_mul.c */ + +/* Start: bn_mp_mul_2.c */ +#include "tommath_private.h" +#ifdef BN_MP_MUL_2_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = a*2 */ +mp_err mp_mul_2(const mp_int *a, mp_int *b) +{ + int x, oldused; + mp_err err; + + /* grow to accomodate result */ + if (b->alloc < (a->used + 1)) { + if ((err = mp_grow(b, a->used + 1)) != MP_OKAY) { + return err; + } + } + + oldused = b->used; + b->used = a->used; + + { + mp_digit r, rr, *tmpa, *tmpb; + + /* alias for source */ + tmpa = a->dp; + + /* alias for dest */ + tmpb = b->dp; + + /* carry */ + r = 0; + for (x = 0; x < a->used; x++) { + + /* get what will be the *next* carry bit from the + * MSB of the current digit + */ + rr = *tmpa >> (mp_digit)(MP_DIGIT_BIT - 1); + + /* now shift up this digit, add in the carry [from the previous] */ + *tmpb++ = ((*tmpa++ << 1uL) | r) & MP_MASK; + + /* copy the carry that would be from the source + * digit into the next iteration + */ + r = rr; + } + + /* new leading digit? */ + if (r != 0u) { + /* add a MSB which is always 1 at this point */ + *tmpb = 1; + ++(b->used); + } + + /* now zero any excess digits on the destination + * that we didn't write to + */ + MP_ZERO_DIGITS(b->dp + b->used, oldused - b->used); + } + b->sign = a->sign; + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_2.c */ + +/* Start: bn_mp_mul_2d.c */ +#include "tommath_private.h" +#ifdef BN_MP_MUL_2D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift left by a certain bit count */ +mp_err mp_mul_2d(const mp_int *a, int b, mp_int *c) +{ + mp_digit d; + mp_err err; + + /* copy */ + if (a != c) { + if ((err = mp_copy(a, c)) != MP_OKAY) { + return err; + } + } + + if (c->alloc < (c->used + (b / MP_DIGIT_BIT) + 1)) { + if ((err = mp_grow(c, c->used + (b / MP_DIGIT_BIT) + 1)) != MP_OKAY) { + return err; + } + } + + /* shift by as many digits in the bit count */ + if (b >= MP_DIGIT_BIT) { + if ((err = mp_lshd(c, b / MP_DIGIT_BIT)) != MP_OKAY) { + return err; + } + } + + /* shift any bit count < MP_DIGIT_BIT */ + d = (mp_digit)(b % MP_DIGIT_BIT); + if (d != 0u) { + mp_digit *tmpc, shift, mask, r, rr; + int x; + + /* bitmask for carries */ + mask = ((mp_digit)1 << d) - (mp_digit)1; + + /* shift for msbs */ + shift = (mp_digit)MP_DIGIT_BIT - d; + + /* alias */ + tmpc = c->dp; + + /* carry */ + r = 0; + for (x = 0; x < c->used; x++) { + /* get the higher bits of the current word */ + rr = (*tmpc >> shift) & mask; + + /* shift the current word and OR in the carry */ + *tmpc = ((*tmpc << d) | r) & MP_MASK; + ++tmpc; + + /* set the carry to the carry bits of the current word */ + r = rr; + } + + /* set final carry */ + if (r != 0u) { + c->dp[(c->used)++] = r; + } + } + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_2d.c */ + +/* Start: bn_mp_mul_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_MUL_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* multiply by a digit */ +mp_err mp_mul_d(const mp_int *a, mp_digit b, mp_int *c) +{ + mp_digit u, *tmpa, *tmpc; + mp_word r; + mp_err err; + int ix, olduse; + + /* make sure c is big enough to hold a*b */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } + + /* get the original destinations used count */ + olduse = c->used; + + /* set the sign */ + c->sign = a->sign; + + /* alias for a->dp [source] */ + tmpa = a->dp; + + /* alias for c->dp [dest] */ + tmpc = c->dp; + + /* zero carry */ + u = 0; + + /* compute columns */ + for (ix = 0; ix < a->used; ix++) { + /* compute product and carry sum for this term */ + r = (mp_word)u + ((mp_word)*tmpa++ * (mp_word)b); + + /* mask off higher bits to get a single digit */ + *tmpc++ = (mp_digit)(r & (mp_word)MP_MASK); + + /* send carry into next iteration */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + + /* store final carry [if any] and increment ix offset */ + *tmpc++ = u; + ++ix; + + /* now zero digits above the top */ + MP_ZERO_DIGITS(tmpc, olduse - ix); + + /* set used count */ + c->used = a->used + 1; + mp_clamp(c); + + return MP_OKAY; +} +#endif + +/* End: bn_mp_mul_d.c */ + +/* Start: bn_mp_mulmod.c */ +#include "tommath_private.h" +#ifdef BN_MP_MULMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* d = a * b (mod c) */ +mp_err mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) +{ + mp_err err; + mp_int t; + + if ((err = mp_init_size(&t, c->used)) != MP_OKAY) { + return err; + } + + if ((err = mp_mul(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_mulmod.c */ + +/* Start: bn_mp_neg.c */ +#include "tommath_private.h" +#ifdef BN_MP_NEG_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* b = -a */ +mp_err mp_neg(const mp_int *a, mp_int *b) +{ + mp_err err; + if (a != b) { + if ((err = mp_copy(a, b)) != MP_OKAY) { + return err; + } + } + + if (!MP_IS_ZERO(b)) { + b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS; + } else { + b->sign = MP_ZPOS; + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_neg.c */ + +/* Start: bn_mp_or.c */ +#include "tommath_private.h" +#ifdef BN_MP_OR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* two complement or */ +mp_err mp_or(const mp_int *a, const mp_int *b, mp_int *c) +{ + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = ((a->sign == MP_NEG) || (b->sign == MP_NEG)) ? MP_NEG : MP_ZPOS; + + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } + + for (i = 0; i < used; i++) { + mp_digit x, y; + + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } + + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x | y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_or.c */ + +/* Start: bn_mp_pack.c */ +#include "tommath_private.h" +#ifdef BN_MP_PACK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* based on gmp's mpz_export. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +mp_err mp_pack(void *rop, size_t maxcount, size_t *written, mp_order order, size_t size, + mp_endian endian, size_t nails, const mp_int *op) +{ + mp_err err; + size_t odd_nails, nail_bytes, i, j, count; + unsigned char odd_nail_mask; + + mp_int t; + + count = mp_pack_count(op, nails, size); + + if (count > maxcount) { + return MP_BUF; + } + + if ((err = mp_init_copy(&t, op)) != MP_OKAY) { + return err; + } + + if (endian == MP_NATIVE_ENDIAN) { + MP_GET_ENDIANNESS(endian); + } + + odd_nails = (nails % 8u); + odd_nail_mask = 0xff; + for (i = 0u; i < odd_nails; ++i) { + odd_nail_mask ^= (unsigned char)(1u << (7u - i)); + } + nail_bytes = nails / 8u; + + for (i = 0u; i < count; ++i) { + for (j = 0u; j < size; ++j) { + unsigned char *byte = (unsigned char *)rop + + (((order == MP_LSB_FIRST) ? i : ((count - 1u) - i)) * size) + + ((endian == MP_LITTLE_ENDIAN) ? j : ((size - 1u) - j)); + + if (j >= (size - nail_bytes)) { + *byte = 0; + continue; + } + + *byte = (unsigned char)((j == ((size - nail_bytes) - 1u)) ? (t.dp[0] & odd_nail_mask) : (t.dp[0] & 0xFFuL)); + + if ((err = mp_div_2d(&t, (j == ((size - nail_bytes) - 1u)) ? (int)(8u - odd_nails) : 8, &t, NULL)) != MP_OKAY) { + goto LBL_ERR; + } + + } + } + + if (written != NULL) { + *written = count; + } + err = MP_OKAY; + +LBL_ERR: + mp_clear(&t); + return err; +} + +#endif + +/* End: bn_mp_pack.c */ + +/* Start: bn_mp_pack_count.c */ +#include "tommath_private.h" +#ifdef BN_MP_PACK_COUNT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +size_t mp_pack_count(const mp_int *a, size_t nails, size_t size) +{ + size_t bits = (size_t)mp_count_bits(a); + return ((bits / ((size * 8u) - nails)) + (((bits % ((size * 8u) - nails)) != 0u) ? 1u : 0u)); +} + +#endif + +/* End: bn_mp_pack_count.c */ + +/* Start: bn_mp_prime_fermat.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_FERMAT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* performs one Fermat test. + * + * If "a" were prime then b**a == b (mod a) since the order of + * the multiplicative sub-group would be phi(a) = a-1. That means + * it would be the same as b**(a mod (a-1)) == b**1 == b (mod a). + * + * Sets result to 1 if the congruence holds, or zero otherwise. + */ +mp_err mp_prime_fermat(const mp_int *a, const mp_int *b, mp_bool *result) +{ + mp_int t; + mp_err err; + + /* default to composite */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1uL) != MP_GT) { + return MP_VAL; + } + + /* init t */ + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + /* compute t = b**a mod a */ + if ((err = mp_exptmod(b, a, a, &t)) != MP_OKAY) { + goto LBL_T; + } + + /* is it equal to b? */ + if (mp_cmp(&t, b) == MP_EQ) { + *result = MP_YES; + } + + err = MP_OKAY; +LBL_T: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_prime_fermat.c */ + +/* Start: bn_mp_prime_frobenius_underwood.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_FROBENIUS_UNDERWOOD_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + * See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details + */ +#ifndef LTM_USE_ONLY_MR + +#ifdef MP_8BIT +/* + * floor of positive solution of + * (2^16)-1 = (a+4)*(2*a+5) + * TODO: Both values are smaller than N^(1/4), would have to use a bigint + * for a instead but any a biger than about 120 are already so rare that + * it is possible to ignore them and still get enough pseudoprimes. + * But it is still a restriction of the set of available pseudoprimes + * which makes this implementation less secure if used stand-alone. + */ +#define LTM_FROBENIUS_UNDERWOOD_A 177 +#else +#define LTM_FROBENIUS_UNDERWOOD_A 32764 +#endif +mp_err mp_prime_frobenius_underwood(const mp_int *N, mp_bool *result) +{ + mp_int T1z, T2z, Np1z, sz, tz; + + int a, ap2, length, i, j; + mp_err err; + + *result = MP_NO; + + if ((err = mp_init_multi(&T1z, &T2z, &Np1z, &sz, &tz, NULL)) != MP_OKAY) { + return err; + } + + for (a = 0; a < LTM_FROBENIUS_UNDERWOOD_A; a++) { + /* TODO: That's ugly! No, really, it is! */ + if ((a==2) || (a==4) || (a==7) || (a==8) || (a==10) || + (a==14) || (a==18) || (a==23) || (a==26) || (a==28)) { + continue; + } + /* (32764^2 - 4) < 2^31, no bigint for >MP_8BIT needed) */ + mp_set_u32(&T1z, (uint32_t)a); + + if ((err = mp_sqr(&T1z, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if ((err = mp_sub_d(&T1z, 4uL, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if ((err = mp_kronecker(&T1z, N, &j)) != MP_OKAY) goto LBL_FU_ERR; + + if (j == -1) { + break; + } + + if (j == 0) { + /* composite */ + goto LBL_FU_ERR; + } + } + /* Tell it a composite and set return value accordingly */ + if (a >= LTM_FROBENIUS_UNDERWOOD_A) { + err = MP_ITER; + goto LBL_FU_ERR; + } + /* Composite if N and (a+4)*(2*a+5) are not coprime */ + mp_set_u32(&T1z, (uint32_t)((a+4)*((2*a)+5))); + + if ((err = mp_gcd(N, &T1z, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + + if (!((T1z.used == 1) && (T1z.dp[0] == 1u))) goto LBL_FU_ERR; + + ap2 = a + 2; + if ((err = mp_add_d(N, 1uL, &Np1z)) != MP_OKAY) goto LBL_FU_ERR; + + mp_set(&sz, 1uL); + mp_set(&tz, 2uL); + length = mp_count_bits(&Np1z); + + for (i = length - 2; i >= 0; i--) { + /* + * temp = (sz*(a*sz+2*tz))%N; + * tz = ((tz-sz)*(tz+sz))%N; + * sz = temp; + */ + if ((err = mp_mul_2(&tz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + + /* a = 0 at about 50% of the cases (non-square and odd input) */ + if (a != 0) { + if ((err = mp_mul_d(&sz, (mp_digit)a, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_add(&T1z, &T2z, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + } + + if ((err = mp_mul(&T2z, &sz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_sub(&tz, &sz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_add(&sz, &tz, &sz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mul(&sz, &T2z, &tz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mod(&tz, N, &tz)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mod(&T1z, N, &sz)) != MP_OKAY) goto LBL_FU_ERR; + if (s_mp_get_bit(&Np1z, (unsigned int)i) == MP_YES) { + /* + * temp = (a+2) * sz + tz + * tz = 2 * tz - sz + * sz = temp + */ + if (a == 0) { + if ((err = mp_mul_2(&sz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + } else { + if ((err = mp_mul_d(&sz, (mp_digit)ap2, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + } + if ((err = mp_add(&T1z, &tz, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_mul_2(&tz, &T2z)) != MP_OKAY) goto LBL_FU_ERR; + if ((err = mp_sub(&T2z, &sz, &tz)) != MP_OKAY) goto LBL_FU_ERR; + mp_exch(&sz, &T1z); + } + } + + mp_set_u32(&T1z, (uint32_t)((2 * a) + 5)); + if ((err = mp_mod(&T1z, N, &T1z)) != MP_OKAY) goto LBL_FU_ERR; + if (MP_IS_ZERO(&sz) && (mp_cmp(&tz, &T1z) == MP_EQ)) { + *result = MP_YES; + } + +LBL_FU_ERR: + mp_clear_multi(&tz, &sz, &Np1z, &T2z, &T1z, NULL); + return err; +} + +#endif +#endif + +/* End: bn_mp_prime_frobenius_underwood.c */ + +/* Start: bn_mp_prime_is_prime.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_IS_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* portable integer log of two with small footprint */ +static unsigned int s_floor_ilog2(int value) +{ + unsigned int r = 0; + while ((value >>= 1) != 0) { + r++; + } + return r; +} + + +mp_err mp_prime_is_prime(const mp_int *a, int t, mp_bool *result) +{ + mp_int b; + int ix, p_max = 0, size_a, len; + mp_bool res; + mp_err err; + unsigned int fips_rand, mask; + + /* default to no */ + *result = MP_NO; + + /* Some shortcuts */ + /* N > 3 */ + if (a->used == 1) { + if ((a->dp[0] == 0u) || (a->dp[0] == 1u)) { + *result = MP_NO; + return MP_OKAY; + } + if (a->dp[0] == 2u) { + *result = MP_YES; + return MP_OKAY; + } + } + + /* N must be odd */ + if (MP_IS_EVEN(a)) { + return MP_OKAY; + } + /* N is not a perfect square: floor(sqrt(N))^2 != N */ + if ((err = mp_is_square(a, &res)) != MP_OKAY) { + return err; + } + if (res != MP_NO) { + return MP_OKAY; + } + + /* is the input equal to one of the primes in the table? */ + for (ix = 0; ix < PRIVATE_MP_PRIME_TAB_SIZE; ix++) { + if (mp_cmp_d(a, s_mp_prime_tab[ix]) == MP_EQ) { + *result = MP_YES; + return MP_OKAY; + } + } +#ifdef MP_8BIT + /* The search in the loop above was exhaustive in this case */ + if ((a->used == 1) && (PRIVATE_MP_PRIME_TAB_SIZE >= 31)) { + return MP_OKAY; + } +#endif + + /* first perform trial division */ + if ((err = s_mp_prime_is_divisible(a, &res)) != MP_OKAY) { + return err; + } + + /* return if it was trivially divisible */ + if (res == MP_YES) { + return MP_OKAY; + } + + /* + Run the Miller-Rabin test with base 2 for the BPSW test. + */ + if ((err = mp_init_set(&b, 2uL)) != MP_OKAY) { + return err; + } + + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + /* + Rumours have it that Mathematica does a second M-R test with base 3. + Other rumours have it that their strong L-S test is slightly different. + It does not hurt, though, beside a bit of extra runtime. + */ + b.dp[0]++; + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + + /* + * Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite + * slow so if speed is an issue, define LTM_USE_ONLY_MR to use M-R tests with + * bases 2, 3 and t random bases. + */ +#ifndef LTM_USE_ONLY_MR + if (t >= 0) { + /* + * Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for + * MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit + * integers but the necesssary analysis is on the todo-list). + */ +#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST) + err = mp_prime_frobenius_underwood(a, &res); + if ((err != MP_OKAY) && (err != MP_ITER)) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } +#else + if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } +#endif + } +#endif + + /* run at least one Miller-Rabin test with a random base */ + if (t == 0) { + t = 1; + } + + /* + Only recommended if the input range is known to be < 3317044064679887385961981 + + It uses the bases necessary for a deterministic M-R test if the input is + smaller than 3317044064679887385961981 + The caller has to check the size. + TODO: can be made a bit finer grained but comparing is not free. + */ + if (t < 0) { + /* + Sorenson, Jonathan; Webster, Jonathan (2015). + "Strong Pseudoprimes to Twelve Prime Bases". + */ + /* 0x437ae92817f9fc85b7e5 = 318665857834031151167461 */ + if ((err = mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) { + goto LBL_B; + } + + if (mp_cmp(a, &b) == MP_LT) { + p_max = 12; + } else { + /* 0x2be6951adc5b22410a5fd = 3317044064679887385961981 */ + if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) { + goto LBL_B; + } + + if (mp_cmp(a, &b) == MP_LT) { + p_max = 13; + } else { + err = MP_VAL; + goto LBL_B; + } + } + + /* we did bases 2 and 3 already, skip them */ + for (ix = 2; ix < p_max; ix++) { + mp_set(&b, s_mp_prime_tab[ix]); + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + } + } + /* + Do "t" M-R tests with random bases between 3 and "a". + See Fips 186.4 p. 126ff + */ + else if (t > 0) { + /* + * The mp_digit's have a defined bit-size but the size of the + * array a.dp is a simple 'int' and this library can not assume full + * compliance to the current C-standard (ISO/IEC 9899:2011) because + * it gets used for small embeded processors, too. Some of those MCUs + * have compilers that one cannot call standard compliant by any means. + * Hence the ugly type-fiddling in the following code. + */ + size_a = mp_count_bits(a); + mask = (1u << s_floor_ilog2(size_a)) - 1u; + /* + Assuming the General Rieman hypothesis (never thought to write that in a + comment) the upper bound can be lowered to 2*(log a)^2. + E. Bach, "Explicit bounds for primality testing and related problems," + Math. Comp. 55 (1990), 355-380. + + size_a = (size_a/10) * 7; + len = 2 * (size_a * size_a); + + E.g.: a number of size 2^2048 would be reduced to the upper limit + + floor(2048/10)*7 = 1428 + 2 * 1428^2 = 4078368 + + (would have been ~4030331.9962 with floats and natural log instead) + That number is smaller than 2^28, the default bit-size of mp_digit. + */ + + /* + How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame + does exactly 1. In words: one. Look at the end of _GMP_is_prime() in + Math-Prime-Util-GMP-0.50/primality.c if you do not believe it. + + The function mp_rand() goes to some length to use a cryptographically + good PRNG. That also means that the chance to always get the same base + in the loop is non-zero, although very low. + If the BPSW test and/or the addtional Frobenious test have been + performed instead of just the Miller-Rabin test with the bases 2 and 3, + a single extra test should suffice, so such a very unlikely event + will not do much harm. + + To preemptivly answer the dangling question: no, a witness does not + need to be prime. + */ + for (ix = 0; ix < t; ix++) { + /* mp_rand() guarantees the first digit to be non-zero */ + if ((err = mp_rand(&b, 1)) != MP_OKAY) { + goto LBL_B; + } + /* + * Reduce digit before casting because mp_digit might be bigger than + * an unsigned int and "mask" on the other side is most probably not. + */ + fips_rand = (unsigned int)(b.dp[0] & (mp_digit) mask); +#ifdef MP_8BIT + /* + * One 8-bit digit is too small, so concatenate two if the size of + * unsigned int allows for it. + */ + if ((MP_SIZEOF_BITS(unsigned int)/2) >= MP_SIZEOF_BITS(mp_digit)) { + if ((err = mp_rand(&b, 1)) != MP_OKAY) { + goto LBL_B; + } + fips_rand <<= MP_SIZEOF_BITS(mp_digit); + fips_rand |= (unsigned int) b.dp[0]; + fips_rand &= mask; + } +#endif + if (fips_rand > (unsigned int)(INT_MAX - MP_DIGIT_BIT)) { + len = INT_MAX / MP_DIGIT_BIT; + } else { + len = (((int)fips_rand + MP_DIGIT_BIT) / MP_DIGIT_BIT); + } + /* Unlikely. */ + if (len < 0) { + ix--; + continue; + } + /* + * As mentioned above, one 8-bit digit is too small and + * although it can only happen in the unlikely case that + * an "unsigned int" is smaller than 16 bit a simple test + * is cheap and the correction even cheaper. + */ +#ifdef MP_8BIT + /* All "a" < 2^8 have been caught before */ + if (len == 1) { + len++; + } +#endif + if ((err = mp_rand(&b, len)) != MP_OKAY) { + goto LBL_B; + } + /* + * That number might got too big and the witness has to be + * smaller than "a" + */ + len = mp_count_bits(&b); + if (len >= size_a) { + len = (len - size_a) + 1; + if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) { + goto LBL_B; + } + } + /* Although the chance for b <= 3 is miniscule, try again. */ + if (mp_cmp_d(&b, 3uL) != MP_GT) { + ix--; + continue; + } + if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) { + goto LBL_B; + } + if (res == MP_NO) { + goto LBL_B; + } + } + } + + /* passed the test */ + *result = MP_YES; +LBL_B: + mp_clear(&b); + return err; +} + +#endif + +/* End: bn_mp_prime_is_prime.c */ + +/* Start: bn_mp_prime_miller_rabin.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_MILLER_RABIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Miller-Rabin test of "a" to the base of "b" as described in + * HAC pp. 139 Algorithm 4.24 + * + * Sets result to 0 if definitely composite or 1 if probably prime. + * Randomly the chance of error is no more than 1/4 and often + * very much lower. + */ +mp_err mp_prime_miller_rabin(const mp_int *a, const mp_int *b, mp_bool *result) +{ + mp_int n1, y, r; + mp_err err; + int s, j; + + /* default */ + *result = MP_NO; + + /* ensure b > 1 */ + if (mp_cmp_d(b, 1uL) != MP_GT) { + return MP_VAL; + } + + /* get n1 = a - 1 */ + if ((err = mp_init_copy(&n1, a)) != MP_OKAY) { + return err; + } + if ((err = mp_sub_d(&n1, 1uL, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* set 2**s * r = n1 */ + if ((err = mp_init_copy(&r, &n1)) != MP_OKAY) { + goto LBL_N1; + } + + /* count the number of least significant bits + * which are zero + */ + s = mp_cnt_lsb(&r); + + /* now divide n - 1 by 2**s */ + if ((err = mp_div_2d(&r, s, &r, NULL)) != MP_OKAY) { + goto LBL_R; + } + + /* compute y = b**r mod a */ + if ((err = mp_init(&y)) != MP_OKAY) { + goto LBL_R; + } + if ((err = mp_exptmod(b, &r, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y != 1 and y != n1 do */ + if ((mp_cmp_d(&y, 1uL) != MP_EQ) && (mp_cmp(&y, &n1) != MP_EQ)) { + j = 1; + /* while j <= s-1 and y != n1 */ + while ((j <= (s - 1)) && (mp_cmp(&y, &n1) != MP_EQ)) { + if ((err = mp_sqrmod(&y, a, &y)) != MP_OKAY) { + goto LBL_Y; + } + + /* if y == 1 then composite */ + if (mp_cmp_d(&y, 1uL) == MP_EQ) { + goto LBL_Y; + } + + ++j; + } + + /* if y != n1 then composite */ + if (mp_cmp(&y, &n1) != MP_EQ) { + goto LBL_Y; + } + } + + /* probably prime now */ + *result = MP_YES; +LBL_Y: + mp_clear(&y); +LBL_R: + mp_clear(&r); +LBL_N1: + mp_clear(&n1); + return err; +} +#endif + +/* End: bn_mp_prime_miller_rabin.c */ + +/* Start: bn_mp_prime_next_prime.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_NEXT_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* finds the next prime after the number "a" using "t" trials + * of Miller-Rabin. + * + * bbs_style = 1 means the prime must be congruent to 3 mod 4 + */ +mp_err mp_prime_next_prime(mp_int *a, int t, int bbs_style) +{ + int x, y; + mp_ord cmp; + mp_err err; + mp_bool res = MP_NO; + mp_digit res_tab[PRIVATE_MP_PRIME_TAB_SIZE], step, kstep; + mp_int b; + + /* force positive */ + a->sign = MP_ZPOS; + + /* simple algo if a is less than the largest prime in the table */ + if (mp_cmp_d(a, s_mp_prime_tab[PRIVATE_MP_PRIME_TAB_SIZE-1]) == MP_LT) { + /* find which prime it is bigger than "a" */ + for (x = 0; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + cmp = mp_cmp_d(a, s_mp_prime_tab[x]); + if (cmp == MP_EQ) { + continue; + } + if (cmp != MP_GT) { + if ((bbs_style == 1) && ((s_mp_prime_tab[x] & 3u) != 3u)) { + /* try again until we get a prime congruent to 3 mod 4 */ + continue; + } else { + mp_set(a, s_mp_prime_tab[x]); + return MP_OKAY; + } + } + } + /* fall through to the sieve */ + } + + /* generate a prime congruent to 3 mod 4 or 1/3 mod 4? */ + if (bbs_style == 1) { + kstep = 4; + } else { + kstep = 2; + } + + /* at this point we will use a combination of a sieve and Miller-Rabin */ + + if (bbs_style == 1) { + /* if a mod 4 != 3 subtract the correct value to make it so */ + if ((a->dp[0] & 3u) != 3u) { + if ((err = mp_sub_d(a, (a->dp[0] & 3u) + 1u, a)) != MP_OKAY) { + return err; + } + } + } else { + if (MP_IS_EVEN(a)) { + /* force odd */ + if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) { + return err; + } + } + } + + /* generate the restable */ + for (x = 1; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + if ((err = mp_mod_d(a, s_mp_prime_tab[x], res_tab + x)) != MP_OKAY) { + return err; + } + } + + /* init temp used for Miller-Rabin Testing */ + if ((err = mp_init(&b)) != MP_OKAY) { + return err; + } + + for (;;) { + /* skip to the next non-trivially divisible candidate */ + step = 0; + do { + /* y == 1 if any residue was zero [e.g. cannot be prime] */ + y = 0; + + /* increase step to next candidate */ + step += kstep; + + /* compute the new residue without using division */ + for (x = 1; x < PRIVATE_MP_PRIME_TAB_SIZE; x++) { + /* add the step to each residue */ + res_tab[x] += kstep; + + /* subtract the modulus [instead of using division] */ + if (res_tab[x] >= s_mp_prime_tab[x]) { + res_tab[x] -= s_mp_prime_tab[x]; + } + + /* set flag if zero */ + if (res_tab[x] == 0u) { + y = 1; + } + } + } while ((y == 1) && (step < (((mp_digit)1 << MP_DIGIT_BIT) - kstep))); + + /* add the step */ + if ((err = mp_add_d(a, step, a)) != MP_OKAY) { + goto LBL_ERR; + } + + /* if didn't pass sieve and step == MP_MAX then skip test */ + if ((y == 1) && (step >= (((mp_digit)1 << MP_DIGIT_BIT) - kstep))) { + continue; + } + + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto LBL_ERR; + } + if (res == MP_YES) { + break; + } + } + + err = MP_OKAY; +LBL_ERR: + mp_clear(&b); + return err; +} + +#endif + +/* End: bn_mp_prime_next_prime.c */ + +/* Start: bn_mp_prime_rabin_miller_trials.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_RABIN_MILLER_TRIALS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +static const struct { + int k, t; +} sizes[] = { + { 80, -1 }, /* Use deterministic algorithm for size <= 80 bits */ + { 81, 37 }, /* max. error = 2^(-96)*/ + { 96, 32 }, /* max. error = 2^(-96)*/ + { 128, 40 }, /* max. error = 2^(-112)*/ + { 160, 35 }, /* max. error = 2^(-112)*/ + { 256, 27 }, /* max. error = 2^(-128)*/ + { 384, 16 }, /* max. error = 2^(-128)*/ + { 512, 18 }, /* max. error = 2^(-160)*/ + { 768, 11 }, /* max. error = 2^(-160)*/ + { 896, 10 }, /* max. error = 2^(-160)*/ + { 1024, 12 }, /* max. error = 2^(-192)*/ + { 1536, 8 }, /* max. error = 2^(-192)*/ + { 2048, 6 }, /* max. error = 2^(-192)*/ + { 3072, 4 }, /* max. error = 2^(-192)*/ + { 4096, 5 }, /* max. error = 2^(-256)*/ + { 5120, 4 }, /* max. error = 2^(-256)*/ + { 6144, 4 }, /* max. error = 2^(-256)*/ + { 8192, 3 }, /* max. error = 2^(-256)*/ + { 9216, 3 }, /* max. error = 2^(-256)*/ + { 10240, 2 } /* For bigger keysizes use always at least 2 Rounds */ +}; + +/* returns # of RM trials required for a given bit size */ +int mp_prime_rabin_miller_trials(int size) +{ + int x; + + for (x = 0; x < (int)(sizeof(sizes)/(sizeof(sizes[0]))); x++) { + if (sizes[x].k == size) { + return sizes[x].t; + } else if (sizes[x].k > size) { + return (x == 0) ? sizes[0].t : sizes[x - 1].t; + } + } + return sizes[x-1].t; +} + + +#endif + +/* End: bn_mp_prime_rabin_miller_trials.c */ + +/* Start: bn_mp_prime_rand.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_RAND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* makes a truly random prime of a given size (bits), + * + * Flags are as follows: + * + * MP_PRIME_BBS - make prime congruent to 3 mod 4 + * MP_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies MP_PRIME_BBS) + * MP_PRIME_2MSB_ON - make the 2nd highest bit one + * + * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can + * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself + * so it can be NULL + * + */ + +/* This is possibly the mother of all prime generation functions, muahahahahaha! */ +mp_err s_mp_prime_random_ex(mp_int *a, int t, int size, int flags, private_mp_prime_callback cb, void *dat) +{ + unsigned char *tmp, maskAND, maskOR_msb, maskOR_lsb; + int bsize, maskOR_msb_offset; + mp_bool res; + mp_err err; + + /* sanity check the input */ + if ((size <= 1) || (t <= 0)) { + return MP_VAL; + } + + /* MP_PRIME_SAFE implies MP_PRIME_BBS */ + if ((flags & MP_PRIME_SAFE) != 0) { + flags |= MP_PRIME_BBS; + } + + /* calc the byte size */ + bsize = (size>>3) + ((size&7)?1:0); + + /* we need a buffer of bsize bytes */ + tmp = (unsigned char *) MP_MALLOC((size_t)bsize); + if (tmp == NULL) { + return MP_MEM; + } + + /* calc the maskAND value for the MSbyte*/ + maskAND = ((size&7) == 0) ? 0xFFu : (unsigned char)(0xFFu >> (8 - (size & 7))); + + /* calc the maskOR_msb */ + maskOR_msb = 0; + maskOR_msb_offset = ((size & 7) == 1) ? 1 : 0; + if ((flags & MP_PRIME_2MSB_ON) != 0) { + maskOR_msb |= (unsigned char)(0x80 >> ((9 - size) & 7)); + } + + /* get the maskOR_lsb */ + maskOR_lsb = 1u; + if ((flags & MP_PRIME_BBS) != 0) { + maskOR_lsb |= 3u; + } + + do { + /* read the bytes */ + if (cb(tmp, bsize, dat) != bsize) { + err = MP_VAL; + goto error; + } + + /* work over the MSbyte */ + tmp[0] &= maskAND; + tmp[0] |= (unsigned char)(1 << ((size - 1) & 7)); + + /* mix in the maskORs */ + tmp[maskOR_msb_offset] |= maskOR_msb; + tmp[bsize-1] |= maskOR_lsb; + + /* read it in */ + /* TODO: casting only for now until all lengths have been changed to the type "size_t"*/ + if ((err = mp_from_ubin(a, tmp, (size_t)bsize)) != MP_OKAY) { + goto error; + } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto error; + } + if (res == MP_NO) { + continue; + } + + if ((flags & MP_PRIME_SAFE) != 0) { + /* see if (a-1)/2 is prime */ + if ((err = mp_sub_d(a, 1uL, a)) != MP_OKAY) { + goto error; + } + if ((err = mp_div_2(a, a)) != MP_OKAY) { + goto error; + } + + /* is it prime? */ + if ((err = mp_prime_is_prime(a, t, &res)) != MP_OKAY) { + goto error; + } + } + } while (res == MP_NO); + + if ((flags & MP_PRIME_SAFE) != 0) { + /* restore a to the original value */ + if ((err = mp_mul_2(a, a)) != MP_OKAY) { + goto error; + } + if ((err = mp_add_d(a, 1uL, a)) != MP_OKAY) { + goto error; + } + } + + err = MP_OKAY; +error: + MP_FREE_BUFFER(tmp, (size_t)bsize); + return err; +} + +static int s_mp_rand_cb(unsigned char *dst, int len, void *dat) +{ + (void)dat; + if (len <= 0) { + return len; + } + if (s_mp_rand_source(dst, (size_t)len) != MP_OKAY) { + return 0; + } + return len; +} + +mp_err mp_prime_rand(mp_int *a, int t, int size, int flags) +{ + return s_mp_prime_random_ex(a, t, size, flags, s_mp_rand_cb, NULL); +} + +#endif + +/* End: bn_mp_prime_rand.c */ + +/* Start: bn_mp_prime_strong_lucas_selfridge.c */ +#include "tommath_private.h" +#ifdef BN_MP_PRIME_STRONG_LUCAS_SELFRIDGE_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* + * See file bn_mp_prime_is_prime.c or the documentation in doc/bn.tex for the details + */ +#ifndef LTM_USE_ONLY_MR + +/* + * 8-bit is just too small. You can try the Frobenius test + * but that frobenius test can fail, too, for the same reason. + */ +#ifndef MP_8BIT + +/* + * multiply bigint a with int d and put the result in c + * Like mp_mul_d() but with a signed long as the small input + */ +static mp_err s_mp_mul_si(const mp_int *a, int32_t d, mp_int *c) +{ + mp_int t; + mp_err err; + + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + /* + * mp_digit might be smaller than a long, which excludes + * the use of mp_mul_d() here. + */ + mp_set_i32(&t, d); + err = mp_mul(a, &t, c); + mp_clear(&t); + return err; +} +/* + Strong Lucas-Selfridge test. + returns MP_YES if it is a strong L-S prime, MP_NO if it is composite + + Code ported from Thomas Ray Nicely's implementation of the BPSW test + at http://www.trnicely.net/misc/bpsw.html + + Freeware copyright (C) 2016 Thomas R. Nicely . + Released into the public domain by the author, who disclaims any legal + liability arising from its use + + The multi-line comments are made by Thomas R. Nicely and are copied verbatim. + Additional comments marked "CZ" (without the quotes) are by the code-portist. + + (If that name sounds familiar, he is the guy who found the fdiv bug in the + Pentium (P5x, I think) Intel processor) +*/ +mp_err mp_prime_strong_lucas_selfridge(const mp_int *a, mp_bool *result) +{ + /* CZ TODO: choose better variable names! */ + mp_int Dz, gcd, Np1, Uz, Vz, U2mz, V2mz, Qmz, Q2mz, Qkdz, T1z, T2z, T3z, T4z, Q2kdz; + /* CZ TODO: Some of them need the full 32 bit, hence the (temporary) exclusion of MP_8BIT */ + int32_t D, Ds, J, sign, P, Q, r, s, u, Nbits; + mp_err err; + mp_bool oddness; + + *result = MP_NO; + /* + Find the first element D in the sequence {5, -7, 9, -11, 13, ...} + such that Jacobi(D,N) = -1 (Selfridge's algorithm). Theory + indicates that, if N is not a perfect square, D will "nearly + always" be "small." Just in case, an overflow trap for D is + included. + */ + + if ((err = mp_init_multi(&Dz, &gcd, &Np1, &Uz, &Vz, &U2mz, &V2mz, &Qmz, &Q2mz, &Qkdz, &T1z, &T2z, &T3z, &T4z, &Q2kdz, + NULL)) != MP_OKAY) { + return err; + } + + D = 5; + sign = 1; + + for (;;) { + Ds = sign * D; + sign = -sign; + mp_set_u32(&Dz, (uint32_t)D); + if ((err = mp_gcd(a, &Dz, &gcd)) != MP_OKAY) goto LBL_LS_ERR; + + /* if 1 < GCD < N then N is composite with factor "D", and + Jacobi(D,N) is technically undefined (but often returned + as zero). */ + if ((mp_cmp_d(&gcd, 1uL) == MP_GT) && (mp_cmp(&gcd, a) == MP_LT)) { + goto LBL_LS_ERR; + } + if (Ds < 0) { + Dz.sign = MP_NEG; + } + if ((err = mp_kronecker(&Dz, a, &J)) != MP_OKAY) goto LBL_LS_ERR; + + if (J == -1) { + break; + } + D += 2; + + if (D > (INT_MAX - 2)) { + err = MP_VAL; + goto LBL_LS_ERR; + } + } + + + + P = 1; /* Selfridge's choice */ + Q = (1 - Ds) / 4; /* Required so D = P*P - 4*Q */ + + /* NOTE: The conditions (a) N does not divide Q, and + (b) D is square-free or not a perfect square, are included by + some authors; e.g., "Prime numbers and computer methods for + factorization," Hans Riesel (2nd ed., 1994, Birkhauser, Boston), + p. 130. For this particular application of Lucas sequences, + these conditions were found to be immaterial. */ + + /* Now calculate N - Jacobi(D,N) = N + 1 (even), and calculate the + odd positive integer d and positive integer s for which + N + 1 = 2^s*d (similar to the step for N - 1 in Miller's test). + The strong Lucas-Selfridge test then returns N as a strong + Lucas probable prime (slprp) if any of the following + conditions is met: U_d=0, V_d=0, V_2d=0, V_4d=0, V_8d=0, + V_16d=0, ..., etc., ending with V_{2^(s-1)*d}=V_{(N+1)/2}=0 + (all equalities mod N). Thus d is the highest index of U that + must be computed (since V_2m is independent of U), compared + to U_{N+1} for the standard Lucas-Selfridge test; and no + index of V beyond (N+1)/2 is required, just as in the + standard Lucas-Selfridge test. However, the quantity Q^d must + be computed for use (if necessary) in the latter stages of + the test. The result is that the strong Lucas-Selfridge test + has a running time only slightly greater (order of 10 %) than + that of the standard Lucas-Selfridge test, while producing + only (roughly) 30 % as many pseudoprimes (and every strong + Lucas pseudoprime is also a standard Lucas pseudoprime). Thus + the evidence indicates that the strong Lucas-Selfridge test is + more effective than the standard Lucas-Selfridge test, and a + Baillie-PSW test based on the strong Lucas-Selfridge test + should be more reliable. */ + + if ((err = mp_add_d(a, 1uL, &Np1)) != MP_OKAY) goto LBL_LS_ERR; + s = mp_cnt_lsb(&Np1); + + /* CZ + * This should round towards zero because + * Thomas R. Nicely used GMP's mpz_tdiv_q_2exp() + * and mp_div_2d() is equivalent. Additionally: + * dividing an even number by two does not produce + * any leftovers. + */ + if ((err = mp_div_2d(&Np1, s, &Dz, NULL)) != MP_OKAY) goto LBL_LS_ERR; + /* We must now compute U_d and V_d. Since d is odd, the accumulated + values U and V are initialized to U_1 and V_1 (if the target + index were even, U and V would be initialized instead to U_0=0 + and V_0=2). The values of U_2m and V_2m are also initialized to + U_1 and V_1; the FOR loop calculates in succession U_2 and V_2, + U_4 and V_4, U_8 and V_8, etc. If the corresponding bits + (1, 2, 3, ...) of t are on (the zero bit having been accounted + for in the initialization of U and V), these values are then + combined with the previous totals for U and V, using the + composition formulas for addition of indices. */ + + mp_set(&Uz, 1uL); /* U=U_1 */ + mp_set(&Vz, (mp_digit)P); /* V=V_1 */ + mp_set(&U2mz, 1uL); /* U_1 */ + mp_set(&V2mz, (mp_digit)P); /* V_1 */ + + mp_set_i32(&Qmz, Q); + if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR; + /* Initializes calculation of Q^d */ + mp_set_i32(&Qkdz, Q); + + Nbits = mp_count_bits(&Dz); + + for (u = 1; u < Nbits; u++) { /* zero bit off, already accounted for */ + /* Formulas for doubling of indices (carried out mod N). Note that + * the indices denoted as "2m" are actually powers of 2, specifically + * 2^(ul-1) beginning each loop and 2^ul ending each loop. + * + * U_2m = U_m*V_m + * V_2m = V_m*V_m - 2*Q^m + */ + + if ((err = mp_mul(&U2mz, &V2mz, &U2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&U2mz, a, &U2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sqr(&V2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sub(&V2mz, &Q2mz, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&V2mz, a, &V2mz)) != MP_OKAY) goto LBL_LS_ERR; + + /* Must calculate powers of Q for use in V_2m, also for Q^d later */ + if ((err = mp_sqr(&Qmz, &Qmz)) != MP_OKAY) goto LBL_LS_ERR; + + /* prevents overflow */ /* CZ still necessary without a fixed prealloc'd mem.? */ + if ((err = mp_mod(&Qmz, a, &Qmz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul_2(&Qmz, &Q2mz)) != MP_OKAY) goto LBL_LS_ERR; + + if (s_mp_get_bit(&Dz, (unsigned int)u) == MP_YES) { + /* Formulas for addition of indices (carried out mod N); + * + * U_(m+n) = (U_m*V_n + U_n*V_m)/2 + * V_(m+n) = (V_m*V_n + D*U_m*U_n)/2 + * + * Be careful with division by 2 (mod N)! + */ + if ((err = mp_mul(&U2mz, &Vz, &T1z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&Uz, &V2mz, &T2z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&V2mz, &Vz, &T3z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul(&U2mz, &Uz, &T4z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = s_mp_mul_si(&T4z, Ds, &T4z)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_add(&T1z, &T2z, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ODD(&Uz)) { + if ((err = mp_add(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + } + /* CZ + * This should round towards negative infinity because + * Thomas R. Nicely used GMP's mpz_fdiv_q_2exp(). + * But mp_div_2() does not do so, it is truncating instead. + */ + oddness = MP_IS_ODD(&Uz) ? MP_YES : MP_NO; + if ((err = mp_div_2(&Uz, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if ((Uz.sign == MP_NEG) && (oddness != MP_NO)) { + if ((err = mp_sub_d(&Uz, 1uL, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + } + if ((err = mp_add(&T3z, &T4z, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ODD(&Vz)) { + if ((err = mp_add(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + } + oddness = MP_IS_ODD(&Vz) ? MP_YES : MP_NO; + if ((err = mp_div_2(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((Vz.sign == MP_NEG) && (oddness != MP_NO)) { + if ((err = mp_sub_d(&Vz, 1uL, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + } + if ((err = mp_mod(&Uz, a, &Uz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + + /* Calculating Q^d for later use */ + if ((err = mp_mul(&Qkdz, &Qmz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + } + } + + /* If U_d or V_d is congruent to 0 mod N, then N is a prime or a + strong Lucas pseudoprime. */ + if (MP_IS_ZERO(&Uz) || MP_IS_ZERO(&Vz)) { + *result = MP_YES; + goto LBL_LS_ERR; + } + + /* NOTE: Ribenboim ("The new book of prime number records," 3rd ed., + 1995/6) omits the condition V0 on p.142, but includes it on + p. 130. The condition is NECESSARY; otherwise the test will + return false negatives---e.g., the primes 29 and 2000029 will be + returned as composite. */ + + /* Otherwise, we must compute V_2d, V_4d, V_8d, ..., V_{2^(s-1)*d} + by repeated use of the formula V_2m = V_m*V_m - 2*Q^m. If any of + these are congruent to 0 mod N, then N is a prime or a strong + Lucas pseudoprime. */ + + /* Initialize 2*Q^(d*2^r) for V_2m */ + if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR; + + for (r = 1; r < s; r++) { + if ((err = mp_sqr(&Vz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_sub(&Vz, &Q2kdz, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Vz, a, &Vz)) != MP_OKAY) goto LBL_LS_ERR; + if (MP_IS_ZERO(&Vz)) { + *result = MP_YES; + goto LBL_LS_ERR; + } + /* Calculate Q^{d*2^r} for next r (final iteration irrelevant). */ + if (r < (s - 1)) { + if ((err = mp_sqr(&Qkdz, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mod(&Qkdz, a, &Qkdz)) != MP_OKAY) goto LBL_LS_ERR; + if ((err = mp_mul_2(&Qkdz, &Q2kdz)) != MP_OKAY) goto LBL_LS_ERR; + } + } +LBL_LS_ERR: + mp_clear_multi(&Q2kdz, &T4z, &T3z, &T2z, &T1z, &Qkdz, &Q2mz, &Qmz, &V2mz, &U2mz, &Vz, &Uz, &Np1, &gcd, &Dz, NULL); + return err; +} +#endif +#endif +#endif + +/* End: bn_mp_prime_strong_lucas_selfridge.c */ + +/* Start: bn_mp_radix_size.c */ +#include "tommath_private.h" +#ifdef BN_MP_RADIX_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* returns size of ASCII representation */ +mp_err mp_radix_size(const mp_int *a, int radix, int *size) +{ + mp_err err; + int digs; + mp_int t; + mp_digit d; + + *size = 0; + + /* make sure the radix is in range */ + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } + + if (MP_IS_ZERO(a)) { + *size = 2; + return MP_OKAY; + } + + /* special case for binary */ + if (radix == 2) { + *size = (mp_count_bits(a) + ((a->sign == MP_NEG) ? 1 : 0) + 1); + return MP_OKAY; + } + + /* digs is the digit count */ + digs = 0; + + /* if it's negative add one for the sign */ + if (a->sign == MP_NEG) { + ++digs; + } + + /* init a copy of the input */ + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } + + /* force temp to positive */ + t.sign = MP_ZPOS; + + /* fetch out all of the digits */ + while (!MP_IS_ZERO(&t)) { + if ((err = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) { + goto LBL_ERR; + } + ++digs; + } + + /* return digs + 1, the 1 is for the NULL byte that would be required. */ + *size = digs + 1; + err = MP_OKAY; + +LBL_ERR: + mp_clear(&t); + return err; +} + +#endif + +/* End: bn_mp_radix_size.c */ + +/* Start: bn_mp_radix_smap.c */ +#include "tommath_private.h" +#ifdef BN_MP_RADIX_SMAP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* chars used in radix conversions */ +const char *const mp_s_rmap = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz+/"; +const uint8_t mp_s_rmap_reverse[] = { + 0xff, 0xff, 0xff, 0x3e, 0xff, 0xff, 0xff, 0x3f, /* ()*+,-./ */ + 0x00, 0x01, 0x02, 0x03, 0x04, 0x05, 0x06, 0x07, /* 01234567 */ + 0x08, 0x09, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* 89:;<=>? */ + 0xff, 0x0a, 0x0b, 0x0c, 0x0d, 0x0e, 0x0f, 0x10, /* @ABCDEFG */ + 0x11, 0x12, 0x13, 0x14, 0x15, 0x16, 0x17, 0x18, /* HIJKLMNO */ + 0x19, 0x1a, 0x1b, 0x1c, 0x1d, 0x1e, 0x1f, 0x20, /* PQRSTUVW */ + 0x21, 0x22, 0x23, 0xff, 0xff, 0xff, 0xff, 0xff, /* XYZ[\]^_ */ + 0xff, 0x24, 0x25, 0x26, 0x27, 0x28, 0x29, 0x2a, /* `abcdefg */ + 0x2b, 0x2c, 0x2d, 0x2e, 0x2f, 0x30, 0x31, 0x32, /* hijklmno */ + 0x33, 0x34, 0x35, 0x36, 0x37, 0x38, 0x39, 0x3a, /* pqrstuvw */ + 0x3b, 0x3c, 0x3d, 0xff, 0xff, 0xff, 0xff, 0xff, /* xyz{|}~. */ +}; +const size_t mp_s_rmap_reverse_sz = sizeof(mp_s_rmap_reverse); +#endif + +/* End: bn_mp_radix_smap.c */ + +/* Start: bn_mp_rand.c */ +#include "tommath_private.h" +#ifdef BN_MP_RAND_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +mp_err(*s_mp_rand_source)(void *out, size_t size) = s_mp_rand_platform; + +void mp_rand_source(mp_err(*source)(void *out, size_t size)) +{ + s_mp_rand_source = (source == NULL) ? s_mp_rand_platform : source; +} + +mp_err mp_rand(mp_int *a, int digits) +{ + int i; + mp_err err; + + mp_zero(a); + + if (digits <= 0) { + return MP_OKAY; + } + + if ((err = mp_grow(a, digits)) != MP_OKAY) { + return err; + } + + if ((err = s_mp_rand_source(a->dp, (size_t)digits * sizeof(mp_digit))) != MP_OKAY) { + return err; + } + + /* TODO: We ensure that the highest digit is nonzero. Should this be removed? */ + while ((a->dp[digits - 1] & MP_MASK) == 0u) { + if ((err = s_mp_rand_source(a->dp + digits - 1, sizeof(mp_digit))) != MP_OKAY) { + return err; + } + } + + a->used = digits; + for (i = 0; i < digits; ++i) { + a->dp[i] &= MP_MASK; + } + + return MP_OKAY; +} +#endif + +/* End: bn_mp_rand.c */ + +/* Start: bn_mp_read_radix.c */ +#include "tommath_private.h" +#ifdef BN_MP_READ_RADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#define MP_TOUPPER(c) ((((c) >= 'a') && ((c) <= 'z')) ? (((c) + 'A') - 'a') : (c)) + +/* read a string [ASCII] in a given radix */ +mp_err mp_read_radix(mp_int *a, const char *str, int radix) +{ + mp_err err; + int y; + mp_sign neg; + unsigned pos; + char ch; + + /* zero the digit bignum */ + mp_zero(a); + + /* make sure the radix is ok */ + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } + + /* if the leading digit is a + * minus set the sign to negative. + */ + if (*str == '-') { + ++str; + neg = MP_NEG; + } else { + neg = MP_ZPOS; + } + + /* set the integer to the default of zero */ + mp_zero(a); + + /* process each digit of the string */ + while (*str != '\0') { + /* if the radix <= 36 the conversion is case insensitive + * this allows numbers like 1AB and 1ab to represent the same value + * [e.g. in hex] + */ + ch = (radix <= 36) ? (char)MP_TOUPPER((int)*str) : *str; + pos = (unsigned)(ch - '('); + if (mp_s_rmap_reverse_sz < pos) { + break; + } + y = (int)mp_s_rmap_reverse[pos]; + + /* if the char was found in the map + * and is less than the given radix add it + * to the number, otherwise exit the loop. + */ + if ((y == 0xff) || (y >= radix)) { + break; + } + if ((err = mp_mul_d(a, (mp_digit)radix, a)) != MP_OKAY) { + return err; + } + if ((err = mp_add_d(a, (mp_digit)y, a)) != MP_OKAY) { + return err; + } + ++str; + } + + /* if an illegal character was found, fail. */ + if (!((*str == '\0') || (*str == '\r') || (*str == '\n'))) { + mp_zero(a); + return MP_VAL; + } + + /* set the sign only if a != 0 */ + if (!MP_IS_ZERO(a)) { + a->sign = neg; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_read_radix.c */ + +/* Start: bn_mp_reduce.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reduces x mod m, assumes 0 < x < m**2, mu is + * precomputed via mp_reduce_setup. + * From HAC pp.604 Algorithm 14.42 + */ +mp_err mp_reduce(mp_int *x, const mp_int *m, const mp_int *mu) +{ + mp_int q; + mp_err err; + int um = m->used; + + /* q = x */ + if ((err = mp_init_copy(&q, x)) != MP_OKAY) { + return err; + } + + /* q1 = x / b**(k-1) */ + mp_rshd(&q, um - 1); + + /* according to HAC this optimization is ok */ + if ((mp_digit)um > ((mp_digit)1 << (MP_DIGIT_BIT - 1))) { + if ((err = mp_mul(&q, mu, &q)) != MP_OKAY) { + goto CLEANUP; + } + } else if (MP_HAS(S_MP_MUL_HIGH_DIGS)) { + if ((err = s_mp_mul_high_digs(&q, mu, &q, um)) != MP_OKAY) { + goto CLEANUP; + } + } else if (MP_HAS(S_MP_MUL_HIGH_DIGS_FAST)) { + if ((err = s_mp_mul_high_digs_fast(&q, mu, &q, um)) != MP_OKAY) { + goto CLEANUP; + } + } else { + err = MP_VAL; + goto CLEANUP; + } + + /* q3 = q2 / b**(k+1) */ + mp_rshd(&q, um + 1); + + /* x = x mod b**(k+1), quick (no division) */ + if ((err = mp_mod_2d(x, MP_DIGIT_BIT * (um + 1), x)) != MP_OKAY) { + goto CLEANUP; + } + + /* q = q * m mod b**(k+1), quick (no division) */ + if ((err = s_mp_mul_digs(&q, m, &q, um + 1)) != MP_OKAY) { + goto CLEANUP; + } + + /* x = x - q */ + if ((err = mp_sub(x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + + /* If x < 0, add b**(k+1) to it */ + if (mp_cmp_d(x, 0uL) == MP_LT) { + mp_set(&q, 1uL); + if ((err = mp_lshd(&q, um + 1)) != MP_OKAY) { + goto CLEANUP; + } + if ((err = mp_add(x, &q, x)) != MP_OKAY) { + goto CLEANUP; + } + } + + /* Back off if it's too big */ + while (mp_cmp(x, m) != MP_LT) { + if ((err = s_mp_sub(x, m, x)) != MP_OKAY) { + goto CLEANUP; + } + } + +CLEANUP: + mp_clear(&q); + + return err; +} +#endif + +/* End: bn_mp_reduce.c */ + +/* Start: bn_mp_reduce_2k.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reduces a modulo n where n is of the form 2**p - d */ +mp_err mp_reduce_2k(mp_int *a, const mp_int *n, mp_digit d) +{ + mp_int q; + mp_err err; + int p; + + if ((err = mp_init(&q)) != MP_OKAY) { + return err; + } + + p = mp_count_bits(n); +top: + /* q = a/2**p, a = a mod 2**p */ + if ((err = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto LBL_ERR; + } + + if (d != 1u) { + /* q = q * d */ + if ((err = mp_mul_d(&q, d, &q)) != MP_OKAY) { + goto LBL_ERR; + } + } + + /* a = a + q */ + if ((err = s_mp_add(a, &q, a)) != MP_OKAY) { + goto LBL_ERR; + } + + if (mp_cmp_mag(a, n) != MP_LT) { + if ((err = s_mp_sub(a, n, a)) != MP_OKAY) { + goto LBL_ERR; + } + goto top; + } + +LBL_ERR: + mp_clear(&q); + return err; +} + +#endif + +/* End: bn_mp_reduce_2k.c */ + +/* Start: bn_mp_reduce_2k_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_2K_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reduces a modulo n where n is of the form 2**p - d + This differs from reduce_2k since "d" can be larger + than a single digit. +*/ +mp_err mp_reduce_2k_l(mp_int *a, const mp_int *n, const mp_int *d) +{ + mp_int q; + mp_err err; + int p; + + if ((err = mp_init(&q)) != MP_OKAY) { + return err; + } + + p = mp_count_bits(n); +top: + /* q = a/2**p, a = a mod 2**p */ + if ((err = mp_div_2d(a, p, &q, a)) != MP_OKAY) { + goto LBL_ERR; + } + + /* q = q * d */ + if ((err = mp_mul(&q, d, &q)) != MP_OKAY) { + goto LBL_ERR; + } + + /* a = a + q */ + if ((err = s_mp_add(a, &q, a)) != MP_OKAY) { + goto LBL_ERR; + } + + if (mp_cmp_mag(a, n) != MP_LT) { + if ((err = s_mp_sub(a, n, a)) != MP_OKAY) { + goto LBL_ERR; + } + goto top; + } + +LBL_ERR: + mp_clear(&q); + return err; +} + +#endif + +/* End: bn_mp_reduce_2k_l.c */ + +/* Start: bn_mp_reduce_2k_setup.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_2K_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines the setup value */ +mp_err mp_reduce_2k_setup(const mp_int *a, mp_digit *d) +{ + mp_err err; + mp_int tmp; + int p; + + if ((err = mp_init(&tmp)) != MP_OKAY) { + return err; + } + + p = mp_count_bits(a); + if ((err = mp_2expt(&tmp, p)) != MP_OKAY) { + mp_clear(&tmp); + return err; + } + + if ((err = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) { + mp_clear(&tmp); + return err; + } + + *d = tmp.dp[0]; + mp_clear(&tmp); + return MP_OKAY; +} +#endif + +/* End: bn_mp_reduce_2k_setup.c */ + +/* Start: bn_mp_reduce_2k_setup_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_2K_SETUP_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines the setup value */ +mp_err mp_reduce_2k_setup_l(const mp_int *a, mp_int *d) +{ + mp_err err; + mp_int tmp; + + if ((err = mp_init(&tmp)) != MP_OKAY) { + return err; + } + + if ((err = mp_2expt(&tmp, mp_count_bits(a))) != MP_OKAY) { + goto LBL_ERR; + } + + if ((err = s_mp_sub(&tmp, a, d)) != MP_OKAY) { + goto LBL_ERR; + } + +LBL_ERR: + mp_clear(&tmp); + return err; +} +#endif + +/* End: bn_mp_reduce_2k_setup_l.c */ + +/* Start: bn_mp_reduce_is_2k.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_IS_2K_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines if mp_reduce_2k can be used */ +mp_bool mp_reduce_is_2k(const mp_int *a) +{ + int ix, iy, iw; + mp_digit iz; + + if (a->used == 0) { + return MP_NO; + } else if (a->used == 1) { + return MP_YES; + } else if (a->used > 1) { + iy = mp_count_bits(a); + iz = 1; + iw = 1; + + /* Test every bit from the second digit up, must be 1 */ + for (ix = MP_DIGIT_BIT; ix < iy; ix++) { + if ((a->dp[iw] & iz) == 0u) { + return MP_NO; + } + iz <<= 1; + if (iz > MP_DIGIT_MAX) { + ++iw; + iz = 1; + } + } + return MP_YES; + } else { + return MP_YES; + } +} + +#endif + +/* End: bn_mp_reduce_is_2k.c */ + +/* Start: bn_mp_reduce_is_2k_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_IS_2K_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines if reduce_2k_l can be used */ +mp_bool mp_reduce_is_2k_l(const mp_int *a) +{ + int ix, iy; + + if (a->used == 0) { + return MP_NO; + } else if (a->used == 1) { + return MP_YES; + } else if (a->used > 1) { + /* if more than half of the digits are -1 we're sold */ + for (iy = ix = 0; ix < a->used; ix++) { + if (a->dp[ix] == MP_DIGIT_MAX) { + ++iy; + } + } + return (iy >= (a->used/2)) ? MP_YES : MP_NO; + } else { + return MP_NO; + } +} + +#endif + +/* End: bn_mp_reduce_is_2k_l.c */ + +/* Start: bn_mp_reduce_setup.c */ +#include "tommath_private.h" +#ifdef BN_MP_REDUCE_SETUP_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* pre-calculate the value required for Barrett reduction + * For a given modulus "b" it calulates the value required in "a" + */ +mp_err mp_reduce_setup(mp_int *a, const mp_int *b) +{ + mp_err err; + if ((err = mp_2expt(a, b->used * 2 * MP_DIGIT_BIT)) != MP_OKAY) { + return err; + } + return mp_div(a, b, a, NULL); +} +#endif + +/* End: bn_mp_reduce_setup.c */ + +/* Start: bn_mp_root_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_ROOT_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* find the n'th root of an integer + * + * Result found such that (c)**b <= a and (c+1)**b > a + * + * This algorithm uses Newton's approximation + * x[i+1] = x[i] - f(x[i])/f'(x[i]) + * which will find the root in log(N) time where + * each step involves a fair bit. + */ +mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c) +{ + mp_int t1, t2, t3, a_; + mp_ord cmp; + int ilog2; + mp_err err; + + /* input must be positive if b is even */ + if (((b & 1u) == 0u) && (a->sign == MP_NEG)) { + return MP_VAL; + } + + if ((err = mp_init_multi(&t1, &t2, &t3, NULL)) != MP_OKAY) { + return err; + } + + /* if a is negative fudge the sign but keep track */ + a_ = *a; + a_.sign = MP_ZPOS; + + /* Compute seed: 2^(log_2(n)/b + 2)*/ + ilog2 = mp_count_bits(a); + + /* + If "b" is larger than INT_MAX it is also larger than + log_2(n) because the bit-length of the "n" is measured + with an int and hence the root is always < 2 (two). + */ + if (b > (uint32_t)(INT_MAX/2)) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + + /* "b" is smaller than INT_MAX, we can cast safely */ + if (ilog2 < (int)b) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + ilog2 = ilog2 / ((int)b); + if (ilog2 == 0) { + mp_set(c, 1uL); + c->sign = a->sign; + err = MP_OKAY; + goto LBL_ERR; + } + /* Start value must be larger than root */ + ilog2 += 2; + if ((err = mp_2expt(&t2,ilog2)) != MP_OKAY) goto LBL_ERR; + do { + /* t1 = t2 */ + if ((err = mp_copy(&t2, &t1)) != MP_OKAY) goto LBL_ERR; + + /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */ + + /* t3 = t1**(b-1) */ + if ((err = mp_expt_u32(&t1, b - 1u, &t3)) != MP_OKAY) goto LBL_ERR; + + /* numerator */ + /* t2 = t1**b */ + if ((err = mp_mul(&t3, &t1, &t2)) != MP_OKAY) goto LBL_ERR; + + /* t2 = t1**b - a */ + if ((err = mp_sub(&t2, &a_, &t2)) != MP_OKAY) goto LBL_ERR; + + /* denominator */ + /* t3 = t1**(b-1) * b */ + if ((err = mp_mul_d(&t3, b, &t3)) != MP_OKAY) goto LBL_ERR; + + /* t3 = (t1**b - a)/(b * t1**(b-1)) */ + if ((err = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&t1, &t3, &t2)) != MP_OKAY) goto LBL_ERR; + + /* + Number of rounds is at most log_2(root). If it is more it + got stuck, so break out of the loop and do the rest manually. + */ + if (ilog2-- == 0) { + break; + } + } while (mp_cmp(&t1, &t2) != MP_EQ); + + /* result can be off by a few so check */ + /* Loop beneath can overshoot by one if found root is smaller than actual root */ + for (;;) { + if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR; + cmp = mp_cmp(&t2, &a_); + if (cmp == MP_EQ) { + err = MP_OKAY; + goto LBL_ERR; + } + if (cmp == MP_LT) { + if ((err = mp_add_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR; + } else { + break; + } + } + /* correct overshoot from above or from recurrence */ + for (;;) { + if ((err = mp_expt_u32(&t1, b, &t2)) != MP_OKAY) goto LBL_ERR; + if (mp_cmp(&t2, &a_) == MP_GT) { + if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto LBL_ERR; + } else { + break; + } + } + + /* set the result */ + mp_exch(&t1, c); + + /* set the sign of the result */ + c->sign = a->sign; + + err = MP_OKAY; + +LBL_ERR: + mp_clear_multi(&t1, &t2, &t3, NULL); + return err; +} + +#endif + +/* End: bn_mp_root_u32.c */ + +/* Start: bn_mp_rshd.c */ +#include "tommath_private.h" +#ifdef BN_MP_RSHD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift right a certain amount of digits */ +void mp_rshd(mp_int *a, int b) +{ + int x; + mp_digit *bottom, *top; + + /* if b <= 0 then ignore it */ + if (b <= 0) { + return; + } + + /* if b > used then simply zero it and return */ + if (a->used <= b) { + mp_zero(a); + return; + } + + /* shift the digits down */ + + /* bottom */ + bottom = a->dp; + + /* top [offset into digits] */ + top = a->dp + b; + + /* this is implemented as a sliding window where + * the window is b-digits long and digits from + * the top of the window are copied to the bottom + * + * e.g. + + b-2 | b-1 | b0 | b1 | b2 | ... | bb | ----> + /\ | ----> + \-------------------/ ----> + */ + for (x = 0; x < (a->used - b); x++) { + *bottom++ = *top++; + } + + /* zero the top digits */ + MP_ZERO_DIGITS(bottom, a->used - x); + + /* remove excess digits */ + a->used -= b; +} +#endif + +/* End: bn_mp_rshd.c */ + +/* Start: bn_mp_sbin_size.c */ +#include "tommath_private.h" +#ifdef BN_MP_SBIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* get the size for an signed equivalent */ +size_t mp_sbin_size(const mp_int *a) +{ + return 1u + mp_ubin_size(a); +} +#endif + +/* End: bn_mp_sbin_size.c */ + +/* Start: bn_mp_set.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* set to a digit */ +void mp_set(mp_int *a, mp_digit b) +{ + a->dp[0] = b & MP_MASK; + a->sign = MP_ZPOS; + a->used = (a->dp[0] != 0u) ? 1 : 0; + MP_ZERO_DIGITS(a->dp + a->used, a->alloc - a->used); +} +#endif + +/* End: bn_mp_set.c */ + +/* Start: bn_mp_set_double.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_DOUBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#if defined(__STDC_IEC_559__) || defined(__GCC_IEC_559) +mp_err mp_set_double(mp_int *a, double b) +{ + uint64_t frac; + int exp; + mp_err err; + union { + double dbl; + uint64_t bits; + } cast; + cast.dbl = b; + + exp = (int)((unsigned)(cast.bits >> 52) & 0x7FFu); + frac = (cast.bits & ((1uLL << 52) - 1uLL)) | (1uLL << 52); + + if (exp == 0x7FF) { /* +-inf, NaN */ + return MP_VAL; + } + exp -= 1023 + 52; + + mp_set_u64(a, frac); + + err = (exp < 0) ? mp_div_2d(a, -exp, a, NULL) : mp_mul_2d(a, exp, a); + if (err != MP_OKAY) { + return err; + } + + if (((cast.bits >> 63) != 0uLL) && !MP_IS_ZERO(a)) { + a->sign = MP_NEG; + } + + return MP_OKAY; +} +#else +/* pragma message() not supported by several compilers (in mostly older but still used versions) */ +# ifdef _MSC_VER +# pragma message("mp_set_double implementation is only available on platforms with IEEE754 floating point format") +# else +# warning "mp_set_double implementation is only available on platforms with IEEE754 floating point format" +# endif +#endif +#endif + +/* End: bn_mp_set_double.c */ + +/* Start: bn_mp_set_i32.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_I32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_i32, mp_set_u32, int32_t, uint32_t) +#endif + +/* End: bn_mp_set_i32.c */ + +/* Start: bn_mp_set_i64.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_I64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_i64, mp_set_u64, int64_t, uint64_t) +#endif + +/* End: bn_mp_set_i64.c */ + +/* Start: bn_mp_set_l.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_L_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_l, mp_set_ul, long, unsigned long) +#endif + +/* End: bn_mp_set_l.c */ + +/* Start: bn_mp_set_ll.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_LL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_SIGNED(mp_set_ll, mp_set_ull, long long, unsigned long long) +#endif + +/* End: bn_mp_set_ll.c */ + +/* Start: bn_mp_set_u32.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_U32_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_u32, uint32_t) +#endif + +/* End: bn_mp_set_u32.c */ + +/* Start: bn_mp_set_u64.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_U64_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_u64, uint64_t) +#endif + +/* End: bn_mp_set_u64.c */ + +/* Start: bn_mp_set_ul.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_UL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_ul, unsigned long) +#endif + +/* End: bn_mp_set_ul.c */ + +/* Start: bn_mp_set_ull.c */ +#include "tommath_private.h" +#ifdef BN_MP_SET_ULL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +MP_SET_UNSIGNED(mp_set_ull, unsigned long long) +#endif + +/* End: bn_mp_set_ull.c */ + +/* Start: bn_mp_shrink.c */ +#include "tommath_private.h" +#ifdef BN_MP_SHRINK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shrink a bignum */ +mp_err mp_shrink(mp_int *a) +{ + mp_digit *tmp; + int alloc = MP_MAX(MP_MIN_PREC, a->used); + if (a->alloc != alloc) { + if ((tmp = (mp_digit *) MP_REALLOC(a->dp, + (size_t)a->alloc * sizeof(mp_digit), + (size_t)alloc * sizeof(mp_digit))) == NULL) { + return MP_MEM; + } + a->dp = tmp; + a->alloc = alloc; + } + return MP_OKAY; +} +#endif + +/* End: bn_mp_shrink.c */ + +/* Start: bn_mp_signed_rsh.c */ +#include "tommath_private.h" +#ifdef BN_MP_SIGNED_RSH_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* shift right by a certain bit count with sign extension */ +mp_err mp_signed_rsh(const mp_int *a, int b, mp_int *c) +{ + mp_err res; + if (a->sign == MP_ZPOS) { + return mp_div_2d(a, b, c, NULL); + } + + res = mp_add_d(a, 1uL, c); + if (res != MP_OKAY) { + return res; + } + + res = mp_div_2d(c, b, c, NULL); + return (res == MP_OKAY) ? mp_sub_d(c, 1uL, c) : res; +} +#endif + +/* End: bn_mp_signed_rsh.c */ + +/* Start: bn_mp_sqr.c */ +#include "tommath_private.h" +#ifdef BN_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes b = a*a */ +mp_err mp_sqr(const mp_int *a, mp_int *b) +{ + mp_err err; + if (MP_HAS(S_MP_TOOM_SQR) && /* use Toom-Cook? */ + (a->used >= MP_TOOM_SQR_CUTOFF)) { + err = s_mp_toom_sqr(a, b); + } else if (MP_HAS(S_MP_KARATSUBA_SQR) && /* Karatsuba? */ + (a->used >= MP_KARATSUBA_SQR_CUTOFF)) { + err = s_mp_karatsuba_sqr(a, b); + } else if (MP_HAS(S_MP_SQR_FAST) && /* can we use the fast comba multiplier? */ + (((a->used * 2) + 1) < MP_WARRAY) && + (a->used < (MP_MAXFAST / 2))) { + err = s_mp_sqr_fast(a, b); + } else if (MP_HAS(S_MP_SQR)) { + err = s_mp_sqr(a, b); + } else { + err = MP_VAL; + } + b->sign = MP_ZPOS; + return err; +} +#endif + +/* End: bn_mp_sqr.c */ + +/* Start: bn_mp_sqrmod.c */ +#include "tommath_private.h" +#ifdef BN_MP_SQRMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* c = a * a (mod b) */ +mp_err mp_sqrmod(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_err err; + mp_int t; + + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + if ((err = mp_sqr(a, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, b, c); + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_sqrmod.c */ + +/* Start: bn_mp_sqrt.c */ +#include "tommath_private.h" +#ifdef BN_MP_SQRT_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* this function is less generic than mp_n_root, simpler and faster */ +mp_err mp_sqrt(const mp_int *arg, mp_int *ret) +{ + mp_err err; + mp_int t1, t2; + + /* must be positive */ + if (arg->sign == MP_NEG) { + return MP_VAL; + } + + /* easy out */ + if (MP_IS_ZERO(arg)) { + mp_zero(ret); + return MP_OKAY; + } + + if ((err = mp_init_copy(&t1, arg)) != MP_OKAY) { + return err; + } + + if ((err = mp_init(&t2)) != MP_OKAY) { + goto E2; + } + + /* First approx. (not very bad for large arg) */ + mp_rshd(&t1, t1.used/2); + + /* t1 > 0 */ + if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) { + goto E1; + } + if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) { + goto E1; + } + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) { + goto E1; + } + /* And now t1 > sqrt(arg) */ + do { + if ((err = mp_div(arg, &t1, &t2, NULL)) != MP_OKAY) { + goto E1; + } + if ((err = mp_add(&t1, &t2, &t1)) != MP_OKAY) { + goto E1; + } + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) { + goto E1; + } + /* t1 >= sqrt(arg) >= t2 at this point */ + } while (mp_cmp_mag(&t1, &t2) == MP_GT); + + mp_exch(&t1, ret); + +E1: + mp_clear(&t2); +E2: + mp_clear(&t1); + return err; +} + +#endif + +/* End: bn_mp_sqrt.c */ + +/* Start: bn_mp_sqrtmod_prime.c */ +#include "tommath_private.h" +#ifdef BN_MP_SQRTMOD_PRIME_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Tonelli-Shanks algorithm + * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm + * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html + * + */ + +mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret) +{ + mp_err err; + int legendre; + mp_int t1, C, Q, S, Z, M, T, R, two; + mp_digit i; + + /* first handle the simple cases */ + if (mp_cmp_d(n, 0uL) == MP_EQ) { + mp_zero(ret); + return MP_OKAY; + } + if (mp_cmp_d(prime, 2uL) == MP_EQ) return MP_VAL; /* prime must be odd */ + if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY) return err; + if (legendre == -1) return MP_VAL; /* quadratic non-residue mod prime */ + + if ((err = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) { + return err; + } + + /* SPECIAL CASE: if prime mod 4 == 3 + * compute directly: err = n^(prime+1)/4 mod prime + * Handbook of Applied Cryptography algorithm 3.36 + */ + if ((err = mp_mod_d(prime, 4uL, &i)) != MP_OKAY) goto cleanup; + if (i == 3u) { + if ((err = mp_add_d(prime, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup; + err = MP_OKAY; + goto cleanup; + } + + /* NOW: Tonelli-Shanks algorithm */ + + /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */ + if ((err = mp_copy(prime, &Q)) != MP_OKAY) goto cleanup; + if ((err = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY) goto cleanup; + /* Q = prime - 1 */ + mp_zero(&S); + /* S = 0 */ + while (MP_IS_EVEN(&Q)) { + if ((err = mp_div_2(&Q, &Q)) != MP_OKAY) goto cleanup; + /* Q = Q / 2 */ + if ((err = mp_add_d(&S, 1uL, &S)) != MP_OKAY) goto cleanup; + /* S = S + 1 */ + } + + /* find a Z such that the Legendre symbol (Z|prime) == -1 */ + mp_set_u32(&Z, 2u); + /* Z = 2 */ + for (;;) { + if ((err = mp_kronecker(&Z, prime, &legendre)) != MP_OKAY) goto cleanup; + if (legendre == -1) break; + if ((err = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY) goto cleanup; + /* Z = Z + 1 */ + } + + if ((err = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY) goto cleanup; + /* C = Z ^ Q mod prime */ + if ((err = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + /* t1 = (Q + 1) / 2 */ + if ((err = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = n ^ ((Q + 1) / 2) mod prime */ + if ((err = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY) goto cleanup; + /* T = n ^ Q mod prime */ + if ((err = mp_copy(&S, &M)) != MP_OKAY) goto cleanup; + /* M = S */ + mp_set_u32(&two, 2u); + + for (;;) { + if ((err = mp_copy(&T, &t1)) != MP_OKAY) goto cleanup; + i = 0; + for (;;) { + if (mp_cmp_d(&t1, 1uL) == MP_EQ) break; + if ((err = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup; + i++; + } + if (i == 0u) { + if ((err = mp_copy(&R, ret)) != MP_OKAY) goto cleanup; + err = MP_OKAY; + goto cleanup; + } + if ((err = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto cleanup; + if ((err = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = 2 ^ (M - i - 1) */ + if ((err = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */ + if ((err = mp_sqrmod(&t1, prime, &C)) != MP_OKAY) goto cleanup; + /* C = (t1 * t1) mod prime */ + if ((err = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = (R * t1) mod prime */ + if ((err = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup; + /* T = (T * C) mod prime */ + mp_set(&M, i); + /* M = i */ + } + +cleanup: + mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL); + return err; +} + +#endif + +/* End: bn_mp_sqrtmod_prime.c */ + +/* Start: bn_mp_sub.c */ +#include "tommath_private.h" +#ifdef BN_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* high level subtraction (handles signs) */ +mp_err mp_sub(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_sign sa = a->sign, sb = b->sign; + mp_err err; + + if (sa != sb) { + /* subtract a negative from a positive, OR */ + /* subtract a positive from a negative. */ + /* In either case, ADD their magnitudes, */ + /* and use the sign of the first number. */ + c->sign = sa; + err = s_mp_add(a, b, c); + } else { + /* subtract a positive from a positive, OR */ + /* subtract a negative from a negative. */ + /* First, take the difference between their */ + /* magnitudes, then... */ + if (mp_cmp_mag(a, b) != MP_LT) { + /* Copy the sign from the first */ + c->sign = sa; + /* The first has a larger or equal magnitude */ + err = s_mp_sub(a, b, c); + } else { + /* The result has the *opposite* sign from */ + /* the first number. */ + c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS; + /* The second has a larger magnitude */ + err = s_mp_sub(b, a, c); + } + } + return err; +} + +#endif + +/* End: bn_mp_sub.c */ + +/* Start: bn_mp_sub_d.c */ +#include "tommath_private.h" +#ifdef BN_MP_SUB_D_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* single digit subtraction */ +mp_err mp_sub_d(const mp_int *a, mp_digit b, mp_int *c) +{ + mp_digit *tmpa, *tmpc; + mp_err err; + int ix, oldused; + + /* grow c as required */ + if (c->alloc < (a->used + 1)) { + if ((err = mp_grow(c, a->used + 1)) != MP_OKAY) { + return err; + } + } + + /* if a is negative just do an unsigned + * addition [with fudged signs] + */ + if (a->sign == MP_NEG) { + mp_int a_ = *a; + a_.sign = MP_ZPOS; + err = mp_add_d(&a_, b, c); + c->sign = MP_NEG; + + /* clamp */ + mp_clamp(c); + + return err; + } + + /* setup regs */ + oldused = c->used; + tmpa = a->dp; + tmpc = c->dp; + + /* if a <= b simply fix the single digit */ + if (((a->used == 1) && (a->dp[0] <= b)) || (a->used == 0)) { + if (a->used == 1) { + *tmpc++ = b - *tmpa; + } else { + *tmpc++ = b; + } + ix = 1; + + /* negative/1digit */ + c->sign = MP_NEG; + c->used = 1; + } else { + mp_digit mu = b; + + /* positive/size */ + c->sign = MP_ZPOS; + c->used = a->used; + + /* subtract digits, mu is carry */ + for (ix = 0; ix < a->used; ix++) { + *tmpc = *tmpa++ - mu; + mu = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); + *tmpc++ &= MP_MASK; + } + } + + /* zero excess digits */ + MP_ZERO_DIGITS(tmpc, oldused - ix); + + mp_clamp(c); + return MP_OKAY; +} + +#endif + +/* End: bn_mp_sub_d.c */ + +/* Start: bn_mp_submod.c */ +#include "tommath_private.h" +#ifdef BN_MP_SUBMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* d = a - b (mod c) */ +mp_err mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) +{ + mp_err err; + mp_int t; + + if ((err = mp_init(&t)) != MP_OKAY) { + return err; + } + + if ((err = mp_sub(a, b, &t)) != MP_OKAY) { + goto LBL_ERR; + } + err = mp_mod(&t, c, d); + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_submod.c */ + +/* Start: bn_mp_to_radix.c */ +#include "tommath_private.h" +#ifdef BN_MP_TO_RADIX_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* stores a bignum as a ASCII string in a given radix (2..64) + * + * Stores upto "size - 1" chars and always a NULL byte, puts the number of characters + * written, including the '\0', in "written". + */ +mp_err mp_to_radix(const mp_int *a, char *str, size_t maxlen, size_t *written, int radix) +{ + size_t digs; + mp_err err; + mp_int t; + mp_digit d; + char *_s = str; + + /* check range of radix and size*/ + if (maxlen < 2u) { + return MP_BUF; + } + if ((radix < 2) || (radix > 64)) { + return MP_VAL; + } + + /* quick out if its zero */ + if (MP_IS_ZERO(a)) { + *str++ = '0'; + *str = '\0'; + if (written != NULL) { + *written = 2u; + } + return MP_OKAY; + } + + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } + + /* if it is negative output a - */ + if (t.sign == MP_NEG) { + /* we have to reverse our digits later... but not the - sign!! */ + ++_s; + + /* store the flag and mark the number as positive */ + *str++ = '-'; + t.sign = MP_ZPOS; + + /* subtract a char */ + --maxlen; + } + digs = 0u; + while (!MP_IS_ZERO(&t)) { + if (--maxlen < 1u) { + /* no more room */ + err = MP_BUF; + goto LBL_ERR; + } + if ((err = mp_div_d(&t, (mp_digit)radix, &t, &d)) != MP_OKAY) { + goto LBL_ERR; + } + *str++ = mp_s_rmap[d]; + ++digs; + } + /* reverse the digits of the string. In this case _s points + * to the first digit [exluding the sign] of the number + */ + s_mp_reverse((unsigned char *)_s, digs); + + /* append a NULL so the string is properly terminated */ + *str = '\0'; + digs++; + + if (written != NULL) { + *written = (a->sign == MP_NEG) ? (digs + 1u): digs; + } + +LBL_ERR: + mp_clear(&t); + return err; +} + +#endif + +/* End: bn_mp_to_radix.c */ + +/* Start: bn_mp_to_sbin.c */ +#include "tommath_private.h" +#ifdef BN_MP_TO_SBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* store in signed [big endian] format */ +mp_err mp_to_sbin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) +{ + mp_err err; + if (maxlen == 0u) { + return MP_BUF; + } + if ((err = mp_to_ubin(a, buf + 1, maxlen - 1u, written)) != MP_OKAY) { + return err; + } + if (written != NULL) { + (*written)++; + } + buf[0] = (a->sign == MP_ZPOS) ? (unsigned char)0 : (unsigned char)1; + return MP_OKAY; +} +#endif + +/* End: bn_mp_to_sbin.c */ + +/* Start: bn_mp_to_ubin.c */ +#include "tommath_private.h" +#ifdef BN_MP_TO_UBIN_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* store in unsigned [big endian] format */ +mp_err mp_to_ubin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) +{ + size_t x, count; + mp_err err; + mp_int t; + + count = mp_ubin_size(a); + if (count > maxlen) { + return MP_BUF; + } + + if ((err = mp_init_copy(&t, a)) != MP_OKAY) { + return err; + } + + for (x = count; x --> 0u;) { +#ifndef MP_8BIT + buf[x] = (unsigned char)(t.dp[0] & 255u); +#else + buf[x] = (unsigned char)(t.dp[0] | ((t.dp[1] & 1u) << 7)); +#endif + if ((err = mp_div_2d(&t, 8, &t, NULL)) != MP_OKAY) { + goto LBL_ERR; + } + } + + if (written != NULL) { + *written = count; + } + +LBL_ERR: + mp_clear(&t); + return err; +} +#endif + +/* End: bn_mp_to_ubin.c */ + +/* Start: bn_mp_ubin_size.c */ +#include "tommath_private.h" +#ifdef BN_MP_UBIN_SIZE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* get the size for an unsigned equivalent */ +size_t mp_ubin_size(const mp_int *a) +{ + size_t size = (size_t)mp_count_bits(a); + return (size / 8u) + (((size & 7u) != 0u) ? 1u : 0u); +} +#endif + +/* End: bn_mp_ubin_size.c */ + +/* Start: bn_mp_unpack.c */ +#include "tommath_private.h" +#ifdef BN_MP_UNPACK_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* based on gmp's mpz_import. + * see http://gmplib.org/manual/Integer-Import-and-Export.html + */ +mp_err mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, + mp_endian endian, size_t nails, const void *op) +{ + mp_err err; + size_t odd_nails, nail_bytes, i, j; + unsigned char odd_nail_mask; + + mp_zero(rop); + + if (endian == MP_NATIVE_ENDIAN) { + MP_GET_ENDIANNESS(endian); + } + + odd_nails = (nails % 8u); + odd_nail_mask = 0xff; + for (i = 0; i < odd_nails; ++i) { + odd_nail_mask ^= (unsigned char)(1u << (7u - i)); + } + nail_bytes = nails / 8u; + + for (i = 0; i < count; ++i) { + for (j = 0; j < (size - nail_bytes); ++j) { + unsigned char byte = *((const unsigned char *)op + + (((order == MP_MSB_FIRST) ? i : ((count - 1u) - i)) * size) + + ((endian == MP_BIG_ENDIAN) ? (j + nail_bytes) : (((size - 1u) - j) - nail_bytes))); + + if ((err = mp_mul_2d(rop, (j == 0u) ? (int)(8u - odd_nails) : 8, rop)) != MP_OKAY) { + return err; + } + + rop->dp[0] |= (j == 0u) ? (mp_digit)(byte & odd_nail_mask) : (mp_digit)byte; + rop->used += 1; + } + } + + mp_clamp(rop); + + return MP_OKAY; +} + +#endif + +/* End: bn_mp_unpack.c */ + +/* Start: bn_mp_xor.c */ +#include "tommath_private.h" +#ifdef BN_MP_XOR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* two complement xor */ +mp_err mp_xor(const mp_int *a, const mp_int *b, mp_int *c) +{ + int used = MP_MAX(a->used, b->used) + 1, i; + mp_err err; + mp_digit ac = 1, bc = 1, cc = 1; + mp_sign csign = (a->sign != b->sign) ? MP_NEG : MP_ZPOS; + + if (c->alloc < used) { + if ((err = mp_grow(c, used)) != MP_OKAY) { + return err; + } + } + + for (i = 0; i < used; i++) { + mp_digit x, y; + + /* convert to two complement if negative */ + if (a->sign == MP_NEG) { + ac += (i >= a->used) ? MP_MASK : (~a->dp[i] & MP_MASK); + x = ac & MP_MASK; + ac >>= MP_DIGIT_BIT; + } else { + x = (i >= a->used) ? 0uL : a->dp[i]; + } + + /* convert to two complement if negative */ + if (b->sign == MP_NEG) { + bc += (i >= b->used) ? MP_MASK : (~b->dp[i] & MP_MASK); + y = bc & MP_MASK; + bc >>= MP_DIGIT_BIT; + } else { + y = (i >= b->used) ? 0uL : b->dp[i]; + } + + c->dp[i] = x ^ y; + + /* convert to to sign-magnitude if negative */ + if (csign == MP_NEG) { + cc += ~c->dp[i] & MP_MASK; + c->dp[i] = cc & MP_MASK; + cc >>= MP_DIGIT_BIT; + } + } + + c->used = used; + c->sign = csign; + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_mp_xor.c */ + +/* Start: bn_mp_zero.c */ +#include "tommath_private.h" +#ifdef BN_MP_ZERO_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* set to zero */ +void mp_zero(mp_int *a) +{ + a->sign = MP_ZPOS; + a->used = 0; + MP_ZERO_DIGITS(a->dp, a->alloc); +} +#endif + +/* End: bn_mp_zero.c */ + +/* Start: bn_prime_tab.c */ +#include "tommath_private.h" +#ifdef BN_PRIME_TAB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +const mp_digit ltm_prime_tab[] = { + 0x0002, 0x0003, 0x0005, 0x0007, 0x000B, 0x000D, 0x0011, 0x0013, + 0x0017, 0x001D, 0x001F, 0x0025, 0x0029, 0x002B, 0x002F, 0x0035, + 0x003B, 0x003D, 0x0043, 0x0047, 0x0049, 0x004F, 0x0053, 0x0059, + 0x0061, 0x0065, 0x0067, 0x006B, 0x006D, 0x0071, 0x007F, +#ifndef MP_8BIT + 0x0083, + 0x0089, 0x008B, 0x0095, 0x0097, 0x009D, 0x00A3, 0x00A7, 0x00AD, + 0x00B3, 0x00B5, 0x00BF, 0x00C1, 0x00C5, 0x00C7, 0x00D3, 0x00DF, + 0x00E3, 0x00E5, 0x00E9, 0x00EF, 0x00F1, 0x00FB, 0x0101, 0x0107, + 0x010D, 0x010F, 0x0115, 0x0119, 0x011B, 0x0125, 0x0133, 0x0137, + + 0x0139, 0x013D, 0x014B, 0x0151, 0x015B, 0x015D, 0x0161, 0x0167, + 0x016F, 0x0175, 0x017B, 0x017F, 0x0185, 0x018D, 0x0191, 0x0199, + 0x01A3, 0x01A5, 0x01AF, 0x01B1, 0x01B7, 0x01BB, 0x01C1, 0x01C9, + 0x01CD, 0x01CF, 0x01D3, 0x01DF, 0x01E7, 0x01EB, 0x01F3, 0x01F7, + 0x01FD, 0x0209, 0x020B, 0x021D, 0x0223, 0x022D, 0x0233, 0x0239, + 0x023B, 0x0241, 0x024B, 0x0251, 0x0257, 0x0259, 0x025F, 0x0265, + 0x0269, 0x026B, 0x0277, 0x0281, 0x0283, 0x0287, 0x028D, 0x0293, + 0x0295, 0x02A1, 0x02A5, 0x02AB, 0x02B3, 0x02BD, 0x02C5, 0x02CF, + + 0x02D7, 0x02DD, 0x02E3, 0x02E7, 0x02EF, 0x02F5, 0x02F9, 0x0301, + 0x0305, 0x0313, 0x031D, 0x0329, 0x032B, 0x0335, 0x0337, 0x033B, + 0x033D, 0x0347, 0x0355, 0x0359, 0x035B, 0x035F, 0x036D, 0x0371, + 0x0373, 0x0377, 0x038B, 0x038F, 0x0397, 0x03A1, 0x03A9, 0x03AD, + 0x03B3, 0x03B9, 0x03C7, 0x03CB, 0x03D1, 0x03D7, 0x03DF, 0x03E5, + 0x03F1, 0x03F5, 0x03FB, 0x03FD, 0x0407, 0x0409, 0x040F, 0x0419, + 0x041B, 0x0425, 0x0427, 0x042D, 0x043F, 0x0443, 0x0445, 0x0449, + 0x044F, 0x0455, 0x045D, 0x0463, 0x0469, 0x047F, 0x0481, 0x048B, + + 0x0493, 0x049D, 0x04A3, 0x04A9, 0x04B1, 0x04BD, 0x04C1, 0x04C7, + 0x04CD, 0x04CF, 0x04D5, 0x04E1, 0x04EB, 0x04FD, 0x04FF, 0x0503, + 0x0509, 0x050B, 0x0511, 0x0515, 0x0517, 0x051B, 0x0527, 0x0529, + 0x052F, 0x0551, 0x0557, 0x055D, 0x0565, 0x0577, 0x0581, 0x058F, + 0x0593, 0x0595, 0x0599, 0x059F, 0x05A7, 0x05AB, 0x05AD, 0x05B3, + 0x05BF, 0x05C9, 0x05CB, 0x05CF, 0x05D1, 0x05D5, 0x05DB, 0x05E7, + 0x05F3, 0x05FB, 0x0607, 0x060D, 0x0611, 0x0617, 0x061F, 0x0623, + 0x062B, 0x062F, 0x063D, 0x0641, 0x0647, 0x0649, 0x064D, 0x0653 +#endif +}; + +#if defined(__GNUC__) && __GNUC__ >= 4 +#pragma GCC diagnostic push +#pragma GCC diagnostic ignored "-Wdeprecated-declarations" +const mp_digit *s_mp_prime_tab = ltm_prime_tab; +#pragma GCC diagnostic pop +#elif defined(_MSC_VER) && _MSC_VER >= 1500 +#pragma warning(push) +#pragma warning(disable: 4996) +const mp_digit *s_mp_prime_tab = ltm_prime_tab; +#pragma warning(pop) +#else +const mp_digit *s_mp_prime_tab = ltm_prime_tab; +#endif + +#endif + +/* End: bn_prime_tab.c */ + +/* Start: bn_s_mp_add.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_ADD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* low level addition, based on HAC pp.594, Algorithm 14.7 */ +mp_err s_mp_add(const mp_int *a, const mp_int *b, mp_int *c) +{ + const mp_int *x; + mp_err err; + int olduse, min, max; + + /* find sizes, we let |a| <= |b| which means we have to sort + * them. "x" will point to the input with the most digits + */ + if (a->used > b->used) { + min = b->used; + max = a->used; + x = a; + } else { + min = a->used; + max = b->used; + x = b; + } + + /* init result */ + if (c->alloc < (max + 1)) { + if ((err = mp_grow(c, max + 1)) != MP_OKAY) { + return err; + } + } + + /* get old used digit count and set new one */ + olduse = c->used; + c->used = max + 1; + + { + mp_digit u, *tmpa, *tmpb, *tmpc; + int i; + + /* alias for digit pointers */ + + /* first input */ + tmpa = a->dp; + + /* second input */ + tmpb = b->dp; + + /* destination */ + tmpc = c->dp; + + /* zero the carry */ + u = 0; + for (i = 0; i < min; i++) { + /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */ + *tmpc = *tmpa++ + *tmpb++ + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> (mp_digit)MP_DIGIT_BIT; + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, that is in A+B + * if A or B has more digits add those in + */ + if (min != max) { + for (; i < max; i++) { + /* T[i] = X[i] + U */ + *tmpc = x->dp[i] + u; + + /* U = carry bit of T[i] */ + u = *tmpc >> (mp_digit)MP_DIGIT_BIT; + + /* take away carry bit from T[i] */ + *tmpc++ &= MP_MASK; + } + } + + /* add carry */ + *tmpc++ = u; + + /* clear digits above oldused */ + MP_ZERO_DIGITS(tmpc, olduse - c->used); + } + + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_add.c */ + +/* Start: bn_s_mp_balance_mul.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_BALANCE_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* single-digit multiplication with the smaller number as the single-digit */ +mp_err s_mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + int count, len_a, len_b, nblocks, i, j, bsize; + mp_int a0, tmp, A, B, r; + mp_err err; + + len_a = a->used; + len_b = b->used; + + nblocks = MP_MAX(a->used, b->used) / MP_MIN(a->used, b->used); + bsize = MP_MIN(a->used, b->used) ; + + if ((err = mp_init_size(&a0, bsize + 2)) != MP_OKAY) { + return err; + } + if ((err = mp_init_multi(&tmp, &r, NULL)) != MP_OKAY) { + mp_clear(&a0); + return err; + } + + /* Make sure that A is the larger one*/ + if (len_a < len_b) { + B = *a; + A = *b; + } else { + A = *a; + B = *b; + } + + for (i = 0, j=0; i < nblocks; i++) { + /* Cut a slice off of a */ + a0.used = 0; + for (count = 0; count < bsize; count++) { + a0.dp[count] = A.dp[ j++ ]; + a0.used++; + } + mp_clamp(&a0); + /* Multiply with b */ + if ((err = mp_mul(&a0, &B, &tmp)) != MP_OKAY) { + goto LBL_ERR; + } + /* Shift tmp to the correct position */ + if ((err = mp_lshd(&tmp, bsize * i)) != MP_OKAY) { + goto LBL_ERR; + } + /* Add to output. No carry needed */ + if ((err = mp_add(&r, &tmp, &r)) != MP_OKAY) { + goto LBL_ERR; + } + } + /* The left-overs; there are always left-overs */ + if (j < A.used) { + a0.used = 0; + for (count = 0; j < A.used; count++) { + a0.dp[count] = A.dp[ j++ ]; + a0.used++; + } + mp_clamp(&a0); + if ((err = mp_mul(&a0, &B, &tmp)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_lshd(&tmp, bsize * i)) != MP_OKAY) { + goto LBL_ERR; + } + if ((err = mp_add(&r, &tmp, &r)) != MP_OKAY) { + goto LBL_ERR; + } + } + + mp_exch(&r,c); +LBL_ERR: + mp_clear_multi(&a0, &tmp, &r,NULL); + return err; +} +#endif + +/* End: bn_s_mp_balance_mul.c */ + +/* Start: bn_s_mp_exptmod.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_EXPTMOD_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifdef MP_LOW_MEM +# define TAB_SIZE 32 +# define MAX_WINSIZE 5 +#else +# define TAB_SIZE 256 +# define MAX_WINSIZE 0 +#endif + +mp_err s_mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) +{ + mp_int M[TAB_SIZE], res, mu; + mp_digit buf; + mp_err err; + int bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + mp_err(*redux)(mp_int *x, const mp_int *m, const mp_int *mu); + + /* find window size */ + x = mp_count_bits(X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + + winsize = MAX_WINSIZE ? MP_MIN(MAX_WINSIZE, winsize) : winsize; + + /* init M array */ + /* init first cell */ + if ((err = mp_init(&M[1])) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init(&M[x])) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear(&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* create mu, used for Barrett reduction */ + if ((err = mp_init(&mu)) != MP_OKAY) goto LBL_M; + + if (redmode == 0) { + if ((err = mp_reduce_setup(&mu, P)) != MP_OKAY) goto LBL_MU; + redux = mp_reduce; + } else { + if ((err = mp_reduce_2k_setup_l(P, &mu)) != MP_OKAY) goto LBL_MU; + redux = mp_reduce_2k_l; + } + + /* create M table + * + * The M table contains powers of the base, + * e.g. M[x] = G**x mod P + * + * The first half of the table is not + * computed though accept for M[0] and M[1] + */ + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) goto LBL_MU; + + /* compute the value at M[1<<(winsize-1)] by squaring + * M[1] (winsize-1) times + */ + if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_MU; + + for (x = 0; x < (winsize - 1); x++) { + /* square it */ + if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)], + &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_MU; + + /* reduce modulo P */ + if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, &mu)) != MP_OKAY) goto LBL_MU; + } + + /* create upper table, that is M[x] = M[x-1] * M[1] (mod P) + * for x = (2**(winsize - 1) + 1) to (2**winsize - 1) + */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) goto LBL_MU; + if ((err = redux(&M[x], P, &mu)) != MP_OKAY) goto LBL_MU; + } + + /* setup result */ + if ((err = mp_init(&res)) != MP_OKAY) goto LBL_MU; + mp_set(&res, 1uL); + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits */ + if (digidx == -1) { + break; + } + /* read next digit and reset the bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)MP_DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (buf >> (mp_digit)(MP_DIGIT_BIT - 1)) & 1uL; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if ((mode == 0) && (y == 0)) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if ((mode == 1) && (y == 0)) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + } + + /* then multiply */ + if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if ((mode == 2) && (bitcpy > 0)) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, &mu)) != MP_OKAY) goto LBL_RES; + } + } + } + + mp_exch(&res, Y); + err = MP_OKAY; +LBL_RES: + mp_clear(&res); +LBL_MU: + mp_clear(&mu); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear(&M[x]); + } + return err; +} +#endif + +/* End: bn_s_mp_exptmod.c */ + +/* Start: bn_s_mp_exptmod_fast.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_EXPTMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes Y == G**X mod P, HAC pp.616, Algorithm 14.85 + * + * Uses a left-to-right k-ary sliding window to compute the modular exponentiation. + * The value of k changes based on the size of the exponent. + * + * Uses Montgomery or Diminished Radix reduction [whichever appropriate] + */ + +#ifdef MP_LOW_MEM +# define TAB_SIZE 32 +# define MAX_WINSIZE 5 +#else +# define TAB_SIZE 256 +# define MAX_WINSIZE 0 +#endif + +mp_err s_mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) +{ + mp_int M[TAB_SIZE], res; + mp_digit buf, mp; + int bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize; + mp_err err; + + /* use a pointer to the reduction algorithm. This allows us to use + * one of many reduction algorithms without modding the guts of + * the code with if statements everywhere. + */ + mp_err(*redux)(mp_int *x, const mp_int *n, mp_digit rho); + + /* find window size */ + x = mp_count_bits(X); + if (x <= 7) { + winsize = 2; + } else if (x <= 36) { + winsize = 3; + } else if (x <= 140) { + winsize = 4; + } else if (x <= 450) { + winsize = 5; + } else if (x <= 1303) { + winsize = 6; + } else if (x <= 3529) { + winsize = 7; + } else { + winsize = 8; + } + + winsize = MAX_WINSIZE ? MP_MIN(MAX_WINSIZE, winsize) : winsize; + + /* init M array */ + /* init first cell */ + if ((err = mp_init_size(&M[1], P->alloc)) != MP_OKAY) { + return err; + } + + /* now init the second half of the array */ + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + if ((err = mp_init_size(&M[x], P->alloc)) != MP_OKAY) { + for (y = 1<<(winsize-1); y < x; y++) { + mp_clear(&M[y]); + } + mp_clear(&M[1]); + return err; + } + } + + /* determine and setup reduction code */ + if (redmode == 0) { + if (MP_HAS(MP_MONTGOMERY_SETUP)) { + /* now setup montgomery */ + if ((err = mp_montgomery_setup(P, &mp)) != MP_OKAY) goto LBL_M; + } else { + err = MP_VAL; + goto LBL_M; + } + + /* automatically pick the comba one if available (saves quite a few calls/ifs) */ + if (MP_HAS(S_MP_MONTGOMERY_REDUCE_FAST) && + (((P->used * 2) + 1) < MP_WARRAY) && + (P->used < MP_MAXFAST)) { + redux = s_mp_montgomery_reduce_fast; + } else if (MP_HAS(MP_MONTGOMERY_REDUCE)) { + /* use slower baseline Montgomery method */ + redux = mp_montgomery_reduce; + } else { + err = MP_VAL; + goto LBL_M; + } + } else if (redmode == 1) { + if (MP_HAS(MP_DR_SETUP) && MP_HAS(MP_DR_REDUCE)) { + /* setup DR reduction for moduli of the form B**k - b */ + mp_dr_setup(P, &mp); + redux = mp_dr_reduce; + } else { + err = MP_VAL; + goto LBL_M; + } + } else if (MP_HAS(MP_REDUCE_2K_SETUP) && MP_HAS(MP_REDUCE_2K)) { + /* setup DR reduction for moduli of the form 2**k - b */ + if ((err = mp_reduce_2k_setup(P, &mp)) != MP_OKAY) goto LBL_M; + redux = mp_reduce_2k; + } else { + err = MP_VAL; + goto LBL_M; + } + + /* setup result */ + if ((err = mp_init_size(&res, P->alloc)) != MP_OKAY) goto LBL_M; + + /* create M table + * + + * + * The first half of the table is not computed though accept for M[0] and M[1] + */ + + if (redmode == 0) { + if (MP_HAS(MP_MONTGOMERY_CALC_NORMALIZATION)) { + /* now we need R mod m */ + if ((err = mp_montgomery_calc_normalization(&res, P)) != MP_OKAY) goto LBL_RES; + + /* now set M[1] to G * R mod m */ + if ((err = mp_mulmod(G, &res, P, &M[1])) != MP_OKAY) goto LBL_RES; + } else { + err = MP_VAL; + goto LBL_RES; + } + } else { + mp_set(&res, 1uL); + if ((err = mp_mod(G, P, &M[1])) != MP_OKAY) goto LBL_RES; + } + + /* compute the value at M[1<<(winsize-1)] by squaring M[1] (winsize-1) times */ + if ((err = mp_copy(&M[1], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_RES; + + for (x = 0; x < (winsize - 1); x++) { + if ((err = mp_sqr(&M[(size_t)1 << (winsize - 1)], &M[(size_t)1 << (winsize - 1)])) != MP_OKAY) goto LBL_RES; + if ((err = redux(&M[(size_t)1 << (winsize - 1)], P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* create upper table */ + for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) { + if ((err = mp_mul(&M[x - 1], &M[1], &M[x])) != MP_OKAY) goto LBL_RES; + if ((err = redux(&M[x], P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* set initial mode and bit cnt */ + mode = 0; + bitcnt = 1; + buf = 0; + digidx = X->used - 1; + bitcpy = 0; + bitbuf = 0; + + for (;;) { + /* grab next digit as required */ + if (--bitcnt == 0) { + /* if digidx == -1 we are out of digits so break */ + if (digidx == -1) { + break; + } + /* read next digit and reset bitcnt */ + buf = X->dp[digidx--]; + bitcnt = (int)MP_DIGIT_BIT; + } + + /* grab the next msb from the exponent */ + y = (mp_digit)(buf >> (MP_DIGIT_BIT - 1)) & 1uL; + buf <<= (mp_digit)1; + + /* if the bit is zero and mode == 0 then we ignore it + * These represent the leading zero bits before the first 1 bit + * in the exponent. Technically this opt is not required but it + * does lower the # of trivial squaring/reductions used + */ + if ((mode == 0) && (y == 0)) { + continue; + } + + /* if the bit is zero and mode == 1 then we square */ + if ((mode == 1) && (y == 0)) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + continue; + } + + /* else we add it to the window */ + bitbuf |= (y << (winsize - ++bitcpy)); + mode = 2; + + if (bitcpy == winsize) { + /* ok window is filled so square as required and multiply */ + /* square first */ + for (x = 0; x < winsize; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* then multiply */ + if ((err = mp_mul(&res, &M[bitbuf], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + + /* empty window and reset */ + bitcpy = 0; + bitbuf = 0; + mode = 1; + } + } + + /* if bits remain then square/multiply */ + if ((mode == 2) && (bitcpy > 0)) { + /* square then multiply if the bit is set */ + for (x = 0; x < bitcpy; x++) { + if ((err = mp_sqr(&res, &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + + /* get next bit of the window */ + bitbuf <<= 1; + if ((bitbuf & (1 << winsize)) != 0) { + /* then multiply */ + if ((err = mp_mul(&res, &M[1], &res)) != MP_OKAY) goto LBL_RES; + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + } + } + + if (redmode == 0) { + /* fixup result if Montgomery reduction is used + * recall that any value in a Montgomery system is + * actually multiplied by R mod n. So we have + * to reduce one more time to cancel out the factor + * of R. + */ + if ((err = redux(&res, P, mp)) != MP_OKAY) goto LBL_RES; + } + + /* swap res with Y */ + mp_exch(&res, Y); + err = MP_OKAY; +LBL_RES: + mp_clear(&res); +LBL_M: + mp_clear(&M[1]); + for (x = 1<<(winsize-1); x < (1 << winsize); x++) { + mp_clear(&M[x]); + } + return err; +} +#endif + +/* End: bn_s_mp_exptmod_fast.c */ + +/* Start: bn_s_mp_get_bit.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_GET_BIT_C + +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Get bit at position b and return MP_YES if the bit is 1, MP_NO if it is 0 */ +mp_bool s_mp_get_bit(const mp_int *a, unsigned int b) +{ + mp_digit bit; + int limb = (int)(b / MP_DIGIT_BIT); + + if (limb >= a->used) { + return MP_NO; + } + + bit = (mp_digit)1 << (b % MP_DIGIT_BIT); + return ((a->dp[limb] & bit) != 0u) ? MP_YES : MP_NO; +} + +#endif + +/* End: bn_s_mp_get_bit.c */ + +/* Start: bn_s_mp_invmod_fast.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_INVMOD_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* computes the modular inverse via binary extended euclidean algorithm, + * that is c = 1/a mod b + * + * Based on slow invmod except this is optimized for the case where b is + * odd as per HAC Note 14.64 on pp. 610 + */ +mp_err s_mp_invmod_fast(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x, y, u, v, B, D; + mp_sign neg; + mp_err err; + + /* 2. [modified] b must be odd */ + if (MP_IS_EVEN(b)) { + return MP_VAL; + } + + /* init all our temps */ + if ((err = mp_init_multi(&x, &y, &u, &v, &B, &D, NULL)) != MP_OKAY) { + return err; + } + + /* x == modulus, y == value to invert */ + if ((err = mp_copy(b, &x)) != MP_OKAY) goto LBL_ERR; + + /* we need y = |a| */ + if ((err = mp_mod(a, b, &y)) != MP_OKAY) goto LBL_ERR; + + /* if one of x,y is zero return an error! */ + if (MP_IS_ZERO(&x) || MP_IS_ZERO(&y)) { + err = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((err = mp_copy(&x, &u)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&y, &v)) != MP_OKAY) goto LBL_ERR; + mp_set(&D, 1uL); + +top: + /* 4. while u is even do */ + while (MP_IS_EVEN(&u)) { + /* 4.1 u = u/2 */ + if ((err = mp_div_2(&u, &u)) != MP_OKAY) goto LBL_ERR; + + /* 4.2 if B is odd then */ + if (MP_IS_ODD(&B)) { + if ((err = mp_sub(&B, &x, &B)) != MP_OKAY) goto LBL_ERR; + } + /* B = B/2 */ + if ((err = mp_div_2(&B, &B)) != MP_OKAY) goto LBL_ERR; + } + + /* 5. while v is even do */ + while (MP_IS_EVEN(&v)) { + /* 5.1 v = v/2 */ + if ((err = mp_div_2(&v, &v)) != MP_OKAY) goto LBL_ERR; + + /* 5.2 if D is odd then */ + if (MP_IS_ODD(&D)) { + /* D = (D-x)/2 */ + if ((err = mp_sub(&D, &x, &D)) != MP_OKAY) goto LBL_ERR; + } + /* D = D/2 */ + if ((err = mp_div_2(&D, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* 6. if u >= v then */ + if (mp_cmp(&u, &v) != MP_LT) { + /* u = u - v, B = B - D */ + if ((err = mp_sub(&u, &v, &u)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&B, &D, &B)) != MP_OKAY) goto LBL_ERR; + } else { + /* v - v - u, D = D - B */ + if ((err = mp_sub(&v, &u, &v)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&D, &B, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* if not zero goto step 4 */ + if (!MP_IS_ZERO(&u)) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d(&v, 1uL) != MP_EQ) { + err = MP_VAL; + goto LBL_ERR; + } + + /* b is now the inverse */ + neg = a->sign; + while (D.sign == MP_NEG) { + if ((err = mp_add(&D, b, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* too big */ + while (mp_cmp_mag(&D, b) != MP_LT) { + if ((err = mp_sub(&D, b, &D)) != MP_OKAY) goto LBL_ERR; + } + + mp_exch(&D, c); + c->sign = neg; + err = MP_OKAY; + +LBL_ERR: + mp_clear_multi(&x, &y, &u, &v, &B, &D, NULL); + return err; +} +#endif + +/* End: bn_s_mp_invmod_fast.c */ + +/* Start: bn_s_mp_invmod_slow.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_INVMOD_SLOW_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* hac 14.61, pp608 */ +mp_err s_mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x, y, u, v, A, B, C, D; + mp_err err; + + /* b cannot be negative */ + if ((b->sign == MP_NEG) || MP_IS_ZERO(b)) { + return MP_VAL; + } + + /* init temps */ + if ((err = mp_init_multi(&x, &y, &u, &v, + &A, &B, &C, &D, NULL)) != MP_OKAY) { + return err; + } + + /* x = a, y = b */ + if ((err = mp_mod(a, b, &x)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(b, &y)) != MP_OKAY) goto LBL_ERR; + + /* 2. [modified] if x,y are both even then return an error! */ + if (MP_IS_EVEN(&x) && MP_IS_EVEN(&y)) { + err = MP_VAL; + goto LBL_ERR; + } + + /* 3. u=x, v=y, A=1, B=0, C=0,D=1 */ + if ((err = mp_copy(&x, &u)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_copy(&y, &v)) != MP_OKAY) goto LBL_ERR; + mp_set(&A, 1uL); + mp_set(&D, 1uL); + +top: + /* 4. while u is even do */ + while (MP_IS_EVEN(&u)) { + /* 4.1 u = u/2 */ + if ((err = mp_div_2(&u, &u)) != MP_OKAY) goto LBL_ERR; + + /* 4.2 if A or B is odd then */ + if (MP_IS_ODD(&A) || MP_IS_ODD(&B)) { + /* A = (A+y)/2, B = (B-x)/2 */ + if ((err = mp_add(&A, &y, &A)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&B, &x, &B)) != MP_OKAY) goto LBL_ERR; + } + /* A = A/2, B = B/2 */ + if ((err = mp_div_2(&A, &A)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2(&B, &B)) != MP_OKAY) goto LBL_ERR; + } + + /* 5. while v is even do */ + while (MP_IS_EVEN(&v)) { + /* 5.1 v = v/2 */ + if ((err = mp_div_2(&v, &v)) != MP_OKAY) goto LBL_ERR; + + /* 5.2 if C or D is odd then */ + if (MP_IS_ODD(&C) || MP_IS_ODD(&D)) { + /* C = (C+y)/2, D = (D-x)/2 */ + if ((err = mp_add(&C, &y, &C)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_sub(&D, &x, &D)) != MP_OKAY) goto LBL_ERR; + } + /* C = C/2, D = D/2 */ + if ((err = mp_div_2(&C, &C)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_div_2(&D, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* 6. if u >= v then */ + if (mp_cmp(&u, &v) != MP_LT) { + /* u = u - v, A = A - C, B = B - D */ + if ((err = mp_sub(&u, &v, &u)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&A, &C, &A)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&B, &D, &B)) != MP_OKAY) goto LBL_ERR; + } else { + /* v - v - u, C = C - A, D = D - B */ + if ((err = mp_sub(&v, &u, &v)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&C, &A, &C)) != MP_OKAY) goto LBL_ERR; + + if ((err = mp_sub(&D, &B, &D)) != MP_OKAY) goto LBL_ERR; + } + + /* if not zero goto step 4 */ + if (!MP_IS_ZERO(&u)) { + goto top; + } + + /* now a = C, b = D, gcd == g*v */ + + /* if v != 1 then there is no inverse */ + if (mp_cmp_d(&v, 1uL) != MP_EQ) { + err = MP_VAL; + goto LBL_ERR; + } + + /* if its too low */ + while (mp_cmp_d(&C, 0uL) == MP_LT) { + if ((err = mp_add(&C, b, &C)) != MP_OKAY) goto LBL_ERR; + } + + /* too big */ + while (mp_cmp_mag(&C, b) != MP_LT) { + if ((err = mp_sub(&C, b, &C)) != MP_OKAY) goto LBL_ERR; + } + + /* C is now the inverse */ + mp_exch(&C, c); + err = MP_OKAY; +LBL_ERR: + mp_clear_multi(&x, &y, &u, &v, &A, &B, &C, &D, NULL); + return err; +} +#endif + +/* End: bn_s_mp_invmod_slow.c */ + +/* Start: bn_s_mp_karatsuba_mul.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_KARATSUBA_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* c = |a| * |b| using Karatsuba Multiplication using + * three half size multiplications + * + * Let B represent the radix [e.g. 2**MP_DIGIT_BIT] and + * let n represent half of the number of digits in + * the min(a,b) + * + * a = a1 * B**n + a0 + * b = b1 * B**n + b0 + * + * Then, a * b => + a1b1 * B**2n + ((a1 + a0)(b1 + b0) - (a0b0 + a1b1)) * B + a0b0 + * + * Note that a1b1 and a0b0 are used twice and only need to be + * computed once. So in total three half size (half # of + * digit) multiplications are performed, a0b0, a1b1 and + * (a1+b1)(a0+b0) + * + * Note that a multiplication of half the digits requires + * 1/4th the number of single precision multiplications so in + * total after one call 25% of the single precision multiplications + * are saved. Note also that the call to mp_mul can end up back + * in this function if the a0, a1, b0, or b1 are above the threshold. + * This is known as divide-and-conquer and leads to the famous + * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than + * the standard O(N**2) that the baseline/comba methods use. + * Generally though the overhead of this method doesn't pay off + * until a certain size (N ~ 80) is reached. + */ +mp_err s_mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int x0, x1, y0, y1, t1, x0y0, x1y1; + int B; + mp_err err = MP_MEM; /* default the return code to an error */ + + /* min # of digits */ + B = MP_MIN(a->used, b->used); + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size(&x0, B) != MP_OKAY) { + goto LBL_ERR; + } + if (mp_init_size(&x1, a->used - B) != MP_OKAY) { + goto X0; + } + if (mp_init_size(&y0, B) != MP_OKAY) { + goto X1; + } + if (mp_init_size(&y1, b->used - B) != MP_OKAY) { + goto Y0; + } + + /* init temps */ + if (mp_init_size(&t1, B * 2) != MP_OKAY) { + goto Y1; + } + if (mp_init_size(&x0y0, B * 2) != MP_OKAY) { + goto T1; + } + if (mp_init_size(&x1y1, B * 2) != MP_OKAY) { + goto X0Y0; + } + + /* now shift the digits */ + x0.used = y0.used = B; + x1.used = a->used - B; + y1.used = b->used - B; + + { + int x; + mp_digit *tmpa, *tmpb, *tmpx, *tmpy; + + /* we copy the digits directly instead of using higher level functions + * since we also need to shift the digits + */ + tmpa = a->dp; + tmpb = b->dp; + + tmpx = x0.dp; + tmpy = y0.dp; + for (x = 0; x < B; x++) { + *tmpx++ = *tmpa++; + *tmpy++ = *tmpb++; + } + + tmpx = x1.dp; + for (x = B; x < a->used; x++) { + *tmpx++ = *tmpa++; + } + + tmpy = y1.dp; + for (x = B; x < b->used; x++) { + *tmpy++ = *tmpb++; + } + } + + /* only need to clamp the lower words since by definition the + * upper words x1/y1 must have a known number of digits + */ + mp_clamp(&x0); + mp_clamp(&y0); + + /* now calc the products x0y0 and x1y1 */ + /* after this x0 is no longer required, free temp [x0==t2]! */ + if (mp_mul(&x0, &y0, &x0y0) != MP_OKAY) { + goto X1Y1; /* x0y0 = x0*y0 */ + } + if (mp_mul(&x1, &y1, &x1y1) != MP_OKAY) { + goto X1Y1; /* x1y1 = x1*y1 */ + } + + /* now calc x1+x0 and y1+y0 */ + if (s_mp_add(&x1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = x1 - x0 */ + } + if (s_mp_add(&y1, &y0, &x0) != MP_OKAY) { + goto X1Y1; /* t2 = y1 - y0 */ + } + if (mp_mul(&t1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = (x1 + x0) * (y1 + y0) */ + } + + /* add x0y0 */ + if (mp_add(&x0y0, &x1y1, &x0) != MP_OKAY) { + goto X1Y1; /* t2 = x0y0 + x1y1 */ + } + if (s_mp_sub(&t1, &x0, &t1) != MP_OKAY) { + goto X1Y1; /* t1 = (x1+x0)*(y1+y0) - (x1y1 + x0y0) */ + } + + /* shift by B */ + if (mp_lshd(&t1, B) != MP_OKAY) { + goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<used; + + /* now divide in two */ + B = B >> 1; + + /* init copy all the temps */ + if (mp_init_size(&x0, B) != MP_OKAY) + goto LBL_ERR; + if (mp_init_size(&x1, a->used - B) != MP_OKAY) + goto X0; + + /* init temps */ + if (mp_init_size(&t1, a->used * 2) != MP_OKAY) + goto X1; + if (mp_init_size(&t2, a->used * 2) != MP_OKAY) + goto T1; + if (mp_init_size(&x0x0, B * 2) != MP_OKAY) + goto T2; + if (mp_init_size(&x1x1, (a->used - B) * 2) != MP_OKAY) + goto X0X0; + + { + int x; + mp_digit *dst, *src; + + src = a->dp; + + /* now shift the digits */ + dst = x0.dp; + for (x = 0; x < B; x++) { + *dst++ = *src++; + } + + dst = x1.dp; + for (x = B; x < a->used; x++) { + *dst++ = *src++; + } + } + + x0.used = B; + x1.used = a->used - B; + + mp_clamp(&x0); + + /* now calc the products x0*x0 and x1*x1 */ + if (mp_sqr(&x0, &x0x0) != MP_OKAY) + goto X1X1; /* x0x0 = x0*x0 */ + if (mp_sqr(&x1, &x1x1) != MP_OKAY) + goto X1X1; /* x1x1 = x1*x1 */ + + /* now calc (x1+x0)**2 */ + if (s_mp_add(&x1, &x0, &t1) != MP_OKAY) + goto X1X1; /* t1 = x1 - x0 */ + if (mp_sqr(&t1, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */ + + /* add x0y0 */ + if (s_mp_add(&x0x0, &x1x1, &t2) != MP_OKAY) + goto X1X1; /* t2 = x0x0 + x1x1 */ + if (s_mp_sub(&t1, &t2, &t1) != MP_OKAY) + goto X1X1; /* t1 = (x1+x0)**2 - (x0x0 + x1x1) */ + + /* shift by B */ + if (mp_lshd(&t1, B) != MP_OKAY) + goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<used > MP_WARRAY) { + return MP_VAL; + } + + /* get old used count */ + olduse = x->used; + + /* grow a as required */ + if (x->alloc < (n->used + 1)) { + if ((err = mp_grow(x, n->used + 1)) != MP_OKAY) { + return err; + } + } + + /* first we have to get the digits of the input into + * an array of double precision words W[...] + */ + { + mp_word *_W; + mp_digit *tmpx; + + /* alias for the W[] array */ + _W = W; + + /* alias for the digits of x*/ + tmpx = x->dp; + + /* copy the digits of a into W[0..a->used-1] */ + for (ix = 0; ix < x->used; ix++) { + *_W++ = *tmpx++; + } + + /* zero the high words of W[a->used..m->used*2] */ + if (ix < ((n->used * 2) + 1)) { + MP_ZERO_BUFFER(_W, sizeof(mp_word) * (size_t)(((n->used * 2) + 1) - ix)); + } + } + + /* now we proceed to zero successive digits + * from the least significant upwards + */ + for (ix = 0; ix < n->used; ix++) { + /* mu = ai * m' mod b + * + * We avoid a double precision multiplication (which isn't required) + * by casting the value down to a mp_digit. Note this requires + * that W[ix-1] have the carry cleared (see after the inner loop) + */ + mp_digit mu; + mu = ((W[ix] & MP_MASK) * rho) & MP_MASK; + + /* a = a + mu * m * b**i + * + * This is computed in place and on the fly. The multiplication + * by b**i is handled by offseting which columns the results + * are added to. + * + * Note the comba method normally doesn't handle carries in the + * inner loop In this case we fix the carry from the previous + * column since the Montgomery reduction requires digits of the + * result (so far) [see above] to work. This is + * handled by fixing up one carry after the inner loop. The + * carry fixups are done in order so after these loops the + * first m->used words of W[] have the carries fixed + */ + { + int iy; + mp_digit *tmpn; + mp_word *_W; + + /* alias for the digits of the modulus */ + tmpn = n->dp; + + /* Alias for the columns set by an offset of ix */ + _W = W + ix; + + /* inner loop */ + for (iy = 0; iy < n->used; iy++) { + *_W++ += (mp_word)mu * (mp_word)*tmpn++; + } + } + + /* now fix carry for next digit, W[ix+1] */ + W[ix + 1] += W[ix] >> (mp_word)MP_DIGIT_BIT; + } + + /* now we have to propagate the carries and + * shift the words downward [all those least + * significant digits we zeroed]. + */ + { + mp_digit *tmpx; + mp_word *_W, *_W1; + + /* nox fix rest of carries */ + + /* alias for current word */ + _W1 = W + ix; + + /* alias for next word, where the carry goes */ + _W = W + ++ix; + + for (; ix < ((n->used * 2) + 1); ix++) { + *_W++ += *_W1++ >> (mp_word)MP_DIGIT_BIT; + } + + /* copy out, A = A/b**n + * + * The result is A/b**n but instead of converting from an + * array of mp_word to mp_digit than calling mp_rshd + * we just copy them in the right order + */ + + /* alias for destination word */ + tmpx = x->dp; + + /* alias for shifted double precision result */ + _W = W + n->used; + + for (ix = 0; ix < (n->used + 1); ix++) { + *tmpx++ = *_W++ & (mp_word)MP_MASK; + } + + /* zero oldused digits, if the input a was larger than + * m->used+1 we'll have to clear the digits + */ + MP_ZERO_DIGITS(tmpx, olduse - ix); + } + + /* set the max used and clamp */ + x->used = n->used + 1; + mp_clamp(x); + + /* if A >= m then A = A - m */ + if (mp_cmp_mag(x, n) != MP_LT) { + return s_mp_sub(x, n, x); + } + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_montgomery_reduce_fast.c */ + +/* Start: bn_s_mp_mul_digs.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* multiplies |a| * |b| and only computes upto digs digits of result + * HAC pp. 595, Algorithm 14.12 Modified so you can control how + * many digits of output are created. + */ +mp_err s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + mp_int t; + mp_err err; + int pa, pb, ix, iy; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ + if ((digs < MP_WARRAY) && + (MP_MIN(a->used, b->used) < MP_MAXFAST)) { + return s_mp_mul_digs_fast(a, b, c, digs); + } + + if ((err = mp_init_size(&t, digs)) != MP_OKAY) { + return err; + } + t.used = digs; + + /* compute the digits of the product directly */ + pa = a->used; + for (ix = 0; ix < pa; ix++) { + /* set the carry to zero */ + u = 0; + + /* limit ourselves to making digs digits of output */ + pb = MP_MIN(b->used, digs - ix); + + /* setup some aliases */ + /* copy of the digit from a used within the nested loop */ + tmpx = a->dp[ix]; + + /* an alias for the destination shifted ix places */ + tmpt = t.dp + ix; + + /* an alias for the digits of b */ + tmpy = b->dp; + + /* compute the columns of the output and propagate the carry */ + for (iy = 0; iy < pb; iy++) { + /* compute the column as a mp_word */ + r = (mp_word)*tmpt + + ((mp_word)tmpx * (mp_word)*tmpy++) + + (mp_word)u; + + /* the new column is the lower part of the result */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); + + /* get the carry word from the result */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + /* set carry if it is placed below digs */ + if ((ix + iy) < digs) { + *tmpt = u; + } + } + + mp_clamp(&t); + mp_exch(&t, c); + + mp_clear(&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_digs.c */ + +/* Start: bn_s_mp_mul_digs_fast.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_DIGS_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Fast (comba) multiplier + * + * This is the fast column-array [comba] multiplier. It is + * designed to compute the columns of the product first + * then handle the carries afterwards. This has the effect + * of making the nested loops that compute the columns very + * simple and schedulable on super-scalar processors. + * + * This has been modified to produce a variable number of + * digits of output so if say only a half-product is required + * you don't have to compute the upper half (a feature + * required for fast Barrett reduction). + * + * Based on Algorithm 14.12 on pp.595 of HAC. + * + */ +mp_err s_mp_mul_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + int olduse, pa, ix, iz; + mp_err err; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + if (c->alloc < digs) { + if ((err = mp_grow(c, digs)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + pa = MP_MIN(digs, a->used + b->used); + + /* clear the carry */ + _W = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty; + int iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MP_MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; ++iz) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + + } + + /* store term */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + _W = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + mp_digit *tmpc; + tmpc = c->dp; + for (ix = 0; ix < pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpc, olduse - ix); + } + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_digs_fast.c */ + +/* Start: bn_s_mp_mul_high_digs.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_HIGH_DIGS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* multiplies |a| * |b| and does not compute the lower digs digits + * [meant to get the higher part of the product] + */ +mp_err s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + mp_int t; + int pa, pb, ix, iy; + mp_err err; + mp_digit u; + mp_word r; + mp_digit tmpx, *tmpt, *tmpy; + + /* can we use the fast multiplier? */ + if (MP_HAS(S_MP_MUL_HIGH_DIGS_FAST) + && ((a->used + b->used + 1) < MP_WARRAY) + && (MP_MIN(a->used, b->used) < MP_MAXFAST)) { + return s_mp_mul_high_digs_fast(a, b, c, digs); + } + + if ((err = mp_init_size(&t, a->used + b->used + 1)) != MP_OKAY) { + return err; + } + t.used = a->used + b->used + 1; + + pa = a->used; + pb = b->used; + for (ix = 0; ix < pa; ix++) { + /* clear the carry */ + u = 0; + + /* left hand side of A[ix] * B[iy] */ + tmpx = a->dp[ix]; + + /* alias to the address of where the digits will be stored */ + tmpt = &(t.dp[digs]); + + /* alias for where to read the right hand side from */ + tmpy = b->dp + (digs - ix); + + for (iy = digs - ix; iy < pb; iy++) { + /* calculate the double precision result */ + r = (mp_word)*tmpt + + ((mp_word)tmpx * (mp_word)*tmpy++) + + (mp_word)u; + + /* get the lower part */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); + + /* carry the carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + *tmpt = u; + } + mp_clamp(&t); + mp_exch(&t, c); + mp_clear(&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_high_digs.c */ + +/* Start: bn_s_mp_mul_high_digs_fast.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_MUL_HIGH_DIGS_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* this is a modified version of fast_s_mul_digs that only produces + * output digits *above* digs. See the comments for fast_s_mul_digs + * to see how it works. + * + * This is used in the Barrett reduction since for one of the multiplications + * only the higher digits were needed. This essentially halves the work. + * + * Based on Algorithm 14.12 on pp.595 of HAC. + */ +mp_err s_mp_mul_high_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) +{ + int olduse, pa, ix, iz; + mp_err err; + mp_digit W[MP_WARRAY]; + mp_word _W; + + /* grow the destination as required */ + pa = a->used + b->used; + if (c->alloc < pa) { + if ((err = mp_grow(c, pa)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + pa = a->used + b->used; + _W = 0; + for (ix = digs; ix < pa; ix++) { + int tx, ty, iy; + mp_digit *tmpx, *tmpy; + + /* get offsets into the two bignums */ + ty = MP_MIN(b->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = b->dp + ty; + + /* this is the number of times the loop will iterrate, essentially its + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + } + + /* store term */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + _W = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = c->used; + c->used = pa; + + { + mp_digit *tmpc; + + tmpc = c->dp + digs; + for (ix = digs; ix < pa; ix++) { + /* now extract the previous digit [below the carry] */ + *tmpc++ = W[ix]; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpc, olduse - ix); + } + mp_clamp(c); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_mul_high_digs_fast.c */ + +/* Start: bn_s_mp_prime_is_divisible.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_PRIME_IS_DIVISIBLE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* determines if an integers is divisible by one + * of the first PRIME_SIZE primes or not + * + * sets result to 0 if not, 1 if yes + */ +mp_err s_mp_prime_is_divisible(const mp_int *a, mp_bool *result) +{ + int ix; + mp_err err; + mp_digit res; + + /* default to not */ + *result = MP_NO; + + for (ix = 0; ix < PRIVATE_MP_PRIME_TAB_SIZE; ix++) { + /* what is a mod LBL_prime_tab[ix] */ + if ((err = mp_mod_d(a, s_mp_prime_tab[ix], &res)) != MP_OKAY) { + return err; + } + + /* is the residue zero? */ + if (res == 0u) { + *result = MP_YES; + return MP_OKAY; + } + } + + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_prime_is_divisible.c */ + +/* Start: bn_s_mp_rand_jenkins.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_RAND_JENKINS_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* Bob Jenkins' http://burtleburtle.net/bob/rand/smallprng.html */ +/* Chosen for speed and a good "mix" */ +typedef struct { + uint64_t a; + uint64_t b; + uint64_t c; + uint64_t d; +} ranctx; + +static ranctx jenkins_x; + +#define rot(x,k) (((x)<<(k))|((x)>>(64-(k)))) +static uint64_t s_rand_jenkins_val(void) +{ + uint64_t e = jenkins_x.a - rot(jenkins_x.b, 7); + jenkins_x.a = jenkins_x.b ^ rot(jenkins_x.c, 13); + jenkins_x.b = jenkins_x.c + rot(jenkins_x.d, 37); + jenkins_x.c = jenkins_x.d + e; + jenkins_x.d = e + jenkins_x.a; + return jenkins_x.d; +} + +void s_mp_rand_jenkins_init(uint64_t seed) +{ + uint64_t i; + jenkins_x.a = 0xf1ea5eedULL; + jenkins_x.b = jenkins_x.c = jenkins_x.d = seed; + for (i = 0uLL; i < 20uLL; ++i) { + (void)s_rand_jenkins_val(); + } +} + +mp_err s_mp_rand_jenkins(void *p, size_t n) +{ + char *q = (char *)p; + while (n > 0u) { + int i; + uint64_t x = s_rand_jenkins_val(); + for (i = 0; (i < 8) && (n > 0u); ++i, --n) { + *q++ = (char)(x & 0xFFuLL); + x >>= 8; + } + } + return MP_OKAY; +} + +#endif + +/* End: bn_s_mp_rand_jenkins.c */ + +/* Start: bn_s_mp_rand_platform.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_RAND_PLATFORM_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* First the OS-specific special cases + * - *BSD + * - Windows + */ +#if defined(__FreeBSD__) || defined(__OpenBSD__) || defined(__NetBSD__) || defined(__DragonFly__) +#define BN_S_READ_ARC4RANDOM_C +static mp_err s_read_arc4random(void *p, size_t n) +{ + arc4random_buf(p, n); + return MP_OKAY; +} +#endif + +#if defined(_WIN32) || defined(_WIN32_WCE) +#define BN_S_READ_WINCSP_C + +#ifndef _WIN32_WINNT +#define _WIN32_WINNT 0x0400 +#endif +#ifdef _WIN32_WCE +#define UNDER_CE +#define ARM +#endif + +#define WIN32_LEAN_AND_MEAN +#include +#include + +static mp_err s_read_wincsp(void *p, size_t n) +{ + static HCRYPTPROV hProv = 0; + if (hProv == 0) { + HCRYPTPROV h = 0; + if (!CryptAcquireContext(&h, NULL, MS_DEF_PROV, PROV_RSA_FULL, + (CRYPT_VERIFYCONTEXT | CRYPT_MACHINE_KEYSET)) && + !CryptAcquireContext(&h, NULL, MS_DEF_PROV, PROV_RSA_FULL, + CRYPT_VERIFYCONTEXT | CRYPT_MACHINE_KEYSET | CRYPT_NEWKEYSET)) { + return MP_ERR; + } + hProv = h; + } + return CryptGenRandom(hProv, (DWORD)n, (BYTE *)p) == TRUE ? MP_OKAY : MP_ERR; +} +#endif /* WIN32 */ + +#if !defined(BN_S_READ_WINCSP_C) && defined(__linux__) && defined(__GLIBC_PREREQ) +#if __GLIBC_PREREQ(2, 25) +#define BN_S_READ_GETRANDOM_C +#include +#include + +static mp_err s_read_getrandom(void *p, size_t n) +{ + char *q = (char *)p; + while (n > 0u) { + ssize_t ret = getrandom(q, n, 0); + if (ret < 0) { + if (errno == EINTR) { + continue; + } + return MP_ERR; + } + q += ret; + n -= (size_t)ret; + } + return MP_OKAY; +} +#endif +#endif + +/* We assume all platforms besides windows provide "/dev/urandom". + * In case yours doesn't, define MP_NO_DEV_URANDOM at compile-time. + */ +#if !defined(BN_S_READ_WINCSP_C) && !defined(MP_NO_DEV_URANDOM) +#define BN_S_READ_URANDOM_C +#ifndef MP_DEV_URANDOM +#define MP_DEV_URANDOM "/dev/urandom" +#endif +#include +#include +#include + +static mp_err s_read_urandom(void *p, size_t n) +{ + int fd; + char *q = (char *)p; + + do { + fd = open(MP_DEV_URANDOM, O_RDONLY); + } while ((fd == -1) && (errno == EINTR)); + if (fd == -1) return MP_ERR; + + while (n > 0u) { + ssize_t ret = read(fd, p, n); + if (ret < 0) { + if (errno == EINTR) { + continue; + } + close(fd); + return MP_ERR; + } + q += ret; + n -= (size_t)ret; + } + + close(fd); + return MP_OKAY; +} +#endif + +#if defined(MP_PRNG_ENABLE_LTM_RNG) +#define BN_S_READ_LTM_RNG +unsigned long (*ltm_rng)(unsigned char *out, unsigned long outlen, void (*callback)(void)); +void (*ltm_rng_callback)(void); + +static mp_err s_read_ltm_rng(void *p, size_t n) +{ + unsigned long res; + if (ltm_rng == NULL) return MP_ERR; + res = ltm_rng(p, n, ltm_rng_callback); + if (res != n) return MP_ERR; + return MP_OKAY; +} +#endif + +mp_err s_read_arc4random(void *p, size_t n); +mp_err s_read_wincsp(void *p, size_t n); +mp_err s_read_getrandom(void *p, size_t n); +mp_err s_read_urandom(void *p, size_t n); +mp_err s_read_ltm_rng(void *p, size_t n); + +mp_err s_mp_rand_platform(void *p, size_t n) +{ + mp_err err = MP_ERR; + if ((err != MP_OKAY) && MP_HAS(S_READ_ARC4RANDOM)) err = s_read_arc4random(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_WINCSP)) err = s_read_wincsp(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_GETRANDOM)) err = s_read_getrandom(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_URANDOM)) err = s_read_urandom(p, n); + if ((err != MP_OKAY) && MP_HAS(S_READ_LTM_RNG)) err = s_read_ltm_rng(p, n); + return err; +} + +#endif + +/* End: bn_s_mp_rand_platform.c */ + +/* Start: bn_s_mp_reverse.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_REVERSE_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* reverse an array, used for radix code */ +void s_mp_reverse(unsigned char *s, size_t len) +{ + size_t ix, iy; + unsigned char t; + + ix = 0u; + iy = len - 1u; + while (ix < iy) { + t = s[ix]; + s[ix] = s[iy]; + s[iy] = t; + ++ix; + --iy; + } +} +#endif + +/* End: bn_s_mp_reverse.c */ + +/* Start: bn_s_mp_sqr.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */ +mp_err s_mp_sqr(const mp_int *a, mp_int *b) +{ + mp_int t; + int ix, iy, pa; + mp_err err; + mp_word r; + mp_digit u, tmpx, *tmpt; + + pa = a->used; + if ((err = mp_init_size(&t, (2 * pa) + 1)) != MP_OKAY) { + return err; + } + + /* default used is maximum possible size */ + t.used = (2 * pa) + 1; + + for (ix = 0; ix < pa; ix++) { + /* first calculate the digit at 2*ix */ + /* calculate double precision result */ + r = (mp_word)t.dp[2*ix] + + ((mp_word)a->dp[ix] * (mp_word)a->dp[ix]); + + /* store lower part in result */ + t.dp[ix+ix] = (mp_digit)(r & (mp_word)MP_MASK); + + /* get the carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + + /* left hand side of A[ix] * A[iy] */ + tmpx = a->dp[ix]; + + /* alias for where to store the results */ + tmpt = t.dp + ((2 * ix) + 1); + + for (iy = ix + 1; iy < pa; iy++) { + /* first calculate the product */ + r = (mp_word)tmpx * (mp_word)a->dp[iy]; + + /* now calculate the double precision result, note we use + * addition instead of *2 since it's easier to optimize + */ + r = (mp_word)*tmpt + r + r + (mp_word)u; + + /* store lower part */ + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); + + /* get carry */ + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + /* propagate upwards */ + while (u != 0uL) { + r = (mp_word)*tmpt + (mp_word)u; + *tmpt++ = (mp_digit)(r & (mp_word)MP_MASK); + u = (mp_digit)(r >> (mp_word)MP_DIGIT_BIT); + } + } + + mp_clamp(&t); + mp_exch(&t, b); + mp_clear(&t); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_sqr.c */ + +/* Start: bn_s_mp_sqr_fast.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_SQR_FAST_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* the jist of squaring... + * you do like mult except the offset of the tmpx [one that + * starts closer to zero] can't equal the offset of tmpy. + * So basically you set up iy like before then you min it with + * (ty-tx) so that it never happens. You double all those + * you add in the inner loop + +After that loop you do the squares and add them in. +*/ + +mp_err s_mp_sqr_fast(const mp_int *a, mp_int *b) +{ + int olduse, pa, ix, iz; + mp_digit W[MP_WARRAY], *tmpx; + mp_word W1; + mp_err err; + + /* grow the destination as required */ + pa = a->used + a->used; + if (b->alloc < pa) { + if ((err = mp_grow(b, pa)) != MP_OKAY) { + return err; + } + } + + /* number of output digits to produce */ + W1 = 0; + for (ix = 0; ix < pa; ix++) { + int tx, ty, iy; + mp_word _W; + mp_digit *tmpy; + + /* clear counter */ + _W = 0; + + /* get offsets into the two bignums */ + ty = MP_MIN(a->used-1, ix); + tx = ix - ty; + + /* setup temp aliases */ + tmpx = a->dp + tx; + tmpy = a->dp + ty; + + /* this is the number of times the loop will iterrate, essentially + while (tx++ < a->used && ty-- >= 0) { ... } + */ + iy = MP_MIN(a->used-tx, ty+1); + + /* now for squaring tx can never equal ty + * we halve the distance since they approach at a rate of 2x + * and we have to round because odd cases need to be executed + */ + iy = MP_MIN(iy, ((ty-tx)+1)>>1); + + /* execute loop */ + for (iz = 0; iz < iy; iz++) { + _W += (mp_word)*tmpx++ * (mp_word)*tmpy--; + } + + /* double the inner product and add carry */ + _W = _W + _W + W1; + + /* even columns have the square term in them */ + if (((unsigned)ix & 1u) == 0u) { + _W += (mp_word)a->dp[ix>>1] * (mp_word)a->dp[ix>>1]; + } + + /* store it */ + W[ix] = (mp_digit)_W & MP_MASK; + + /* make next carry */ + W1 = _W >> (mp_word)MP_DIGIT_BIT; + } + + /* setup dest */ + olduse = b->used; + b->used = a->used+a->used; + + { + mp_digit *tmpb; + tmpb = b->dp; + for (ix = 0; ix < pa; ix++) { + *tmpb++ = W[ix] & MP_MASK; + } + + /* clear unused digits [that existed in the old copy of c] */ + MP_ZERO_DIGITS(tmpb, olduse - ix); + } + mp_clamp(b); + return MP_OKAY; +} +#endif + +/* End: bn_s_mp_sqr_fast.c */ + +/* Start: bn_s_mp_sub.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_SUB_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */ +mp_err s_mp_sub(const mp_int *a, const mp_int *b, mp_int *c) +{ + int olduse, min, max; + mp_err err; + + /* find sizes */ + min = b->used; + max = a->used; + + /* init result */ + if (c->alloc < max) { + if ((err = mp_grow(c, max)) != MP_OKAY) { + return err; + } + } + olduse = c->used; + c->used = max; + + { + mp_digit u, *tmpa, *tmpb, *tmpc; + int i; + + /* alias for digit pointers */ + tmpa = a->dp; + tmpb = b->dp; + tmpc = c->dp; + + /* set carry to zero */ + u = 0; + for (i = 0; i < min; i++) { + /* T[i] = A[i] - B[i] - U */ + *tmpc = (*tmpa++ - *tmpb++) - u; + + /* U = carry bit of T[i] + * Note this saves performing an AND operation since + * if a carry does occur it will propagate all the way to the + * MSB. As a result a single shift is enough to get the carry + */ + u = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* now copy higher words if any, e.g. if A has more digits than B */ + for (; i < max; i++) { + /* T[i] = A[i] - U */ + *tmpc = *tmpa++ - u; + + /* U = carry bit of T[i] */ + u = *tmpc >> (MP_SIZEOF_BITS(mp_digit) - 1u); + + /* Clear carry from T[i] */ + *tmpc++ &= MP_MASK; + } + + /* clear digits above used (since we may not have grown result above) */ + MP_ZERO_DIGITS(tmpc, olduse - c->used); + } + + mp_clamp(c); + return MP_OKAY; +} + +#endif + +/* End: bn_s_mp_sub.c */ + +/* Start: bn_s_mp_toom_mul.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_TOOM_MUL_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* multiplication using the Toom-Cook 3-way algorithm + * + * Much more complicated than Karatsuba but has a lower + * asymptotic running time of O(N**1.464). This algorithm is + * only particularly useful on VERY large inputs + * (we're talking 1000s of digits here...). +*/ + +/* + This file contains code from J. Arndt's book "Matters Computational" + and the accompanying FXT-library with permission of the author. +*/ + +/* + Setup from + + Chung, Jaewook, and M. Anwar Hasan. "Asymmetric squaring formulae." + 18th IEEE Symposium on Computer Arithmetic (ARITH'07). IEEE, 2007. + + The interpolation from above needed one temporary variable more + than the interpolation here: + + Bodrato, Marco, and Alberto Zanoni. "What about Toom-Cook matrices optimality." + Centro Vito Volterra Universita di Roma Tor Vergata (2006) +*/ + +mp_err s_mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c) +{ + mp_int S1, S2, T1, a0, a1, a2, b0, b1, b2; + int B, count; + mp_err err; + + /* init temps */ + if ((err = mp_init_multi(&S1, &S2, &T1, NULL)) != MP_OKAY) { + return err; + } + + /* B */ + B = MP_MIN(a->used, b->used) / 3; + + /** a = a2 * x^2 + a1 * x + a0; */ + if ((err = mp_init_size(&a0, B)) != MP_OKAY) goto LBL_ERRa0; + + for (count = 0; count < B; count++) { + a0.dp[count] = a->dp[count]; + a0.used++; + } + mp_clamp(&a0); + if ((err = mp_init_size(&a1, B)) != MP_OKAY) goto LBL_ERRa1; + for (; count < (2 * B); count++) { + a1.dp[count - B] = a->dp[count]; + a1.used++; + } + mp_clamp(&a1); + if ((err = mp_init_size(&a2, B + (a->used - (3 * B)))) != MP_OKAY) goto LBL_ERRa2; + for (; count < a->used; count++) { + a2.dp[count - (2 * B)] = a->dp[count]; + a2.used++; + } + mp_clamp(&a2); + + /** b = b2 * x^2 + b1 * x + b0; */ + if ((err = mp_init_size(&b0, B)) != MP_OKAY) goto LBL_ERRb0; + for (count = 0; count < B; count++) { + b0.dp[count] = b->dp[count]; + b0.used++; + } + mp_clamp(&b0); + if ((err = mp_init_size(&b1, B)) != MP_OKAY) goto LBL_ERRb1; + for (; count < (2 * B); count++) { + b1.dp[count - B] = b->dp[count]; + b1.used++; + } + mp_clamp(&b1); + if ((err = mp_init_size(&b2, B + (b->used - (3 * B)))) != MP_OKAY) goto LBL_ERRb2; + for (; count < b->used; count++) { + b2.dp[count - (2 * B)] = b->dp[count]; + b2.used++; + } + mp_clamp(&b2); + + /** \\ S1 = (a2+a1+a0) * (b2+b1+b0); */ + /** T1 = a2 + a1; */ + if ((err = mp_add(&a2, &a1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = T1 + a0; */ + if ((err = mp_add(&T1, &a0, &S2)) != MP_OKAY) goto LBL_ERR; + + /** c = b2 + b1; */ + if ((err = mp_add(&b2, &b1, c)) != MP_OKAY) goto LBL_ERR; + + /** S1 = c + b0; */ + if ((err = mp_add(c, &b0, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 * S2; */ + if ((err = mp_mul(&S1, &S2, &S1)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = (4*a2+2*a1+a0) * (4*b2+2*b1+b0); */ + /** T1 = T1 + a2; */ + if ((err = mp_add(&T1, &a2, &T1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = T1 << 1; */ + if ((err = mp_mul_2(&T1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = T1 + a0; */ + if ((err = mp_add(&T1, &a0, &T1)) != MP_OKAY) goto LBL_ERR; + + /** c = c + b2; */ + if ((err = mp_add(c, &b2, c)) != MP_OKAY) goto LBL_ERR; + + /** c = c << 1; */ + if ((err = mp_mul_2(c, c)) != MP_OKAY) goto LBL_ERR; + + /** c = c + b0; */ + if ((err = mp_add(c, &b0, c)) != MP_OKAY) goto LBL_ERR; + + /** S2 = T1 * c; */ + if ((err = mp_mul(&T1, c, &S2)) != MP_OKAY) goto LBL_ERR; + + /** \\S3 = (a2-a1+a0) * (b2-b1+b0); */ + /** a1 = a2 - a1; */ + if ((err = mp_sub(&a2, &a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 + a0; */ + if ((err = mp_add(&a1, &a0, &a1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = b2 - b1; */ + if ((err = mp_sub(&b2, &b1, &b1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = b1 + b0; */ + if ((err = mp_add(&b1, &b0, &b1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 * b1; */ + if ((err = mp_mul(&a1, &b1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** b1 = a2 * b2; */ + if ((err = mp_mul(&a2, &b2, &b1)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = (S2 - S3)/3; */ + /** S2 = S2 - a1; */ + if ((err = mp_sub(&S2, &a1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 / 3; \\ this is an exact division */ + if ((err = mp_div_3(&S2, &S2, NULL)) != MP_OKAY) goto LBL_ERR; + + /** a1 = S1 - a1; */ + if ((err = mp_sub(&S1, &a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 >> 1; */ + if ((err = mp_div_2(&a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** a0 = a0 * b0; */ + if ((err = mp_mul(&a0, &b0, &a0)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - a0; */ + if ((err = mp_sub(&S1, &a0, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 - S1; */ + if ((err = mp_sub(&S2, &S1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 >> 1; */ + if ((err = mp_div_2(&S2, &S2)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - a1; */ + if ((err = mp_sub(&S1, &a1, &S1)) != MP_OKAY) goto LBL_ERR; + + /** S1 = S1 - b1; */ + if ((err = mp_sub(&S1, &b1, &S1)) != MP_OKAY) goto LBL_ERR; + + /** T1 = b1 << 1; */ + if ((err = mp_mul_2(&b1, &T1)) != MP_OKAY) goto LBL_ERR; + + /** S2 = S2 - T1; */ + if ((err = mp_sub(&S2, &T1, &S2)) != MP_OKAY) goto LBL_ERR; + + /** a1 = a1 - S2; */ + if ((err = mp_sub(&a1, &S2, &a1)) != MP_OKAY) goto LBL_ERR; + + + /** P = b1*x^4+ S2*x^3+ S1*x^2+ a1*x + a0; */ + if ((err = mp_lshd(&b1, 4 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&S2, 3 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &S2, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&S1, 2 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &S1, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a1, 1 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &a1, &b1)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&b1, &a0, c)) != MP_OKAY) goto LBL_ERR; + + /** a * b - P */ + + +LBL_ERR: + mp_clear(&b2); +LBL_ERRb2: + mp_clear(&b1); +LBL_ERRb1: + mp_clear(&b0); +LBL_ERRb0: + mp_clear(&a2); +LBL_ERRa2: + mp_clear(&a1); +LBL_ERRa1: + mp_clear(&a0); +LBL_ERRa0: + mp_clear_multi(&S1, &S2, &T1, NULL); + return err; +} + +#endif + +/* End: bn_s_mp_toom_mul.c */ + +/* Start: bn_s_mp_toom_sqr.c */ +#include "tommath_private.h" +#ifdef BN_S_MP_TOOM_SQR_C +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +/* squaring using Toom-Cook 3-way algorithm */ + +/* + This file contains code from J. Arndt's book "Matters Computational" + and the accompanying FXT-library with permission of the author. +*/ + +/* squaring using Toom-Cook 3-way algorithm */ +/* + Setup and interpolation from algorithm SQR_3 in + + Chung, Jaewook, and M. Anwar Hasan. "Asymmetric squaring formulae." + 18th IEEE Symposium on Computer Arithmetic (ARITH'07). IEEE, 2007. + +*/ +mp_err s_mp_toom_sqr(const mp_int *a, mp_int *b) +{ + mp_int S0, a0, a1, a2; + mp_digit *tmpa, *tmpc; + int B, count; + mp_err err; + + + /* init temps */ + if ((err = mp_init(&S0)) != MP_OKAY) { + return err; + } + + /* B */ + B = a->used / 3; + + /** a = a2 * x^2 + a1 * x + a0; */ + if ((err = mp_init_size(&a0, B)) != MP_OKAY) goto LBL_ERRa0; + + a0.used = B; + if ((err = mp_init_size(&a1, B)) != MP_OKAY) goto LBL_ERRa1; + a1.used = B; + if ((err = mp_init_size(&a2, B + (a->used - (3 * B)))) != MP_OKAY) goto LBL_ERRa2; + + tmpa = a->dp; + tmpc = a0.dp; + for (count = 0; count < B; count++) { + *tmpc++ = *tmpa++; + } + tmpc = a1.dp; + for (; count < (2 * B); count++) { + *tmpc++ = *tmpa++; + } + tmpc = a2.dp; + for (; count < a->used; count++) { + *tmpc++ = *tmpa++; + a2.used++; + } + mp_clamp(&a0); + mp_clamp(&a1); + mp_clamp(&a2); + + /** S0 = a0^2; */ + if ((err = mp_sqr(&a0, &S0)) != MP_OKAY) goto LBL_ERR; + + /** \\S1 = (a2 + a1 + a0)^2 */ + /** \\S2 = (a2 - a1 + a0)^2 */ + /** \\S1 = a0 + a2; */ + /** a0 = a0 + a2; */ + if ((err = mp_add(&a0, &a2, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S1 - a1; */ + /** b = a0 - a1; */ + if ((err = mp_sub(&a0, &a1, b)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1 + a1; */ + /** a0 = a0 + a1; */ + if ((err = mp_add(&a0, &a1, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1^2; */ + /** a0 = a0^2; */ + if ((err = mp_sqr(&a0, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S2^2; */ + /** b = b^2; */ + if ((err = mp_sqr(b, b)) != MP_OKAY) goto LBL_ERR; + + /** \\ S3 = 2 * a1 * a2 */ + /** \\S3 = a1 * a2; */ + /** a1 = a1 * a2; */ + if ((err = mp_mul(&a1, &a2, &a1)) != MP_OKAY) goto LBL_ERR; + /** \\S3 = S3 << 1; */ + /** a1 = a1 << 1; */ + if ((err = mp_mul_2(&a1, &a1)) != MP_OKAY) goto LBL_ERR; + + /** \\S4 = a2^2; */ + /** a2 = a2^2; */ + if ((err = mp_sqr(&a2, &a2)) != MP_OKAY) goto LBL_ERR; + + /** \\ tmp = (S1 + S2)/2 */ + /** \\tmp = S1 + S2; */ + /** b = a0 + b; */ + if ((err = mp_add(&a0, b, b)) != MP_OKAY) goto LBL_ERR; + /** \\tmp = tmp >> 1; */ + /** b = b >> 1; */ + if ((err = mp_div_2(b, b)) != MP_OKAY) goto LBL_ERR; + + /** \\ S1 = S1 - tmp - S3 */ + /** \\S1 = S1 - tmp; */ + /** a0 = a0 - b; */ + if ((err = mp_sub(&a0, b, &a0)) != MP_OKAY) goto LBL_ERR; + /** \\S1 = S1 - S3; */ + /** a0 = a0 - a1; */ + if ((err = mp_sub(&a0, &a1, &a0)) != MP_OKAY) goto LBL_ERR; + + /** \\S2 = tmp - S4 -S0 */ + /** \\S2 = tmp - S4; */ + /** b = b - a2; */ + if ((err = mp_sub(b, &a2, b)) != MP_OKAY) goto LBL_ERR; + /** \\S2 = S2 - S0; */ + /** b = b - S0; */ + if ((err = mp_sub(b, &S0, b)) != MP_OKAY) goto LBL_ERR; + + + /** \\P = S4*x^4 + S3*x^3 + S2*x^2 + S1*x + S0; */ + /** P = a2*x^4 + a1*x^3 + b*x^2 + a0*x + S0; */ + + if ((err = mp_lshd(&a2, 4 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a1, 3 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(b, 2 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_lshd(&a0, 1 * B)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&a2, &a1, &a2)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(&a2, b, b)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(b, &a0, b)) != MP_OKAY) goto LBL_ERR; + if ((err = mp_add(b, &S0, b)) != MP_OKAY) goto LBL_ERR; + /** a^2 - P */ + + +LBL_ERR: + mp_clear(&a2); +LBL_ERRa2: + mp_clear(&a1); +LBL_ERRa1: + mp_clear(&a0); +LBL_ERRa0: + mp_clear(&S0); + + return err; +} + +#endif + +/* End: bn_s_mp_toom_sqr.c */ + + +/* EOF */ diff --git a/lib/hcrypto/libtommath/pretty.build b/lib/hcrypto/libtommath/pretty.build deleted file mode 100644 index 0f5081902..000000000 --- a/lib/hcrypto/libtommath/pretty.build +++ /dev/null @@ -1,66 +0,0 @@ -#!/bin/perl -w -# -# Cute little builder for perl -# Total waste of development time... -# -# This will build all the object files and then the archive .a file -# requires GCC, GNU make and a sense of humour. -# -# Tom St Denis -use strict; - -my $count = 0; -my $starttime = time; -my $rate = 0; -print "Scanning for source files...\n"; -foreach my $filename (glob "*.c") { - ++$count; -} -print "Source files to build: $count\nBuilding...\n"; -my $i = 0; -my $lines = 0; -my $filesbuilt = 0; -foreach my $filename (glob "*.c") { - printf("Building %3.2f%%, ", (++$i/$count)*100.0); - if ($i % 4 == 0) { print "/, "; } - if ($i % 4 == 1) { print "-, "; } - if ($i % 4 == 2) { print "\\, "; } - if ($i % 4 == 3) { print "|, "; } - if ($rate > 0) { - my $tleft = ($count - $i) / $rate; - my $tsec = $tleft%60; - my $tmin = ($tleft/60)%60; - my $thour = ($tleft/3600)%60; - printf("%2d:%02d:%02d left, ", $thour, $tmin, $tsec); - } - my $cnt = ($i/$count)*30.0; - my $x = 0; - print "["; - for (; $x < $cnt; $x++) { print "#"; } - for (; $x < 30; $x++) { print " "; } - print "]\r"; - my $tmp = $filename; - $tmp =~ s/\.c/".o"/ge; - if (open(SRC, "<$tmp")) { - close SRC; - } else { - !system("make $tmp > /dev/null 2>/dev/null") or die "\nERROR: Failed to make $tmp!!!\n"; - open( SRC, "<$filename" ) or die "Couldn't open $filename for reading: $!"; - ++$lines while (); - close SRC or die "Error closing $filename after reading: $!"; - ++$filesbuilt; - } - - # update timer - if (time != $starttime) { - my $delay = time - $starttime; - $rate = $i/$delay; - } -} - -# finish building the library -printf("\nFinished building source (%d seconds, %3.2f files per second).\n", time - $starttime, $rate); -print "Compiled approximately $filesbuilt files and $lines lines of code.\n"; -print "Doing final make (building archive...)\n"; -!system("make > /dev/null 2>/dev/null") or die "\nERROR: Failed to perform last make command!!!\n"; -print "done.\n"; diff --git a/lib/hcrypto/libtommath/testme.sh b/lib/hcrypto/libtommath/testme.sh new file mode 100755 index 000000000..40fa32d50 --- /dev/null +++ b/lib/hcrypto/libtommath/testme.sh @@ -0,0 +1,394 @@ +#!/bin/bash +# +# return values of this script are: +# 0 success +# 128 a test failed +# >0 the number of timed-out tests +# 255 parsing of parameters failed + +set -e + +if [ -f /proc/cpuinfo ] +then + MAKE_JOBS=$(( ($(cat /proc/cpuinfo | grep -E '^processor[[:space:]]*:' | tail -n -1 | cut -d':' -f2) + 1) * 2 + 1 )) +else + MAKE_JOBS=8 +fi + +ret=0 +TEST_CFLAGS="" + +_help() +{ + echo "Usage options for $(basename $0) [--with-cc=arg [other options]]" + echo + echo "Executing this script without any parameter will only run the default" + echo "configuration that has automatically been determined for the" + echo "architecture you're running." + echo + echo " --with-cc=* The compiler(s) to use for the tests" + echo " This is an option that will be iterated." + echo + echo " --test-vs-mtest=* Run test vs. mtest for '*' operations." + echo " Only the first of each options will be" + echo " taken into account." + echo + echo "To be able to specify options a compiler has to be given with" + echo "the option --with-cc=compilername" + echo "All other options will be tested with all MP_xBIT configurations." + echo + echo " --with-{m64,m32,mx32} The architecture(s) to build and test" + echo " for, e.g. --with-mx32." + echo " This is an option that will be iterated," + echo " multiple selections are possible." + echo " The mx32 architecture is not supported" + echo " by clang and will not be executed." + echo + echo " --cflags=* Give an option to the compiler," + echo " e.g. --cflags=-g" + echo " This is an option that will always be" + echo " passed as parameter to CC." + echo + echo " --make-option=* Give an option to make," + echo " e.g. --make-option=\"-f makefile.shared\"" + echo " This is an option that will always be" + echo " passed as parameter to make." + echo + echo " --with-low-mp Also build&run tests with -DMP_{8,16,32}BIT." + echo + echo " --mtest-real-rand Use real random data when running mtest." + echo + echo " --with-valgrind" + echo " --with-valgrind=* Run in valgrind (slow!)." + echo + echo " --with-travis-valgrind Run with valgrind on Travis on specific branches." + echo + echo " --valgrind-options Additional Valgrind options" + echo " Some of the options like e.g.:" + echo " --track-origins=yes add a lot of extra" + echo " runtime and may trigger the 30 minutes" + echo " timeout." + echo + echo "Godmode:" + echo + echo " --all Choose all architectures and gcc and clang" + echo " as compilers but does not run valgrind." + echo + echo " --format Runs the various source-code formatters" + echo " and generators and checks if the sources" + echo " are clean." + echo + echo " -h" + echo " --help This message" + echo + echo " -v" + echo " --version Prints the version. It is just the number" + echo " of git commits to this file, no deeper" + echo " meaning attached" + exit 0 +} + +_die() +{ + echo "error $2 while $1" + if [ "$2" != "124" ] + then + exit 128 + else + echo "assuming timeout while running test - continue" + local _tail="" + which tail >/dev/null && _tail="tail -n 1 test_${suffix}.log" && \ + echo "last line of test_"${suffix}".log was:" && $_tail && echo "" + ret=$(( $ret + 1 )) + fi +} + +_make() +{ + echo -ne " Compile $1 $2" + suffix=$(echo ${1}${2} | tr ' ' '_') + CC="$1" CFLAGS="$2 $TEST_CFLAGS" make -j$MAKE_JOBS $3 $MAKE_OPTIONS > /dev/null 2>gcc_errors_${suffix}.log + errcnt=$(wc -l < gcc_errors_${suffix}.log) + if [[ ${errcnt} -gt 1 ]]; then + echo " failed" + cat gcc_errors_${suffix}.log + exit 128 + fi +} + + +_runtest() +{ + make clean > /dev/null + local _timeout="" + which timeout >/dev/null && _timeout="timeout --foreground 90" + if [[ "$MAKE_OPTIONS" =~ "tune" ]] + then + # "make tune" will run "tune_it.sh" automatically, hence "autotune", but it cannot + # get switched off without some effort, so we just let it run twice for testing purposes + echo -e "\rRun autotune $1 $2" + _make "$1" "$2" "" + $_timeout $TUNE_CMD > test_${suffix}.log || _die "running autotune" $? + else + _make "$1" "$2" "test" + echo -e "\rRun test $1 $2" + $_timeout ./test > test_${suffix}.log || _die "running tests" $? + fi +} + +# This is not much more of a C&P of _runtest with a different timeout +# and the additional valgrind call. +# TODO: merge +_runvalgrind() +{ + make clean > /dev/null + local _timeout="" + # 30 minutes? Yes. Had it at 20 minutes and the Valgrind run needed over 25 minutes. + # A bit too close for comfort. + which timeout >/dev/null && _timeout="timeout --foreground 1800" +echo "MAKE_OPTIONS = \"$MAKE_OPTIONS\"" + if [[ "$MAKE_OPTIONS" =~ "tune" ]] + then +echo "autotune branch" + _make "$1" "$2" "" + # The shell used for /bin/sh is DASH 0.5.7-4ubuntu1 on the author's machine which fails valgrind, so + # we just run on instance of etc/tune with the same options as in etc/tune_it.sh + echo -e "\rRun etc/tune $1 $2 once inside valgrind" + $_timeout $VALGRIND_BIN $VALGRIND_OPTS $TUNE_CMD > test_${suffix}.log || _die "running etc/tune" $? + else + _make "$1" "$2" "test" + echo -e "\rRun test $1 $2 inside valgrind" + $_timeout $VALGRIND_BIN $VALGRIND_OPTS ./test > test_${suffix}.log || _die "running tests" $? + fi +} + + +_banner() +{ + echo "uname="$(uname -a) + [[ "$#" != "0" ]] && (echo $1=$($1 -dumpversion)) || true +} + +_exit() +{ + if [ "$ret" == "0" ] + then + echo "Tests successful" + else + echo "$ret tests timed out" + fi + + exit $ret +} + +ARCHFLAGS="" +COMPILERS="" +CFLAGS="" +WITH_LOW_MP="" +TEST_VS_MTEST="" +MTEST_RAND="" +# timed with an AMD A8-6600K +# 25 minutes +#VALGRIND_OPTS=" --track-origins=yes --leak-check=full --show-leak-kinds=all --error-exitcode=1 " +# 9 minutes (14 minutes with --test-vs-mtest=333333 --mtest-real-rand) +VALGRIND_OPTS=" --leak-check=full --show-leak-kinds=all --error-exitcode=1 " +#VALGRIND_OPTS="" +VALGRIND_BIN="" +CHECK_FORMAT="" +TUNE_CMD="./etc/tune -t -r 10 -L 3" + +alive_pid=0 + +function kill_alive() { + disown $alive_pid || true + kill $alive_pid 2>/dev/null +} + +function start_alive_printing() { + [ "$alive_pid" == "0" ] || return 0; + for i in `seq 1 10` ; do sleep 300 && echo "Tests still in Progress..."; done & + alive_pid=$! + trap kill_alive EXIT +} + +while [ $# -gt 0 ]; +do + case $1 in + "--with-m64" | "--with-m32" | "--with-mx32") + ARCHFLAGS="$ARCHFLAGS ${1:6}" + ;; + --with-cc=*) + COMPILERS="$COMPILERS ${1#*=}" + ;; + --cflags=*) + CFLAGS="$CFLAGS ${1#*=}" + ;; + --valgrind-options=*) + VALGRIND_OPTS="$VALGRIND_OPTS ${1#*=}" + ;; + --with-valgrind*) + if [[ ${1#*d} != "" ]] + then + VALGRIND_BIN="${1#*=}" + else + VALGRIND_BIN="valgrind" + fi + start_alive_printing + ;; + --with-travis-valgrind*) + if [[ ("$TRAVIS_BRANCH" == "develop" && "$TRAVIS_PULL_REQUEST" == "false") || "$TRAVIS_BRANCH" == *"valgrind"* || "$TRAVIS_COMMIT_MESSAGE" == *"valgrind"* ]] + then + if [[ ${1#*d} != "" ]] + then + VALGRIND_BIN="${1#*=}" + else + VALGRIND_BIN="valgrind" + fi + start_alive_printing + fi + ;; + --make-option=*) + MAKE_OPTIONS="$MAKE_OPTIONS ${1#*=}" + ;; + --with-low-mp) + WITH_LOW_MP="1" + ;; + --test-vs-mtest=*) + TEST_VS_MTEST="${1#*=}" + if ! [ "$TEST_VS_MTEST" -eq "$TEST_VS_MTEST" ] 2> /dev/null + then + echo "--test-vs-mtest Parameter has to be int" + exit 255 + fi + start_alive_printing + ;; + --mtest-real-rand) + MTEST_RAND="-DLTM_MTEST_REAL_RAND" + ;; + --format) + CHECK_FORMAT="1" + ;; + --all) + COMPILERS="gcc clang" + ARCHFLAGS="-m64 -m32 -mx32" + ;; + --help | -h) + _help + ;; + --version | -v) + echo $(git rev-list HEAD --count -- testme.sh) || echo "Unknown. Please run in original libtommath git repository." + exit 0 + ;; + *) + echo "Ignoring option ${1}" + ;; + esac + shift +done + +function _check_git() { + git update-index --refresh >/dev/null || true + git diff-index --quiet HEAD -- . || ( echo "FAILURE: $*" && exit 1 ) +} + +if [[ "$CHECK_FORMAT" == "1" ]] +then + make astyle + _check_git "make astyle" + perl helper.pl --update-files + _check_git "helper.pl --update-files" + perl helper.pl --check-all + _check_git "helper.pl --check-all" + exit $? +fi + +[[ "$VALGRIND_BIN" == "" ]] && VALGRIND_OPTS="" + +# default to CC environment variable if no compiler is defined but some other options +if [[ "$COMPILERS" == "" ]] && [[ "$ARCHFLAGS$MAKE_OPTIONS$CFLAGS" != "" ]] +then + COMPILERS="$CC" +# default to CC environment variable and run only default config if no option is given +elif [[ "$COMPILERS" == "" ]] +then + _banner "$CC" + if [[ "$VALGRIND_BIN" != "" ]] + then + _runvalgrind "$CC" "" + else + _runtest "$CC" "" + fi + _exit +fi + + +archflags=( $ARCHFLAGS ) +compilers=( $COMPILERS ) + +# choosing a compiler without specifying an architecture will use the default architecture +if [ "${#archflags[@]}" == "0" ] +then + archflags[0]=" " +fi + +_banner + +if [[ "$TEST_VS_MTEST" != "" ]] +then + make clean > /dev/null + _make "${compilers[0]} ${archflags[0]}" "$CFLAGS" "mtest_opponent" + echo + _make "gcc" "$MTEST_RAND" "mtest" + echo + echo "Run test vs. mtest for $TEST_VS_MTEST iterations" + _timeout="" + which timeout >/dev/null && _timeout="timeout --foreground 1800" + $_timeout ./mtest/mtest $TEST_VS_MTEST | $VALGRIND_BIN $VALGRIND_OPTS ./mtest_opponent > valgrind_test.log 2> test_vs_mtest_err.log + retval=$? + head -n 5 valgrind_test.log + tail -n 2 valgrind_test.log + exit $retval +fi + +for i in "${compilers[@]}" +do + if [ -z "$(which $i)" ] + then + echo "Skipped compiler $i, file not found" + continue + fi + compiler_version=$(echo "$i="$($i -dumpversion)) + if [ "$compiler_version" == "clang=4.2.1" ] + then + # one of my versions of clang complains about some stuff in stdio.h and stdarg.h ... + TEST_CFLAGS="-Wno-typedef-redefinition" + else + TEST_CFLAGS="" + fi + echo $compiler_version + + for a in "${archflags[@]}" + do + if [[ $(expr "$i" : "clang") -ne 0 && "$a" == "-mx32" ]] + then + echo "clang -mx32 tests skipped" + continue + fi + if [[ "$VALGRIND_BIN" != "" ]] + then + _runvalgrind "$i $a" "$CFLAGS" + [ "$WITH_LOW_MP" != "1" ] && continue + _runvalgrind "$i $a" "-DMP_8BIT $CFLAGS" + _runvalgrind "$i $a" "-DMP_16BIT $CFLAGS" + _runvalgrind "$i $a" "-DMP_32BIT $CFLAGS" + else + _runtest "$i $a" "$CFLAGS" + [ "$WITH_LOW_MP" != "1" ] && continue + _runtest "$i $a" "-DMP_8BIT $CFLAGS" + _runtest "$i $a" "-DMP_16BIT $CFLAGS" + _runtest "$i $a" "-DMP_32BIT $CFLAGS" + fi + done +done + +_exit diff --git a/lib/hcrypto/libtommath/tombc/grammar.txt b/lib/hcrypto/libtommath/tombc/grammar.txt deleted file mode 100644 index a780e759d..000000000 --- a/lib/hcrypto/libtommath/tombc/grammar.txt +++ /dev/null @@ -1,35 +0,0 @@ -program := program statement | statement | empty -statement := { statement } | - identifier = numexpression; | - identifier[numexpression] = numexpression; | - function(expressionlist); | - for (identifer = numexpression; numexpression; identifier = numexpression) { statement } | - while (numexpression) { statement } | - if (numexpresion) { statement } elif | - break; | - continue; - -elif := else statement | empty -function := abs | countbits | exptmod | jacobi | print | isprime | nextprime | issquare | readinteger | exit -expressionlist := expressionlist, expression | expression - -// LR(1) !!!? -expression := string | numexpression -numexpression := cmpexpr && cmpexpr | cmpexpr \|\| cmpexpr | cmpexpr -cmpexpr := boolexpr < boolexpr | boolexpr > boolexpr | boolexpr == boolexpr | - boolexpr <= boolexpr | boolexpr >= boolexpr | boolexpr -boolexpr := shiftexpr & shiftexpr | shiftexpr ^ shiftexpr | shiftexpr \| shiftexpr | shiftexpr -shiftexpr := addsubexpr << addsubexpr | addsubexpr >> addsubexpr | addsubexpr -addsubexpr := mulexpr + mulexpr | mulexpr - mulexpr | mulexpr -mulexpr := expr * expr | expr / expr | expr % expr | expr -expr := -nexpr | nexpr -nexpr := integer | identifier | ( numexpression ) | identifier[numexpression] - -identifier := identifer digits | identifier alpha | alpha -alpha := a ... z | A ... Z -integer := hexnumber | digits -hexnumber := 0xhexdigits -hexdigits := hexdigits hexdigit | hexdigit -hexdigit := 0 ... 9 | a ... f | A ... F -digits := digits digit | digit -digit := 0 ... 9 diff --git a/lib/hcrypto/libtommath/tommath.def b/lib/hcrypto/libtommath/tommath.def new file mode 100644 index 000000000..229fae49a --- /dev/null +++ b/lib/hcrypto/libtommath/tommath.def @@ -0,0 +1,145 @@ +; libtommath +; +; Use this command to produce a 32-bit .lib file, for use in any MSVC version +; lib -machine:X86 -name:libtommath.dll -def:tommath.def -out:tommath.lib +; Use this command to produce a 64-bit .lib file, for use in any MSVC version +; lib -machine:X64 -name:libtommath.dll -def:tommath.def -out:tommath.lib +; +EXPORTS + mp_2expt + mp_abs + mp_add + mp_add_d + mp_addmod + mp_and + mp_clamp + mp_clear + mp_clear_multi + mp_cmp + mp_cmp_d + mp_cmp_mag + mp_cnt_lsb + mp_complement + mp_copy + mp_count_bits + mp_decr + mp_div + mp_div_2 + mp_div_2d + mp_div_3 + mp_div_d + mp_dr_is_modulus + mp_dr_reduce + mp_dr_setup + mp_error_to_string + mp_exch + mp_expt_u32 + mp_exptmod + mp_exteuclid + mp_fread + mp_from_sbin + mp_from_ubin + mp_fwrite + mp_gcd + mp_get_double + mp_get_i32 + mp_get_i64 + mp_get_int + mp_get_l + mp_get_ll + mp_get_long + mp_get_long_long + mp_get_mag_u32 + mp_get_mag_u64 + mp_get_mag_ul + mp_get_mag_ull + mp_grow + mp_incr + mp_init + mp_init_copy + mp_init_i32 + mp_init_i64 + mp_init_l + mp_init_ll + mp_init_multi + mp_init_set + mp_init_set_int + mp_init_size + mp_init_u32 + mp_init_u64 + mp_init_ul + mp_init_ull + mp_invmod + mp_is_square + mp_iseven + mp_isodd + mp_kronecker + mp_lcm + mp_log_u32 + mp_lshd + mp_mod + mp_mod_2d + mp_mod_d + mp_montgomery_calc_normalization + mp_montgomery_reduce + mp_montgomery_setup + mp_mul + mp_mul_2 + mp_mul_2d + mp_mul_d + mp_mulmod + mp_neg + mp_or + mp_pack + mp_pack_count + mp_prime_fermat + mp_prime_frobenius_underwood + mp_prime_is_prime + mp_prime_miller_rabin + mp_prime_next_prime + mp_prime_rabin_miller_trials + mp_prime_rand + mp_prime_strong_lucas_selfridge + mp_radix_size + mp_rand + mp_read_radix + mp_reduce + mp_reduce_2k + mp_reduce_2k_l + mp_reduce_2k_setup + mp_reduce_2k_setup_l + mp_reduce_is_2k + mp_reduce_is_2k_l + mp_reduce_setup + mp_root_u32 + mp_rshd + mp_sbin_size + mp_set + mp_set_double + mp_set_i32 + mp_set_i64 + mp_set_int + mp_set_l + mp_set_ll + mp_set_long + mp_set_long_long + mp_set_u32 + mp_set_u64 + mp_set_ul + mp_set_ull + mp_shrink + mp_signed_rsh + mp_sqr + mp_sqrmod + mp_sqrt + mp_sqrtmod_prime + mp_sub + mp_sub_d + mp_submod + mp_to_radix + mp_to_sbin + mp_to_ubin + mp_ubin_size + mp_unpack + mp_xor + mp_zero diff --git a/lib/hcrypto/libtommath/tommath.h b/lib/hcrypto/libtommath/tommath.h index e5d8d5367..e87bb086c 100644 --- a/lib/hcrypto/libtommath/tommath.h +++ b/lib/hcrypto/libtommath/tommath.h @@ -1,497 +1,692 @@ -/* LibTomMath, multiple-precision integer library -- Tom St Denis - * - * LibTomMath is a library that provides multiple-precision - * integer arithmetic as well as number theoretic functionality. - * - * The library was designed directly after the MPI library by - * Michael Fromberger but has been written from scratch with - * additional optimizations in place. - * - * The library is free for all purposes without any express - * guarantee it works. - * - * Tom St Denis, tomstdenis@gmail.com, http://math.libtomcrypt.com - */ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + #ifndef BN_H_ #define BN_H_ -#include -#include -#include -#include +#include +#include #include -#include - -#ifndef MIN - #define MIN(x,y) ((x)<(y)?(x):(y)) +#ifdef LTM_NO_FILE +# warning LTM_NO_FILE has been deprecated, use MP_NO_FILE. +# define MP_NO_FILE #endif -#ifndef MAX - #define MAX(x,y) ((x)>(y)?(x):(y)) +#ifndef MP_NO_FILE +# include +#endif + +#ifdef MP_8BIT +# ifdef _MSC_VER +# pragma message("8-bit (MP_8BIT) support is deprecated and will be dropped completely in the next version.") +# else +# warning "8-bit (MP_8BIT) support is deprecated and will be dropped completely in the next version." +# endif #endif #ifdef __cplusplus extern "C" { - -/* C++ compilers don't like assigning void * to mp_digit * */ -#define OPT_CAST(x) (x *) - -#else - -/* C on the other hand doesn't care */ -#define OPT_CAST(x) - #endif +/* MS Visual C++ doesn't have a 128bit type for words, so fall back to 32bit MPI's (where words are 64bit) */ +#if (defined(_MSC_VER) || defined(__LLP64__) || defined(__e2k__) || defined(__LCC__)) && !defined(MP_64BIT) +# define MP_32BIT +#endif /* detect 64-bit mode if possible */ -#if defined(__x86_64__) && !defined(__ILP32__) - #if !(defined(MP_64BIT) && defined(MP_16BIT) && defined(MP_8BIT)) - #define MP_64BIT - #endif +#if defined(__x86_64__) || defined(_M_X64) || defined(_M_AMD64) || \ + defined(__powerpc64__) || defined(__ppc64__) || defined(__PPC64__) || \ + defined(__s390x__) || defined(__arch64__) || defined(__aarch64__) || \ + defined(__sparcv9) || defined(__sparc_v9__) || defined(__sparc64__) || \ + defined(__ia64) || defined(__ia64__) || defined(__itanium__) || defined(_M_IA64) || \ + defined(__LP64__) || defined(_LP64) || defined(__64BIT__) +# if !(defined(MP_64BIT) || defined(MP_32BIT) || defined(MP_16BIT) || defined(MP_8BIT)) +# if defined(__GNUC__) && !defined(__hppa) +/* we support 128bit integers only via: __attribute__((mode(TI))) */ +# define MP_64BIT +# else +/* otherwise we fall back to MP_32BIT even on 64bit platforms */ +# define MP_32BIT +# endif +# endif +#endif + +#ifdef MP_DIGIT_BIT +# error Defining MP_DIGIT_BIT is disallowed, use MP_8/16/31/32/64BIT #endif /* some default configurations. * - * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits - * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits + * A "mp_digit" must be able to hold MP_DIGIT_BIT + 1 bits + * A "mp_word" must be able to hold 2*MP_DIGIT_BIT + 1 bits * * At the very least a mp_digit must be able to hold 7 bits * [any size beyond that is ok provided it doesn't overflow the data type] */ + #ifdef MP_8BIT - typedef unsigned char mp_digit; - typedef unsigned short mp_word; +typedef uint8_t mp_digit; +typedef uint16_t private_mp_word; +# define MP_DIGIT_BIT 7 #elif defined(MP_16BIT) - typedef unsigned short mp_digit; - typedef unsigned long mp_word; +typedef uint16_t mp_digit; +typedef uint32_t private_mp_word; +# define MP_DIGIT_BIT 15 #elif defined(MP_64BIT) - /* for GCC only on supported platforms */ -#ifndef CRYPT - typedef unsigned long long ulong64; - typedef signed long long long64; +/* for GCC only on supported platforms */ +typedef uint64_t mp_digit; +#if defined(__GNUC__) +typedef unsigned long private_mp_word __attribute__((mode(TI))); #endif - - typedef unsigned long mp_digit; - typedef unsigned long mp_word __attribute__ ((mode(TI))); - - #define DIGIT_BIT 60 +# define MP_DIGIT_BIT 60 #else - /* this is the default case, 28-bit digits */ - - /* this is to make porting into LibTomCrypt easier :-) */ -#ifndef CRYPT - #if defined(_MSC_VER) || defined(__BORLANDC__) - typedef unsigned __int64 ulong64; - typedef signed __int64 long64; - #else - typedef unsigned long long ulong64; - typedef signed long long long64; - #endif +typedef uint32_t mp_digit; +typedef uint64_t private_mp_word; +# ifdef MP_31BIT +/* + * This is an extension that uses 31-bit digits. + * Please be aware that not all functions support this size, especially s_mp_mul_digs_fast + * will be reduced to work on small numbers only: + * Up to 8 limbs, 248 bits instead of up to 512 limbs, 15872 bits with MP_28BIT. + */ +# define MP_DIGIT_BIT 31 +# else +/* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ +# define MP_DIGIT_BIT 28 +# define MP_28BIT +# endif #endif - typedef unsigned long mp_digit; - typedef ulong64 mp_word; +/* mp_word is a private type */ +#define mp_word MP_DEPRECATED_PRAGMA("mp_word has been made private") private_mp_word -#ifdef MP_31BIT - /* this is an extension that uses 31-bit digits */ - #define DIGIT_BIT 31 -#else - /* default case is 28-bit digits, defines MP_28BIT as a handy macro to test */ - #define DIGIT_BIT 28 - #define MP_28BIT -#endif -#endif +#define MP_SIZEOF_MP_DIGIT (MP_DEPRECATED_PRAGMA("MP_SIZEOF_MP_DIGIT has been deprecated, use sizeof (mp_digit)") sizeof (mp_digit)) -/* define heap macros */ -#ifndef CRYPT - /* default to libc stuff */ - #ifndef XMALLOC - #define XMALLOC malloc - #define XFREE free - #define XREALLOC realloc - #define XCALLOC calloc - #else - /* prototypes for our heap functions */ - extern void *XMALLOC(size_t n); - extern void *XREALLOC(void *p, size_t n); - extern void *XCALLOC(size_t n, size_t s); - extern void XFREE(void *p); - #endif -#endif - - -/* otherwise the bits per digit is calculated automatically from the size of a mp_digit */ -#ifndef DIGIT_BIT - #define DIGIT_BIT ((int)((CHAR_BIT * sizeof(mp_digit) - 1))) /* bits per digit */ -#endif - -#define MP_DIGIT_BIT DIGIT_BIT -#define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)1)) +#define MP_MASK ((((mp_digit)1)<<((mp_digit)MP_DIGIT_BIT))-((mp_digit)1)) #define MP_DIGIT_MAX MP_MASK -/* equalities */ +/* Primality generation flags */ +#define MP_PRIME_BBS 0x0001 /* BBS style prime */ +#define MP_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */ +#define MP_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */ + +#define LTM_PRIME_BBS (MP_DEPRECATED_PRAGMA("LTM_PRIME_BBS has been deprecated, use MP_PRIME_BBS") MP_PRIME_BBS) +#define LTM_PRIME_SAFE (MP_DEPRECATED_PRAGMA("LTM_PRIME_SAFE has been deprecated, use MP_PRIME_SAFE") MP_PRIME_SAFE) +#define LTM_PRIME_2MSB_ON (MP_DEPRECATED_PRAGMA("LTM_PRIME_2MSB_ON has been deprecated, use MP_PRIME_2MSB_ON") MP_PRIME_2MSB_ON) + +#ifdef MP_USE_ENUMS +typedef enum { + MP_ZPOS = 0, /* positive */ + MP_NEG = 1 /* negative */ +} mp_sign; +typedef enum { + MP_LT = -1, /* less than */ + MP_EQ = 0, /* equal */ + MP_GT = 1 /* greater than */ +} mp_ord; +typedef enum { + MP_NO = 0, + MP_YES = 1 +} mp_bool; +typedef enum { + MP_OKAY = 0, /* no error */ + MP_ERR = -1, /* unknown error */ + MP_MEM = -2, /* out of mem */ + MP_VAL = -3, /* invalid input */ + MP_ITER = -4, /* maximum iterations reached */ + MP_BUF = -5 /* buffer overflow, supplied buffer too small */ +} mp_err; +typedef enum { + MP_LSB_FIRST = -1, + MP_MSB_FIRST = 1 +} mp_order; +typedef enum { + MP_LITTLE_ENDIAN = -1, + MP_NATIVE_ENDIAN = 0, + MP_BIG_ENDIAN = 1 +} mp_endian; +#else +typedef int mp_sign; +#define MP_ZPOS 0 /* positive integer */ +#define MP_NEG 1 /* negative */ +typedef int mp_ord; #define MP_LT -1 /* less than */ #define MP_EQ 0 /* equal to */ #define MP_GT 1 /* greater than */ - -#define MP_ZPOS 0 /* positive integer */ -#define MP_NEG 1 /* negative */ - -#define MP_OKAY 0 /* ok result */ +typedef int mp_bool; +#define MP_YES 1 +#define MP_NO 0 +typedef int mp_err; +#define MP_OKAY 0 /* no error */ +#define MP_ERR -1 /* unknown error */ #define MP_MEM -2 /* out of mem */ #define MP_VAL -3 /* invalid input */ -#define MP_RANGE MP_VAL +#define MP_RANGE (MP_DEPRECATED_PRAGMA("MP_RANGE has been deprecated in favor of MP_VAL") MP_VAL) +#define MP_ITER -4 /* maximum iterations reached */ +#define MP_BUF -5 /* buffer overflow, supplied buffer too small */ +typedef int mp_order; +#define MP_LSB_FIRST -1 +#define MP_MSB_FIRST 1 +typedef int mp_endian; +#define MP_LITTLE_ENDIAN -1 +#define MP_NATIVE_ENDIAN 0 +#define MP_BIG_ENDIAN 1 +#endif -#define MP_YES 1 /* yes response */ -#define MP_NO 0 /* no response */ +/* tunable cutoffs */ -/* Primality generation flags */ -#define LTM_PRIME_BBS 0x0001 /* BBS style prime */ -#define LTM_PRIME_SAFE 0x0002 /* Safe prime (p-1)/2 == prime */ -#define LTM_PRIME_2MSB_ON 0x0008 /* force 2nd MSB to 1 */ - -typedef int mp_err; - -/* you'll have to tune these... */ -extern int KARATSUBA_MUL_CUTOFF, - KARATSUBA_SQR_CUTOFF, - TOOM_MUL_CUTOFF, - TOOM_SQR_CUTOFF; +#ifndef MP_FIXED_CUTOFFS +extern int +KARATSUBA_MUL_CUTOFF, +KARATSUBA_SQR_CUTOFF, +TOOM_MUL_CUTOFF, +TOOM_SQR_CUTOFF; +#endif /* define this to use lower memory usage routines (exptmods mostly) */ /* #define MP_LOW_MEM */ /* default precision */ #ifndef MP_PREC - #ifndef MP_LOW_MEM - #define MP_PREC 32 /* default digits of precision */ - #else - #define MP_PREC 8 /* default digits of precision */ - #endif +# ifndef MP_LOW_MEM +# define PRIVATE_MP_PREC 32 /* default digits of precision */ +# elif defined(MP_8BIT) +# define PRIVATE_MP_PREC 16 /* default digits of precision */ +# else +# define PRIVATE_MP_PREC 8 /* default digits of precision */ +# endif +# define MP_PREC (MP_DEPRECATED_PRAGMA("MP_PREC is an internal macro") PRIVATE_MP_PREC) #endif /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER_DIGIT*2) */ -#define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGIT_BIT + 1)) +#define PRIVATE_MP_WARRAY (int)(1uLL << (((CHAR_BIT * sizeof(private_mp_word)) - (2 * MP_DIGIT_BIT)) + 1)) +#define MP_WARRAY (MP_DEPRECATED_PRAGMA("MP_WARRAY is an internal macro") PRIVATE_MP_WARRAY) + +#if defined(__GNUC__) && __GNUC__ >= 4 +# define MP_NULL_TERMINATED __attribute__((sentinel)) +#else +# define MP_NULL_TERMINATED +#endif + +/* + * MP_WUR - warn unused result + * --------------------------- + * + * The result of functions annotated with MP_WUR must be + * checked and cannot be ignored. + * + * Most functions in libtommath return an error code. + * This error code must be checked in order to prevent crashes or invalid + * results. + * + * If you still want to avoid the error checks for quick and dirty programs + * without robustness guarantees, you can `#define MP_WUR` before including + * tommath.h, disabling the warnings. + */ +#ifndef MP_WUR +# if defined(__GNUC__) && __GNUC__ >= 4 +# define MP_WUR __attribute__((warn_unused_result)) +# else +# define MP_WUR +# endif +#endif + +#if defined(__GNUC__) && (__GNUC__ * 100 + __GNUC_MINOR__ >= 405) +# define MP_DEPRECATED(x) __attribute__((deprecated("replaced by " #x))) +# define PRIVATE_MP_DEPRECATED_PRAGMA(s) _Pragma(#s) +# define MP_DEPRECATED_PRAGMA(s) PRIVATE_MP_DEPRECATED_PRAGMA(GCC warning s) +#elif defined(_MSC_VER) && _MSC_VER >= 1500 +# define MP_DEPRECATED(x) __declspec(deprecated("replaced by " #x)) +# define MP_DEPRECATED_PRAGMA(s) __pragma(message(s)) +#else +# define MP_DEPRECATED(s) +# define MP_DEPRECATED_PRAGMA(s) +#endif + +#define DIGIT_BIT (MP_DEPRECATED_PRAGMA("DIGIT_BIT macro is deprecated, MP_DIGIT_BIT instead") MP_DIGIT_BIT) +#define USED(m) (MP_DEPRECATED_PRAGMA("USED macro is deprecated, use z->used instead") (m)->used) +#define DIGIT(m, k) (MP_DEPRECATED_PRAGMA("DIGIT macro is deprecated, use z->dp instead") (m)->dp[(k)]) +#define SIGN(m) (MP_DEPRECATED_PRAGMA("SIGN macro is deprecated, use z->sign instead") (m)->sign) /* the infamous mp_int structure */ typedef struct { - int used, alloc, sign; - mp_digit *dp; + int used, alloc; + mp_sign sign; + mp_digit *dp; } mp_int; /* callback for mp_prime_random, should fill dst with random bytes and return how many read [upto len] */ -typedef int ltm_prime_callback(unsigned char *dst, int len, void *dat); +typedef int private_mp_prime_callback(unsigned char *dst, int len, void *dat); +typedef private_mp_prime_callback MP_DEPRECATED(mp_rand_source) ltm_prime_callback; - -#define USED(m) ((m)->used) -#define DIGIT(m,k) ((m)->dp[(k)]) -#define SIGN(m) ((m)->sign) - -/* error code to const char* string */ -const char *mp_error_to_string(int code); +/* error code to char* string */ +const char *mp_error_to_string(mp_err code) MP_WUR; /* ---> init and deinit bignum functions <--- */ /* init a bignum */ -int mp_init(mp_int *a); +mp_err mp_init(mp_int *a) MP_WUR; /* free a bignum */ void mp_clear(mp_int *a); /* init a null terminated series of arguments */ -int mp_init_multi(mp_int *mp, ...); +mp_err mp_init_multi(mp_int *mp, ...) MP_NULL_TERMINATED MP_WUR; /* clear a null terminated series of arguments */ -void mp_clear_multi(mp_int *mp, ...); +void mp_clear_multi(mp_int *mp, ...) MP_NULL_TERMINATED; /* exchange two ints */ void mp_exch(mp_int *a, mp_int *b); /* shrink ram required for a bignum */ -int mp_shrink(mp_int *a); +mp_err mp_shrink(mp_int *a) MP_WUR; /* grow an int to a given size */ -int mp_grow(mp_int *a, int size); +mp_err mp_grow(mp_int *a, int size) MP_WUR; /* init to a given number of digits */ -int mp_init_size(mp_int *a, int size); +mp_err mp_init_size(mp_int *a, int size) MP_WUR; /* ---> Basic Manipulations <--- */ #define mp_iszero(a) (((a)->used == 0) ? MP_YES : MP_NO) -#define mp_iseven(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 0)) ? MP_YES : MP_NO) -#define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? MP_YES : MP_NO) -#define mp_isneg(a) (((a)->sign) ? MP_YES : MP_NO) +mp_bool mp_iseven(const mp_int *a) MP_WUR; +mp_bool mp_isodd(const mp_int *a) MP_WUR; +#define mp_isneg(a) (((a)->sign != MP_ZPOS) ? MP_YES : MP_NO) /* set to zero */ void mp_zero(mp_int *a); -/* set to zero, multi */ -void mp_zero_multi(mp_int *a, ...); +/* get and set doubles */ +double mp_get_double(const mp_int *a) MP_WUR; +mp_err mp_set_double(mp_int *a, double b) MP_WUR; -/* set to a digit */ +/* get integer, set integer and init with integer (int32_t) */ +int32_t mp_get_i32(const mp_int *a) MP_WUR; +void mp_set_i32(mp_int *a, int32_t b); +mp_err mp_init_i32(mp_int *a, int32_t b) MP_WUR; + +/* get integer, set integer and init with integer, behaves like two complement for negative numbers (uint32_t) */ +#define mp_get_u32(a) ((uint32_t)mp_get_i32(a)) +void mp_set_u32(mp_int *a, uint32_t b); +mp_err mp_init_u32(mp_int *a, uint32_t b) MP_WUR; + +/* get integer, set integer and init with integer (int64_t) */ +int64_t mp_get_i64(const mp_int *a) MP_WUR; +void mp_set_i64(mp_int *a, int64_t b); +mp_err mp_init_i64(mp_int *a, int64_t b) MP_WUR; + +/* get integer, set integer and init with integer, behaves like two complement for negative numbers (uint64_t) */ +#define mp_get_u64(a) ((uint64_t)mp_get_i64(a)) +void mp_set_u64(mp_int *a, uint64_t b); +mp_err mp_init_u64(mp_int *a, uint64_t b) MP_WUR; + +/* get magnitude */ +uint32_t mp_get_mag_u32(const mp_int *a) MP_WUR; +uint64_t mp_get_mag_u64(const mp_int *a) MP_WUR; +unsigned long mp_get_mag_ul(const mp_int *a) MP_WUR; +unsigned long long mp_get_mag_ull(const mp_int *a) MP_WUR; + +/* get integer, set integer (long) */ +long mp_get_l(const mp_int *a) MP_WUR; +void mp_set_l(mp_int *a, long b); +mp_err mp_init_l(mp_int *a, long b) MP_WUR; + +/* get integer, set integer (unsigned long) */ +#define mp_get_ul(a) ((unsigned long)mp_get_l(a)) +void mp_set_ul(mp_int *a, unsigned long b); +mp_err mp_init_ul(mp_int *a, unsigned long b) MP_WUR; + +/* get integer, set integer (long long) */ +long long mp_get_ll(const mp_int *a) MP_WUR; +void mp_set_ll(mp_int *a, long long b); +mp_err mp_init_ll(mp_int *a, long long b) MP_WUR; + +/* get integer, set integer (unsigned long long) */ +#define mp_get_ull(a) ((unsigned long long)mp_get_ll(a)) +void mp_set_ull(mp_int *a, unsigned long long b); +mp_err mp_init_ull(mp_int *a, unsigned long long b) MP_WUR; + +/* set to single unsigned digit, up to MP_DIGIT_MAX */ void mp_set(mp_int *a, mp_digit b); +mp_err mp_init_set(mp_int *a, mp_digit b) MP_WUR; -/* set a 32-bit const */ -int mp_set_int(mp_int *a, unsigned long b); - -/* get a 32-bit value */ -unsigned long mp_get_int(mp_int * a); - -/* initialize and set a digit */ -int mp_init_set (mp_int * a, mp_digit b); - -/* initialize and set 32-bit value */ -int mp_init_set_int (mp_int * a, unsigned long b); +/* get integer, set integer and init with integer (deprecated) */ +MP_DEPRECATED(mp_get_mag_u32/mp_get_u32) unsigned long mp_get_int(const mp_int *a) MP_WUR; +MP_DEPRECATED(mp_get_mag_ul/mp_get_ul) unsigned long mp_get_long(const mp_int *a) MP_WUR; +MP_DEPRECATED(mp_get_mag_ull/mp_get_ull) unsigned long long mp_get_long_long(const mp_int *a) MP_WUR; +MP_DEPRECATED(mp_set_ul) mp_err mp_set_int(mp_int *a, unsigned long b); +MP_DEPRECATED(mp_set_ul) mp_err mp_set_long(mp_int *a, unsigned long b); +MP_DEPRECATED(mp_set_ull) mp_err mp_set_long_long(mp_int *a, unsigned long long b); +MP_DEPRECATED(mp_init_ul) mp_err mp_init_set_int(mp_int *a, unsigned long b) MP_WUR; /* copy, b = a */ -int mp_copy(mp_int *a, mp_int *b); +mp_err mp_copy(const mp_int *a, mp_int *b) MP_WUR; /* inits and copies, a = b */ -int mp_init_copy(mp_int *a, mp_int *b); +mp_err mp_init_copy(mp_int *a, const mp_int *b) MP_WUR; /* trim unused digits */ void mp_clamp(mp_int *a); + +/* export binary data */ +MP_DEPRECATED(mp_pack) mp_err mp_export(void *rop, size_t *countp, int order, size_t size, + int endian, size_t nails, const mp_int *op) MP_WUR; + +/* import binary data */ +MP_DEPRECATED(mp_unpack) mp_err mp_import(mp_int *rop, size_t count, int order, + size_t size, int endian, size_t nails, + const void *op) MP_WUR; + +/* unpack binary data */ +mp_err mp_unpack(mp_int *rop, size_t count, mp_order order, size_t size, mp_endian endian, + size_t nails, const void *op) MP_WUR; + +/* pack binary data */ +size_t mp_pack_count(const mp_int *a, size_t nails, size_t size) MP_WUR; +mp_err mp_pack(void *rop, size_t maxcount, size_t *written, mp_order order, size_t size, + mp_endian endian, size_t nails, const mp_int *op) MP_WUR; + /* ---> digit manipulation <--- */ /* right shift by "b" digits */ void mp_rshd(mp_int *a, int b); /* left shift by "b" digits */ -int mp_lshd(mp_int *a, int b); +mp_err mp_lshd(mp_int *a, int b) MP_WUR; -/* c = a / 2**b */ -int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d); +/* c = a / 2**b, implemented as c = a >> b */ +mp_err mp_div_2d(const mp_int *a, int b, mp_int *c, mp_int *d) MP_WUR; /* b = a/2 */ -int mp_div_2(mp_int *a, mp_int *b); +mp_err mp_div_2(const mp_int *a, mp_int *b) MP_WUR; -/* c = a * 2**b */ -int mp_mul_2d(mp_int *a, int b, mp_int *c); +/* a/3 => 3c + d == a */ +mp_err mp_div_3(const mp_int *a, mp_int *c, mp_digit *d) MP_WUR; + +/* c = a * 2**b, implemented as c = a << b */ +mp_err mp_mul_2d(const mp_int *a, int b, mp_int *c) MP_WUR; /* b = a*2 */ -int mp_mul_2(mp_int *a, mp_int *b); +mp_err mp_mul_2(const mp_int *a, mp_int *b) MP_WUR; -/* c = a mod 2**d */ -int mp_mod_2d(mp_int *a, int b, mp_int *c); +/* c = a mod 2**b */ +mp_err mp_mod_2d(const mp_int *a, int b, mp_int *c) MP_WUR; /* computes a = 2**b */ -int mp_2expt(mp_int *a, int b); +mp_err mp_2expt(mp_int *a, int b) MP_WUR; /* Counts the number of lsbs which are zero before the first zero bit */ -int mp_cnt_lsb(mp_int *a); +int mp_cnt_lsb(const mp_int *a) MP_WUR; /* I Love Earth! */ -/* makes a pseudo-random int of a given size */ -int mp_rand(mp_int *a, int digits); +/* makes a pseudo-random mp_int of a given size */ +mp_err mp_rand(mp_int *a, int digits) MP_WUR; +/* makes a pseudo-random small int of a given size */ +MP_DEPRECATED(mp_rand) mp_err mp_rand_digit(mp_digit *r) MP_WUR; +/* use custom random data source instead of source provided the platform */ +void mp_rand_source(mp_err(*source)(void *out, size_t size)); + +#ifdef MP_PRNG_ENABLE_LTM_RNG +# warning MP_PRNG_ENABLE_LTM_RNG has been deprecated, use mp_rand_source instead. +/* A last resort to provide random data on systems without any of the other + * implemented ways to gather entropy. + * It is compatible with `rng_get_bytes()` from libtomcrypt so you could + * provide that one and then set `ltm_rng = rng_get_bytes;` */ +extern unsigned long (*ltm_rng)(unsigned char *out, unsigned long outlen, void (*callback)(void)); +extern void (*ltm_rng_callback)(void); +#endif /* ---> binary operations <--- */ -/* c = a XOR b */ -int mp_xor(mp_int *a, mp_int *b, mp_int *c); -/* c = a OR b */ -int mp_or(mp_int *a, mp_int *b, mp_int *c); +/* Checks the bit at position b and returns MP_YES + * if the bit is 1, MP_NO if it is 0 and MP_VAL + * in case of error + */ +MP_DEPRECATED(s_mp_get_bit) int mp_get_bit(const mp_int *a, int b) MP_WUR; -/* c = a AND b */ -int mp_and(mp_int *a, mp_int *b, mp_int *c); +/* c = a XOR b (two complement) */ +MP_DEPRECATED(mp_xor) mp_err mp_tc_xor(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +mp_err mp_xor(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; + +/* c = a OR b (two complement) */ +MP_DEPRECATED(mp_or) mp_err mp_tc_or(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +mp_err mp_or(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; + +/* c = a AND b (two complement) */ +MP_DEPRECATED(mp_and) mp_err mp_tc_and(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +mp_err mp_and(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; + +/* b = ~a (bitwise not, two complement) */ +mp_err mp_complement(const mp_int *a, mp_int *b) MP_WUR; + +/* right shift with sign extension */ +MP_DEPRECATED(mp_signed_rsh) mp_err mp_tc_div_2d(const mp_int *a, int b, mp_int *c) MP_WUR; +mp_err mp_signed_rsh(const mp_int *a, int b, mp_int *c) MP_WUR; /* ---> Basic arithmetic <--- */ /* b = -a */ -int mp_neg(mp_int *a, mp_int *b); +mp_err mp_neg(const mp_int *a, mp_int *b) MP_WUR; /* b = |a| */ -int mp_abs(mp_int *a, mp_int *b); +mp_err mp_abs(const mp_int *a, mp_int *b) MP_WUR; /* compare a to b */ -int mp_cmp(mp_int *a, mp_int *b); +mp_ord mp_cmp(const mp_int *a, const mp_int *b) MP_WUR; /* compare |a| to |b| */ -int mp_cmp_mag(mp_int *a, mp_int *b); +mp_ord mp_cmp_mag(const mp_int *a, const mp_int *b) MP_WUR; /* c = a + b */ -int mp_add(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_add(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* c = a - b */ -int mp_sub(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_sub(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* c = a * b */ -int mp_mul(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_mul(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* b = a*a */ -int mp_sqr(mp_int *a, mp_int *b); +mp_err mp_sqr(const mp_int *a, mp_int *b) MP_WUR; /* a/b => cb + d == a */ -int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d); +mp_err mp_div(const mp_int *a, const mp_int *b, mp_int *c, mp_int *d) MP_WUR; /* c = a mod b, 0 <= c < b */ -int mp_mod(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_mod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; + +/* Increment "a" by one like "a++". Changes input! */ +mp_err mp_incr(mp_int *a) MP_WUR; + +/* Decrement "a" by one like "a--". Changes input! */ +mp_err mp_decr(mp_int *a) MP_WUR; /* ---> single digit functions <--- */ /* compare against a single digit */ -int mp_cmp_d(mp_int *a, mp_digit b); +mp_ord mp_cmp_d(const mp_int *a, mp_digit b) MP_WUR; /* c = a + b */ -int mp_add_d(mp_int *a, mp_digit b, mp_int *c); +mp_err mp_add_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR; /* c = a - b */ -int mp_sub_d(mp_int *a, mp_digit b, mp_int *c); +mp_err mp_sub_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR; /* c = a * b */ -int mp_mul_d(mp_int *a, mp_digit b, mp_int *c); +mp_err mp_mul_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR; /* a/b => cb + d == a */ -int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d); - -/* a/3 => 3c + d == a */ -int mp_div_3(mp_int *a, mp_int *c, mp_digit *d); - -/* c = a**b */ -int mp_expt_d(mp_int *a, mp_digit b, mp_int *c); +mp_err mp_div_d(const mp_int *a, mp_digit b, mp_int *c, mp_digit *d) MP_WUR; /* c = a mod b, 0 <= c < b */ -int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c); +mp_err mp_mod_d(const mp_int *a, mp_digit b, mp_digit *c) MP_WUR; /* ---> number theory <--- */ /* d = a + b (mod c) */ -int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); +mp_err mp_addmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR; /* d = a - b (mod c) */ -int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); +mp_err mp_submod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR; /* d = a * b (mod c) */ -int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); +mp_err mp_mulmod(const mp_int *a, const mp_int *b, const mp_int *c, mp_int *d) MP_WUR; /* c = a * a (mod b) */ -int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_sqrmod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* c = 1/a (mod b) */ -int mp_invmod(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_invmod(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* c = (a, b) */ -int mp_gcd(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_gcd(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* produces value such that U1*a + U2*b = U3 */ -int mp_exteuclid(mp_int *a, mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3); +mp_err mp_exteuclid(const mp_int *a, const mp_int *b, mp_int *U1, mp_int *U2, mp_int *U3) MP_WUR; /* c = [a, b] or (a*b)/(a, b) */ -int mp_lcm(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_lcm(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; /* finds one of the b'th root of a, such that |c|**b <= |a| * * returns error if a < 0 and b is even */ -int mp_n_root(mp_int *a, mp_digit b, mp_int *c); +mp_err mp_root_u32(const mp_int *a, uint32_t b, mp_int *c) MP_WUR; +MP_DEPRECATED(mp_root_u32) mp_err mp_n_root(const mp_int *a, mp_digit b, mp_int *c) MP_WUR; +MP_DEPRECATED(mp_root_u32) mp_err mp_n_root_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) MP_WUR; /* special sqrt algo */ -int mp_sqrt(mp_int *arg, mp_int *ret); +mp_err mp_sqrt(const mp_int *arg, mp_int *ret) MP_WUR; + +/* special sqrt (mod prime) */ +mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret) MP_WUR; /* is number a square? */ -int mp_is_square(mp_int *arg, int *ret); +mp_err mp_is_square(const mp_int *arg, mp_bool *ret) MP_WUR; /* computes the jacobi c = (a | n) (or Legendre if b is prime) */ -int mp_jacobi(mp_int *a, mp_int *n, int *c); +MP_DEPRECATED(mp_kronecker) mp_err mp_jacobi(const mp_int *a, const mp_int *n, int *c) MP_WUR; + +/* computes the Kronecker symbol c = (a | p) (like jacobi() but with {a,p} in Z */ +mp_err mp_kronecker(const mp_int *a, const mp_int *p, int *c) MP_WUR; /* used to setup the Barrett reduction for a given modulus b */ -int mp_reduce_setup(mp_int *a, mp_int *b); +mp_err mp_reduce_setup(mp_int *a, const mp_int *b) MP_WUR; /* Barrett Reduction, computes a (mod b) with a precomputed value c * - * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely - * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code]. + * Assumes that 0 < x <= m*m, note if 0 > x > -(m*m) then you can merely + * compute the reduction as -1 * mp_reduce(mp_abs(x)) [pseudo code]. */ -int mp_reduce(mp_int *a, mp_int *b, mp_int *c); +mp_err mp_reduce(mp_int *x, const mp_int *m, const mp_int *mu) MP_WUR; /* setups the montgomery reduction */ -int mp_montgomery_setup(mp_int *a, mp_digit *mp); +mp_err mp_montgomery_setup(const mp_int *n, mp_digit *rho) MP_WUR; /* computes a = B**n mod b without division or multiplication useful for * normalizing numbers in a Montgomery system. */ -int mp_montgomery_calc_normalization(mp_int *a, mp_int *b); +mp_err mp_montgomery_calc_normalization(mp_int *a, const mp_int *b) MP_WUR; /* computes x/R == x (mod N) via Montgomery Reduction */ -int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); +mp_err mp_montgomery_reduce(mp_int *x, const mp_int *n, mp_digit rho) MP_WUR; /* returns 1 if a is a valid DR modulus */ -int mp_dr_is_modulus(mp_int *a); +mp_bool mp_dr_is_modulus(const mp_int *a) MP_WUR; /* sets the value of "d" required for mp_dr_reduce */ -void mp_dr_setup(mp_int *a, mp_digit *d); +void mp_dr_setup(const mp_int *a, mp_digit *d); -/* reduces a modulo b using the Diminished Radix method */ -int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp); +/* reduces a modulo n using the Diminished Radix method */ +mp_err mp_dr_reduce(mp_int *x, const mp_int *n, mp_digit k) MP_WUR; /* returns true if a can be reduced with mp_reduce_2k */ -int mp_reduce_is_2k(mp_int *a); +mp_bool mp_reduce_is_2k(const mp_int *a) MP_WUR; /* determines k value for 2k reduction */ -int mp_reduce_2k_setup(mp_int *a, mp_digit *d); +mp_err mp_reduce_2k_setup(const mp_int *a, mp_digit *d) MP_WUR; /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ -int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit d); +mp_err mp_reduce_2k(mp_int *a, const mp_int *n, mp_digit d) MP_WUR; /* returns true if a can be reduced with mp_reduce_2k_l */ -int mp_reduce_is_2k_l(mp_int *a); +mp_bool mp_reduce_is_2k_l(const mp_int *a) MP_WUR; /* determines k value for 2k reduction */ -int mp_reduce_2k_setup_l(mp_int *a, mp_int *d); +mp_err mp_reduce_2k_setup_l(const mp_int *a, mp_int *d) MP_WUR; /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */ -int mp_reduce_2k_l(mp_int *a, mp_int *n, mp_int *d); +mp_err mp_reduce_2k_l(mp_int *a, const mp_int *n, const mp_int *d) MP_WUR; -/* d = a**b (mod c) */ -int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d); +/* Y = G**X (mod P) */ +mp_err mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y) MP_WUR; /* ---> Primes <--- */ /* number of primes */ #ifdef MP_8BIT - #define PRIME_SIZE 31 +# define PRIVATE_MP_PRIME_TAB_SIZE 31 #else - #define PRIME_SIZE 256 +# define PRIVATE_MP_PRIME_TAB_SIZE 256 #endif +#define PRIME_SIZE (MP_DEPRECATED_PRAGMA("PRIME_SIZE has been made internal") PRIVATE_MP_PRIME_TAB_SIZE) /* table of first PRIME_SIZE primes */ -extern const mp_digit ltm_prime_tab[]; +MP_DEPRECATED(internal) extern const mp_digit ltm_prime_tab[PRIVATE_MP_PRIME_TAB_SIZE]; /* result=1 if a is divisible by one of the first PRIME_SIZE primes */ -int mp_prime_is_divisible(mp_int *a, int *result); +MP_DEPRECATED(mp_prime_is_prime) mp_err mp_prime_is_divisible(const mp_int *a, mp_bool *result) MP_WUR; /* performs one Fermat test of "a" using base "b". * Sets result to 0 if composite or 1 if probable prime */ -int mp_prime_fermat(mp_int *a, mp_int *b, int *result); +mp_err mp_prime_fermat(const mp_int *a, const mp_int *b, mp_bool *result) MP_WUR; /* performs one Miller-Rabin test of "a" using base "b". * Sets result to 0 if composite or 1 if probable prime */ -int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result); +mp_err mp_prime_miller_rabin(const mp_int *a, const mp_int *b, mp_bool *result) MP_WUR; /* This gives [for a given bit size] the number of trials required * such that Miller-Rabin gives a prob of failure lower than 2^-96 */ -int mp_prime_rabin_miller_trials(int size); +int mp_prime_rabin_miller_trials(int size) MP_WUR; -/* performs t rounds of Miller-Rabin on "a" using the first - * t prime bases. Also performs an initial sieve of trial +/* performs one strong Lucas-Selfridge test of "a". + * Sets result to 0 if composite or 1 if probable prime + */ +mp_err mp_prime_strong_lucas_selfridge(const mp_int *a, mp_bool *result) MP_WUR; + +/* performs one Frobenius test of "a" as described by Paul Underwood. + * Sets result to 0 if composite or 1 if probable prime + */ +mp_err mp_prime_frobenius_underwood(const mp_int *N, mp_bool *result) MP_WUR; + +/* performs t random rounds of Miller-Rabin on "a" additional to + * bases 2 and 3. Also performs an initial sieve of trial * division. Determines if "a" is prime with probability * of error no more than (1/4)**t. + * Both a strong Lucas-Selfridge to complete the BPSW test + * and a separate Frobenius test are available at compile time. + * With t<0 a deterministic test is run for primes up to + * 318665857834031151167461. With t<13 (abs(t)-13) additional + * tests with sequential small primes are run starting at 43. + * Is Fips 186.4 compliant if called with t as computed by + * mp_prime_rabin_miller_trials(); * * Sets result to 1 if probably prime, 0 otherwise */ -int mp_prime_is_prime(mp_int *a, int t, int *result); +mp_err mp_prime_is_prime(const mp_int *a, int t, mp_bool *result) MP_WUR; /* finds the next prime after the number "a" using "t" trials * of Miller-Rabin. * * bbs_style = 1 means the prime must be congruent to 3 mod 4 */ -int mp_prime_next_prime(mp_int *a, int t, int bbs_style); +mp_err mp_prime_next_prime(mp_int *a, int t, int bbs_style) MP_WUR; /* makes a truly random prime of a given size (bytes), * call with bbs = 1 if you want it to be congruent to 3 mod 4 @@ -502,89 +697,85 @@ int mp_prime_next_prime(mp_int *a, int t, int bbs_style); * * The prime generated will be larger than 2^(8*size). */ -#define mp_prime_random(a, t, size, bbs, cb, dat) mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?LTM_PRIME_BBS:0, cb, dat) +#define mp_prime_random(a, t, size, bbs, cb, dat) (MP_DEPRECATED_PRAGMA("mp_prime_random has been deprecated, use mp_prime_rand instead") mp_prime_random_ex(a, t, ((size) * 8) + 1, (bbs==1)?MP_PRIME_BBS:0, cb, dat)) /* makes a truly random prime of a given size (bits), * * Flags are as follows: * - * LTM_PRIME_BBS - make prime congruent to 3 mod 4 - * LTM_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies LTM_PRIME_BBS) - * LTM_PRIME_2MSB_OFF - make the 2nd highest bit zero - * LTM_PRIME_2MSB_ON - make the 2nd highest bit one + * MP_PRIME_BBS - make prime congruent to 3 mod 4 + * MP_PRIME_SAFE - make sure (p-1)/2 is prime as well (implies MP_PRIME_BBS) + * MP_PRIME_2MSB_ON - make the 2nd highest bit one * * You have to supply a callback which fills in a buffer with random bytes. "dat" is a parameter you can * have passed to the callback (e.g. a state or something). This function doesn't use "dat" itself * so it can be NULL * */ -int mp_prime_random_ex(mp_int *a, int t, int size, int flags, ltm_prime_callback cb, void *dat); +MP_DEPRECATED(mp_prime_rand) mp_err mp_prime_random_ex(mp_int *a, int t, int size, int flags, + private_mp_prime_callback cb, void *dat) MP_WUR; +mp_err mp_prime_rand(mp_int *a, int t, int size, int flags) MP_WUR; -int mp_find_prime(mp_int *a, int t); +/* Integer logarithm to integer base */ +mp_err mp_log_u32(const mp_int *a, uint32_t base, uint32_t *c) MP_WUR; + +/* c = a**b */ +mp_err mp_expt_u32(const mp_int *a, uint32_t b, mp_int *c) MP_WUR; +MP_DEPRECATED(mp_expt_u32) mp_err mp_expt_d(const mp_int *a, mp_digit b, mp_int *c) MP_WUR; +MP_DEPRECATED(mp_expt_u32) mp_err mp_expt_d_ex(const mp_int *a, mp_digit b, mp_int *c, int fast) MP_WUR; /* ---> radix conversion <--- */ -int mp_count_bits(mp_int *a); +int mp_count_bits(const mp_int *a) MP_WUR; -int mp_unsigned_bin_size(mp_int *a); -int mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c); -int mp_to_unsigned_bin(mp_int *a, unsigned char *b); -int mp_to_unsigned_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen); -int mp_signed_bin_size(mp_int *a); -int mp_read_signed_bin(mp_int *a, const unsigned char *b, int c); -int mp_to_signed_bin(mp_int *a, unsigned char *b); -int mp_to_signed_bin_n (mp_int * a, unsigned char *b, unsigned long *outlen); +MP_DEPRECATED(mp_ubin_size) int mp_unsigned_bin_size(const mp_int *a) MP_WUR; +MP_DEPRECATED(mp_from_ubin) mp_err mp_read_unsigned_bin(mp_int *a, const unsigned char *b, int c) MP_WUR; +MP_DEPRECATED(mp_to_ubin) mp_err mp_to_unsigned_bin(const mp_int *a, unsigned char *b) MP_WUR; +MP_DEPRECATED(mp_to_ubin) mp_err mp_to_unsigned_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) MP_WUR; -int mp_read_radix(mp_int *a, const char *str, int radix); -int mp_toradix(mp_int *a, char *str, int radix); -int mp_toradix_n(mp_int * a, char *str, int radix, int maxlen); -int mp_radix_size(mp_int *a, int radix, int *size); +MP_DEPRECATED(mp_sbin_size) int mp_signed_bin_size(const mp_int *a) MP_WUR; +MP_DEPRECATED(mp_from_sbin) mp_err mp_read_signed_bin(mp_int *a, const unsigned char *b, int c) MP_WUR; +MP_DEPRECATED(mp_to_sbin) mp_err mp_to_signed_bin(const mp_int *a, unsigned char *b) MP_WUR; +MP_DEPRECATED(mp_to_sbin) mp_err mp_to_signed_bin_n(const mp_int *a, unsigned char *b, unsigned long *outlen) MP_WUR; -int mp_fread(mp_int *a, int radix, FILE *stream); -int mp_fwrite(mp_int *a, int radix, FILE *stream); +size_t mp_ubin_size(const mp_int *a) MP_WUR; +mp_err mp_from_ubin(mp_int *a, const unsigned char *buf, size_t size) MP_WUR; +mp_err mp_to_ubin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) MP_WUR; -#define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len)) -#define mp_raw_size(mp) mp_signed_bin_size(mp) -#define mp_toraw(mp, str) mp_to_signed_bin((mp), (str)) -#define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len)) -#define mp_mag_size(mp) mp_unsigned_bin_size(mp) -#define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str)) +size_t mp_sbin_size(const mp_int *a) MP_WUR; +mp_err mp_from_sbin(mp_int *a, const unsigned char *buf, size_t size) MP_WUR; +mp_err mp_to_sbin(const mp_int *a, unsigned char *buf, size_t maxlen, size_t *written) MP_WUR; -#define mp_tobinary(M, S) mp_toradix((M), (S), 2) -#define mp_tooctal(M, S) mp_toradix((M), (S), 8) -#define mp_todecimal(M, S) mp_toradix((M), (S), 10) -#define mp_tohex(M, S) mp_toradix((M), (S), 16) +mp_err mp_read_radix(mp_int *a, const char *str, int radix) MP_WUR; +MP_DEPRECATED(mp_to_radix) mp_err mp_toradix(const mp_int *a, char *str, int radix) MP_WUR; +MP_DEPRECATED(mp_to_radix) mp_err mp_toradix_n(const mp_int *a, char *str, int radix, int maxlen) MP_WUR; +mp_err mp_to_radix(const mp_int *a, char *str, size_t maxlen, size_t *written, int radix) MP_WUR; +mp_err mp_radix_size(const mp_int *a, int radix, int *size) MP_WUR; -/* lowlevel functions, do not call! */ -int s_mp_add(mp_int *a, mp_int *b, mp_int *c); -int s_mp_sub(mp_int *a, mp_int *b, mp_int *c); -#define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1) -int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs); -int fast_s_mp_sqr(mp_int *a, mp_int *b); -int s_mp_sqr(mp_int *a, mp_int *b); -int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c); -int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c); -int mp_karatsuba_sqr(mp_int *a, mp_int *b); -int mp_toom_sqr(mp_int *a, mp_int *b); -int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c); -int mp_invmod_slow (mp_int * a, mp_int * b, mp_int * c); -int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp); -int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode); -int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int mode); -void bn_reverse(unsigned char *s, int len); +#ifndef MP_NO_FILE +mp_err mp_fread(mp_int *a, int radix, FILE *stream) MP_WUR; +mp_err mp_fwrite(const mp_int *a, int radix, FILE *stream) MP_WUR; +#endif -extern const char *mp_s_rmap; +#define mp_read_raw(mp, str, len) (MP_DEPRECATED_PRAGMA("replaced by mp_read_signed_bin") mp_read_signed_bin((mp), (str), (len))) +#define mp_raw_size(mp) (MP_DEPRECATED_PRAGMA("replaced by mp_signed_bin_size") mp_signed_bin_size(mp)) +#define mp_toraw(mp, str) (MP_DEPRECATED_PRAGMA("replaced by mp_to_signed_bin") mp_to_signed_bin((mp), (str))) +#define mp_read_mag(mp, str, len) (MP_DEPRECATED_PRAGMA("replaced by mp_read_unsigned_bin") mp_read_unsigned_bin((mp), (str), (len)) +#define mp_mag_size(mp) (MP_DEPRECATED_PRAGMA("replaced by mp_unsigned_bin_size") mp_unsigned_bin_size(mp)) +#define mp_tomag(mp, str) (MP_DEPRECATED_PRAGMA("replaced by mp_to_unsigned_bin") mp_to_unsigned_bin((mp), (str))) + +#define mp_tobinary(M, S) (MP_DEPRECATED_PRAGMA("replaced by mp_to_binary") mp_toradix((M), (S), 2)) +#define mp_tooctal(M, S) (MP_DEPRECATED_PRAGMA("replaced by mp_to_octal") mp_toradix((M), (S), 8)) +#define mp_todecimal(M, S) (MP_DEPRECATED_PRAGMA("replaced by mp_to_decimal") mp_toradix((M), (S), 10)) +#define mp_tohex(M, S) (MP_DEPRECATED_PRAGMA("replaced by mp_to_hex") mp_toradix((M), (S), 16)) + +#define mp_to_binary(M, S, N) mp_to_radix((M), (S), (N), NULL, 2) +#define mp_to_octal(M, S, N) mp_to_radix((M), (S), (N), NULL, 8) +#define mp_to_decimal(M, S, N) mp_to_radix((M), (S), (N), NULL, 10) +#define mp_to_hex(M, S, N) mp_to_radix((M), (S), (N), NULL, 16) #ifdef __cplusplus - } +} #endif #endif - - -/* $Source: /cvs/libtom/libtommath/tommath.h,v $ */ -/* $Revision: 1.8 $ */ -/* $Date: 2006/03/31 14:18:44 $ */ diff --git a/lib/hcrypto/libtommath/tommath.out b/lib/hcrypto/libtommath/tommath.out deleted file mode 100644 index 9f6261727..000000000 --- a/lib/hcrypto/libtommath/tommath.out +++ /dev/null @@ -1,139 +0,0 @@ -\BOOKMARK [0][-]{chapter.1}{Introduction}{} -\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1} -\BOOKMARK [2][-]{subsection.1.1.1}{What is Multiple Precision Arithmetic?}{section.1.1} -\BOOKMARK [2][-]{subsection.1.1.2}{The Need for Multiple Precision Arithmetic}{section.1.1} -\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1} -\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1} -\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1} -\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.2}{Precision Notation}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.3}{Algorithm Inputs and Outputs}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.4}{Mathematical Expressions}{section.1.3} -\BOOKMARK [2][-]{subsection.1.3.5}{Work Effort}{section.1.3} -\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1} -\BOOKMARK [1][-]{section.1.5}{Introduction to LibTomMath}{chapter.1} -\BOOKMARK [2][-]{subsection.1.5.1}{What is LibTomMath?}{section.1.5} -\BOOKMARK [2][-]{subsection.1.5.2}{Goals of LibTomMath}{section.1.5} -\BOOKMARK [1][-]{section.1.6}{Choice of LibTomMath}{chapter.1} -\BOOKMARK [2][-]{subsection.1.6.1}{Code Base}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.2}{API Simplicity}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.3}{Optimizations}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.4}{Portability and Stability}{section.1.6} -\BOOKMARK [2][-]{subsection.1.6.5}{Choice}{section.1.6} -\BOOKMARK [0][-]{chapter.2}{Getting Started}{} -\BOOKMARK [1][-]{section.2.1}{Library Basics}{chapter.2} -\BOOKMARK [1][-]{section.2.2}{What is a Multiple Precision Integer?}{chapter.2} -\BOOKMARK [2][-]{subsection.2.2.1}{The mp\137int Structure}{section.2.2} -\BOOKMARK [1][-]{section.2.3}{Argument Passing}{chapter.2} -\BOOKMARK [1][-]{section.2.4}{Return Values}{chapter.2} -\BOOKMARK [1][-]{section.2.5}{Initialization and Clearing}{chapter.2} -\BOOKMARK [2][-]{subsection.2.5.1}{Initializing an mp\137int}{section.2.5} -\BOOKMARK [2][-]{subsection.2.5.2}{Clearing an mp\137int}{section.2.5} -\BOOKMARK [1][-]{section.2.6}{Maintenance Algorithms}{chapter.2} -\BOOKMARK [2][-]{subsection.2.6.1}{Augmenting an mp\137int's Precision}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.2}{Initializing Variable Precision mp\137ints}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.3}{Multiple Integer Initializations and Clearings}{section.2.6} -\BOOKMARK [2][-]{subsection.2.6.4}{Clamping Excess Digits}{section.2.6} -\BOOKMARK [0][-]{chapter.3}{Basic Operations}{} -\BOOKMARK [1][-]{section.3.1}{Introduction}{chapter.3} -\BOOKMARK [1][-]{section.3.2}{Assigning Values to mp\137int Structures}{chapter.3} -\BOOKMARK [2][-]{subsection.3.2.1}{Copying an mp\137int}{section.3.2} -\BOOKMARK [2][-]{subsection.3.2.2}{Creating a Clone}{section.3.2} -\BOOKMARK [1][-]{section.3.3}{Zeroing an Integer}{chapter.3} -\BOOKMARK [1][-]{section.3.4}{Sign Manipulation}{chapter.3} -\BOOKMARK [2][-]{subsection.3.4.1}{Absolute Value}{section.3.4} -\BOOKMARK [2][-]{subsection.3.4.2}{Integer Negation}{section.3.4} -\BOOKMARK [1][-]{section.3.5}{Small Constants}{chapter.3} -\BOOKMARK [2][-]{subsection.3.5.1}{Setting Small Constants}{section.3.5} -\BOOKMARK [2][-]{subsection.3.5.2}{Setting Large Constants}{section.3.5} -\BOOKMARK [1][-]{section.3.6}{Comparisons}{chapter.3} -\BOOKMARK [2][-]{subsection.3.6.1}{Unsigned Comparisions}{section.3.6} -\BOOKMARK [2][-]{subsection.3.6.2}{Signed Comparisons}{section.3.6} -\BOOKMARK [0][-]{chapter.4}{Basic Arithmetic}{} -\BOOKMARK [1][-]{section.4.1}{Introduction}{chapter.4} -\BOOKMARK [1][-]{section.4.2}{Addition and Subtraction}{chapter.4} -\BOOKMARK [2][-]{subsection.4.2.1}{Low Level Addition}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.2}{Low Level Subtraction}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.3}{High Level Addition}{section.4.2} -\BOOKMARK [2][-]{subsection.4.2.4}{High Level Subtraction}{section.4.2} -\BOOKMARK [1][-]{section.4.3}{Bit and Digit Shifting}{chapter.4} -\BOOKMARK [2][-]{subsection.4.3.1}{Multiplication by Two}{section.4.3} -\BOOKMARK [2][-]{subsection.4.3.2}{Division by Two}{section.4.3} -\BOOKMARK [1][-]{section.4.4}{Polynomial Basis Operations}{chapter.4} -\BOOKMARK [2][-]{subsection.4.4.1}{Multiplication by x}{section.4.4} -\BOOKMARK [2][-]{subsection.4.4.2}{Division by x}{section.4.4} -\BOOKMARK [1][-]{section.4.5}{Powers of Two}{chapter.4} -\BOOKMARK [2][-]{subsection.4.5.1}{Multiplication by Power of Two}{section.4.5} -\BOOKMARK [2][-]{subsection.4.5.2}{Division by Power of Two}{section.4.5} -\BOOKMARK [2][-]{subsection.4.5.3}{Remainder of Division by Power of Two}{section.4.5} -\BOOKMARK [0][-]{chapter.5}{Multiplication and Squaring}{} -\BOOKMARK [1][-]{section.5.1}{The Multipliers}{chapter.5} -\BOOKMARK [1][-]{section.5.2}{Multiplication}{chapter.5} -\BOOKMARK [2][-]{subsection.5.2.1}{The Baseline Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.2}{Faster Multiplication by the ``Comba'' Method}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.3}{Polynomial Basis Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.4}{Karatsuba Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.5}{Toom-Cook 3-Way Multiplication}{section.5.2} -\BOOKMARK [2][-]{subsection.5.2.6}{Signed Multiplication}{section.5.2} -\BOOKMARK [1][-]{section.5.3}{Squaring}{chapter.5} -\BOOKMARK [2][-]{subsection.5.3.1}{The Baseline Squaring Algorithm}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.2}{Faster Squaring by the ``Comba'' Method}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.3}{Polynomial Basis Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.4}{Karatsuba Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.5}{Toom-Cook Squaring}{section.5.3} -\BOOKMARK [2][-]{subsection.5.3.6}{High Level Squaring}{section.5.3} -\BOOKMARK [0][-]{chapter.6}{Modular Reduction}{} -\BOOKMARK [1][-]{section.6.1}{Basics of Modular Reduction}{chapter.6} -\BOOKMARK [1][-]{section.6.2}{The Barrett Reduction}{chapter.6} -\BOOKMARK [2][-]{subsection.6.2.1}{Fixed Point Arithmetic}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.2}{Choosing a Radix Point}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.3}{Trimming the Quotient}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.4}{Trimming the Residue}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.5}{The Barrett Algorithm}{section.6.2} -\BOOKMARK [2][-]{subsection.6.2.6}{The Barrett Setup Algorithm}{section.6.2} -\BOOKMARK [1][-]{section.6.3}{The Montgomery Reduction}{chapter.6} -\BOOKMARK [2][-]{subsection.6.3.1}{Digit Based Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.2}{Baseline Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.3}{Faster ``Comba'' Montgomery Reduction}{section.6.3} -\BOOKMARK [2][-]{subsection.6.3.4}{Montgomery Setup}{section.6.3} -\BOOKMARK [1][-]{section.6.4}{The Diminished Radix Algorithm}{chapter.6} -\BOOKMARK [2][-]{subsection.6.4.1}{Choice of Moduli}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.2}{Choice of k}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.3}{Restricted Diminished Radix Reduction}{section.6.4} -\BOOKMARK [2][-]{subsection.6.4.4}{Unrestricted Diminished Radix Reduction}{section.6.4} -\BOOKMARK [1][-]{section.6.5}{Algorithm Comparison}{chapter.6} -\BOOKMARK [0][-]{chapter.7}{Exponentiation}{} -\BOOKMARK [1][-]{section.7.1}{Exponentiation Basics}{chapter.7} -\BOOKMARK [2][-]{subsection.7.1.1}{Single Digit Exponentiation}{section.7.1} -\BOOKMARK [1][-]{section.7.2}{k-ary Exponentiation}{chapter.7} -\BOOKMARK [2][-]{subsection.7.2.1}{Optimal Values of k}{section.7.2} -\BOOKMARK [2][-]{subsection.7.2.2}{Sliding-Window Exponentiation}{section.7.2} -\BOOKMARK [1][-]{section.7.3}{Modular Exponentiation}{chapter.7} -\BOOKMARK [2][-]{subsection.7.3.1}{Barrett Modular Exponentiation}{section.7.3} -\BOOKMARK [1][-]{section.7.4}{Quick Power of Two}{chapter.7} -\BOOKMARK [0][-]{chapter.8}{Higher Level Algorithms}{} -\BOOKMARK [1][-]{section.8.1}{Integer Division with Remainder}{chapter.8} -\BOOKMARK [2][-]{subsection.8.1.1}{Quotient Estimation}{section.8.1} -\BOOKMARK [2][-]{subsection.8.1.2}{Normalized Integers}{section.8.1} -\BOOKMARK [2][-]{subsection.8.1.3}{Radix- Division with Remainder}{section.8.1} -\BOOKMARK [1][-]{section.8.2}{Single Digit Helpers}{chapter.8} -\BOOKMARK [2][-]{subsection.8.2.1}{Single Digit Addition and Subtraction}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.2}{Single Digit Multiplication}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.3}{Single Digit Division}{section.8.2} -\BOOKMARK [2][-]{subsection.8.2.4}{Single Digit Root Extraction}{section.8.2} -\BOOKMARK [1][-]{section.8.3}{Random Number Generation}{chapter.8} -\BOOKMARK [1][-]{section.8.4}{Formatted Representations}{chapter.8} -\BOOKMARK [2][-]{subsection.8.4.1}{Reading Radix-n Input}{section.8.4} -\BOOKMARK [2][-]{subsection.8.4.2}{Generating Radix-n Output}{section.8.4} -\BOOKMARK [0][-]{chapter.9}{Number Theoretic Algorithms}{} -\BOOKMARK [1][-]{section.9.1}{Greatest Common Divisor}{chapter.9} -\BOOKMARK [2][-]{subsection.9.1.1}{Complete Greatest Common Divisor}{section.9.1} -\BOOKMARK [1][-]{section.9.2}{Least Common Multiple}{chapter.9} -\BOOKMARK [1][-]{section.9.3}{Jacobi Symbol Computation}{chapter.9} -\BOOKMARK [2][-]{subsection.9.3.1}{Jacobi Symbol}{section.9.3} -\BOOKMARK [1][-]{section.9.4}{Modular Inverse}{chapter.9} -\BOOKMARK [2][-]{subsection.9.4.1}{General Case}{section.9.4} -\BOOKMARK [1][-]{section.9.5}{Primality Tests}{chapter.9} -\BOOKMARK [2][-]{subsection.9.5.1}{Trial Division}{section.9.5} -\BOOKMARK [2][-]{subsection.9.5.2}{The Fermat Test}{section.9.5} -\BOOKMARK [2][-]{subsection.9.5.3}{The Miller-Rabin Test}{section.9.5} diff --git a/lib/hcrypto/libtommath/tommath.pdf b/lib/hcrypto/libtommath/tommath.pdf deleted file mode 100644 index 33994c35a..000000000 Binary files a/lib/hcrypto/libtommath/tommath.pdf and /dev/null differ diff --git a/lib/hcrypto/libtommath/tommath.src b/lib/hcrypto/libtommath/tommath.src deleted file mode 100644 index 40658222c..000000000 --- a/lib/hcrypto/libtommath/tommath.src +++ /dev/null @@ -1,6350 +0,0 @@ -\documentclass[b5paper]{book} -\usepackage{hyperref} -\usepackage{makeidx} -\usepackage{amssymb} -\usepackage{color} -\usepackage{alltt} -\usepackage{graphicx} -\usepackage{layout} -\def\union{\cup} -\def\intersect{\cap} -\def\getsrandom{\stackrel{\rm R}{\gets}} -\def\cross{\times} -\def\cat{\hspace{0.5em} \| \hspace{0.5em}} -\def\catn{$\|$} -\def\divides{\hspace{0.3em} | \hspace{0.3em}} -\def\nequiv{\not\equiv} -\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} -\def\lcm{{\rm lcm}} -\def\gcd{{\rm gcd}} -\def\log{{\rm log}} -\def\ord{{\rm ord}} -\def\abs{{\mathit abs}} -\def\rep{{\mathit rep}} -\def\mod{{\mathit\ mod\ }} -\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} -\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} -\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} -\def\Or{{\rm\ or\ }} -\def\And{{\rm\ and\ }} -\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} -\def\implies{\Rightarrow} -\def\undefined{{\rm ``undefined"}} -\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} -\let\oldphi\phi -\def\phi{\varphi} -\def\Pr{{\rm Pr}} -\newcommand{\str}[1]{{\mathbf{#1}}} -\def\F{{\mathbb F}} -\def\N{{\mathbb N}} -\def\Z{{\mathbb Z}} -\def\R{{\mathbb R}} -\def\C{{\mathbb C}} -\def\Q{{\mathbb Q}} -\definecolor{DGray}{gray}{0.5} -\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} -\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} -\def\gap{\vspace{0.5ex}} -\makeindex -\begin{document} -\frontmatter -\pagestyle{empty} -\title{Multi--Precision Math} -\author{\mbox{ -%\begin{small} -\begin{tabular}{c} -Tom St Denis \\ -Algonquin College \\ -\\ -Mads Rasmussen \\ -Open Communications Security \\ -\\ -Greg Rose \\ -QUALCOMM Australia \\ -\end{tabular} -%\end{small} -} -} -\maketitle -This text has been placed in the public domain. This text corresponds to the v0.39 release of the -LibTomMath project. - -\begin{alltt} -Tom St Denis -111 Banning Rd -Ottawa, Ontario -K2L 1C3 -Canada - -Phone: 1-613-836-3160 -Email: tomstdenis@gmail.com -\end{alltt} - -This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} -{\em book} macro package and the Perl {\em booker} package. - -\tableofcontents -\listoffigures -\chapter*{Prefaces} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which -perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give -back to society in the form of tools and knowledge that can help others in their endeavours. - -I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source -code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not -explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works -itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality -of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far -from relatively straightforward algebra and I hope that this book can be a valuable learning asset. - -This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora -of kind people donating their time, resources and kind words to help support my work. Writing a text of significant -length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. - -To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I -honour your kind gestures with this project. - -Open Source. Open Academia. Open Minds. - -\begin{flushright} Tom St Denis \end{flushright} - -\newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also -contribute to educate others facing the problem of having to handle big number mathematical calculations. - -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about -the layout and language used. - -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. - -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular -reductions, for example, become painfully slow. - -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks -themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' - -\begin{flushright} -Mads Rasmussen - -S\~{a}o Paulo - SP - -Brazil -\end{flushright} - -\newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not -really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. - -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the -sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. -Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. - -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, -and I'm pleased to be involved with it. - -\begin{flushright} -Greg Rose, Sydney, Australia, June 2003. -\end{flushright} - -\mainmatter -\pagestyle{headings} -\chapter{Introduction} -\section{Multiple Precision Arithmetic} - -\subsection{What is Multiple Precision Arithmetic?} -When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple -precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. - -By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in -the decimal system with fixed precision $6 \cdot 7 = 2$. - -Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. - -\subsection{The Need for Multiple Precision Arithmetic} -The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and -Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{|r|c|} -\hline \textbf{Data Type} & \textbf{Range} \\ -\hline char & $-128 \ldots 127$ \\ -\hline short & $-32768 \ldots 32767$ \\ -\hline long & $-2147483648 \ldots 2147483647$ \\ -\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ -\hline -\end{tabular} -\end{center} -\caption{Typical Data Types for the C Programming Language} -\label{fig:ISOC} -\end{figure} - -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by -extending the range of representable integers while using single precision data types. - -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and -deployment of efficient algorithms. - -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small -(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where -scientific applications must minimize the total output error over long calculations. - -Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). -In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. - -\subsection{Benefits of Multiple Precision Arithmetic} -\index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system -would truncate excess bits to maintain a fixed level of precision. - -It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard -processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not -normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. - -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the -overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to -be written and tested once. - -\section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning -the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not -discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). - -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. - -To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used -to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field -tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text -discusses a very large portion of the inner workings of the library. - -The algorithms that are presented will always include at least one ``pseudo-code'' description followed -by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same -algorithm in other programming languages as the reader sees fit. - -This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing -the reader how the algorithms fit together as well as where to start on various taskings. - -\section{Discussion and Notation} -\subsection{Notation} -A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. - -\index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. - -For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These -algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. - -\subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the -carry. Since all modern computers are binary, it is assumed that $q$ is two. - -\index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to -the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. -Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single -precision data type. - -For example, if $\beta = 10^2$ a single precision data type may represent a value in the -range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let -$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written -as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. - -\subsection{Algorithm Inputs and Outputs} -Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. - -\subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression -itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. - -The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. - -\subsection{Work Effort} -\index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to -complete. While this is generally not true in practice, it will simplify the discussions considerably. - -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a -result small constant factors in the work effort will make an observable difference in algorithm efficiency. - -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how -various optimizations will help pay off in the long run. - -\section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. - -That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular -are encouraged to verify they can answer the problems correctly before moving on. - -Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring system used. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|l|} -\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ -\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ - & time usage. Usually requires a program to be written to \\ - & solve the problem. \\ -\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ - & of work. Usually involves trivial research and development of \\ - & new theory from the perspective of a student. \\ -\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ - & of work and research, the solution to which will demonstrate \\ - & a higher mastery of the subject matter. \\ -\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ - & novice to solve. Solutions to these problems will demonstrate a \\ - & complete mastery of the given subject. \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Exercise Scoring System} -\end{figure} - -Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level -are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. - -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always -involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can -answer these questions will feel comfortable with the concepts behind the topic at hand. - -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. - -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a -mastery of the subject matter at hand. - -Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader -is encouraged to answer the follow-up problems and try to draw the relevance of problems. - -\section{Introduction to LibTomMath} - -\subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. - -The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such -as public key cryptosystems and still maintain a relatively small footprint. - -\subsection{Goals of LibTomMath} - -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. - -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. - -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should -be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument -passing conventions, it has been written from scratch by Tom St Denis. - -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. - -\section{Choice of LibTomMath} -LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for -reasons that will be explained in the following sub-sections. - -\subsection{Code Base} -The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional -segments of code littered throughout the source. This clean and uncluttered approach to the library means that a -developer can more readily discern the true intent of a given section of source code without trying to keep track of -what conditional code will be used. - -The code base of LibTomMath is well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source -file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. - -When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about -$50$KiB) but LibTomMath is also much faster and more complete than MPI. - -\subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. - -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. - -The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors -are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely -undersireable in many situations. - -\subsection{Optimizations} -While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP -and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few -of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. - -LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually -slower than the best libraries such as GMP and OpenSSL by only a small factor. - -\subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. - -GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active -development and are very stable across a variety of platforms. - -\subsection{Choice} -LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. - -\chapter{Getting Started} -\section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the -inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written -as portable source code that is reasonably efficient across several different computer platforms. - -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, -before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is -highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. - -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I -implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development -for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. - -FIGU,design_process,Design Flow of the First Few Original LibTomMath Functions. - -Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. - -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. -This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. - -\section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. - -As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed -precision computer words with the exception that a different radix is used. - -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. - -The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. -Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. - -\subsection{The mp\_int Structure} -\label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition -used within LibTomMath. - -\index{mp\_int} -\begin{figure}[here] -\begin{center} -\begin{small} -%\begin{verbatim} -\begin{tabular}{|l|} -\hline -typedef struct \{ \\ -\hspace{3mm}int used, alloc, sign;\\ -\hspace{3mm}mp\_digit *dp;\\ -\} \textbf{mp\_int}; \\ -\hline -\end{tabular} -%\end{verbatim} -\end{small} -\caption{The mp\_int Structure} -\label{fig:mpint} -\end{center} -\end{figure} - -The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. - -\begin{enumerate} -\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. - -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. - -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least -significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ - -\index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). -\end{enumerate} - -\subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. -The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). - -\begin{enumerate} -\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated -array of digits. -\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, -leading zero digits in the most significant positions must be trimmed. - \begin{enumerate} - \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. - \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; -this represents the mp\_int value of zero. -\end{enumerate} - -\section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. -Consider the following examples. - -\begin{verbatim} - mp_mul(&a, &b, &c); /* c = a * b */ - mp_add(&a, &b, &a); /* a = a + b */ - mp_sqr(&a, &b); /* b = a * a */ -\end{verbatim} - -The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the -functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. - -Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the -source is fully read. - -\section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end -developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may -fault by dereferencing memory not owned by the application. - -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an -\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). - -\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} -\begin{figure}[here] -\begin{center} -\begin{tabular}{|l|l|} -\hline \textbf{Value} & \textbf{Meaning} \\ -\hline \textbf{MP\_OKAY} & The function was successful \\ -\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ -\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ -\hline -\end{tabular} -\end{center} -\caption{LibTomMath Error Codes} -\label{fig:errcodes} -\end{figure} - -When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the -function was called. Error checking with this style of API is fairly simple. - -\begin{verbatim} - int err; - if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { - printf("Error: %s\n", mp_error_to_string(err)); - exit(EXIT_FAILURE); - } -\end{verbatim} - -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal -and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. - -\section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and -clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. - -Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of -the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though -the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations -would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. - -If the memory for the digits has been successfully allocated then the rest of the members of the structure must -be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set -to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. - -\subsection{Initializing an mp\_int} -An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the -structure are set to valid values. The mp\_init algorithm will perform such an action. - -\index{mp\_init} -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ -\hline \\ -1. Allocate memory for \textbf{MP\_PREC} digits. \\ -2. If the allocation failed return(\textit{MP\_MEM}) \\ -3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$\\ -4. $a.sign \leftarrow MP\_ZPOS$\\ -5. $a.used \leftarrow 0$\\ -6. $a.alloc \leftarrow MP\_PREC$\\ -7. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init} -\end{figure} - -\textbf{Algorithm mp\_init.} -The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly -manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. - -Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} -used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest -precision number you'll be working with. - -Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack -memory and the number of heap operations will be trivial. - -Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and -\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless -of the original condition of the input. - -\textbf{Remark.} -This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally -when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that -a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each -iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate -decrementally. - -EXAM,bn_mp_init.c - -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. - -Here we see (line @23,XMALLOC@) the memory allocation is performed first. This allows us to exit cleanly and quickly -if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there -was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function -but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in -memory allocation routine. - -In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a -portable fashion you have to actually assign the value. The for loop (line @28,for@) performs this required -operation. - -After the memory has been successfully initialized the remainder of the members are initialized -(lines @29,used@ through @31,sign@) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. - -\subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be -returned to the application's memory pool with the mp\_clear algorithm. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clear}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. The memory for $a$ shall be deallocated. \\ -\hline \\ -1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ -3. Free the memory allocated for the digits of $a$. \\ -4. $a.used \leftarrow 0$ \\ -5. $a.alloc \leftarrow 0$ \\ -6. $a.sign \leftarrow MP\_ZPOS$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_clear} -\end{figure} - -\textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that -if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal -is to free the allocated memory. - -The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid -digit pointer \textbf{dp} setting. - -Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm -with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. - -EXAM,bn_mp_clear.c - -The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line @23,a->dp != NULL@) -checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be -\textbf{NULL} in which case the if statement will evaluate to true. - -The digits of the mp\_int are cleared by the for loop (line @25,for@) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. - -The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to -a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line @33,NULL@). - -Now that the digits have been cleared and deallocated the other members are set to their final values (lines @34,= 0@ and @35,ZPOS@). - -\section{Maintenance Algorithms} - -The previous sections describes how to initialize and clear an mp\_int structure. To further support operations -that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. - -These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level -algorithms such as addition, multiplication and modular exponentiation. - -\subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it -must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_grow}. \\ -\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ -\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ -\hline \\ -1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ -2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ -3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -4. Re-allocate the array of digits $a$ to size $v$ \\ -5. If the allocation failed then return(\textit{MP\_MEM}). \\ -6. for n from a.alloc to $v - 1$ do \\ -\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.alloc \leftarrow v$ \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_grow} -\end{figure} - -\textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. - -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. - -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are -assumed to contain undefined values they are initially set to zero. - -EXAM,bn_mp_grow.c - -A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line @24,alloc@) checks -if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} -the function skips the re-allocation part thus saving time. - -When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is -padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line @25, size@). The XREALLOC function is used -to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc -function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before -the re-allocation. All that is left is to clear the newly allocated digits and return. - -Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return -an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. - -\subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_size}. \\ -\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ -\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ -\hline \\ -1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ -2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -3. Allocate $v$ digits. \\ -4. for $n$ from $0$ to $v - 1$ do \\ -\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ -5. $a.sign \leftarrow MP\_ZPOS$\\ -6. $a.used \leftarrow 0$\\ -7. $a.alloc \leftarrow v$\\ -8. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_init\_size} -\end{figure} - -\textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial -allocations from becoming a bottleneck in the rest of the algorithms. - -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This -particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is -correct no further memory re-allocations are required to work with the mp\_int. - -EXAM,bn_mp_init_size.c - -The number of digits $b$ requested is padded (line @22,MP_PREC@) by first augmenting it to the next multiple of -\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the -mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be -returned (line @27,return@). - -The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The -\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set -to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines @29,used@, @30,alloc@ and @31,sign@). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the -functions to work with. - -\subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. -The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single -statement. It is essentially a shortcut to multiple initializations. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_multi}. \\ -\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ -\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ -\hline \\ -1. for $n$ from 0 to $k - 1$ do \\ -\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ -\hspace{+3mm}1.2. If initialization failed then do \\ -\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ -\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ -\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ -2. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_multi} -\end{figure} - -\textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' -initialization which allows for quick recovery from runtime errors. - -EXAM,bn_mp_init_multi.c - -This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. - -The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count -$n$ of succesfully initialized mp\_int structures is maintained (line @47,n++@) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines @27,if@ to @46,}@). - - -\subsection{Clamping Excess Digits} -When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of -the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a -$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ -though, with no final carry into the last position. However, suppose the destination had to be first expanded -(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. -That would be a considerable waste of time since heap operations are relatively slow. - -The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function -terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked -there would be an excess high order zero digit. - -For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit -will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would -accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very -low the representation is excessively large. - -The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the -\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a -positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to -\textbf{MP\_ZPOS}. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clamp}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ -\hline \\ -1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ -\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ -2. if $a.used = 0$ then do \\ -\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ -\hline \\ -\end{tabular} -\end{center} -\caption{Algorithm mp\_clamp} -\end{figure} - -\textbf{Algorithm mp\_clamp.} -As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at -the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for -when all of the digits are zero to ensure that the mp\_int is valid at all times. - -EXAM,bn_mp_clamp.c - -Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming -language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is -important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously -undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ - & \\ -$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ - & encryption when $\beta = 2^{28}$. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ - & \\ -$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ - & \\ -\end{tabular} - - -%%% -% CHAPTER FOUR -%%% - -\chapter{Basic Operations} - -\section{Introduction} -In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low -level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they -work before proceeding since these algorithms will be used almost intrinsically in the following chapters. - -The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of -mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. - -\section{Assigning Values to mp\_int Structures} -\subsection{Copying an mp\_int} -Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making -a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$. \\ -\textbf{Output}. Store a copy of $a$ in $b$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ -3. for $n$ from $a.used$ to $b.used - 1$ do \\ -\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $b.sign \leftarrow a.sign$ \\ -6. return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_copy} -\end{figure} - -\textbf{Algorithm mp\_copy.} -This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the -mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. - -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow -algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two -and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of -$b$. - -\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is -limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to -implement the pseudo-code. - -EXAM,bn_mp_copy.c - -Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line @24,a == b@). - -The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than -$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines @29,alloc@ to @33,}@). In order to -simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits -of the mp\_ints $a$ and $b$ respectively. These aliases (lines @42,tmpa@ and @45,tmpb@) allow the compiler to access the digits without first dereferencing the -mp\_int pointers and then subsequently the pointer to the digits. - -After the aliases are established the digits from $a$ are copied into $b$ (lines @48,for@ to @50,}@) and then the excess -digits of $b$ are set to zero (lines @53,for@ to @55,}@). Both ``for'' loops make use of the pointer aliases and in -fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization -allows the alias to stay in a machine register fairly easy between the two loops. - -\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will -be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the -number of pointer dereferencing operations required to access data. For example, a for loop may resemble - -\begin{alltt} -for (x = 0; x < 100; x++) \{ - a->num[4]->dp[x] = 0; -\} -\end{alltt} - -This could be re-written using aliases as - -\begin{alltt} -mp_digit *tmpa; -a = a->num[4]->dp; -for (x = 0; x < 100; x++) \{ - *a++ = 0; -\} -\end{alltt} - -In this case an alias is used to access the -array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required -as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. - -The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations -may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may -work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer -aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code -stands a better chance of being faster. - -The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' -loop of the function mp\_copy() re-written to not use pointer aliases. - -\begin{alltt} - /* copy all the digits */ - for (n = 0; n < a->used; n++) \{ - b->dp[n] = a->dp[n]; - \} -\end{alltt} - -Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more -complicated as there are four variables within the statement instead of just two. - -\subsubsection{Nested Statements} -Another commonly used technique in the source routines is that certain sections of code are nested. This is used in -particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) -will typically have three different phases. First the temporaries are initialized, then the columns calculated and -finally the carries are propagated. In this example the middle column production phase will typically be nested as it -uses temporary variables and aliases the most. - -The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result -the various temporary variables required do not propagate into other sections of code. - - -\subsection{Creating a Clone} -Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int -and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is -useful within functions that need to modify an argument but do not wish to actually modify the original copy. The -mp\_init\_copy algorithm has been designed to help perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$\\ -\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ -\hline \\ -1. Init $a$. (\textit{mp\_init}) \\ -2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ -3. Return the status of the copy operation. \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_copy} -\end{figure} - -\textbf{Algorithm mp\_init\_copy.} -This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As -such this algorithm will perform two operations in one step. - -EXAM,bn_mp_init_copy.c - -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that -\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. - -\section{Zeroing an Integer} -Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to -perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_zero}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Zero the contents of $a$ \\ -\hline \\ -1. $a.used \leftarrow 0$ \\ -2. $a.sign \leftarrow$ MP\_ZPOS \\ -3. for $n$ from 0 to $a.alloc - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_zero} -\end{figure} - -\textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. - -EXAM,bn_mp_zero.c - -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the -\textbf{sign} variable is set to \textbf{MP\_ZPOS}. - -\section{Sign Manipulation} -\subsection{Absolute Value} -With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute -the absolute value of an mp\_int. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_abs}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = \vert a \vert$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. $b.sign \leftarrow MP\_ZPOS$ \\ -4. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_abs} -\end{figure} - -\textbf{Algorithm mp\_abs.} -This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an -algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition -logic to handle it. - -EXAM,bn_mp_abs.c - -This fairly trivial algorithm first eliminates non--required duplications (line @27,a != b@) and then sets the -\textbf{sign} flag to \textbf{MP\_ZPOS}. - -\subsection{Integer Negation} -With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute -the negative of an mp\_int input. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_neg}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = -a$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ -4. If $a.sign = MP\_ZPOS$ then do \\ -\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ -5. else do \\ -\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ -6. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_neg} -\end{figure} - -\textbf{Algorithm mp\_neg.} -This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if -$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return -zero as negative. - -EXAM,bn_mp_neg.c - -Like mp\_abs() this function avoids non--required duplications (line @21,a != b@) and then sets the sign. We -have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero -than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. - -\section{Small Constants} -\subsection{Setting Small Constants} -Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set}. \\ -\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}). \\ -2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ -3. $a.used \leftarrow \left \lbrace \begin{array}{ll} - 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 - \end{array} \right .$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set} -\end{figure} - -\textbf{Algorithm mp\_set.} -This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The -single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. - -EXAM,bn_mp_set.c - -First we zero (line @21,mp_zero@) the mp\_int to make sure that the other members are initialized for a -small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line @22,MP_MASK@). After this step we have to -check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise -to zero. - -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with -$2^k - 1$ will perform the same operation. - -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses -this function should take that into account. Only trivially small constants can be set using this function. - -\subsection{Setting Large Constants} -To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' -data type as input and will always treat it as a 32-bit integer. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set\_int}. \\ -\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}) \\ -2. for $n$ from 0 to 7 do \\ -\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ -\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ -\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ -\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ -3. Clamp excess used digits (\textit{mp\_clamp}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set\_int} -\end{figure} - -\textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the -mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is -incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have -zero digits used and the newly added four bits would be ignored. - -Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. - -EXAM,bn_mp_set_int.c - -This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not -seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@ -as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps -the number of used digits low. - -\section{Comparisons} -\subsection{Unsigned Comparisions} -Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, -to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. - -The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the -signs are known to agree in advance. - -To facilitate working with the results of the comparison functions three constants are required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|r|l|} -\hline \textbf{Constant} & \textbf{Meaning} \\ -\hline \textbf{MP\_GT} & Greater Than \\ -\hline \textbf{MP\_EQ} & Equal To \\ -\hline \textbf{MP\_LT} & Less Than \\ -\hline -\end{tabular} -\end{center} -\caption{Comparison Return Codes} -\end{figure} - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp\_mag}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$. \\ -\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ -\hline \\ -1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ -2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ -3. for n from $a.used - 1$ to 0 do \\ -\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ -\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ -4. Return(\textit{MP\_EQ}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp\_mag} -\end{figure} - -\textbf{Algorithm mp\_cmp\_mag.} -By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. - -By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to -the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. - -EXAM,bn_mp_cmp_mag.c - -The two if statements (lines @24,if@ and @28,if@) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be -smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. - - - -\subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude -comparison a trivial signed comparison algorithm can be written. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ -\hline \\ -1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ -2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ -3. if $a.sign = MP\_NEG$ then \\ -\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ -4 Otherwise \\ -\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp} -\end{figure} - -\textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then -$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. - -EXAM,bn_mp_cmp.c - -The two if statements (lines @22,if@ and @26,if@) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line @30,if@) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line @31,mp_cmp_mag@). Otherwise, the signs are assumed to -be both positive and a forward direction unsigned comparison is performed. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ - & \\ -$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ - & of two random digits (of equal magnitude) before a difference is found. \\ - & \\ -$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ - & on the observations made in the previous problem. \\ - & -\end{tabular} - -\chapter{Basic Arithmetic} -\section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. - -MARK,SHIFTS -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. -For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. - -One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. - -\section{Addition and Subtraction} -In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. -As a result subtraction can be performed with a trivial series of logical operations and an addition. - -However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or -subtraction algorithms with the sign fixed up appropriately. - -The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of -the integers respectively. - -\subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. -Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. - -\newpage -\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ -\hline \\ -1. if $a.used > b.used$ then \\ -\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ -\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ -\hspace{+3mm}1.3 $x \leftarrow a$ \\ -2. else \\ -\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ -\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ -\hspace{+3mm}2.3 $x \leftarrow b$ \\ -3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max + 1$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ -\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min \ne max$ then do \\ -\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ -\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. $c_{max} \leftarrow u$ \\ -10. if $olduse > max$ then \\ -\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ -\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ -11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_add} -\end{figure} - -\textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the -MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. - -The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic -will simply add all of the smallest input to the largest input and store that first part of the result in the -destination. Then it will apply a simpler addition loop to excess digits of the larger input. - -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two -inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. - -At this point the first addition loop will go through as many digit positions that both inputs have. The carry -variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce -one digit of the summand. First -two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored -in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. - -Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias -for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits -and the carry to the destination. - -The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. - - -EXAM,bn_s_mp_add.c - -We first sort (lines @27,if@ to @35,}@) the inputs based on magnitude and determine the $min$ and $max$ variables. -Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (@37,init@ to @42,}@) ensure that it can accomodate the result of the addition. - -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on -lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the -compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. - -The initial carry $u$ will be cleared (line @65,u = 0@), note that $u$ is of type mp\_digit which ensures type -compatibility within the implementation. The initial addition (line @66,for@ to @75,}@) adds digits from -both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line @81,for@ to @90,}@) adds the remaining digits from the larger of the two inputs. The addition is finished -with the final carry being stored in $tmpc$ (line @94,tmpc++@). Note the ``++'' operator within the same expression. -After line @94,tmpc++@, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop (line @97,for@ to @99,}@) which set any old upper digits to zero. - -\subsection{Low Level Subtraction} -The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. -This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. - -MARK,GAMMA - -For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). - -For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' -data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ -\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ -\hline \\ -1. $min \leftarrow b.used$ \\ -2. $max \leftarrow a.used$ \\ -3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ -\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min < max$ then do \\ -\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ -\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. if $oldused > max$ then do \\ -\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ -\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ -10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_sub} -\end{figure} - -\textbf{Algorithm s\_mp\_sub.} -This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when -passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This -algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case -of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. - -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and -set to the maximal count for the operation. - -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. - -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. - -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most -significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. - -If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step -10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. - -EXAM,bn_s_mp_sub.c - -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines @24,min@ and @25,max@). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used -within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized -(lines @42,tmpa@, @43,tmpb@ and @44,tmpc@) for $a$, $b$ and $c$ respectively. - -The first subtraction loop (lines @47,u = 0@ through @61,}@) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line @57, >>@). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This -optimization only works on twos compliment machines which is a safe assumption to make. - -If $a$ has a larger magnitude than $b$ an additional loop (lines @64,for@ through @73,}@) is required to propagate -the carry through $a$ and copy the result to $c$. - -\subsection{High Level Addition} -Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. - -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} -flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed addition $c = a + b$. \\ -\hline \\ -1. if $a.sign = b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_add} -\end{figure} - -\textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly -straightforward but restricted since subtraction can only produce positive results. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&&\\ - -\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ -\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ - -\hline &&&&\\ - -\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ - -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Addition Guide Chart} -\label{fig:AddChart} -\end{figure} - -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best -follows how the implementation actually was achieved. - -Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms -s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} -to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. - -For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. - -EXAM,bn_mp_add.c - -The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which -is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without -explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower -level functions do so. Returning their return code is sufficient. - -\subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed subtraction $c = a - b$. \\ -\hline \\ -1. if $a.sign \ne b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_sub} -\end{figure} - -\textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or -\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and -the operations required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Subtraction Guide Chart} -\label{fig:SubChart} -\end{figure} - -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. - -EXAM,bn_mp_sub.c - -Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. - -\section{Bit and Digit Shifting} -MARK,POLY -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. - -In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift -the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. - -\subsection{Multiplication by Two} - -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = 2a$. \\ -\hline \\ -1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ -2. $oldused \leftarrow b.used$ \\ -3. $b.used \leftarrow a.used$ \\ -4. $r \leftarrow 0$ \\ -5. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ -\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.3 $r \leftarrow rr$ \\ -6. If $r \ne 0$ then do \\ -\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ -\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2} -\end{figure} - -\textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. - -Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count -is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. - -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together -are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to -obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with -forwarding the carry to the next iteration. - -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. -Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. - -EXAM,bn_mp_mul_2.c - -This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling. - -\subsection{Division by Two} -A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = a/2$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ -2. If the reallocation failed return(\textit{MP\_MEM}). \\ -3. $oldused \leftarrow b.used$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $r \leftarrow 0$ \\ -6. for $n$ from $b.used - 1$ to $0$ do \\ -\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ -\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}6.3 $r \leftarrow rr$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ -10. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2} -\end{figure} - -\textbf{Algorithm mp\_div\_2.} -This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition -core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm -could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent -reading past the end of the array of digits. - -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. - -EXAM,bn_mp_div_2.c - -\section{Polynomial Basis Operations} -Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single -place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. - -Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that -$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. - -\subsection{Multiplication by $x$} - -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one -degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ -2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. $a.used \leftarrow a.used + b$ \\ -5. $i \leftarrow a.used - 1$ \\ -6. $j \leftarrow a.used - 1 - b$ \\ -7. for $n$ from $a.used - 1$ to $b$ do \\ -\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ -\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ -\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ -8. for $n$ from 0 to $b - 1$ do \\ -\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lshd} -\end{figure} - -\textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs -from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The -motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally -different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. - -First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on -step 8 sets the lower $b$ digits to zero. - -\newpage -FIGU,sliding_window,Sliding Window Movement - -EXAM,bn_mp_lshd.c - -The if statement (line @24,if@) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates -the need for an additional variable in the for loop. The variable $top$ (line @42,top@) is an alias -for the leading digit while $bottom$ (line @45,bottom@) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. - -\subsection{Division by $x$} - -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return. \\ -2. If $a.used \le b$ then do \\ -\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ -\hspace{3mm}2.2 Return. \\ -3. $i \leftarrow 0$ \\ -4. $j \leftarrow b$ \\ -5. for $n$ from 0 to $a.used - b - 1$ do \\ -\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ -\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ -\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ -6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ -\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.used \leftarrow a.used - b$ \\ -8. Return. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rshd} -\end{figure} - -\textbf{Algorithm mp\_rshd.} -This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. - -If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal -to the shift count $b$ then it will simply zero the input and return. - -After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. -Also the digits are copied from the leading to the trailing edge. - -Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. - -EXAM,bn_mp_rshd.c - -The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we -form a sliding window except we copy in the other direction. After the window (line @59,for (;@) we then zero -the upper digits of the input to make sure the result is correct. - -\section{Powers of Two} - -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For -example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. - -\subsection{Multiplication by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ -\hline \\ -1. $c \leftarrow a$. (\textit{mp\_copy}) \\ -2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. If $b \ge lg(\beta)$ then \\ -\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ -\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ -5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $d \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -\hspace{3mm}6.4 If $r > 0$ then do \\ -\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ -\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2d} -\end{figure} - -\textbf{Algorithm mp\_mul\_2d.} -This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to -quickly compute the product. - -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ -left. - -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. -Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ -variable is used to extract the upper $d$ bits to form the carry for the next iteration. - -This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to -complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. - -EXAM,bn_mp_mul_2d.c - -The shifting is performed in--place which means the first step (line @24,a != c@) is to copy the input to the -destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then -has to be grown (line @31,grow@) to accomodate the result. - -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift -loop (lines @45,if@ to @76,}@) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. - -\subsection{Division by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow a$ \\ -3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -4. If $b \ge lg(\beta)$ then do \\ -\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ -5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $k \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2d} -\end{figure} - -\textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm -mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division -by using algorithm mp\_mod\_2d. - -EXAM,bn_mp_div_2d.c - -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the -result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before -the quotient is obtained. - -The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is -the direction of the shifts. - -\subsection{Remainder of Division by Power of Two} - -The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mod\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b > a.used \cdot lg(\beta)$ then do \\ -\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}2.2 Return the result of step 2.1. \\ -3. $c \leftarrow a$ \\ -4. If step 3 failed return(\textit{MP\_MEM}). \\ -5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ -\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ -6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ -8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mod\_2d} -\end{figure} - -\textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ -is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. - -EXAM,bn_mp_mod_2d.c - -We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger -than the input we just mp\_copy() the input and return right away. After this point we know we must actually -perform some work to produce the remainder. - -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line @41,for@). Next we reduce the -leading digit of both (line @45,&=@) and then mp\_clamp(). - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ - & in $O(n)$ time. \\ - &\\ -$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ - & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ - & upto $64$ with a hamming weight less than three. \\ - &\\ -$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ - & $2^k - 1$ as well. \\ - &\\ -$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ - & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ - & any $n$-bit input. Note that the time of addition is ignored in the \\ - & calculation. \\ - & \\ -$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ - & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ - & the cost of addition. \\ - & \\ -$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ - & for $n = 64 \ldots 1024$ in steps of $64$. \\ - & \\ -$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ - & calculating the result of a signed comparison. \\ - & -\end{tabular} - -\chapter{Multiplication and Squaring} -\section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. - -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular -exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision -multiplications. - -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. - -\section{Multiplication} -\subsection{The Baseline Multiplication} -\label{sec:basemult} -\index{baseline multiplication} -Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The -constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}). - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -1. If min$(a.used, b.used) < \delta$ then do \\ -\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ -\hspace{3mm}1.2 Return the result of step 1.1 \\ -\\ -Allocate and initialize a temporary mp\_int. \\ -2. Init $t$ to be of size $digs$ \\ -3. If step 2 failed return(\textit{MP\_MEM}). \\ -4. $t.used \leftarrow digs$ \\ -\\ -Compute the product. \\ -5. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}5.1 $u \leftarrow 0$ \\ -\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ -\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ -\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ -\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ -\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ -\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ -6. Clamp excess digits of $t$. \\ -7. Swap $c$ with $t$ \\ -8. Clear $t$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_mul\_digs} -\end{figure} - -\textbf{Algorithm s\_mp\_mul\_digs.} -This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the -inputs. - -The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. - -All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable -is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. - -For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best -visualized in the following table. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|l|} -\hline && & 5 & 7 & 6 & \\ -\hline $\times$&& & 2 & 4 & 1 & \\ -\hline &&&&&&\\ - && & 5 & 7 & 6 & $10^0(1)(576)$ \\ - &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ - 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline -\end{tabular} -\end{center} -\caption{Long-Hand Multiplication Diagram} -\end{figure} - -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate -count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. - -Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step -is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a -double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. - -At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th -digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result -exceed the precision requested. - -EXAM,bn_s_mp_mul_digs.c - -First we determine (line @30,if@) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is -set to $\delta$ but can be reduced when memory is at a premium. - -If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line @36,init@) to the exact size of the output to avoid further re--allocations. At this point we now -begin the $O(n^2)$ loop. - -This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line @48,MIN@) to the maximum -number of inner loop iterations. - -Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that -is required for the product. In x86 terms for example, this means using the MUL instruction. - -Each digit of the product is stored in turn (line @68,tmpt@) and the carry propagated (line @71,>>@) to the -next iteration. - -\subsection{Faster Multiplication by the ``Comba'' Method} -MARK,COMBA - -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written -five years before. - -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. - -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. - -\begin{equation} -\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace -\end{equation} - -Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. - -\newpage\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} - \hline & & 5 & 7 & 6 & First Input\\ - \hline $\times$ & & 2 & 4 & 1 & Second Input\\ -\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ - & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ - $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Comba Multiplication Diagram} -\end{figure} - -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. -Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is -congruent to adding a leading zero digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Comba Fixup}. \\ -\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ -\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ -\hline \\ -1. for $n$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ -\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ -2. Return($\vec x$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Comba Fixup} -\end{figure} - -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case -$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more -efficient than the baseline algorithm why not simply always use this algorithm? - -\subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output -independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix -the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of -three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is -min$(m, n)$ which is fairly obvious. - -The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall -from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these -two quantities we must not violate the following - -\begin{equation} -k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} -\end{equation} - -Which reduces to - -\begin{equation} -k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} -\end{equation} - -Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is -found. - -\begin{equation} -k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} -\end{equation} - -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ -1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ -2. If step 1 failed return(\textit{MP\_MEM}).\\ -\\ -3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ -\\ -4. $\_ \hat W \leftarrow 0$ \\ -5. for $ix$ from 0 to $pa - 1$ do \\ -\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ -\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ -\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ -\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ -\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow c.used$ \\ -7. $c.used \leftarrow digs$ \\ -8. for $ix$ from $0$ to $pa$ do \\ -\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -\\ -10. Clamp $c$. \\ -11. Return MP\_OKAY. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_mul\_digs} -\label{fig:COMBAMULT} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_mul\_digs.} -This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. - -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the -loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and -reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. - -The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than -$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. - -The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time -means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until -$tx \ge a.used$ or $ty < 0$ occurs. - -After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator -into the next round by dividing $\_ \hat W$ by $\beta$. - -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, -the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. - -EXAM,bn_fast_s_mp_mul_digs.c - -As per the pseudo--code we first calculate $pa$ (line @47,MIN@) as the number of digits to output. Next we begin the outer loop -to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines @61,tmpx@, @62,tmpy@) to point -inside the two multiplicands quickly. - -The inner loop (lines @70,for@ to @72,}@) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the -compiler has aliased $\_ \hat W$ to a CPU register. - -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines @75,W[ix]@, @78,>>@) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line @82,W[ix]@) as the last digit of the product. - -\subsection{Polynomial Basis Multiplication} -To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and -$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - -The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will -directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients -requires $O(n^2)$ time and would in practice be slower than the Comba technique. - -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. - -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required -by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. - -When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product -$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the -points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. - -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For -example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. - -\begin{eqnarray} -\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ -16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) -\end{eqnarray} - -Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. - -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is -$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} -summarizes the exponents for various values of $n$. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ -\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ -\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ -\hline $4$ & $1.403677461$ &\\ -\hline $5$ & $1.365212389$ &\\ -\hline $10$ & $1.278753601$ &\\ -\hline $100$ & $1.149426538$ &\\ -\hline $1000$ & $1.100270931$ &\\ -\hline $10000$ & $1.075252070$ &\\ -\hline -\end{tabular} -\end{center} -\caption{Asymptotic Running Time of Polynomial Basis Multiplication} -\label{fig:exponent} -\end{figure} - -At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead -of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. - -\subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, -the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the -polynomial basis approach more costly to use with small inputs. - -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. - -The exact location of $y$ depends on several key architectural elements of the computer platform in question. - -\begin{enumerate} -\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example -on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. - -\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits -grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This -directly reflects on the ratio previous mentioned. - -\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an -influence over the cutoff point. - -\end{enumerate} - -A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point -is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. - -\subsection{Karatsuba Multiplication} -Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with -light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. - -\begin{equation} -f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd -\end{equation} - -Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points -$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. - -\begin{center} -\begin{tabular}{rcrcrcrc} -$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ -$\zeta_{1}$ & $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\ -$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ -\end{tabular} -\end{center} - -By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity -of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ -\hline \\ -1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ -2. If step 2 failed then return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ -3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ -6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ -7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ -\\ -Calculate the three products. \\ -8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ -9. $x1y1 \leftarrow x1 \cdot y1$ \\ -10. $t1 \leftarrow x1 + x0$ (\textit{mp\_add}) \\ -11. $x0 \leftarrow y1 + y0$ \\ -12. $t1 \leftarrow t1 \cdot x0$ \\ -\\ -Calculate the middle term. \\ -13. $x0 \leftarrow x0y0 + x1y1$ \\ -14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\ -\\ -Calculate the final product. \\ -15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ -16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ -17. $t1 \leftarrow x0y0 + t1$ \\ -18. $c \leftarrow t1 + x1y1$ \\ -19. Clear all of the temporary variables. \\ -20. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_mul} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_mul.} -This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description -from Knuth \cite[pp. 294-295]{TAOCPV2}. - -\index{radix point} -In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must -be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the -smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 -compute the lower halves. Step 6 and 7 computer the upper halves. - -After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products -$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead -of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. - -The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. - -EXAM,bn_mp_karatsuba_mul.c - -The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional -wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables -required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only -the temporaries that have been successfully allocated so far. - -The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the -additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective -number of digits for the next section of code. - -The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd -to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and -\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it -is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and -$y1$ respectively. - -By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. - -When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. - -\subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients -of the $W(x)$. - -With the five relations that Toom-Cook specifies, the following system of equations is formed. - -\begin{center} -\begin{tabular}{rcrcrcrcrcr} -$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ -$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ -$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ -$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ -$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ -\end{tabular} -\end{center} - -A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power -of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that -the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ -1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ -2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -\\ -Find the five equations for $w_0, w_1, ..., w_4$. \\ -8. $w_0 \leftarrow a_0 \cdot b_0$ \\ -9. $w_4 \leftarrow a_2 \cdot b_2$ \\ -10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ -11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ -13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ -14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ -15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ -16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ -\\ -Continued on the next page.\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Now solve the system of equations. \\ -18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ -19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ -20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ -21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ -23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ -24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ -\\ -Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ -26. for $n$ from $1$ to $4$ do \\ -\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ -27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ -28. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul (continued)} -\end{figure} - -\textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this -algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this -description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across -any given step. - -The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller -integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. - -The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond -to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find -$f(y)$ and $g(y)$ which significantly speeds up the algorithm. - -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients -$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of -the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. - -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer -result $a \cdot b$ is produced. - -EXAM,bn_mp_toom_mul.c - -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very -large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with -Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this -algorithm is not practical as Karatsuba has a much lower cutoff point. - -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines @40,mod@ to @69,rshd@) with -combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. - -Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so -we get those out of the way first (lines @72,mul@ and @77,mul@). Next we compute $w1, w2$ and $w3$ using Horners method. - -After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. - -\subsection{Signed Multiplication} -Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b$ \\ -\hline \\ -1. If $a.sign = b.sign$ then \\ -\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ -2. else \\ -\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ -3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ -\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ -4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ -\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ -5. else \\ -\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ -\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ -\hspace{3mm}5.3 else \\ -\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ -6. $c.sign \leftarrow sign$ \\ -7. Return the result of the unsigned multiplication performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul} -\end{figure} - -\textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms -available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. - -EXAM,bn_mp_mul.c - -The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?'' -operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. - -\section{Squaring} -\label{sec:basesquare} - -Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization -available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. - -For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ -required for multiplication. The following diagram gives an example of the operations required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{ccccc|c} -&&1&2&3&\\ -$\times$ &&1&2&3&\\ -\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ - & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ - $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ -\end{tabular} -\end{center} -\caption{Squaring Optimization Diagram} -\end{figure} - -MARK,SQUARE -Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. - -The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). - -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. -Column two of row one is a square and column three is the first unique column. - -\subsection{The Baseline Squaring Algorithm} -The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ -2. If step 1 failed return(\textit{MP\_MEM}) \\ -3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ -4. For $ix$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}Calculate the square. \\ -\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ -\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}Calculate the double products after the square. \\ -\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ -\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ -\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}Set the last carry. \\ -\hspace{3mm}4.5 While $u > 0$ do \\ -\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ -\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ -6. Exchange $b$ and $t$. \\ -7. Clear $t$ (\textit{mp\_clear}) \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm s\_mp\_sqr.} -This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the -destination mp\_int to be the same as the source mp\_int. - -The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while -the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. - -The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this -very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that -when it is multiplied by two, it can be properly represented by a mp\_word. - -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. - -EXAM,bn_s_mp_sqr.c - -Inside the outer loop (line @32,for@) the square term is calculated on line @35,r =@. The carry (line @42,>>@) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized -(lines @45,tmpx@ and @48,tmpt@) to simplify the inner loop. The doubling is performed using two -additions (line @57,r + r@) since it is usually faster than shifting, if not at least as fast. - -The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops -get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. - -\subsection{Faster Squaring by the ``Comba'' Method} -A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional -drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these -performance hazards. - -The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry -propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact -that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. - -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and -carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ -1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ -2. If step 1 failed return(\textit{MP\_MEM}). \\ -\\ -3. $pa \leftarrow 2 \cdot a.used$ \\ -4. $\hat W1 \leftarrow 0$ \\ -5. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ -\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ -\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ -\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ -\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ -\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ -\hspace{3mm}5.8 if $ix$ is even then \\ -\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ -\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ -\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow b.used$ \\ -7. $b.used \leftarrow 2 \cdot a.used$ \\ -8. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ -10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. -This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. - -First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop -products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an -addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal -$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum -of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform -fewer multiplications and the routine ends up being faster. - -Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square -only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. - -EXAM,bn_fast_s_mp_sqr.c - -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. - -\subsection{Polynomial Basis Squaring} -The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception -is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. - -\subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a -number with the following equation. - -\begin{equation} -h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 -\end{equation} - -Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in -Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of -$O \left ( n^{lg(3)} \right )$. - -If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm -instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the -time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff -point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. - -Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. -The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication -were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ -2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1\beta^B + x0$ \\ -3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ -\\ -Calculate the three squares. \\ -6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ -7. $x1x1 \leftarrow x1^2$ \\ -8. $t1 \leftarrow x1 + x0$ (\textit{s\_mp\_add}) \\ -9. $t1 \leftarrow t1^2$ \\ -\\ -Compute the middle term. \\ -10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ -11. $t1 \leftarrow t1 - t2$ \\ -\\ -Compute final product. \\ -12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ -13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ -14. $t1 \leftarrow t1 + x0x0$ \\ -15. $b \leftarrow t1 + x1x1$ \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_sqr} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_sqr.} -This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based -multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. - -The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is -placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ -as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. - -By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$. -Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then -this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. - -Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or -machine clock cycles.}. - -\begin{equation} -5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 -\end{equation} - -For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. -\begin{center} -\begin{tabular}{rcl} -${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ -${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ -${13 \over 9}$ & $<$ & $n$ \\ -\end{tabular} -\end{center} - -This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors -where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On -the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a -ratio of 1:7. } than simpler operations such as addition. - -EXAM,bn_mp_karatsuba_sqr.c - -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and -shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used} -count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. - -By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point -is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 -it is actually below the Comba limit (\textit{at 110 digits}). - -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and -mp\_clears are executed normally. - -\subsection{Toom-Cook Squaring} -The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. - -\subsection{High Level Squaring} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ -\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ -2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ -\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ -3. else \\ -\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ -\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ -\hspace{3mm}3.3 else \\ -\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ -4. $b.sign \leftarrow MP\_ZPOS$ \\ -5. Return the result of the unsigned squaring performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_sqr} -\end{figure} - -\textbf{Algorithm mp\_sqr.} -This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least -\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. - -EXAM,bn_mp_sqr.c - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ - & that have different number of digits in Karatsuba multiplication. \\ - & \\ -$\left [ 2 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\ - & of double products and at most one square is stated. Prove this statement. \\ - & \\ -$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ - & \\ -$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ -$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ - & required for equation $6.7$ to be true. \\ - & \\ -$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ - & compute subsets of the columns in each thread. Determine a cutoff point where \\ - & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ - &\\ -$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ - & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ - & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ - & \\ -\end{tabular} - -\chapter{Modular Reduction} -MARK,REDUCTION -\section{Basics of Modular Reduction} -\index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, -such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered -in~\ref{sec:division}. - -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the -``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. - -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. - -\section{The Barrett Reduction} -The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to - -\begin{equation} -c = a - b \cdot \lfloor a/b \rfloor -\end{equation} - -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. -It would take another common optimization to optimize the algorithm. - -\subsection{Fixed Point Arithmetic} -The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). - -In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. - -This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer -$a$ by another integer $b$ can be achieved with the following expression. - -\begin{equation} -\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with -modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. - -Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which -leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and -the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally -larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach -to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ -variable also helps re-inforce the idea that it is meant to be computed once and re-used. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor -\end{equation} - -Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett -reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. - -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. - -For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing -$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. -By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. - -\subsection{Choosing a Radix Point} -Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. -See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. - -Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the -$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then -${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient -is bound by $0 \le {a' \over b} < 1$. - -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits -``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input -with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation - -\begin{equation} -c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor -\end{equation} - -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off -by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting -$b$ once or twice the residue is found. - -The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. -This is considerably faster than the original attempt. - -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. - -\subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As -it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. - -After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. - -The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision -multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. - -\subsection{Trimming the Residue} -After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the -result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. - -The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full -$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. - -With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. - -\subsection{The Barrett Algorithm} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ -\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ -\hline \\ -Let $m$ represent the number of digits in $b$. \\ -1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ -2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ -\\ -Produce the quotient. \\ -3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ -4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ -\\ -Subtract the multiple of modulus from the input. \\ -5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ -7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ -\\ -Add $\beta^{m+1}$ if a carry occured. \\ -8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ -\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ -\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ -\hspace{3mm}8.3 $a \leftarrow a + q$ \\ -\\ -Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ -9. While $a \ge b$ do (\textit{mp\_cmp}) \\ -\hspace{3mm}9.1 $c \leftarrow a - b$ \\ -10. Clear $q$. \\ -11. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce} -\end{figure} - -\textbf{Algorithm mp\_reduce.} -This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must -be adhered to for the algorithm to work. - -First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting -a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order -for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. - -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called -$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that -instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. - -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. - -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is -performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. - -EXAM,bn_mp_reduce.c - -The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves -the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits -in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. - -\subsection{The Barrett Setup Algorithm} -In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_setup}. \\ -\textbf{Input}. mp\_int $a$ ($a > 1$) \\ -\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ -\hline \\ -1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ -2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_setup.} -This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which -is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. - -EXAM,bn_mp_reduce_setup.c - -This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. - -\section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. - -Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of -$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input -is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. - -\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. - -\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. - -From these two simple facts the following simple algorithm can be derived. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction}. \\ -\textbf{Input}. Integer $x$, $n$ and $k$ \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If $x$ is odd then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ -\hspace{3mm}1.2 $x \leftarrow x/2$ \\ -2. Return $x$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction} -\end{figure} - -The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is -added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to -$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ -\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ -\hline $2$ & $x/2 = 1453$ \\ -\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ -\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ -\hline $5$ & $x/2 = 278$ \\ -\hline $6$ & $x/2 = 139$ \\ -\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ -\hline $8$ & $x/2 = 99$ \\ -\hline $9$ & $x + n = 356$, $x/2 = 178$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (I)} -\label{fig:MONT1} -\end{figure} - -Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of -the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue -$r \equiv 158$ is produced. - -Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts -and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. -Fortunately there exists an alternative representation of the algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($2^k > n$) \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ -2. Return $x/2^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified I)} -\end{figure} - -This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single -precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|r|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ -\hline -- & $5555$ & $1010110110011$ \\ -\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ -\hline $2$ & $5812$ & $1011010110100$ \\ -\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ -\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ -\hline $5$ & $8896$ & $10001011000000$ \\ -\hline $6$ & $8896$ & $10001011000000$ \\ -\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ -\hline $8$ & $25344$ & $110001100000000$ \\ -\hline $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\ -\hline -- & $x/2^k = 178$ & \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (II)} -\label{fig:MONT2} -\end{figure} - -Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. -With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the -loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is -zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. - -\subsection{Digit Based Montgomery Reduction} -Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the -previous algorithm re-written to compute the Montgomery reduction in this new fashion. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($\beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ -2. Return $x/\beta^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified II)} -\end{figure} - -The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of -the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This -problem breaks down to solving the following congruency. - -\begin{center} -\begin{tabular}{rcl} -$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\end{tabular} -\end{center} - -In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used -extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. - -For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ -represent the value to reduce. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ -\hline -- & $33$ & --\\ -\hline $0$ & $33 + \mu n = 50$ & $1$ \\ -\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Montgomery Reduction} -\end{figure} - -The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ -which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in -the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and -the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. - -\subsection{Baseline Montgomery Reduction} -The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for -Montgomery reductions. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. $digs \leftarrow 2n.used + 1$ \\ -2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ -\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ -\\ -Setup $x$ for the reduction. \\ -3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ -4. $x.used \leftarrow digs$ \\ -\\ -Eliminate the lower $k$ digits. \\ -5. For $ix$ from $0$ to $k - 1$ do \\ -\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.2 $u \leftarrow 0$ \\ -\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ -\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ -\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.4 While $u > 0$ do \\ -\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ -\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ -\\ -Divide by $\beta^k$ and fix up as required. \\ -6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ -7. If $x \ge n$ then \\ -\hspace{3mm}7.1 $x \leftarrow x - n$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_reduce.} -This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based -on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The -restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as -for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in -advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. - -Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on -the size of the input. This algorithm is discussed in ~COMBARED~. - -Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop -calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and -multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. - -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications -in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. - -EXAM,bn_mp_montgomery_reduce.c - -This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based -routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop. - -The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. - -\subsection{Faster ``Comba'' Montgomery Reduction} -MARK,COMBARED - -The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial -nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba -technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. - -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. - -With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ -1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ -Copy the digits of $x$ into the array $\hat W$ \\ -2. For $ix$ from $0$ to $x.used - 1$ do \\ -\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ -3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ -\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ -Elimiate the lower $k$ digits. \\ -4. for $ix$ from $0$ to $n.used - 1$ do \\ -\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ -\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ -\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Propagate carries upwards. \\ -5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ -\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Shift right and reduce modulo $\beta$ simultaneously. \\ -6. for $ix$ from $0$ to $n.used + 1$ do \\ -\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ -Zero excess digits and fixup $x$. \\ -7. if $x.used > n.used + 1$ then do \\ -\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ -\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ -8. $x.used \leftarrow n.used + 1$ \\ -9. Clamp excessive digits of $x$. \\ -10. If $x \ge n$ then \\ -\hspace{3mm}10.1 $x \leftarrow x - n$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm fast\_mp\_montgomery\_reduce.} -This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly -faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the -the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. - -As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the -contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step -4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such -as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing -a single precision multiplication instead half the amount of time is spent. - -Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step -4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note -how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no -point. - -Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. - -EXAM,bn_fast_mp_montgomery_reduce.c - -The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share -the same alias variables to make the code easier to read. - -The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This -forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry -for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. - -The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns -modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. - -\subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ -\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ -\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\hline \\ -1. $b \leftarrow n_0$ \\ -2. If $b$ is even return(\textit{MP\_VAL}) \\ -3. $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\ -4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ -\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ -5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_setup} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. - -EXAM,bn_mp_montgomery_setup.c - -This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. - -\section{The Diminished Radix Algorithm} -The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett -or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. - -\begin{equation} -(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} -\end{equation} - -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that -then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof -of the above equation is very simple. First write $x$ in the product form. - -\begin{equation} -x = qn + r -\end{equation} - -Now reduce both sides modulo $(n - k)$. - -\begin{equation} -x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} -\end{equation} - -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ -into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Diminished Radix Reduction}. \\ -\textbf{Input}. Integer $x$, $n$, $k$ \\ -\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ -\hline \\ -1. $q \leftarrow \lfloor x / n \rfloor$ \\ -2. $q \leftarrow k \cdot q$ \\ -3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ -4. $x \leftarrow x + q$ \\ -5. If $x \ge (n - k)$ then \\ -\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ -\hspace{3mm}5.2 Goto step 1. \\ -6. Return $x$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Diminished Radix Reduction} -\label{fig:DR} -\end{figure} - -This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always -once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. - -\begin{equation} -0 \le x < n^2 + k^2 - 2nk -\end{equation} - -The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. - -\begin{equation} -q < n - 2k - k^2/n -\end{equation} - -Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by - -\begin{equation} -0 \le q + x < (k + 1)n - 2k^2 - 1 -\end{equation} - -With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. - -\begin{figure} -\begin{small} -\begin{center} -\begin{tabular}{|l|} -\hline -$x = 123456789, n = 256, k = 3$ \\ -\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ -$q \leftarrow q*k = 1446759$ \\ -$x \leftarrow x \mbox{ mod } n = 21$ \\ -$x \leftarrow x + q = 1446780$ \\ -$x \leftarrow x - (n - k) = 1446527$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ -$q \leftarrow q*k = 16950$ \\ -$x \leftarrow x \mbox{ mod } n = 127$ \\ -$x \leftarrow x + q = 17077$ \\ -$x \leftarrow x - (n - k) = 16824$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 65$ \\ -$q \leftarrow q*k = 195$ \\ -$x \leftarrow x \mbox{ mod } n = 184$ \\ -$x \leftarrow x + q = 379$ \\ -$x \leftarrow x - (n - k) = 126$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example Diminished Radix Reduction} -\label{fig:EXDR} -\end{figure} - -Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ -is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only -three passes were required to find the residue $x \equiv 126$. - - -\subsection{Choice of Moduli} -On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other -modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. - -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. - -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. - -Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. - -\subsection{Choice of $k$} -Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ -in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. - -\subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce -an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular -exponentiations are performed. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_reduce}. \\ -\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ -\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ -\textbf{Output}. $x \mbox{ mod } n$ \\ -\hline \\ -1. $m \leftarrow n.used$ \\ -2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ -3. $\mu \leftarrow 0$ \\ -4. for $i$ from $0$ to $m - 1$ do \\ -\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ -\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. $x_{m} \leftarrow \mu$ \\ -6. for $i$ from $m + 1$ to $x.used - 1$ do \\ -\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ -7. Clamp excess digits of $x$. \\ -8. If $x \ge n$ then \\ -\hspace{3mm}8.1 $x \leftarrow x - n$ \\ -\hspace{3mm}8.2 Goto step 3. \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_reduce} -\end{figure} - -\textbf{Algorithm mp\_dr\_reduce.} -This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. - -This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ -and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing -the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. - -At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. - -EXAM,bn_mp_dr_reduce.c - -The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where -the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of -the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. - -The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits -a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) -in this algorithm. - -By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the -same pointer will point to the $m+1$'th digit where the zeroes will be placed. - -Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. -With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used -as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code -does not need to be checked. - -\subsubsection{Setup} -To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for -completeness. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = \beta - n_0$ \\ -\hline \\ -1. $k \leftarrow \beta - n_0$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_setup} -\end{figure} - -EXAM,bn_mp_dr_setup.c - -\subsubsection{Modulus Detection} -Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be -of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ -\hline -1. If $n.used < 2$ then return($0$). \\ -2. for $ix$ from $1$ to $n.used - 1$ do \\ -\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ -3. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_is\_modulus} -\end{figure} - -\textbf{Algorithm mp\_dr\_is\_modulus.} -This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are -in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to -step 3 then $n$ must be of Diminished Radix form. - -EXAM,bn_mp_dr_is_modulus.c - -\subsection{Unrestricted Diminished Radix Reduction} -The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm -is a straightforward adaptation of algorithm~\ref{fig:DR}. - -In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new -algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k}. \\ -\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ -\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ -\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. While $a \ge n$ do \\ -\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ -\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ -\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.5 If $a \ge n$ then do \\ -\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k.} -This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. - -EXAM,bn_mp_reduce_2k.c - -The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d -on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size -is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. - -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. - -\subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = 2^p - n$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ -3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ -4. $k \leftarrow x_0$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k\_setup.} -This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. - -EXAM,bn_mp_reduce_2k_setup.c - -\subsubsection{Unrestricted Detection} -An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. - -\begin{enumerate} -\item The number has only one digit. -\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. -\end{enumerate} - -If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only -one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact -that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most -significant bit. The resulting sum will be a power of two. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ -\hline -1. If $n.used = 0$ then return($0$). \\ -2. If $n.used = 1$ then return($1$). \\ -3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -4. for $x$ from $lg(\beta)$ to $p$ do \\ -\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ -5. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_is\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. - -EXAM,bn_mp_reduce_is_2k.c - - - -\section{Algorithm Comparison} -So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses -that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. - -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ -\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ -\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ -\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ -\hline -\end{tabular} -\end{small} -\end{center} - -In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery -reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of -calling the half precision multipliers, addition and division by $\beta$ algorithms. - -For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly -shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms -primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in -modular exponentiation to greatly speed up the operation. - - - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ - & calculates the correct value of $\rho$. \\ - & \\ -$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ - & \\ -$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ - & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ - & terminate within $1 \le k \le 10$ iterations. \\ - & \\ -\end{tabular} - - -\chapter{Exponentiation} -Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key -cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any -such cryptosystem and many methods have been sought to speed it up. - -\section{Exponentiation Basics} -A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size -the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature -with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. - -Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least -significant bit. If $b$ is a $k$-bit integer than the following equation is true. - -\begin{equation} -a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} -\end{equation} - -By taking the base $a$ logarithm of both sides of the equation the following equation is the result. - -\begin{equation} -b = \sum_{i=0}^{k-1}2^i \cdot b_i -\end{equation} - -The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to -$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average -$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. - -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to -be computed in an auxilary variable. Consider the following equivalent algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Left to Right Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$ and $k$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $k - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Left to Right Exponentiation} -\label{fig:LTOR} -\end{figure} - -This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is -multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. - -For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|} -\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ -\hline - & $1$ \\ -\hline $5$ & $a$ \\ -\hline $4$ & $a^2$ \\ -\hline $3$ & $a^4 \cdot a$ \\ -\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ -\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ -\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Left to Right Exponentiation} -\end{figure} - -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. - -\subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_expt\_d}. \\ -\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ -2. $c \leftarrow 1$ (\textit{mp\_set}) \\ -3. for $x$ from 1 to $lg(\beta)$ do \\ -\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ -\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ -\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ -4. Clear $g$. \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_expt\_d} -\end{figure} - -\textbf{Algorithm mp\_expt\_d.} -This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. - -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of -$1$ in the subsequent step. - -Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared -on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value -of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each -iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. - -EXAM,bn_mp_expt_d.c - -Line @29,mp_set@ sets the initial value of the result to $1$. Next the loop on line @31,for@ steps through each bit of the exponent starting from -the most significant down towards the least significant. The invariant squaring operation placed on line @333,mp_sqr@ is performed first. After -the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -@47,<<@ moves all of the bits of the exponent upwards towards the most significant location. - -\section{$k$-ary Exponentiation} -When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor -slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to -the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} -computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a -portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ -\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ -\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{$k$-ary Exponentiation} -\label{fig:KARY} -\end{figure} - -The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been -precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. -However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. - -Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The -original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings -has increased slightly but the number of multiplications has nearly halved. - -\subsection{Optimal Values of $k$} -An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest -approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ -\hline $16$ & $2$ & $27$ & $24$ \\ -\hline $32$ & $3$ & $49$ & $48$ \\ -\hline $64$ & $3$ & $92$ & $96$ \\ -\hline $128$ & $4$ & $175$ & $192$ \\ -\hline $256$ & $4$ & $335$ & $384$ \\ -\hline $512$ & $5$ & $645$ & $768$ \\ -\hline $1024$ & $6$ & $1257$ & $1536$ \\ -\hline $2048$ & $6$ & $2452$ & $3072$ \\ -\hline $4096$ & $7$ & $4808$ & $6144$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for $k$-ary Exponentiation} -\label{fig:OPTK} -\end{figure} - -\subsection{Sliding-Window Exponentiation} -A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. - -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ -\hline $16$ & $3$ & $24$ & $27$ \\ -\hline $32$ & $3$ & $45$ & $49$ \\ -\hline $64$ & $4$ & $87$ & $92$ \\ -\hline $128$ & $4$ & $167$ & $175$ \\ -\hline $256$ & $5$ & $322$ & $335$ \\ -\hline $512$ & $6$ & $628$ & $645$ \\ -\hline $1024$ & $6$ & $1225$ & $1257$ \\ -\hline $2048$ & $7$ & $2403$ & $2452$ \\ -\hline $4096$ & $8$ & $4735$ & $4808$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for Sliding Window Exponentiation} -\label{fig:OPTK2} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ -\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ -\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ -\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ -\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Sliding Window $k$-ary Exponentiation} -\end{figure} - -Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this -algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. - -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the -exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. - -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. - -\section{Modular Exponentiation} - -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. - -This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in ~REDUCTION~. - -Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm -will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The -value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. If $b.sign = MP\_NEG$ then \\ -\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ -\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ -\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ -3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ -\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ -4. else \\ -\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_exptmod} -\end{figure} - -\textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation -except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). - -EXAM,bn_mp_exptmod.c - -In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input. If the exponent is -negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned -the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive -exponent. - -If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix -form. If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one -of three values. - -\begin{enumerate} -\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. -\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. -\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. -\end{enumerate} - -Line @69,if@ determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. - -\subsection{Barrett Modular Exponentiation} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. $k \leftarrow lg(x)$ \\ -2. $winsize \leftarrow \left \lbrace \begin{array}{ll} - 2 & \mbox{if }k \le 7 \\ - 3 & \mbox{if }7 < k \le 36 \\ - 4 & \mbox{if }36 < k \le 140 \\ - 5 & \mbox{if }140 < k \le 450 \\ - 6 & \mbox{if }450 < k \le 1303 \\ - 7 & \mbox{if }1303 < k \le 3529 \\ - 8 & \mbox{if }3529 < k \\ - \end{array} \right .$ \\ -3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ -4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ -5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ -\\ -Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ -6. $k \leftarrow 2^{winsize - 1}$ \\ -7. $M_{k} \leftarrow M_1$ \\ -8. for $ix$ from 0 to $winsize - 2$ do \\ -\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ -\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ -\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -10. $res \leftarrow 1$ \\ -\\ -Start Sliding Window. \\ -11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ -12. Loop \\ -\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ -\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ -\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ -\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ -\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ -\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ -Continued on next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ -\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ -\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ -\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ -\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ -\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.6.3 Goto step 12. \\ -\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ -\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ -\hspace{3mm}12.9 $mode \leftarrow 2$ \\ -\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ -\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ -\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ -\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ -\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ -\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}Reset the window. \\ -\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ -\\ -No more windows left. Check for residual bits of exponent. \\ -13. If $mode = 2$ and $bitcpy > 0$ then do \\ -\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ -\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ -\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ -\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ -\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ -\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -14. $y \leftarrow res$ \\ -15. Clear $res$, $mu$ and the $M$ array. \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod (continued)} -\end{figure} - -\textbf{Algorithm s\_mp\_exptmod.} -This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction -algorithm to keep the product small throughout the algorithm. - -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the -larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. - -After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make -the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ -times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. - -Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. -\begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. -\begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. - \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit - downwards. -\end{enumerate} -\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit - is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. -\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. -\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and - the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. -\end{enumerate} - -All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step -inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. - -After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to -trailing edges the entire exponent is read from most significant bit to least significant bit. - -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the -algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. - -FIGU,expt_state,Sliding Window State Diagram - -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. - -EXAM,bn_s_mp_exptmod.c - -Lines @31,if@ through @45,}@ determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line @37,if@ the value of $x$ is already known to be greater than $140$. - -The conditional piece of code beginning on line @42,ifdef@ allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. - -The for loop on line @60,for@ initializes the $M$ array while lines @71,mp_init@ and @75,mp_reduce@ through @85,}@ initialize the reduction -function that will be used for this modulus. - --- More later. - -\section{Quick Power of Two} -Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is -equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_2expt}. \\ -\textbf{Input}. integer $b$ \\ -\textbf{Output}. $a \leftarrow 2^b$ \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ -3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ -4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_2expt} -\end{figure} - -\textbf{Algorithm mp\_2expt.} - -EXAM,bn_mp_2expt.c - -\chapter{Higher Level Algorithms} - -This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. - -The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate -various representations of integers. For example, converting from an mp\_int to a string of character. - -\section{Integer Division with Remainder} -\label{sec:division} - -Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication -the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and -let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ -\textbf{Input}. integer $x$ and $y$ \\ -\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ -\hline \\ -1. $q \leftarrow 0$ \\ -2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ -3. for $t$ from $n$ down to $0$ do \\ -\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ -\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ -\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ -4. $r \leftarrow y$ \\ -5. Return($q, r$) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Radix-$\beta$ Integer Division} -\label{fig:raddiv} -\end{figure} - -As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which -their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. - -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and -simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method -used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. -As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. - -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder -$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. - -\subsection{Quotient Estimation} -\label{sec:divest} -As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading -digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically -speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. - -The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ -represent the most significant digits of the dividend and divisor respectively. - -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to -$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other -cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of -inequalities will prove the hypothesis. - -\begin{equation} -y - \hat k x \le y - \hat k x_s\beta^s -\end{equation} - -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. - -\begin{equation} -y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) -\end{equation} - -By simplifying the previous inequality the following inequality is formed. - -\begin{equation} -y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s -\end{equation} - -Subsequently, - -\begin{equation} -y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x -\end{equation} - -Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} - - -\subsection{Normalized Integers} -For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both -$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original -remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. - -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} -\end{equation} - -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. - -\subsection{Radix-$\beta$ Division with Remainder} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div}. \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -1. If $b = 0$ return(\textit{MP\_VAL}). \\ -2. If $\vert a \vert < \vert b \vert$ then do \\ -\hspace{3mm}2.1 $d \leftarrow a$ \\ -\hspace{3mm}2.2 $c \leftarrow 0$ \\ -\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ -\\ -Setup the quotient to receive the digits. \\ -3. Grow $q$ to $a.used + 2$ digits. \\ -4. $q \leftarrow 0$ \\ -5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ -6. $sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = b.sign \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\\ -Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ -7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ -8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ -\\ -Find the leading digit of the quotient. \\ -9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ -10. $y \leftarrow y \cdot \beta^{n - t}$ \\ -11. While ($x \ge y$) do \\ -\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ -\hspace{3mm}11.2 $x \leftarrow x - y$ \\ -12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ -\\ -Continued on the next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div} (continued). \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -Now find the remainder fo the digits. \\ -13. for $i$ from $n$ down to $(t + 1)$ do \\ -\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ -\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ -\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ -\hspace{3mm}13.3 else \\ -\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ -\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ -\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ -\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ -\\ -Fixup quotient estimation. \\ -\hspace{3mm}13.5 Loop \\ -\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ -\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ -\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ -\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ -\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ -\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ -\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ -\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ -\hspace{6mm}13.10 t$1 \leftarrow y$ \\ -\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ -\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\\ -Finalize the result. \\ -14. Clamp excess digits of $q$ \\ -15. $c \leftarrow q, c.sign \leftarrow sign$ \\ -16. $x.sign \leftarrow a.sign$ \\ -17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ -18. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div (continued)} -\end{figure} -\textbf{Algorithm mp\_div.} -This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed -division and will produce a fully qualified quotient and remainder. - -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. - -After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the -divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. - -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is -$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the -shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. - -Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly -accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. - -Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is -to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher -order approximation to adjust the quotient digit. - -After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced -by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. - -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the -remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie -outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. - -EXAM,bn_mp_div.c - -The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or -remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is - -\begin{verbatim} -mp_div(&a, &b, &c, NULL); /* c = [a/b] */ -\end{verbatim} - -Lines @108,if@ and @113,if@ handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line @147,neg@ determines the sign of -the quotient and line @148,sign@ ensures that both $x$ and $y$ are positive. - -The number of bits in the leading digit is calculated on line @151,norm@. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits -of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is -exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting -them to the left by $lg(\beta) - 1 - k$ bits. - -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the -leading digit of the quotient. The loop beginning on line @184,for@ will produce the remainder of the quotient digits. - -The conditional ``continue'' on line @186,continue@ is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the -algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. - -Lines @214,t1@, @216,t1@ and @222,t2@ through @225,t2@ manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. - -\section{Single Digit Helpers} - -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of -the helper functions assume the single digit input is positive and will treat them as such. - -\subsection{Single Digit Addition and Subtraction} - -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction -algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = a + b$ \\ -\hline \\ -1. $t \leftarrow b$ (\textit{mp\_set}) \\ -2. $c \leftarrow a + t$ \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_add\_d} -\end{figure} - -\textbf{Algorithm mp\_add\_d.} -This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. - -EXAM,bn_mp_add_d.c - -Clever use of the letter 't'. - -\subsubsection{Subtraction} -The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. - -\subsection{Single Digit Multiplication} -Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline -multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands -only has one digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = ab$ \\ -\hline \\ -1. $pa \leftarrow a.used$ \\ -2. Grow $c$ to at least $pa + 1$ digits. \\ -3. $oldused \leftarrow c.used$ \\ -4. $c.used \leftarrow pa + 1$ \\ -5. $c.sign \leftarrow a.sign$ \\ -6. $\mu \leftarrow 0$ \\ -7. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ -\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -8. $c_{pa} \leftarrow \mu$ \\ -9. for $ix$ from $pa + 1$ to $oldused$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -10. Clamp excess digits of $c$. \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_d} -\end{figure} -\textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. - -EXAM,bn_mp_mul_d.c - -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. - -\subsection{Single Digit Division} -Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ -\hline \\ -1. If $b = 0$ then return(\textit{MP\_VAL}).\\ -2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ -3. Init $q$ to $a.used$ digits. \\ -4. $q.used \leftarrow a.used$ \\ -5. $q.sign \leftarrow a.sign$ \\ -6. $\hat w \leftarrow 0$ \\ -7. for $ix$ from $a.used - 1$ down to $0$ do \\ -\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ -\hspace{3mm}7.2 If $\hat w \ge b$ then \\ -\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ -\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}7.3 else\\ -\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ -\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ -8. $d \leftarrow \hat w$ \\ -9. Clamp excess digits of $q$. \\ -10. $c \leftarrow q$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_d} -\end{figure} -\textbf{Algorithm mp\_div\_d.} -This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the -algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. - -If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with -a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. - -EXAM,bn_mp_div_d.c - -Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to -indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. - -The division and remainder on lines @44,/@ and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. - -\subsection{Single Digit Root Extraction} - -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. - -\begin{equation} -x_{i+1} = x_i - {f(x_i) \over f'(x_i)} -\label{eqn:newton} -\end{equation} - -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is -simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the -algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_n\_root}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c^b \le a$ \\ -\hline \\ -1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. $sign \leftarrow a.sign$ \\ -3. $a.sign \leftarrow MP\_ZPOS$ \\ -4. t$2 \leftarrow 2$ \\ -5. Loop \\ -\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ -\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ -\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ -\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ -\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ -\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ -\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ -\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ -6. Loop \\ -\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ -\hspace{3mm}6.2 If t$2 > a$ then \\ -\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ -\hspace{6mm}6.2.2 Goto step 6. \\ -7. $a.sign \leftarrow sign$ \\ -8. $c \leftarrow $ t$1$ \\ -9. $c.sign \leftarrow sign$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_n\_root} -\end{figure} -\textbf{Algorithm mp\_n\_root.} -This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation -that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding -$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ -multiplications by t$1$ inside the loop. - -The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the -root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. - -EXAM,bn_mp_n_root.c - -\section{Random Number Generation} - -Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho -factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented -is solely for simulations and not intended for cryptographic use. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rand}. \\ -\textbf{Input}. An integer $b$ \\ -\textbf{Output}. A pseudo-random number of $b$ digits \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ -3. Pick a non-zero random digit $d$. \\ -4. $a \leftarrow a + d$ \\ -5. for $ix$ from 1 to $d - 1$ do \\ -\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ -\hspace{3mm}5.2 Pick a random digit $d$. \\ -\hspace{3mm}5.3 $a \leftarrow a + d$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rand} -\end{figure} -\textbf{Algorithm mp\_rand.} -This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the -final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of -the integers from $0$ to $\beta - 1$. - -EXAM,bn_mp_rand.c - -\section{Formatted Representations} -The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to -be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers -into a program. - -\subsection{Reading Radix-n Input} -For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to -printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the -map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen -such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary -mediums. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{cc|cc|cc|cc} -\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ -\hline -0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ -4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ -8 & 8 & 9 & 9 & 10 & A & 11 & B \\ -12 & C & 13 & D & 14 & E & 15 & F \\ -16 & G & 17 & H & 18 & I & 19 & J \\ -20 & K & 21 & L & 22 & M & 23 & N \\ -24 & O & 25 & P & 26 & Q & 27 & R \\ -28 & S & 29 & T & 30 & U & 31 & V \\ -32 & W & 33 & X & 34 & Y & 35 & Z \\ -36 & a & 37 & b & 38 & c & 39 & d \\ -40 & e & 41 & f & 42 & g & 43 & h \\ -44 & i & 45 & j & 46 & k & 47 & l \\ -48 & m & 49 & n & 50 & o & 51 & p \\ -52 & q & 53 & r & 54 & s & 55 & t \\ -56 & u & 57 & v & 58 & w & 59 & x \\ -60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ -\hline -\end{tabular} -\end{center} -\caption{Lower ASCII Map} -\label{fig:ASC} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_read\_radix}. \\ -\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ -\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. $ix \leftarrow 0$ \\ -3. If $str_0 =$ ``-'' then do \\ -\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ -\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ -4. else \\ -\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ -5. $a \leftarrow 0$ \\ -6. for $iy$ from $ix$ to $sn - 1$ do \\ -\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ -\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ -\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ -\hspace{3mm}6.4 $a \leftarrow a + y$ \\ -7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_read\_radix} -\end{figure} -\textbf{Algorithm mp\_read\_radix.} -This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the -string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input -and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded -as part of larger input without any significant problem. - -EXAM,bn_mp_read_radix.c - -\subsection{Generating Radix-$n$ Output} -Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toradix}. \\ -\textbf{Input}. A mp\_int $a$ and an integer $r$\\ -\textbf{Output}. The radix-$r$ representation of $a$ \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ -3. $t \leftarrow a$ \\ -4. $str \leftarrow$ ``'' \\ -5. if $t.sign = MP\_NEG$ then \\ -\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ -\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ -6. While ($t \ne 0$) do \\ -\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ -\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ -\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ -\hspace{3mm}6.4 $str \leftarrow str + y$ \\ -7. If $str_0 = $``$-$'' then \\ -\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ -8. Otherwise \\ -\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toradix} -\end{figure} -\textbf{Algorithm mp\_toradix.} -This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing -successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in -each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions -are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order -(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ -\hline $1234$ & -- & -- \\ -\hline $123$ & $4$ & ``4'' \\ -\hline $12$ & $3$ & ``43'' \\ -\hline $1$ & $2$ & ``432'' \\ -\hline $0$ & $1$ & ``4321'' \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Algorithm mp\_toradix.} -\label{fig:mpradix} -\end{figure} - -EXAM,bn_mp_toradix.c - -\chapter{Number Theoretic Algorithms} -This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi -symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and -various Sieve based factoring algorithms. - -\section{Greatest Common Divisor} -The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of -both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur -simultaneously. - -The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then -$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}1.2 $a \leftarrow b$ \\ -\hspace{3mm}1.3 $b \leftarrow r$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (I)} -\label{fig:gcd1} -\end{figure} - -This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are -relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of -greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. -In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}1.2 $b \leftarrow b - a$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (II)} -\label{fig:gcd2} -\end{figure} - -\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} -The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other -words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always -divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the -second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. - -As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that -$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does -not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by -the greatest common divisor. - -However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. -Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. $k \leftarrow 0$ \\ -2. While $a$ and $b$ are both divisible by $p$ do \\ -\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ -\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ -3. While $a$ is divisible by $p$ do \\ -\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -4. While $b$ is divisible by $p$ do \\ -\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -5. While ($b > 0$) do \\ -\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}5.2 $b \leftarrow b - a$ \\ -\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ -\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -6. Return($a \cdot p^k$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (III)} -\label{fig:gcd3} -\end{figure} - -This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ -decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common -divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely -divided out of the difference $b - a$ so long as the division leaves no remainder. - -In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy -to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by -step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the -largest of the pair. - -\subsection{Complete Greatest Common Divisor} -The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly -and will produce the greatest common divisor. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_gcd}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b = 0$ then \\ -\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ -4. $k \leftarrow 0$ \\ -5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -8. While $v.used > 0$ \\ -\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\ -\hspace{6mm}8.1.1 Swap $u$ and $v$. \\ -\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ -\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -9. $c \leftarrow u \cdot 2^k$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_gcd} -\end{figure} -\textbf{Algorithm mp\_gcd.} -This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of -Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as -Algorithm B and in practice this appears to be true. - -The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the -largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of -$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. - -Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a -factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step -six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since -they cannot both be even. - -By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to -or greater than $u$. This ensures that the subtraction on step 8.2 will always produce a positive and even result. Step 8.3 removes any -factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. - -After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result -must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. - -EXAM,bn_mp_gcd.c - -This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the -integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise -it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three -trivial cases of inputs are handled on lines @23,zero@ through @29,}@. After those lines the inputs are assumed to be non-zero. - -Lines @32,if@ and @36,if@ make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two -must be divided out of the two inputs. The block starting at line @43,common@ removes common factors of two by first counting the number of trailing -zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that -the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than -entries than are accessible by an ``int'' so this is not a limitation.}. - -At this point there are no more common factors of two in the two values. The divisions by a power of two on lines @60,div_2d@ and @67,div_2d@ remove -any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop -on line @72, while@ performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in -place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. - -\section{Least Common Multiple} -The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the -least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ -and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. - -The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will -collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on -Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). -Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lcm}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The least common multiple $c = [a, b]$. \\ -\hline \\ -1. $c \leftarrow (a, b)$ \\ -2. $t \leftarrow a \cdot b$ \\ -3. $c \leftarrow \lfloor t / c \rfloor$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lcm} -\end{figure} -\textbf{Algorithm mp\_lcm.} -This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by -dividing the product of the two inputs by their greatest common divisor. - -EXAM,bn_mp_lcm.c - -\section{Jacobi Symbol Computation} -To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is -defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is -equivalent to equation \ref{eqn:legendre}. - -\textit{-- Tom, don't be an ass, cite your source here...!} - -\begin{equation} -a^{(p-1)/2} \equiv \begin{array}{rl} - -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ - 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ - 1 & \mbox{if }a\mbox{ is a quadratic residue}. - \end{array} \mbox{ (mod }p\mbox{)} -\label{eqn:legendre} -\end{equation} - -\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} -An integer $a$ is a quadratic residue if the following equation has a solution. - -\begin{equation} -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\label{eqn:root} -\end{equation} - -Consider the following equation. - -\begin{equation} -0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} -\label{eqn:rooti} -\end{equation} - -Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ -then the quantity in the braces must be zero. By reduction, - -\begin{eqnarray} -\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ -\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\end{eqnarray} - -As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ -is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since -\begin{equation} -0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} -\end{equation} -One of the terms on the right hand side must be zero. \textbf{QED} - -\subsection{Jacobi Symbol} -The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then -the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) -\end{equation} - -By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for -further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the -following are true. - -\begin{enumerate} -\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. -\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. -\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. -\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. -\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically -$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. -\end{enumerate} - -Using these facts if $a = 2^k \cdot a'$ then - -\begin{eqnarray} -\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ - = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) -\label{eqn:jacobi} -\end{eqnarray} - -By fact five, - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} -\end{equation} - -The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of -$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the -factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the -Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_jacobi}. \\ -\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ -\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow 0$ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $a = 1$ then \\ -\hspace{3mm}2.1 $c \leftarrow 1$ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $a' \leftarrow a$ \\ -4. $k \leftarrow 0$ \\ -5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ -6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ -\hspace{3mm}6.1 $s \leftarrow 1$ \\ -7. else \\ -\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ -\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ -\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ -\hspace{3mm}7.3 else \\ -\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ -8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ -\hspace{3mm}8.1 $s \leftarrow -s$ \\ -9. If $a' \ne 1$ then \\ -\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ -\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ -10. $c \leftarrow s$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_jacobi} -\end{figure} -\textbf{Algorithm mp\_jacobi.} -This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm -is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. - -Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the -input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one -if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled -the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ -are congruent to one modulo four, otherwise it evaluates to negative one. - -By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute -$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. - -EXAM,bn_mp_jacobi.c - -As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C -variable name character. - -The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm -has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since -the values it may obtain are merely $-1$, $0$ and $1$. - -After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant -bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same -processor requirements and neither is faster than the other. - -Line @59, if@ through @70, }@ determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than -$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of -$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines @73, if@ through @75, }@. - -Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. - -\textit{-- Comment about default $s$ and such...} - -\section{Modular Inverse} -\label{sec:modinv} -The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there -exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is -denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and -fields of integers. However, the former will be the matter of discussion. - -The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the -order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. - -\begin{equation} -ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} -\end{equation} - -However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite -requires all of the prime factors. This approach also is very slow as the size of $p$ grows. - -A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear -Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. - -\begin{equation} -ab + pq = 1 -\end{equation} - -Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of -$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. -However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The -binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine -equation. - -\subsection{General Case} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_invmod}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ -\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ -2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ -3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ -4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ -5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ -\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ -\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ -\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ -\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ -\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ -\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ -8. If $u \ge v$ then \\ -\hspace{3mm}8.1 $u \leftarrow u - v$ \\ -\hspace{3mm}8.2 $A \leftarrow A - C$ \\ -\hspace{3mm}8.3 $B \leftarrow B - D$ \\ -9. else \\ -\hspace{3mm}9.1 $v \leftarrow v - u$ \\ -\hspace{3mm}9.2 $C \leftarrow C - A$ \\ -\hspace{3mm}9.3 $D \leftarrow D - B$ \\ -10. If $u \ne 0$ goto step 6. \\ -11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ -12. While $C \le 0$ do \\ -\hspace{3mm}12.1 $C \leftarrow C + b$ \\ -13. While $C \ge b$ do \\ -\hspace{3mm}13.1 $C \leftarrow C - b$ \\ -14. $c \leftarrow C$ \\ -15. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\end{figure} -\textbf{Algorithm mp\_invmod.} -This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the -extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete -Diophantine solution. - -If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative -inverse for $a$ and the error is reported. - -The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case -the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is - -\begin{equation} -Ca + Db = v -\end{equation} - -If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ -is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie -within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ -then only a couple of additions or subtractions will be required to adjust the inverse. - -EXAM,bn_mp_invmod.c - -\subsubsection{Odd Moduli} - -When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve -the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. - -The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This -optimization will halve the time required to compute the modular inverse. - -\section{Primality Tests} - -A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime -since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. - -Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or -not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all -probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is -prime the algorithm may be incorrect. - -As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as -well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. - -\subsection{Trial Division} - -Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously -cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test -would require a prohibitive amount of time as $n$ grows. - -Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset -of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. - -The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be -discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by -$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range -$3 \le q \le 100$. - -At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to -be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate -approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The -array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ -\hline \\ -1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ -\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ -\hspace{3mm}1.2 If $d = 0$ then \\ -\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ -\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow 0$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_is\_divisible} -\end{figure} -\textbf{Algorithm mp\_prime\_is\_divisible.} -This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. - -EXAM,bn_mp_prime_is_divisible.c - -The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a -mp\_digit. The table \_\_prime\_tab is defined in the following file. - -EXAM,bn_prime_tab.c - -Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes -upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. - -\subsection{The Fermat Test} -The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in -fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of -the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to -$a^1 = a$. - -If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case -it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order -of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several -integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows -in size. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_fermat}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ -\hline \\ -1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ -2. If $t = b$ then \\ -\hspace{3mm}2.1 $c = 1$ \\ -3. else \\ -\hspace{3mm}3.1 $c = 0$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_fermat} -\end{figure} -\textbf{Algorithm mp\_prime\_fermat.} -This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to -determine the result. - -EXAM,bn_mp_prime_fermat.c - -\subsection{The Miller-Rabin Test} -The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen -candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the -value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that -some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ -\hline -1. $a' \leftarrow a - 1$ \\ -2. $r \leftarrow n1$ \\ -3. $c \leftarrow 0, s \leftarrow 0$ \\ -4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ -\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ -5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ -6. If $y \nequiv \pm 1$ then \\ -\hspace{3mm}6.1 $j \leftarrow 1$ \\ -\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ -\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ -\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ -\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ -\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ -7. $c \leftarrow 1$\\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_miller\_rabin} -\end{figure} -\textbf{Algorithm mp\_prime\_miller\_rabin.} -This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine -if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. - -If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will -square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ -is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably -composite then it is \textit{probably} prime. - -EXAM,bn_mp_prime_miller_rabin.c - - - - -\backmatter -\appendix -\begin{thebibliography}{ABCDEF} -\bibitem[1]{TAOCPV2} -Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 - -\bibitem[2]{HAC} -A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 - -\bibitem[3]{ROSE} -Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 - -\bibitem[4]{COMBA} -Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) - -\bibitem[5]{KARA} -A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 - -\bibitem[6]{KARAP} -Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 - -\bibitem[7]{BARRETT} -Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. - -\bibitem[8]{MONT} -P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. - -\bibitem[9]{DRMET} -Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories - -\bibitem[10]{MMB} -J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 - -\bibitem[11]{RSAREF} -R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} - -\bibitem[12]{DHREF} -Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 - -\bibitem[13]{IEEE} -IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) - -\bibitem[14]{GMP} -GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} - -\bibitem[15]{MPI} -Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} - -\bibitem[16]{OPENSSL} -OpenSSL Cryptographic Toolkit, \url{http://openssl.org} - -\bibitem[17]{LIP} -Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} - -\bibitem[18]{ISOC} -JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' - -\bibitem[19]{JAVA} -The Sun Java Website, \url{http://java.sun.com/} - -\end{thebibliography} - -\input{tommath.ind} - -\end{document} diff --git a/lib/hcrypto/libtommath/tommath.tex b/lib/hcrypto/libtommath/tommath.tex deleted file mode 100644 index 17d6041d5..000000000 --- a/lib/hcrypto/libtommath/tommath.tex +++ /dev/null @@ -1,6692 +0,0 @@ -\documentclass[b5paper]{book} -\usepackage{hyperref} -\usepackage{makeidx} -\usepackage{amssymb} -\usepackage{color} -\usepackage{alltt} -\usepackage{graphicx} -\usepackage{layout} -\def\union{\cup} -\def\intersect{\cap} -\def\getsrandom{\stackrel{\rm R}{\gets}} -\def\cross{\times} -\def\cat{\hspace{0.5em} \| \hspace{0.5em}} -\def\catn{$\|$} -\def\divides{\hspace{0.3em} | \hspace{0.3em}} -\def\nequiv{\not\equiv} -\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}} -\def\lcm{{\rm lcm}} -\def\gcd{{\rm gcd}} -\def\log{{\rm log}} -\def\ord{{\rm ord}} -\def\labs{{\mathit labs}} -\def\abs{{\mathit abs}} -\def\rep{{\mathit rep}} -\def\mod{{\mathit\ mod\ }} -\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})} -\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor} -\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil} -\def\Or{{\rm\ or\ }} -\def\And{{\rm\ and\ }} -\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}} -\def\implies{\Rightarrow} -\def\undefined{{\rm ``undefined"}} -\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}} -\let\oldphi\phi -\def\phi{\varphi} -\def\Pr{{\rm Pr}} -\newcommand{\str}[1]{{\mathbf{#1}}} -\def\F{{\mathbb F}} -\def\N{{\mathbb N}} -\def\Z{{\mathbb Z}} -\def\R{{\mathbb R}} -\def\C{{\mathbb C}} -\def\Q{{\mathbb Q}} -\definecolor{DGray}{gray}{0.5} -\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}} -\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}} -\def\gap{\vspace{0.5ex}} -\makeindex -\begin{document} -\frontmatter -\pagestyle{empty} -\title{Multi--Precision Math} -\author{\mbox{ -%\begin{small} -\begin{tabular}{c} -Tom St Denis \\ -Algonquin College \\ -\\ -Mads Rasmussen \\ -Open Communications Security \\ -\\ -Greg Rose \\ -QUALCOMM Australia \\ -\end{tabular} -%\end{small} -} -} -\maketitle -This text has been placed in the public domain. This text corresponds to the v0.39 release of the -LibTomMath project. - -\begin{alltt} -Tom St Denis -111 Banning Rd -Ottawa, Ontario -K2L 1C3 -Canada - -Phone: 1-613-836-3160 -Email: tomstdenis@gmail.com -\end{alltt} - -This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{} -{\em book} macro package and the Perl {\em booker} package. - -\tableofcontents -\listoffigures -\chapter*{Prefaces} -When I tell people about my LibTom projects and that I release them as public domain they are often puzzled. -They ask why I did it and especially why I continue to work on them for free. The best I can explain it is ``Because I can.'' -Which seems odd and perhaps too terse for adult conversation. I often qualify it with ``I am able, I am willing.'' which -perhaps explains it better. I am the first to admit there is not anything that special with what I have done. Perhaps -others can see that too and then we would have a society to be proud of. My LibTom projects are what I am doing to give -back to society in the form of tools and knowledge that can help others in their endeavours. - -I started writing this book because it was the most logical task to further my goal of open academia. The LibTomMath source -code itself was written to be easy to follow and learn from. There are times, however, where pure C source code does not -explain the algorithms properly. Hence this book. The book literally starts with the foundation of the library and works -itself outwards to the more complicated algorithms. The use of both pseudo--code and verbatim source code provides a duality -of ``theory'' and ``practice'' that the computer science students of the world shall appreciate. I never deviate too far -from relatively straightforward algebra and I hope that this book can be a valuable learning asset. - -This book and indeed much of the LibTom projects would not exist in their current form if it was not for a plethora -of kind people donating their time, resources and kind words to help support my work. Writing a text of significant -length (along with the source code) is a tiresome and lengthy process. Currently the LibTom project is four years old, -comprises of literally thousands of users and over 100,000 lines of source code, TeX and other material. People like Mads and Greg -were there at the beginning to encourage me to work well. It is amazing how timely validation from others can boost morale to -continue the project. Definitely my parents were there for me by providing room and board during the many months of work in 2003. - -To my many friends whom I have met through the years I thank you for the good times and the words of encouragement. I hope I -honour your kind gestures with this project. - -Open Source. Open Academia. Open Minds. - -\begin{flushright} Tom St Denis \end{flushright} - -\newpage -I found the opportunity to work with Tom appealing for several reasons, not only could I broaden my own horizons, but also -contribute to educate others facing the problem of having to handle big number mathematical calculations. - -This book is Tom's child and he has been caring and fostering the project ever since the beginning with a clear mind of -how he wanted the project to turn out. I have helped by proofreading the text and we have had several discussions about -the layout and language used. - -I hold a masters degree in cryptography from the University of Southern Denmark and have always been interested in the -practical aspects of cryptography. - -Having worked in the security consultancy business for several years in S\~{a}o Paulo, Brazil, I have been in touch with a -great deal of work in which multiple precision mathematics was needed. Understanding the possibilities for speeding up -multiple precision calculations is often very important since we deal with outdated machine architecture where modular -reductions, for example, become painfully slow. - -This text is for people who stop and wonder when first examining algorithms such as RSA for the first time and asks -themselves, ``You tell me this is only secure for large numbers, fine; but how do you implement these numbers?'' - -\begin{flushright} -Mads Rasmussen - -S\~{a}o Paulo - SP - -Brazil -\end{flushright} - -\newpage -It's all because I broke my leg. That just happened to be at about the same time that Tom asked for someone to review the section of the book about -Karatsuba multiplication. I was laid up, alone and immobile, and thought ``Why not?'' I vaguely knew what Karatsuba multiplication was, but not -really, so I thought I could help, learn, and stop myself from watching daytime cable TV, all at once. - -At the time of writing this, I've still not met Tom or Mads in meatspace. I've been following Tom's progress since his first splash on the -sci.crypt Usenet news group. I watched him go from a clueless newbie, to the cryptographic equivalent of a reformed smoker, to a real -contributor to the field, over a period of about two years. I've been impressed with his obvious intelligence, and astounded by his productivity. -Of course, he's young enough to be my own child, so he doesn't have my problems with staying awake. - -When I reviewed that single section of the book, in its very earliest form, I was very pleasantly surprised. So I decided to collaborate more fully, -and at least review all of it, and perhaps write some bits too. There's still a long way to go with it, and I have watched a number of close -friends go through the mill of publication, so I think that the way to go is longer than Tom thinks it is. Nevertheless, it's a good effort, -and I'm pleased to be involved with it. - -\begin{flushright} -Greg Rose, Sydney, Australia, June 2003. -\end{flushright} - -\mainmatter -\pagestyle{headings} -\chapter{Introduction} -\section{Multiple Precision Arithmetic} - -\subsection{What is Multiple Precision Arithmetic?} -When we think of long-hand arithmetic such as addition or multiplication we rarely consider the fact that we instinctively -raise or lower the precision of the numbers we are dealing with. For example, in decimal we almost immediate can -reason that $7$ times $6$ is $42$. However, $42$ has two digits of precision as opposed to one digit we started with. -Further multiplications of say $3$ result in a larger precision result $126$. In these few examples we have multiple -precisions for the numbers we are working with. Despite the various levels of precision a single subset\footnote{With the occasional optimization.} - of algorithms can be designed to accomodate them. - -By way of comparison a fixed or single precision operation would lose precision on various operations. For example, in -the decimal system with fixed precision $6 \cdot 7 = 2$. - -Essentially at the heart of computer based multiple precision arithmetic are the same long-hand algorithms taught in -schools to manually add, subtract, multiply and divide. - -\subsection{The Need for Multiple Precision Arithmetic} -The most prevalent need for multiple precision arithmetic, often referred to as ``bignum'' math, is within the implementation -of public-key cryptography algorithms. Algorithms such as RSA \cite{RSAREF} and Diffie-Hellman \cite{DHREF} require -integers of significant magnitude to resist known cryptanalytic attacks. For example, at the time of this writing a -typical RSA modulus would be at least greater than $10^{309}$. However, modern programming languages such as ISO C \cite{ISOC} and -Java \cite{JAVA} only provide instrinsic support for integers which are relatively small and single precision. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{|r|c|} -\hline \textbf{Data Type} & \textbf{Range} \\ -\hline char & $-128 \ldots 127$ \\ -\hline short & $-32768 \ldots 32767$ \\ -\hline long & $-2147483648 \ldots 2147483647$ \\ -\hline long long & $-9223372036854775808 \ldots 9223372036854775807$ \\ -\hline -\end{tabular} -\end{center} -\caption{Typical Data Types for the C Programming Language} -\label{fig:ISOC} -\end{figure} - -The largest data type guaranteed to be provided by the ISO C programming -language\footnote{As per the ISO C standard. However, each compiler vendor is allowed to augment the precision as they -see fit.} can only represent values up to $10^{19}$ as shown in figure \ref{fig:ISOC}. On its own the C language is -insufficient to accomodate the magnitude required for the problem at hand. An RSA modulus of magnitude $10^{19}$ could be -trivially factored\footnote{A Pollard-Rho factoring would take only $2^{16}$ time.} on the average desktop computer, -rendering any protocol based on the algorithm insecure. Multiple precision algorithms solve this very problem by -extending the range of representable integers while using single precision data types. - -Most advancements in fast multiple precision arithmetic stem from the need for faster and more efficient cryptographic -primitives. Faster modular reduction and exponentiation algorithms such as Barrett's algorithm, which have appeared in -various cryptographic journals, can render algorithms such as RSA and Diffie-Hellman more efficient. In fact, several -major companies such as RSA Security, Certicom and Entrust have built entire product lines on the implementation and -deployment of efficient algorithms. - -However, cryptography is not the only field of study that can benefit from fast multiple precision integer routines. -Another auxiliary use of multiple precision integers is high precision floating point data types. -The basic IEEE \cite{IEEE} standard floating point type is made up of an integer mantissa $q$, an exponent $e$ and a sign bit $s$. -Numbers are given in the form $n = q \cdot b^e \cdot -1^s$ where $b = 2$ is the most common base for IEEE. Since IEEE -floating point is meant to be implemented in hardware the precision of the mantissa is often fairly small -(\textit{23, 48 and 64 bits}). The mantissa is merely an integer and a multiple precision integer could be used to create -a mantissa of much larger precision than hardware alone can efficiently support. This approach could be useful where -scientific applications must minimize the total output error over long calculations. - -Yet another use for large integers is within arithmetic on polynomials of large characteristic (i.e. $GF(p)[x]$ for large $p$). -In fact the library discussed within this text has already been used to form a polynomial basis library\footnote{See \url{http://poly.libtomcrypt.org} for more details.}. - -\subsection{Benefits of Multiple Precision Arithmetic} -\index{precision} -The benefit of multiple precision representations over single or fixed precision representations is that -no precision is lost while representing the result of an operation which requires excess precision. For example, -the product of two $n$-bit integers requires at least $2n$ bits of precision to be represented faithfully. A multiple -precision algorithm would augment the precision of the destination to accomodate the result while a single precision system -would truncate excess bits to maintain a fixed level of precision. - -It is possible to implement algorithms which require large integers with fixed precision algorithms. For example, elliptic -curve cryptography (\textit{ECC}) is often implemented on smartcards by fixing the precision of the integers to the maximum -size the system will ever need. Such an approach can lead to vastly simpler algorithms which can accomodate the -integers required even if the host platform cannot natively accomodate them\footnote{For example, the average smartcard -processor has an 8 bit accumulator.}. However, as efficient as such an approach may be, the resulting source code is not -normally very flexible. It cannot, at runtime, accomodate inputs of higher magnitude than the designer anticipated. - -Multiple precision algorithms have the most overhead of any style of arithmetic. For the the most part the -overhead can be kept to a minimum with careful planning, but overall, it is not well suited for most memory starved -platforms. However, multiple precision algorithms do offer the most flexibility in terms of the magnitude of the -inputs. That is, the same algorithms based on multiple precision integers can accomodate any reasonable size input -without the designer's explicit forethought. This leads to lower cost of ownership for the code as it only has to -be written and tested once. - -\section{Purpose of This Text} -The purpose of this text is to instruct the reader regarding how to implement efficient multiple precision algorithms. -That is to not only explain a limited subset of the core theory behind the algorithms but also the various ``house keeping'' -elements that are neglected by authors of other texts on the subject. Several well reknowned texts \cite{TAOCPV2,HAC} -give considerably detailed explanations of the theoretical aspects of algorithms and often very little information -regarding the practical implementation aspects. - -In most cases how an algorithm is explained and how it is actually implemented are two very different concepts. For -example, the Handbook of Applied Cryptography (\textit{HAC}), algorithm 14.7 on page 594, gives a relatively simple -algorithm for performing multiple precision integer addition. However, the description lacks any discussion concerning -the fact that the two integer inputs may be of differing magnitudes. As a result the implementation is not as simple -as the text would lead people to believe. Similarly the division routine (\textit{algorithm 14.20, pp. 598}) does not -discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{step \#3}). - -Both texts also do not discuss several key optimal algorithms required such as ``Comba'' and Karatsuba multipliers -and fast modular inversion, which we consider practical oversights. These optimal algorithms are vital to achieve -any form of useful performance in non-trivial applications. - -To solve this problem the focus of this text is on the practical aspects of implementing a multiple precision integer -package. As a case study the ``LibTomMath''\footnote{Available at \url{http://math.libtomcrypt.com}} package is used -to demonstrate algorithms with real implementations\footnote{In the ISO C programming language.} that have been field -tested and work very well. The LibTomMath library is freely available on the Internet for all uses and this text -discusses a very large portion of the inner workings of the library. - -The algorithms that are presented will always include at least one ``pseudo-code'' description followed -by the actual C source code that implements the algorithm. The pseudo-code can be used to implement the same -algorithm in other programming languages as the reader sees fit. - -This text shall also serve as a walkthrough of the creation of multiple precision algorithms from scratch. Showing -the reader how the algorithms fit together as well as where to start on various taskings. - -\section{Discussion and Notation} -\subsection{Notation} -A multiple precision integer of $n$-digits shall be denoted as $x = (x_{n-1}, \ldots, x_1, x_0)_{ \beta }$ and represent -the integer $x \equiv \sum_{i=0}^{n-1} x_i\beta^i$. The elements of the array $x$ are said to be the radix $\beta$ digits -of the integer. For example, $x = (1,2,3)_{10}$ would represent the integer -$1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$. - -\index{mp\_int} -The term ``mp\_int'' shall refer to a composite structure which contains the digits of the integer it represents, as well -as auxilary data required to manipulate the data. These additional members are discussed further in section -\ref{sec:MPINT}. For the purposes of this text a ``multiple precision integer'' and an ``mp\_int'' are assumed to be -synonymous. When an algorithm is specified to accept an mp\_int variable it is assumed the various auxliary data members -are present as well. An expression of the type \textit{variablename.item} implies that it should evaluate to the -member named ``item'' of the variable. For example, a string of characters may have a member ``length'' which would -evaluate to the number of characters in the string. If the string $a$ equals ``hello'' then it follows that -$a.length = 5$. - -For certain discussions more generic algorithms are presented to help the reader understand the final algorithm used -to solve a given problem. When an algorithm is described as accepting an integer input it is assumed the input is -a plain integer with no additional multiple-precision members. That is, algorithms that use integers as opposed to -mp\_ints as inputs do not concern themselves with the housekeeping operations required such as memory management. These -algorithms will be used to establish the relevant theory which will subsequently be used to describe a multiple -precision algorithm to solve the same problem. - -\subsection{Precision Notation} -The variable $\beta$ represents the radix of a single digit of a multiple precision integer and -must be of the form $q^p$ for $q, p \in \Z^+$. A single precision variable must be able to represent integers in -the range $0 \le x < q \beta$ while a double precision variable must be able to represent integers in the range -$0 \le x < q \beta^2$. The extra radix-$q$ factor allows additions and subtractions to proceed without truncation of the -carry. Since all modern computers are binary, it is assumed that $q$ is two. - -\index{mp\_digit} \index{mp\_word} -Within the source code that will be presented for each algorithm, the data type \textbf{mp\_digit} will represent -a single precision integer type, while, the data type \textbf{mp\_word} will represent a double precision integer type. In -several algorithms (notably the Comba routines) temporary results will be stored in arrays of double precision mp\_words. -For the purposes of this text $x_j$ will refer to the $j$'th digit of a single precision array and $\hat x_j$ will refer to -the $j$'th digit of a double precision array. Whenever an expression is to be assigned to a double precision -variable it is assumed that all single precision variables are promoted to double precision during the evaluation. -Expressions that are assigned to a single precision variable are truncated to fit within the precision of a single -precision data type. - -For example, if $\beta = 10^2$ a single precision data type may represent a value in the -range $0 \le x < 10^3$, while a double precision data type may represent a value in the range $0 \le x < 10^5$. Let -$a = 23$ and $b = 49$ represent two single precision variables. The single precision product shall be written -as $c \leftarrow a \cdot b$ while the double precision product shall be written as $\hat c \leftarrow a \cdot b$. -In this particular case, $\hat c = 1127$ and $c = 127$. The most significant digit of the product would not fit -in a single precision data type and as a result $c \ne \hat c$. - -\subsection{Algorithm Inputs and Outputs} -Within the algorithm descriptions all variables are assumed to be scalars of either single or double precision -as indicated. The only exception to this rule is when variables have been indicated to be of type mp\_int. This -distinction is important as scalars are often used as array indicies and various other counters. - -\subsection{Mathematical Expressions} -The $\lfloor \mbox{ } \rfloor$ brackets imply an expression truncated to an integer not greater than the expression -itself. For example, $\lfloor 5.7 \rfloor = 5$. Similarly the $\lceil \mbox{ } \rceil$ brackets imply an expression -rounded to an integer not less than the expression itself. For example, $\lceil 5.1 \rceil = 6$. Typically when -the $/$ division symbol is used the intention is to perform an integer division with truncation. For example, -$5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When an expression is written as a -fraction a real value division is implied, for example ${5 \over 2} = 2.5$. - -The norm of a multiple precision integer, for example $\vert \vert x \vert \vert$, will be used to represent the number of digits in the representation -of the integer. For example, $\vert \vert 123 \vert \vert = 3$ and $\vert \vert 79452 \vert \vert = 5$. - -\subsection{Work Effort} -\index{big-Oh} -To measure the efficiency of the specified algorithms, a modified big-Oh notation is used. In this system all -single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}. -That is a single precision addition, multiplication and division are assumed to take the same time to -complete. While this is generally not true in practice, it will simplify the discussions considerably. - -Some algorithms have slight advantages over others which is why some constants will not be removed in -the notation. For example, a normal baseline multiplication (section \ref{sec:basemult}) requires $O(n^2)$ work while a -baseline squaring (section \ref{sec:basesquare}) requires $O({{n^2 + n}\over 2})$ work. In standard big-Oh notation these -would both be said to be equivalent to $O(n^2)$. However, -in the context of the this text this is not the case as the magnitude of the inputs will typically be rather small. As a -result small constant factors in the work effort will make an observable difference in algorithm efficiency. - -All of the algorithms presented in this text have a polynomial time work level. That is, of the form -$O(n^k)$ for $n, k \in \Z^{+}$. This will help make useful comparisons in terms of the speed of the algorithms and how -various optimizations will help pay off in the long run. - -\section{Exercises} -Within the more advanced chapters a section will be set aside to give the reader some challenging exercises related to -the discussion at hand. These exercises are not designed to be prize winning problems, but instead to be thought -provoking. Wherever possible the problems are forward minded, stating problems that will be answered in subsequent -chapters. The reader is encouraged to finish the exercises as they appear to get a better understanding of the -subject material. - -That being said, the problems are designed to affirm knowledge of a particular subject matter. Students in particular -are encouraged to verify they can answer the problems correctly before moving on. - -Similar to the exercises of \cite[pp. ix]{TAOCPV2} these exercises are given a scoring system based on the difficulty of -the problem. However, unlike \cite{TAOCPV2} the problems do not get nearly as hard. The scoring of these -exercises ranges from one (the easiest) to five (the hardest). The following table sumarizes the -scoring system used. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|l|} -\hline $\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\ - & minutes to solve. Usually does not involve much computer time \\ - & to solve. \\ -\hline $\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\ - & time usage. Usually requires a program to be written to \\ - & solve the problem. \\ -\hline $\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\ - & of work. Usually involves trivial research and development of \\ - & new theory from the perspective of a student. \\ -\hline $\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\ - & of work and research, the solution to which will demonstrate \\ - & a higher mastery of the subject matter. \\ -\hline $\left [ 5 \right ]$ & A hard problem that involves concepts that are difficult for a \\ - & novice to solve. Solutions to these problems will demonstrate a \\ - & complete mastery of the given subject. \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Exercise Scoring System} -\end{figure} - -Problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or -devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level -are also designed to be easy but will require a program or algorithm to be implemented to arrive at the answer. These -two levels are essentially entry level questions. - -Problems at the third level are meant to be a bit more difficult than the first two levels. The answer is often -fairly obvious but arriving at an exacting solution requires some thought and skill. These problems will almost always -involve devising a new algorithm or implementing a variation of another algorithm previously presented. Readers who can -answer these questions will feel comfortable with the concepts behind the topic at hand. - -Problems at the fourth level are meant to be similar to those of the level three questions except they will require -additional research to be completed. The reader will most likely not know the answer right away, nor will the text provide -the exact details of the answer until a subsequent chapter. - -Problems at the fifth level are meant to be the hardest -problems relative to all the other problems in the chapter. People who can correctly answer fifth level problems have a -mastery of the subject matter at hand. - -Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader -is encouraged to answer the follow-up problems and try to draw the relevance of problems. - -\section{Introduction to LibTomMath} - -\subsection{What is LibTomMath?} -LibTomMath is a free and open source multiple precision integer library written entirely in portable ISO C. By portable it -is meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on -any given platform. - -The library has been successfully tested under numerous operating systems including Unix\footnote{All of these -trademarks belong to their respective rightful owners.}, MacOS, Windows, Linux, PalmOS and on standalone hardware such -as the Gameboy Advance. The library is designed to contain enough functionality to be able to develop applications such -as public key cryptosystems and still maintain a relatively small footprint. - -\subsection{Goals of LibTomMath} - -Libraries which obtain the most efficiency are rarely written in a high level programming language such as C. However, -even though this library is written entirely in ISO C, considerable care has been taken to optimize the algorithm implementations within the -library. Specifically the code has been written to work well with the GNU C Compiler (\textit{GCC}) on both x86 and ARM -processors. Wherever possible, highly efficient algorithms, such as Karatsuba multiplication, sliding window -exponentiation and Montgomery reduction have been provided to make the library more efficient. - -Even with the nearly optimal and specialized algorithms that have been included the Application Programing Interface -(\textit{API}) has been kept as simple as possible. Often generic place holder routines will make use of specialized -algorithms automatically without the developer's specific attention. One such example is the generic multiplication -algorithm \textbf{mp\_mul()} which will automatically use Toom--Cook, Karatsuba, Comba or baseline multiplication -based on the magnitude of the inputs and the configuration of the library. - -Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should -be source compatible with another popular library which makes it more attractive for developers to use. In this case the -MPI library was used as a API template for all the basic functions. MPI was chosen because it is another library that fits -in the same niche as LibTomMath. Even though LibTomMath uses MPI as the template for the function names and argument -passing conventions, it has been written from scratch by Tom St Denis. - -The project is also meant to act as a learning tool for students, the logic being that no easy-to-follow ``bignum'' -library exists which can be used to teach computer science students how to perform fast and reliable multiple precision -integer arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. - -\section{Choice of LibTomMath} -LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but -for more worthy reasons. Other libraries such as GMP \cite{GMP}, MPI \cite{MPI}, LIP \cite{LIP} and OpenSSL -\cite{OPENSSL} have multiple precision integer arithmetic routines but would not be ideal for this text for -reasons that will be explained in the following sub-sections. - -\subsection{Code Base} -The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional -segments of code littered throughout the source. This clean and uncluttered approach to the library means that a -developer can more readily discern the true intent of a given section of source code without trying to keep track of -what conditional code will be used. - -The code base of LibTomMath is well organized. Each function is in its own separate source code file -which allows the reader to find a given function very quickly. On average there are $76$ lines of code per source -file which makes the source very easily to follow. By comparison MPI and LIP are single file projects making code tracing -very hard. GMP has many conditional code segments which also hinder tracing. - -When compiled with GCC for the x86 processor and optimized for speed the entire library is approximately $100$KiB\footnote{The notation ``KiB'' means $2^{10}$ octets, similarly ``MiB'' means $2^{20}$ octets.} - which is fairly small compared to GMP (over $250$KiB). LibTomMath is slightly larger than MPI (which compiles to about -$50$KiB) but LibTomMath is also much faster and more complete than MPI. - -\subsection{API Simplicity} -LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build -with LibTomMath without change. The function names correlate directly to the action they perform. Almost all of the -functions share the same parameter passing convention. The learning curve is fairly shallow with the API provided -which is an extremely valuable benefit for the student and developer alike. - -The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to -illegible short hand. LibTomMath does not share this characteristic. - -The GMP library also does not return error codes. Instead it uses a POSIX.1 \cite{POSIX1} signal system where errors -are signaled to the host application. This happens to be the fastest approach but definitely not the most versatile. In -effect a math error (i.e. invalid input, heap error, etc) can cause a program to stop functioning which is definitely -undersireable in many situations. - -\subsection{Optimizations} -While LibTomMath is certainly not the fastest library (GMP often beats LibTomMath by a factor of two) it does -feature a set of optimal algorithms for tasks such as modular reduction, exponentiation, multiplication and squaring. GMP -and LIP also feature such optimizations while MPI only uses baseline algorithms with no optimizations. GMP lacks a few -of the additional modular reduction optimizations that LibTomMath features\footnote{At the time of this writing GMP -only had Barrett and Montgomery modular reduction algorithms.}. - -LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular -exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually -slower than the best libraries such as GMP and OpenSSL by only a small factor. - -\subsection{Portability and Stability} -LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler -(\textit{GCC}). This means that without changes the library will build without configuration or setting up any -variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of -MPI has recently stopped working on his library and LIP has long since been discontinued. - -GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active -development and are very stable across a variety of platforms. - -\subsection{Choice} -LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for -the case study of this text. Various source files from the LibTomMath project will be included within the text. However, -the reader is encouraged to download their own copy of the library to actually be able to work with the library. - -\chapter{Getting Started} -\section{Library Basics} -The trick to writing any useful library of source code is to build a solid foundation and work outwards from it. First, -a problem along with allowable solution parameters should be identified and analyzed. In this particular case the -inability to accomodate multiple precision integers is the problem. Futhermore, the solution must be written -as portable source code that is reasonably efficient across several different computer platforms. - -After a foundation is formed the remainder of the library can be designed and implemented in a hierarchical fashion. -That is, to implement the lowest level dependencies first and work towards the most abstract functions last. For example, -before implementing a modular exponentiation algorithm one would implement a modular reduction algorithm. -By building outwards from a base foundation instead of using a parallel design methodology the resulting project is -highly modular. Being highly modular is a desirable property of any project as it often means the resulting product -has a small footprint and updates are easy to perform. - -Usually when I start a project I will begin with the header files. I define the data types I think I will need and -prototype the initial functions that are not dependent on other functions (within the library). After I -implement these base functions I prototype more dependent functions and implement them. The process repeats until -I implement all of the functions I require. For example, in the case of LibTomMath I implemented functions such as -mp\_init() well before I implemented mp\_mul() and even further before I implemented mp\_exptmod(). As an example as to -why this design works note that the Karatsuba and Toom-Cook multipliers were written \textit{after} the -dependent function mp\_exptmod() was written. Adding the new multiplication algorithms did not require changes to the -mp\_exptmod() function itself and lowered the total cost of ownership (\textit{so to speak}) and of development -for new algorithms. This methodology allows new algorithms to be tested in a complete framework with relative ease. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/design_process.ps} -\caption{Design Flow of the First Few Original LibTomMath Functions.} -\label{pic:design_process} -\end{figure} -\end{center} - -Only after the majority of the functions were in place did I pursue a less hierarchical approach to auditing and optimizing -the source code. For example, one day I may audit the multipliers and the next day the polynomial basis functions. - -It only makes sense to begin the text with the preliminary data types and support algorithms required as well. -This chapter discusses the core algorithms of the library which are the dependents for every other algorithm. - -\section{What is a Multiple Precision Integer?} -Recall that most programming languages, in particular ISO C \cite{ISOC}, only have fixed precision data types that on their own cannot -be used to represent values larger than their precision will allow. The purpose of multiple precision algorithms is -to use fixed precision data types to create and manipulate multiple precision integers which may represent values -that are very large. - -As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system -the largest single digit value is $9$. However, by concatenating digits together larger numbers may be represented. Newly prepended digits -(\textit{to the left}) are said to be in a different power of ten column. That is, the number $123$ can be described as having a $1$ in the hundreds -column, $2$ in the tens column and $3$ in the ones column. Or more formally $123 = 1 \cdot 10^2 + 2 \cdot 10^1 + 3 \cdot 10^0$. Computer based -multiple precision arithmetic is essentially the same concept. Larger integers are represented by adjoining fixed -precision computer words with the exception that a different radix is used. - -What most people probably do not think about explicitly are the various other attributes that describe a multiple precision -integer. For example, the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, -that is the sign of this particular integer is positive as opposed to negative. Second, the integer has three digits in -its representation. There is an additional property that the integer posesses that does not concern pencil-and-paper -arithmetic. The third property is how many digits placeholders are available to hold the integer. - -The human analogy of this third property is ensuring there is enough space on the paper to write the integer. For example, -if one starts writing a large number too far to the right on a piece of paper they will have to erase it and move left. -Similarly, computer algorithms must maintain strict control over memory usage to ensure that the digits of an integer -will not exceed the allowed boundaries. These three properties make up what is known as a multiple precision -integer or mp\_int for short. - -\subsection{The mp\_int Structure} -\label{sec:MPINT} -The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for -any such data type but it does provide for making composite data types known as structures. The following is the structure definition -used within LibTomMath. - -\index{mp\_int} -\begin{figure}[here] -\begin{center} -\begin{small} -%\begin{verbatim} -\begin{tabular}{|l|} -\hline -typedef struct \{ \\ -\hspace{3mm}int used, alloc, sign;\\ -\hspace{3mm}mp\_digit *dp;\\ -\} \textbf{mp\_int}; \\ -\hline -\end{tabular} -%\end{verbatim} -\end{small} -\caption{The mp\_int Structure} -\label{fig:mpint} -\end{center} -\end{figure} - -The mp\_int structure (fig. \ref{fig:mpint}) can be broken down as follows. - -\begin{enumerate} -\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent -a given integer. The \textbf{used} count must be positive (or zero) and may not exceed the \textbf{alloc} count. - -\item The \textbf{alloc} parameter denotes how -many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count -of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the -array to accommodate the precision of the result. - -\item The pointer \textbf{dp} points to a dynamically allocated array of digits that represent the given multiple -precision integer. It is padded with $(\textbf{alloc} - \textbf{used})$ zero digits. The array is maintained in a least -significant digit order. As a pencil and paper analogy the array is organized such that the right most digits are stored -first starting at the location indexed by zero\footnote{In C all arrays begin at zero.} in the array. For example, -if \textbf{dp} contains $\lbrace a, b, c, \ldots \rbrace$ where \textbf{dp}$_0 = a$, \textbf{dp}$_1 = b$, \textbf{dp}$_2 = c$, $\ldots$ then -it would represent the integer $a + b\beta + c\beta^2 + \ldots$ - -\index{MP\_ZPOS} \index{MP\_NEG} -\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}). -\end{enumerate} - -\subsubsection{Valid mp\_int Structures} -Several rules are placed on the state of an mp\_int structure and are assumed to be followed for reasons of efficiency. -The only exceptions are when the structure is passed to initialization functions such as mp\_init() and mp\_init\_copy(). - -\begin{enumerate} -\item The value of \textbf{alloc} may not be less than one. That is \textbf{dp} always points to a previously allocated -array of digits. -\item The value of \textbf{used} may not exceed \textbf{alloc} and must be greater than or equal to zero. -\item The value of \textbf{used} implies the digit at index $(used - 1)$ of the \textbf{dp} array is non-zero. That is, -leading zero digits in the most significant positions must be trimmed. - \begin{enumerate} - \item Digits in the \textbf{dp} array at and above the \textbf{used} location must be zero. - \end{enumerate} -\item The value of \textbf{sign} must be \textbf{MP\_ZPOS} if \textbf{used} is zero; -this represents the mp\_int value of zero. -\end{enumerate} - -\section{Argument Passing} -A convention of argument passing must be adopted early on in the development of any library. Making the function -prototypes consistent will help eliminate many headaches in the future as the library grows to significant complexity. -In LibTomMath the multiple precision integer functions accept parameters from left to right as pointers to mp\_int -structures. That means that the source (input) operands are placed on the left and the destination (output) on the right. -Consider the following examples. - -\begin{verbatim} - mp_mul(&a, &b, &c); /* c = a * b */ - mp_add(&a, &b, &a); /* a = a + b */ - mp_sqr(&a, &b); /* b = a * a */ -\end{verbatim} - -The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the -functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''. - -Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around, to mimic the order -of assignment expressions. That is, the destination (output) is on the left and arguments (inputs) are on the right. In -truth, it is entirely a matter of preference. In the case of LibTomMath the convention from the MPI library has been -adopted. - -Another very useful design consideration, provided for in LibTomMath, is whether to allow argument sources to also be a -destination. For example, the second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important -feature to implement since it allows the calling functions to cut down on the number of variables it must maintain. -However, to implement this feature specific care has to be given to ensure the destination is not modified before the -source is fully read. - -\section{Return Values} -A well implemented application, no matter what its purpose, should trap as many runtime errors as possible and return them -to the caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. However, the end -developer can still manage to cause a library to crash. For example, by passing an invalid pointer an application may -fault by dereferencing memory not owned by the application. - -In the case of LibTomMath the only errors that are checked for are related to inappropriate inputs (division by zero for -instance) and memory allocation errors. It will not check that the mp\_int passed to any function is valid nor -will it check pointers for validity. Any function that can cause a runtime error will return an error code as an -\textbf{int} data type with one of the following values (fig \ref{fig:errcodes}). - -\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM} -\begin{figure}[here] -\begin{center} -\begin{tabular}{|l|l|} -\hline \textbf{Value} & \textbf{Meaning} \\ -\hline \textbf{MP\_OKAY} & The function was successful \\ -\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\ -\hline \textbf{MP\_MEM} & The function ran out of heap memory \\ -\hline -\end{tabular} -\end{center} -\caption{LibTomMath Error Codes} -\label{fig:errcodes} -\end{figure} - -When an error is detected within a function it should free any memory it allocated, often during the initialization of -temporary mp\_ints, and return as soon as possible. The goal is to leave the system in the same state it was when the -function was called. Error checking with this style of API is fairly simple. - -\begin{verbatim} - int err; - if ((err = mp_add(&a, &b, &c)) != MP_OKAY) { - printf("Error: %s\n", mp_error_to_string(err)); - exit(EXIT_FAILURE); - } -\end{verbatim} - -The GMP \cite{GMP} library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal -and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases. - -\section{Initialization and Clearing} -The logical starting point when actually writing multiple precision integer functions is the initialization and -clearing of the mp\_int structures. These two algorithms will be used by the majority of the higher level algorithms. - -Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of -the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even though -the initial integer will represent zero. If only a single digit were allocated quite a few subsequent re-allocations -would occur when operations are performed on the integers. There is a tradeoff between how many default digits to allocate -and how many re-allocations are tolerable. Obviously allocating an excessive amount of digits initially will waste -memory and become unmanageable. - -If the memory for the digits has been successfully allocated then the rest of the members of the structure must -be initialized. Since the initial state of an mp\_int is to represent the zero integer, the allocated digits must be set -to zero. The \textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}. - -\subsection{Initializing an mp\_int} -An mp\_int is said to be initialized if it is set to a valid, preferably default, state such that all of the members of the -structure are set to valid values. The mp\_init algorithm will perform such an action. - -\index{mp\_init} -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Allocate memory and initialize $a$ to a known valid mp\_int state. \\ -\hline \\ -1. Allocate memory for \textbf{MP\_PREC} digits. \\ -2. If the allocation failed return(\textit{MP\_MEM}) \\ -3. for $n$ from $0$ to $MP\_PREC - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$\\ -4. $a.sign \leftarrow MP\_ZPOS$\\ -5. $a.used \leftarrow 0$\\ -6. $a.alloc \leftarrow MP\_PREC$\\ -7. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init} -\end{figure} - -\textbf{Algorithm mp\_init.} -The purpose of this function is to initialize an mp\_int structure so that the rest of the library can properly -manipulte it. It is assumed that the input may not have had any of its members previously initialized which is certainly -a valid assumption if the input resides on the stack. - -Before any of the members such as \textbf{sign}, \textbf{used} or \textbf{alloc} are initialized the memory for -the digits is allocated. If this fails the function returns before setting any of the other members. The \textbf{MP\_PREC} -name represents a constant\footnote{Defined in the ``tommath.h'' header file within LibTomMath.} -used to dictate the minimum precision of newly initialized mp\_int integers. Ideally, it is at least equal to the smallest -precision number you'll be working with. - -Allocating a block of digits at first instead of a single digit has the benefit of lowering the number of usually slow -heap operations later functions will have to perform in the future. If \textbf{MP\_PREC} is set correctly the slack -memory and the number of heap operations will be trivial. - -Once the allocation has been made the digits have to be set to zero as well as the \textbf{used}, \textbf{sign} and -\textbf{alloc} members initialized. This ensures that the mp\_int will always represent the default state of zero regardless -of the original condition of the input. - -\textbf{Remark.} -This function introduces the idiosyncrasy that all iterative loops, commonly initiated with the ``for'' keyword, iterate incrementally -when the ``to'' keyword is placed between two expressions. For example, ``for $a$ from $b$ to $c$ do'' means that -a subsequent expression (or body of expressions) are to be evaluated upto $c - b$ times so long as $b \le c$. In each -iteration the variable $a$ is substituted for a new integer that lies inclusively between $b$ and $c$. If $b > c$ occured -the loop would not iterate. By contrast if the ``downto'' keyword were used in place of ``to'' the loop would iterate -decrementally. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -One immediate observation of this initializtion function is that it does not return a pointer to a mp\_int structure. It -is assumed that the caller has already allocated memory for the mp\_int structure, typically on the application stack. The -call to mp\_init() is used only to initialize the members of the structure to a known default state. - -Here we see (line 24) the memory allocation is performed first. This allows us to exit cleanly and quickly -if there is an error. If the allocation fails the routine will return \textbf{MP\_MEM} to the caller to indicate there -was a memory error. The function XMALLOC is what actually allocates the memory. Technically XMALLOC is not a function -but a macro defined in ``tommath.h``. By default, XMALLOC will evaluate to malloc() which is the C library's built--in -memory allocation routine. - -In order to assure the mp\_int is in a known state the digits must be set to zero. On most platforms this could have been -accomplished by using calloc() instead of malloc(). However, to correctly initialize a integer type to a given value in a -portable fashion you have to actually assign the value. The for loop (line 30) performs this required -operation. - -After the memory has been successfully initialized the remainder of the members are initialized -(lines 34 through 35) to their respective default states. At this point the algorithm has succeeded and -a success code is returned to the calling function. If this function returns \textbf{MP\_OKAY} it is safe to assume the -mp\_int structure has been properly initialized and is safe to use with other functions within the library. - -\subsection{Clearing an mp\_int} -When an mp\_int is no longer required by the application, the memory that has been allocated for its digits must be -returned to the application's memory pool with the mp\_clear algorithm. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clear}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. The memory for $a$ shall be deallocated. \\ -\hline \\ -1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $a_n \leftarrow 0$ \\ -3. Free the memory allocated for the digits of $a$. \\ -4. $a.used \leftarrow 0$ \\ -5. $a.alloc \leftarrow 0$ \\ -6. $a.sign \leftarrow MP\_ZPOS$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_clear} -\end{figure} - -\textbf{Algorithm mp\_clear.} -This algorithm accomplishes two goals. First, it clears the digits and the other mp\_int members. This ensures that -if a developer accidentally re-uses a cleared structure it is less likely to cause problems. The second goal -is to free the allocated memory. - -The logic behind the algorithm is extended by marking cleared mp\_int structures so that subsequent calls to this -algorithm will not try to free the memory multiple times. Cleared mp\_ints are detectable by having a pre-defined invalid -digit pointer \textbf{dp} setting. - -Once an mp\_int has been cleared the mp\_int structure is no longer in a valid state for any other algorithm -with the exception of algorithms mp\_init, mp\_init\_copy, mp\_init\_size and mp\_clear. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm only operates on the mp\_int if it hasn't been previously cleared. The if statement (line 25) -checks to see if the \textbf{dp} member is not \textbf{NULL}. If the mp\_int is a valid mp\_int then \textbf{dp} cannot be -\textbf{NULL} in which case the if statement will evaluate to true. - -The digits of the mp\_int are cleared by the for loop (line 27) which assigns a zero to every digit. Similar to mp\_init() -the digits are assigned zero instead of using block memory operations (such as memset()) since this is more portable. - -The digits are deallocated off the heap via the XFREE macro. Similar to XMALLOC the XFREE macro actually evaluates to -a standard C library function. In this case the free() function. Since free() only deallocates the memory the pointer -still has to be reset to \textbf{NULL} manually (line 35). - -Now that the digits have been cleared and deallocated the other members are set to their final values (lines 36 and 37). - -\section{Maintenance Algorithms} - -The previous sections describes how to initialize and clear an mp\_int structure. To further support operations -that are to be performed on mp\_int structures (such as addition and multiplication) the dependent algorithms must be -able to augment the precision of an mp\_int and -initialize mp\_ints with differing initial conditions. - -These algorithms complete the set of low level algorithms required to work with mp\_int structures in the higher level -algorithms such as addition, multiplication and modular exponentiation. - -\subsection{Augmenting an mp\_int's Precision} -When storing a value in an mp\_int structure, a sufficient number of digits must be available to accomodate the entire -result of an operation without loss of precision. Quite often the size of the array given by the \textbf{alloc} member -is large enough to simply increase the \textbf{used} digit count. However, when the size of the array is too small it -must be re-sized appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_grow}. \\ -\textbf{Input}. An mp\_int $a$ and an integer $b$. \\ -\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\ -\hline \\ -1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\ -2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\ -3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -4. Re-allocate the array of digits $a$ to size $v$ \\ -5. If the allocation failed then return(\textit{MP\_MEM}). \\ -6. for n from a.alloc to $v - 1$ do \\ -\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.alloc \leftarrow v$ \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_grow} -\end{figure} - -\textbf{Algorithm mp\_grow.} -It is ideal to prevent re-allocations from being performed if they are not required (step one). This is useful to -prevent mp\_ints from growing excessively in code that erroneously calls mp\_grow. - -The requested digit count is padded up to next multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} (steps two and three). -This helps prevent many trivial reallocations that would grow an mp\_int by trivially small values. - -It is assumed that the reallocation (step four) leaves the lower $a.alloc$ digits of the mp\_int intact. This is much -akin to how the \textit{realloc} function from the standard C library works. Since the newly allocated digits are -assumed to contain undefined values they are initially set to zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -A quick optimization is to first determine if a memory re-allocation is required at all. The if statement (line 24) checks -if the \textbf{alloc} member of the mp\_int is smaller than the requested digit count. If the count is not larger than \textbf{alloc} -the function skips the re-allocation part thus saving time. - -When a re-allocation is performed it is turned into an optimal request to save time in the future. The requested digit count is -padded upwards to 2nd multiple of \textbf{MP\_PREC} larger than \textbf{alloc} (line 25). The XREALLOC function is used -to re-allocate the memory. As per the other functions XREALLOC is actually a macro which evaluates to realloc by default. The realloc -function leaves the base of the allocation intact which means the first \textbf{alloc} digits of the mp\_int are the same as before -the re-allocation. All that is left is to clear the newly allocated digits and return. - -Note that the re-allocation result is actually stored in a temporary pointer $tmp$. This is to allow this function to return -an error with a valid pointer. Earlier releases of the library stored the result of XREALLOC into the mp\_int $a$. That would -result in a memory leak if XREALLOC ever failed. - -\subsection{Initializing Variable Precision mp\_ints} -Occasionally the number of digits required will be known in advance of an initialization, based on, for example, the size -of input mp\_ints to a given algorithm. The purpose of algorithm mp\_init\_size is similar to mp\_init except that it -will allocate \textit{at least} a specified number of digits. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_size}. \\ -\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$. \\ -\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\ -\hline \\ -1. $u \leftarrow b \mbox{ (mod }MP\_PREC\mbox{)}$ \\ -2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\ -3. Allocate $v$ digits. \\ -4. for $n$ from $0$ to $v - 1$ do \\ -\hspace{3mm}4.1 $a_n \leftarrow 0$ \\ -5. $a.sign \leftarrow MP\_ZPOS$\\ -6. $a.used \leftarrow 0$\\ -7. $a.alloc \leftarrow v$\\ -8. Return(\textit{MP\_OKAY})\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_init\_size} -\end{figure} - -\textbf{Algorithm mp\_init\_size.} -This algorithm will initialize an mp\_int structure $a$ like algorithm mp\_init with the exception that the number of -digits allocated can be controlled by the second input argument $b$. The input size is padded upwards so it is a -multiple of \textbf{MP\_PREC} plus an additional \textbf{MP\_PREC} digits. This padding is used to prevent trivial -allocations from becoming a bottleneck in the rest of the algorithms. - -Like algorithm mp\_init, the mp\_int structure is initialized to a default state representing the integer zero. This -particular algorithm is useful if it is known ahead of time the approximate size of the input. If the approximation is -correct no further memory re-allocations are required to work with the mp\_int. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The number of digits $b$ requested is padded (line 24) by first augmenting it to the next multiple of -\textbf{MP\_PREC} and then adding \textbf{MP\_PREC} to the result. If the memory can be successfully allocated the -mp\_int is placed in a default state representing the integer zero. Otherwise, the error code \textbf{MP\_MEM} will be -returned (line 29). - -The digits are allocated and set to zero at the same time with the calloc() function (line @25,XCALLOC@). The -\textbf{used} count is set to zero, the \textbf{alloc} count set to the padded digit count and the \textbf{sign} flag set -to \textbf{MP\_ZPOS} to achieve a default valid mp\_int state (lines 33, 34 and 35). If the function -returns succesfully then it is correct to assume that the mp\_int structure is in a valid state for the remainder of the -functions to work with. - -\subsection{Multiple Integer Initializations and Clearings} -Occasionally a function will require a series of mp\_int data types to be made available simultaneously. -The purpose of algorithm mp\_init\_multi is to initialize a variable length array of mp\_int structures in a single -statement. It is essentially a shortcut to multiple initializations. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_multi}. \\ -\textbf{Input}. Variable length array $V_k$ of mp\_int variables of length $k$. \\ -\textbf{Output}. The array is initialized such that each mp\_int of $V_k$ is ready to use. \\ -\hline \\ -1. for $n$ from 0 to $k - 1$ do \\ -\hspace{+3mm}1.1. Initialize the mp\_int $V_n$ (\textit{mp\_init}) \\ -\hspace{+3mm}1.2. If initialization failed then do \\ -\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\ -\hspace{+9mm}1.2.1.1. Free the mp\_int $V_j$ (\textit{mp\_clear}) \\ -\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\ -2. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_multi} -\end{figure} - -\textbf{Algorithm mp\_init\_multi.} -The algorithm will initialize the array of mp\_int variables one at a time. If a runtime error has been detected -(\textit{step 1.2}) all of the previously initialized variables are cleared. The goal is an ``all or nothing'' -initialization which allows for quick recovery from runtime errors. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_multi.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function intializes a variable length list of mp\_int structure pointers. However, instead of having the mp\_int -structures in an actual C array they are simply passed as arguments to the function. This function makes use of the -``...'' argument syntax of the C programming language. The list is terminated with a final \textbf{NULL} argument -appended on the right. - -The function uses the ``stdarg.h'' \textit{va} functions to step portably through the arguments to the function. A count -$n$ of succesfully initialized mp\_int structures is maintained (line 48) such that if a failure does occur, -the algorithm can backtrack and free the previously initialized structures (lines 28 to 47). - - -\subsection{Clamping Excess Digits} -When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of -the function instead of checking during the computation. For example, a multiplication of a $i$ digit number by a -$j$ digit produces a result of at most $i + j$ digits. It is entirely possible that the result is $i + j - 1$ -though, with no final carry into the last position. However, suppose the destination had to be first expanded -(\textit{via mp\_grow}) to accomodate $i + j - 1$ digits than further expanded to accomodate the final carry. -That would be a considerable waste of time since heap operations are relatively slow. - -The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function -terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked -there would be an excess high order zero digit. - -For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit -will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would -accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very -low the representation is excessively large. - -The mp\_clamp algorithm is designed to solve this very problem. It will trim high-order zeros by decrementing the -\textbf{used} count until a non-zero most significant digit is found. Also in this system, zero is considered to be a -positive number which means that if the \textbf{used} count is decremented to zero, the sign must be set to -\textbf{MP\_ZPOS}. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_clamp}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Any excess leading zero digits of $a$ are removed \\ -\hline \\ -1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\ -\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\ -2. if $a.used = 0$ then do \\ -\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\ -\hline \\ -\end{tabular} -\end{center} -\caption{Algorithm mp\_clamp} -\end{figure} - -\textbf{Algorithm mp\_clamp.} -As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at -the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for -when all of the digits are zero to ensure that the mp\_int is valid at all times. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Note on line 28 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming -language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is -important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously -undesirable. The parenthesis on line 31 is used to make sure the \textbf{used} count is decremented and not -the pointer ``a''. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\ - & \\ -$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\ - & encryption when $\beta = 2^{28}$. \\ - & \\ -$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\ - & \\ -$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\ - & \\ -\end{tabular} - - -%%% -% CHAPTER FOUR -%%% - -\chapter{Basic Operations} - -\section{Introduction} -In the previous chapter a series of low level algorithms were established that dealt with initializing and maintaining -mp\_int structures. This chapter will discuss another set of seemingly non-algebraic algorithms which will form the low -level basis of the entire library. While these algorithm are relatively trivial it is important to understand how they -work before proceeding since these algorithms will be used almost intrinsically in the following chapters. - -The algorithms in this chapter deal primarily with more ``programmer'' related tasks such as creating copies of -mp\_int structures, assigning small values to mp\_int structures and comparisons of the values mp\_int structures -represent. - -\section{Assigning Values to mp\_int Structures} -\subsection{Copying an mp\_int} -Assigning the value that a given mp\_int structure represents to another mp\_int structure shall be known as making -a copy for the purposes of this text. The copy of the mp\_int will be a separate entity that represents the same -value as the mp\_int it was copied from. The mp\_copy algorithm provides this functionality. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$. \\ -\textbf{Output}. Store a copy of $a$ in $b$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\ -2. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}2.1 $b_{n} \leftarrow a_{n}$ \\ -3. for $n$ from $a.used$ to $b.used - 1$ do \\ -\hspace{3mm}3.1 $b_{n} \leftarrow 0$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $b.sign \leftarrow a.sign$ \\ -6. return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_copy} -\end{figure} - -\textbf{Algorithm mp\_copy.} -This algorithm copies the mp\_int $a$ such that upon succesful termination of the algorithm the mp\_int $b$ will -represent the same integer as the mp\_int $a$. The mp\_int $b$ shall be a complete and distinct copy of the -mp\_int $a$ meaing that the mp\_int $a$ can be modified and it shall not affect the value of the mp\_int $b$. - -If $b$ does not have enough room for the digits of $a$ it must first have its precision augmented via the mp\_grow -algorithm. The digits of $a$ are copied over the digits of $b$ and any excess digits of $b$ are set to zero (step two -and three). The \textbf{used} and \textbf{sign} members of $a$ are finally copied over the respective members of -$b$. - -\textbf{Remark.} This algorithm also introduces a new idiosyncrasy that will be used throughout the rest of the -text. The error return codes of other algorithms are not explicitly checked in the pseudo-code presented. For example, in -step one of the mp\_copy algorithm the return of mp\_grow is not explicitly checked to ensure it succeeded. Text space is -limited so it is assumed that if a algorithm fails it will clear all temporarily allocated mp\_ints and return -the error code itself. However, the C code presented will demonstrate all of the error handling logic required to -implement the pseudo-code. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Occasionally a dependent algorithm may copy an mp\_int effectively into itself such as when the input and output -mp\_int structures passed to a function are one and the same. For this case it is optimal to return immediately without -copying digits (line 25). - -The mp\_int $b$ must have enough digits to accomodate the used digits of the mp\_int $a$. If $b.alloc$ is less than -$a.used$ the algorithm mp\_grow is used to augment the precision of $b$ (lines 30 to 33). In order to -simplify the inner loop that copies the digits from $a$ to $b$, two aliases $tmpa$ and $tmpb$ point directly at the digits -of the mp\_ints $a$ and $b$ respectively. These aliases (lines 43 and 46) allow the compiler to access the digits without first dereferencing the -mp\_int pointers and then subsequently the pointer to the digits. - -After the aliases are established the digits from $a$ are copied into $b$ (lines 49 to 51) and then the excess -digits of $b$ are set to zero (lines 54 to 56). Both ``for'' loops make use of the pointer aliases and in -fact the alias for $b$ is carried through into the second ``for'' loop to clear the excess digits. This optimization -allows the alias to stay in a machine register fairly easy between the two loops. - -\textbf{Remarks.} The use of pointer aliases is an implementation methodology first introduced in this function that will -be used considerably in other functions. Technically, a pointer alias is simply a short hand alias used to lower the -number of pointer dereferencing operations required to access data. For example, a for loop may resemble - -\begin{alltt} -for (x = 0; x < 100; x++) \{ - a->num[4]->dp[x] = 0; -\} -\end{alltt} - -This could be re-written using aliases as - -\begin{alltt} -mp_digit *tmpa; -a = a->num[4]->dp; -for (x = 0; x < 100; x++) \{ - *a++ = 0; -\} -\end{alltt} - -In this case an alias is used to access the -array of digits within an mp\_int structure directly. It may seem that a pointer alias is strictly not required -as a compiler may optimize out the redundant pointer operations. However, there are two dominant reasons to use aliases. - -The first reason is that most compilers will not effectively optimize pointer arithmetic. For example, some optimizations -may work for the Microsoft Visual C++ compiler (MSVC) and not for the GNU C Compiler (GCC). Also some optimizations may -work for GCC and not MSVC. As such it is ideal to find a common ground for as many compilers as possible. Pointer -aliases optimize the code considerably before the compiler even reads the source code which means the end compiled code -stands a better chance of being faster. - -The second reason is that pointer aliases often can make an algorithm simpler to read. Consider the first ``for'' -loop of the function mp\_copy() re-written to not use pointer aliases. - -\begin{alltt} - /* copy all the digits */ - for (n = 0; n < a->used; n++) \{ - b->dp[n] = a->dp[n]; - \} -\end{alltt} - -Whether this code is harder to read depends strongly on the individual. However, it is quantifiably slightly more -complicated as there are four variables within the statement instead of just two. - -\subsubsection{Nested Statements} -Another commonly used technique in the source routines is that certain sections of code are nested. This is used in -particular with the pointer aliases to highlight code phases. For example, a Comba multiplier (discussed in chapter six) -will typically have three different phases. First the temporaries are initialized, then the columns calculated and -finally the carries are propagated. In this example the middle column production phase will typically be nested as it -uses temporary variables and aliases the most. - -The nesting also simplies the source code as variables that are nested are only valid for their scope. As a result -the various temporary variables required do not propagate into other sections of code. - - -\subsection{Creating a Clone} -Another common operation is to make a local temporary copy of an mp\_int argument. To initialize an mp\_int -and then copy another existing mp\_int into the newly intialized mp\_int will be known as creating a clone. This is -useful within functions that need to modify an argument but do not wish to actually modify the original copy. The -mp\_init\_copy algorithm has been designed to help perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_init\_copy}. \\ -\textbf{Input}. An mp\_int $a$ and $b$\\ -\textbf{Output}. $a$ is initialized to be a copy of $b$. \\ -\hline \\ -1. Init $a$. (\textit{mp\_init}) \\ -2. Copy $b$ to $a$. (\textit{mp\_copy}) \\ -3. Return the status of the copy operation. \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_init\_copy} -\end{figure} - -\textbf{Algorithm mp\_init\_copy.} -This algorithm will initialize an mp\_int variable and copy another previously initialized mp\_int variable into it. As -such this algorithm will perform two operations in one step. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that -\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call -and \textbf{a} will be left intact. - -\section{Zeroing an Integer} -Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to -perform this task. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_zero}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Zero the contents of $a$ \\ -\hline \\ -1. $a.used \leftarrow 0$ \\ -2. $a.sign \leftarrow$ MP\_ZPOS \\ -3. for $n$ from 0 to $a.alloc - 1$ do \\ -\hspace{3mm}3.1 $a_n \leftarrow 0$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_zero} -\end{figure} - -\textbf{Algorithm mp\_zero.} -This algorithm simply resets a mp\_int to the default state. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the -\textbf{sign} variable is set to \textbf{MP\_ZPOS}. - -\section{Sign Manipulation} -\subsection{Absolute Value} -With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute -the absolute value of an mp\_int. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_abs}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = \vert a \vert$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. $b.sign \leftarrow MP\_ZPOS$ \\ -4. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_abs} -\end{figure} - -\textbf{Algorithm mp\_abs.} -This algorithm computes the absolute of an mp\_int input. First it copies $a$ over $b$. This is an example of an -algorithm where the check in mp\_copy that determines if the source and destination are equal proves useful. This allows, -for instance, the developer to pass the same mp\_int as the source and destination to this function without addition -logic to handle it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This fairly trivial algorithm first eliminates non--required duplications (line 28) and then sets the -\textbf{sign} flag to \textbf{MP\_ZPOS}. - -\subsection{Integer Negation} -With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute -the negative of an mp\_int input. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_neg}. \\ -\textbf{Input}. An mp\_int $a$ \\ -\textbf{Output}. Computes $b = -a$ \\ -\hline \\ -1. Copy $a$ to $b$. (\textit{mp\_copy}) \\ -2. If the copy failed return(\textit{MP\_MEM}). \\ -3. If $a.used = 0$ then return(\textit{MP\_OKAY}). \\ -4. If $a.sign = MP\_ZPOS$ then do \\ -\hspace{3mm}4.1 $b.sign = MP\_NEG$. \\ -5. else do \\ -\hspace{3mm}5.1 $b.sign = MP\_ZPOS$. \\ -6. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_neg} -\end{figure} - -\textbf{Algorithm mp\_neg.} -This algorithm computes the negation of an input. First it copies $a$ over $b$. If $a$ has no used digits then -the algorithm returns immediately. Otherwise it flips the sign flag and stores the result in $b$. Note that if -$a$ had no digits then it must be positive by definition. Had step three been omitted then the algorithm would return -zero as negative. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like mp\_abs() this function avoids non--required duplications (line 22) and then sets the sign. We -have to make sure that only non--zero values get a \textbf{sign} of \textbf{MP\_NEG}. If the mp\_int is zero -than the \textbf{sign} is hard--coded to \textbf{MP\_ZPOS}. - -\section{Small Constants} -\subsection{Setting Small Constants} -Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set}. \\ -\textbf{Input}. An mp\_int $a$ and a digit $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}). \\ -2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\ -3. $a.used \leftarrow \left \lbrace \begin{array}{ll} - 1 & \mbox{if }a_0 > 0 \\ - 0 & \mbox{if }a_0 = 0 - \end{array} \right .$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set} -\end{figure} - -\textbf{Algorithm mp\_set.} -This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The -single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -First we zero (line 21) the mp\_int to make sure that the other members are initialized for a -small positive constant. mp\_zero() ensures that the \textbf{sign} is positive and the \textbf{used} count -is zero. Next we set the digit and reduce it modulo $\beta$ (line 22). After this step we have to -check if the resulting digit is zero or not. If it is not then we set the \textbf{used} count to one, otherwise -to zero. - -We can quickly reduce modulo $\beta$ since it is of the form $2^k$ and a quick binary AND operation with -$2^k - 1$ will perform the same operation. - -One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses -this function should take that into account. Only trivially small constants can be set using this function. - -\subsection{Setting Large Constants} -To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is ideal. It accepts a ``long'' -data type as input and will always treat it as a 32-bit integer. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_set\_int}. \\ -\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\ -\textbf{Output}. Make $a$ equivalent to $b$ \\ -\hline \\ -1. Zero $a$ (\textit{mp\_zero}) \\ -2. for $n$ from 0 to 7 do \\ -\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\ -\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\ -\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\ -\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\ -3. Clamp excess used digits (\textit{mp\_clamp}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_set\_int} -\end{figure} - -\textbf{Algorithm mp\_set\_int.} -The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the -mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the -next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is -incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have -zero digits used and the newly added four bits would be ignored. - -Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird -addition on line 39 ensures that the newly added in bits are added to the number of digits. While it may not -seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 28 -as well as the call to mp\_clamp() on line 41. Both functions will clamp excess leading digits which keeps -the number of used digits low. - -\section{Comparisons} -\subsection{Unsigned Comparisions} -Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example, -to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$ -to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude -positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater. - -The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two -mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the -signs are known to agree in advance. - -To facilitate working with the results of the comparison functions three constants are required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|r|l|} -\hline \textbf{Constant} & \textbf{Meaning} \\ -\hline \textbf{MP\_GT} & Greater Than \\ -\hline \textbf{MP\_EQ} & Equal To \\ -\hline \textbf{MP\_LT} & Less Than \\ -\hline -\end{tabular} -\end{center} -\caption{Comparison Return Codes} -\end{figure} - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp\_mag}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$. \\ -\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\ -\hline \\ -1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\ -2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\ -3. for n from $a.used - 1$ to 0 do \\ -\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\ -\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\ -4. Return(\textit{MP\_EQ}) \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp\_mag} -\end{figure} - -\textbf{Algorithm mp\_cmp\_mag.} -By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return -\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$. -Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is. -If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit. - -By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to -the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The two if statements (lines 25 and 29) compare the number of digits in the two inputs. These two are -performed before all of the digits are compared since it is a very cheap test to perform and can potentially save -considerable time. The implementation given is also not valid without those two statements. $b.alloc$ may be -smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the array of digits. - - - -\subsection{Signed Comparisons} -Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude -comparison a trivial signed comparison algorithm can be written. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_cmp}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\ -\hline \\ -1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\ -2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\ -3. if $a.sign = MP\_NEG$ then \\ -\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\ -4 Otherwise \\ -\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_cmp} -\end{figure} - -\textbf{Algorithm mp\_cmp.} -The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate -comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step -three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then -$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The two if statements (lines 23 and 24) perform the initial sign comparison. If the signs are not the equal then which ever -has the positive sign is larger. The inputs are compared (line 32) based on magnitudes. If the signs were both -negative then the unsigned comparison is performed in the opposite direction (line 34). Otherwise, the signs are assumed to -be both positive and a forward direction unsigned comparison is performed. - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\ - & \\ -$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\ - & of two random digits (of equal magnitude) before a difference is found. \\ - & \\ -$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\ - & on the observations made in the previous problem. \\ - & -\end{tabular} - -\chapter{Basic Arithmetic} -\section{Introduction} -At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been -established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These -algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important -that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms -which easily places them at $O(n^2)$ or even $O(n^3)$ work levels. - -All of the algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right -logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real -number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $\beta^2 = 10^2$}). -Algebraically a binary logical shift is equivalent to a division or multiplication by a power of two. -For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$. - -One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed -from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the -result is $110_2$. - -\section{Addition and Subtraction} -In common twos complement fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers -$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$. -As a result subtraction can be performed with a trivial series of logical operations and an addition. - -However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the -sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or -subtraction algorithms with the sign fixed up appropriately. - -The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of -the integers respectively. - -\subsection{Low Level Addition} -An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the -trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix. -Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely. - -\newpage -\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\ -\hline \\ -1. if $a.used > b.used$ then \\ -\hspace{+3mm}1.1 $min \leftarrow b.used$ \\ -\hspace{+3mm}1.2 $max \leftarrow a.used$ \\ -\hspace{+3mm}1.3 $x \leftarrow a$ \\ -2. else \\ -\hspace{+3mm}2.1 $min \leftarrow a.used$ \\ -\hspace{+3mm}2.2 $max \leftarrow b.used$ \\ -\hspace{+3mm}2.3 $x \leftarrow b$ \\ -3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max + 1$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{+3mm}7.1 $c_n \leftarrow a_n + b_n + u$ \\ -\hspace{+3mm}7.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min \ne max$ then do \\ -\hspace{+3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{+6mm}8.1.1 $c_n \leftarrow x_n + u$ \\ -\hspace{+6mm}8.1.2 $u \leftarrow c_n >> lg(\beta)$ \\ -\hspace{+6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. $c_{max} \leftarrow u$ \\ -10. if $olduse > max$ then \\ -\hspace{+3mm}10.1 for $n$ from $max + 1$ to $oldused - 1$ do \\ -\hspace{+6mm}10.1.1 $c_n \leftarrow 0$ \\ -11. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\ -12. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_add} -\end{figure} - -\textbf{Algorithm s\_mp\_add.} -This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes. -Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the -MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes. - -The first thing that has to be accomplished is to sort out which of the two inputs is the largest. The addition logic -will simply add all of the smallest input to the largest input and store that first part of the result in the -destination. Then it will apply a simpler addition loop to excess digits of the larger input. - -The first two steps will handle sorting the inputs such that $min$ and $max$ hold the digit counts of the two -inputs. The variable $x$ will be an mp\_int alias for the largest input or the second input $b$ if they have the -same number of digits. After the inputs are sorted the destination $c$ is grown as required to accomodate the sum -of the two inputs. The original \textbf{used} count of $c$ is copied and set to the new used count. - -At this point the first addition loop will go through as many digit positions that both inputs have. The carry -variable $\mu$ is set to zero outside the loop. Inside the loop an ``addition'' step requires three statements to produce -one digit of the summand. First -two digits from $a$ and $b$ are added together along with the carry $\mu$. The carry of this step is extracted and stored -in $\mu$ and finally the digit of the result $c_n$ is truncated within the range $0 \le c_n < \beta$. - -Now all of the digit positions that both inputs have in common have been exhausted. If $min \ne max$ then $x$ is an alias -for one of the inputs that has more digits. A simplified addition loop is then used to essentially copy the remaining digits -and the carry to the destination. - -The final carry is stored in $c_{max}$ and digits above $max$ upto $oldused$ are zeroed which completes the addition. - - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -We first sort (lines 28 to 36) the inputs based on magnitude and determine the $min$ and $max$ variables. -Note that $x$ is a pointer to an mp\_int assigned to the largest input, in effect it is a local alias. Next we -grow the destination (38 to 42) ensure that it can accomodate the result of the addition. - -Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on -lines 56, 59 and 62 represent the two inputs and destination variables respectively. These aliases are used to ensure the -compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int. - -The initial carry $u$ will be cleared (line 65), note that $u$ is of type mp\_digit which ensures type -compatibility within the implementation. The initial addition (line 66 to 75) adds digits from -both inputs until the smallest input runs out of digits. Similarly the conditional addition loop -(line 81 to 90) adds the remaining digits from the larger of the two inputs. The addition is finished -with the final carry being stored in $tmpc$ (line 94). Note the ``++'' operator within the same expression. -After line 94, $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful -for the next loop (line 97 to 99) which set any old upper digits to zero. - -\subsection{Low Level Subtraction} -The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the -unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must -be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly. -This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms. - - -For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent -the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For -this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a -mp\_digit (\textit{this implies $2^{\gamma} > \beta$}). - -For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long'' -data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma \ge 32$. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{small} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\ -\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\ -\hline \\ -1. $min \leftarrow b.used$ \\ -2. $max \leftarrow a.used$ \\ -3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\ -4. $oldused \leftarrow c.used$ \\ -5. $c.used \leftarrow max$ \\ -6. $u \leftarrow 0$ \\ -7. for $n$ from $0$ to $min - 1$ do \\ -\hspace{3mm}7.1 $c_n \leftarrow a_n - b_n - u$ \\ -\hspace{3mm}7.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{3mm}7.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -8. if $min < max$ then do \\ -\hspace{3mm}8.1 for $n$ from $min$ to $max - 1$ do \\ -\hspace{6mm}8.1.1 $c_n \leftarrow a_n - u$ \\ -\hspace{6mm}8.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\ -\hspace{6mm}8.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\ -9. if $oldused > max$ then do \\ -\hspace{3mm}9.1 for $n$ from $max$ to $oldused - 1$ do \\ -\hspace{6mm}9.1.1 $c_n \leftarrow 0$ \\ -10. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Algorithm s\_mp\_sub} -\end{figure} - -\textbf{Algorithm s\_mp\_sub.} -This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when -passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This -algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case -of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude. - -The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2 -set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at -most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and -set to the maximal count for the operation. - -The subtraction loop that begins on step seven is essentially the same as the addition loop of algorithm s\_mp\_add except single precision -subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction -loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry. - -For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to -the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the -third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the -way to the most significant bit. - -Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most -significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that -is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the -carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed. - -If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step -10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like low level addition we ``sort'' the inputs. Except in this case the sorting is hardcoded -(lines 25 and 26). In reality the $min$ and $max$ variables are only aliases and are only -used to make the source code easier to read. Again the pointer alias optimization is used -within this algorithm. The aliases $tmpa$, $tmpb$ and $tmpc$ are initialized -(lines 42, 43 and 44) for $a$, $b$ and $c$ respectively. - -The first subtraction loop (lines 47 through 61) subtract digits from both inputs until the smaller of -the two inputs has been exhausted. As remarked earlier there is an implementation reason for using the ``awkward'' -method of extracting the carry (line 57). The traditional method for extracting the carry would be to shift -by $lg(\beta)$ positions and logically AND the least significant bit. The AND operation is required because all of -the bits above the $\lg(\beta)$'th bit will be set to one after a carry occurs from subtraction. This carry -extraction requires two relatively cheap operations to extract the carry. The other method is to simply shift the -most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This -optimization only works on twos compliment machines which is a safe assumption to make. - -If $a$ has a larger magnitude than $b$ an additional loop (lines 64 through 73) is required to propagate -the carry through $a$ and copy the result to $c$. - -\subsection{High Level Addition} -Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be -established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data -types. - -Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign} -flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases. - -\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed addition $c = a + b$. \\ -\hline \\ -1. if $a.sign = b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_add} -\end{figure} - -\textbf{Algorithm mp\_add.} -This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from -either \cite{TAOCPV2} or \cite{HAC} since they both only provide unsigned operations. The algorithm is fairly -straightforward but restricted since subtraction can only produce positive results. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&&\\ - -\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\ -\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\ - -\hline &&&&\\ - -\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ - -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Addition Guide Chart} -\label{fig:AddChart} -\end{figure} - -Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three -specific cases need to be handled. The return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are -forwarded to step three to check for errors. This simplifies the description of the algorithm considerably and best -follows how the implementation actually was achieved. - -Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms -s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign} -to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero. - -For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would -produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp -within algorithm s\_mp\_add will force $-0$ to become $0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which -is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without -explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower -level functions do so. Returning their return code is sufficient. - -\subsection{High Level Subtraction} -The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm. - -\newpage\begin{figure}[!here] -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sub}. \\ -\textbf{Input}. Two mp\_ints $a$ and $b$ \\ -\textbf{Output}. The signed subtraction $c = a - b$. \\ -\hline \\ -1. if $a.sign \ne b.sign$ then do \\ -\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\ -2. else do \\ -\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\ -\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\ -\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\caption{Algorithm mp\_sub} -\end{figure} - -\textbf{Algorithm mp\_sub.} -This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or -\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. Chart \ref{fig:SubChart} lists the eight possible inputs and -the operations required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|} -\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\ -\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\ -\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\ -\hline &&&& \\ -\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Subtraction Guide Chart} -\label{fig:SubChart} -\end{figure} - -Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the -algorithm from producing $-a - -a = -0$ as a result. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations -and forward it to the end of the function. On line 39 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a -``greater than or equal to'' comparison. - -\section{Bit and Digit Shifting} -It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$. -This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring. - -In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift -the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations -are on radix-$\beta$ digits. - -\subsection{Multiplication by Two} - -In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient -operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = 2a$. \\ -\hline \\ -1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\ -2. $oldused \leftarrow b.used$ \\ -3. $b.used \leftarrow a.used$ \\ -4. $r \leftarrow 0$ \\ -5. for $n$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}5.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\ -\hspace{3mm}5.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.3 $r \leftarrow rr$ \\ -6. If $r \ne 0$ then do \\ -\hspace{3mm}6.1 $b_{n + 1} \leftarrow r$ \\ -\hspace{3mm}6.2 $b.used \leftarrow b.used + 1$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2} -\end{figure} - -\textbf{Algorithm mp\_mul\_2.} -This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such -an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since -it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$. - -Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count -is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment. - -Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together -are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to -obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus -the previous carry. Recall from section 4.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with -forwarding the carry to the next iteration. - -Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$. -Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference -is the use of the logical shift operator on line 52 to perform a single precision doubling. - -\subsection{Division by Two} -A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2}. \\ -\textbf{Input}. One mp\_int $a$ \\ -\textbf{Output}. $b = a/2$. \\ -\hline \\ -1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\ -2. If the reallocation failed return(\textit{MP\_MEM}). \\ -3. $oldused \leftarrow b.used$ \\ -4. $b.used \leftarrow a.used$ \\ -5. $r \leftarrow 0$ \\ -6. for $n$ from $b.used - 1$ to $0$ do \\ -\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\ -\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}6.3 $r \leftarrow rr$ \\ -7. If $b.used < oldused - 1$ then do \\ -\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\ -\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\ -8. $b.sign \leftarrow a.sign$ \\ -9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\ -10. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2} -\end{figure} - -\textbf{Algorithm mp\_div\_2.} -This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition -core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm -could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent -reading past the end of the array of digits. - -Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the -least significant bit not the most significant bit. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Polynomial Basis Operations} -Recall from section 4.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as -the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single -place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer -division and Karatsuba multiplication. - -Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that -$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the -polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$. - -\subsection{Multiplication by $x$} - -Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one -degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to -multiplying by the integer $\beta$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\ -2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. $a.used \leftarrow a.used + b$ \\ -5. $i \leftarrow a.used - 1$ \\ -6. $j \leftarrow a.used - 1 - b$ \\ -7. for $n$ from $a.used - 1$ to $b$ do \\ -\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\ -\hspace{3mm}7.2 $i \leftarrow i - 1$ \\ -\hspace{3mm}7.3 $j \leftarrow j - 1$ \\ -8. for $n$ from 0 to $b - 1$ do \\ -\hspace{3mm}8.1 $a_n \leftarrow 0$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lshd} -\end{figure} - -\textbf{Algorithm mp\_lshd.} -This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs -from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The -motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally -different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is -typically used on values where the original value is no longer required. The algorithm will return success immediately if -$b \le 0$ since the rest of algorithm is only valid when $b > 0$. - -First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over -the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}). -The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on -step 8 sets the lower $b$ digits to zero. - -\newpage -\begin{center} -\begin{figure}[here] -\includegraphics{pics/sliding_window.ps} -\caption{Sliding Window Movement} -\label{pic:sliding_window} -\end{figure} -\end{center} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The if statement (line 24) ensures that the $b$ variable is greater than zero since we do not interpret negative -shift counts properly. The \textbf{used} count is incremented by $b$ before the copy loop begins. This elminates -the need for an additional variable in the for loop. The variable $top$ (line 42) is an alias -for the leading digit while $bottom$ (line 45) is an alias for the trailing edge. The aliases form a -window of exactly $b$ digits over the input. - -\subsection{Division by $x$} - -Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rshd}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\ -\hline \\ -1. If $b \le 0$ then return. \\ -2. If $a.used \le b$ then do \\ -\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\ -\hspace{3mm}2.2 Return. \\ -3. $i \leftarrow 0$ \\ -4. $j \leftarrow b$ \\ -5. for $n$ from 0 to $a.used - b - 1$ do \\ -\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\ -\hspace{3mm}5.2 $i \leftarrow i + 1$ \\ -\hspace{3mm}5.3 $j \leftarrow j + 1$ \\ -6. for $n$ from $a.used - b$ to $a.used - 1$ do \\ -\hspace{3mm}6.1 $a_n \leftarrow 0$ \\ -7. $a.used \leftarrow a.used - b$ \\ -8. Return. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rshd} -\end{figure} - -\textbf{Algorithm mp\_rshd.} -This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since -it does not require single precision division. This algorithm does not actually return an error code as it cannot fail. - -If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal -to the shift count $b$ then it will simply zero the input and return. - -After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that -is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit. -Also the digits are copied from the leading to the trailing edge. - -Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The only noteworthy element of this routine is the lack of a return type since it cannot fail. Like mp\_lshd() we -form a sliding window except we copy in the other direction. After the window (line 60) we then zero -the upper digits of the input to make sure the result is correct. - -\section{Powers of Two} - -Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For -example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single -shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed. - -\subsection{Multiplication by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\ -\hline \\ -1. $c \leftarrow a$. (\textit{mp\_copy}) \\ -2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\ -3. If the reallocation failed return(\textit{MP\_MEM}). \\ -4. If $b \ge lg(\beta)$ then \\ -\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\ -\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\ -5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $d \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -\hspace{3mm}6.4 If $r > 0$ then do \\ -\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\ -\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\ -7. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_2d} -\end{figure} - -\textbf{Algorithm mp\_mul\_2d.} -This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to -quickly compute the product. - -First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than -$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$ -left. - -After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts -required. If it is non-zero a modified shift loop is used to calculate the remaining product. -Essentially the loop is a generic version of algorithm mp\_mul\_2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$ -variable is used to extract the upper $d$ bits to form the carry for the next iteration. - -This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to -complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The shifting is performed in--place which means the first step (line 25) is to copy the input to the -destination. We avoid calling mp\_copy() by making sure the mp\_ints are different. The destination then -has to be grown (line 32) to accomodate the result. - -If the shift count $b$ is larger than $lg(\beta)$ then a call to mp\_lshd() is used to handle all of the multiples -of $lg(\beta)$. Leaving only a remaining shift of $lg(\beta) - 1$ or fewer bits left. Inside the actual shift -loop (lines 46 to 76) we make use of pre--computed values $shift$ and $mask$. These are used to -extract the carry bit(s) to pass into the next iteration of the loop. The $r$ and $rr$ variables form a -chain between consecutive iterations to propagate the carry. - -\subsection{Division by Power of Two} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow a$ \\ -3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -4. If $b \ge lg(\beta)$ then do \\ -\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\ -5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -6. If $k \ne 0$ then do \\ -\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\ -\hspace{3mm}6.2 $r \leftarrow 0$ \\ -\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\ -\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\ -\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\ -\hspace{6mm}6.3.3 $r \leftarrow rr$ \\ -7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_2d} -\end{figure} - -\textbf{Algorithm mp\_div\_2d.} -This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm -mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division -by using algorithm mp\_mod\_2d. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally -ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the -result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before -the quotient is obtained. - -The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. The only significant difference is -the direction of the shifts. - -\subsection{Remainder of Division by Power of Two} - -The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This -algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mod\_2d}. \\ -\textbf{Input}. One mp\_int $a$ and an integer $b$ \\ -\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then do \\ -\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b > a.used \cdot lg(\beta)$ then do \\ -\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\ -\hspace{3mm}2.2 Return the result of step 2.1. \\ -3. $c \leftarrow a$ \\ -4. If step 3 failed return(\textit{MP\_MEM}). \\ -5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\ -\hspace{3mm}5.1 $c_n \leftarrow 0$ \\ -6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\ -7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\ -8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mod\_2d} -\end{figure} - -\textbf{Algorithm mp\_mod\_2d.} -This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the -result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$ -is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -We first avoid cases of $b \le 0$ by simply mp\_zero()'ing the destination in such cases. Next if $2^b$ is larger -than the input we just mp\_copy() the input and return right away. After this point we know we must actually -perform some work to produce the remainder. - -Recalling that reducing modulo $2^k$ and a binary ``and'' with $2^k - 1$ are numerically equivalent we can quickly reduce -the number. First we zero any digits above the last digit in $2^b$ (line 42). Next we reduce the -leading digit of both (line 46) and then mp\_clamp(). - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\ - & in $O(n)$ time. \\ - &\\ -$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\ - & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\ - & upto $64$ with a hamming weight less than three. \\ - &\\ -$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\ - & $2^k - 1$ as well. \\ - &\\ -$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\ - & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\ - & any $n$-bit input. Note that the time of addition is ignored in the \\ - & calculation. \\ - & \\ -$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\ - & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\ - & the cost of addition. \\ - & \\ -$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\ - & for $n = 64 \ldots 1024$ in steps of $64$. \\ - & \\ -$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\ - & calculating the result of a signed comparison. \\ - & -\end{tabular} - -\chapter{Multiplication and Squaring} -\section{The Multipliers} -For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of -algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction -where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication -and squaring, leaving modular reductions for the subsequent chapter. - -The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular -exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular -exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions, -35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision -multiplications. - -For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied -against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the -overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in -1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach. -This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions. - -\section{Multiplication} -\subsection{The Baseline Multiplication} -\label{sec:basemult} -\index{baseline multiplication} -Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication -algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algorithm since for two $n$-digit inputs $n^2$ single precision -multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To -simplify most discussions, it will be assumed that the inputs have comparable number of digits. - -The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be -used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important -facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this -modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product -will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product. - -Recall from sub-section 4.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to -include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The -constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 5.2.2 for more information}). - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -1. If min$(a.used, b.used) < \delta$ then do \\ -\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\ -\hspace{3mm}1.2 Return the result of step 1.1 \\ -\\ -Allocate and initialize a temporary mp\_int. \\ -2. Init $t$ to be of size $digs$ \\ -3. If step 2 failed return(\textit{MP\_MEM}). \\ -4. $t.used \leftarrow digs$ \\ -\\ -Compute the product. \\ -5. for $ix$ from $0$ to $a.used - 1$ do \\ -\hspace{3mm}5.1 $u \leftarrow 0$ \\ -\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\ -\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\ -\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\ -\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\ -\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.5 if $ix + pb < digs$ then do \\ -\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\ -6. Clamp excess digits of $t$. \\ -7. Swap $c$ with $t$ \\ -8. Clear $t$ \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_mul\_digs} -\end{figure} - -\textbf{Algorithm s\_mp\_mul\_digs.} -This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem -a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent -algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}. -Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the -inputs. - -The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either -input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A -temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to -compute products when either $a = c$ or $b = c$ without overwriting the inputs. - -All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable -is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm -will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the -innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$. - -For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best -visualized in the following table. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|l|} -\hline && & 5 & 7 & 6 & \\ -\hline $\times$&& & 2 & 4 & 1 & \\ -\hline &&&&&&\\ - && & 5 & 7 & 6 & $10^0(1)(576)$ \\ - &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\ - 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\ -\hline -\end{tabular} -\end{center} -\caption{Long-Hand Multiplication Diagram} -\end{figure} - -Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate -count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult. - -Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step -is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a -double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step -5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit -$t_{ix+iy}$ and the result would be lost. - -At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th -digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result -exceed the precision requested. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -First we determine (line 31) if the Comba method can be used first since it's faster. The conditions for -sing the Comba routine are that min$(a.used, b.used) < \delta$ and the number of digits of output is less than -\textbf{MP\_WARRAY}. This new constant is used to control the stack usage in the Comba routines. By default it is -set to $\delta$ but can be reduced when memory is at a premium. - -If we cannot use the Comba method we proceed to setup the baseline routine. We allocate the the destination mp\_int -$t$ (line 37) to the exact size of the output to avoid further re--allocations. At this point we now -begin the $O(n^2)$ loop. - -This implementation of multiplication has the caveat that it can be trimmed to only produce a variable number of -digits as output. In each iteration of the outer loop the $pb$ variable is set (line 49) to the maximum -number of inner loop iterations. - -Inside the inner loop we calculate $\hat r$ as the mp\_word product of the two mp\_digits and the addition of the -carry from the previous iteration. A particularly important observation is that most modern optimizing -C compilers (GCC for instance) can recognize that a $N \times N \rightarrow 2N$ multiplication is all that -is required for the product. In x86 terms for example, this means using the MUL instruction. - -Each digit of the product is stored in turn (line 69) and the carry propagated (line 72) to the -next iteration. - -\subsection{Faster Multiplication by the ``Comba'' Method} - -One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be -computed and propagated upwards. This makes the nested loop very sequential and hard to unroll and implement -in parallel. The ``Comba'' \cite{COMBA} method is named after little known (\textit{in cryptographic venues}) Paul G. -Comba who described a method of implementing fast multipliers that do not require nested carry fixup operations. As an -interesting aside it seems that Paul Barrett describes a similar technique in his 1986 paper \cite{BARRETT} written -five years before. - -At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight -twist is placed on how the columns of the result are produced. In the standard long-hand algorithm rows of products -are produced then added together to form the final result. In the baseline algorithm the columns are added together -after each iteration to get the result instantaneously. - -In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at -the $O(n^2)$ level a simple multiplication and addition step is performed. The carries of the columns are propagated -after the nested loop to reduce the amount of work requiored. Succintly the first step of the algorithm is to compute -the product vector $\vec x$ as follows. - -\begin{equation} -\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace -\end{equation} - -Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication -of $576$ and $241$. - -\newpage\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|c|c|c|c|c|} - \hline & & 5 & 7 & 6 & First Input\\ - \hline $\times$ & & 2 & 4 & 1 & Second Input\\ -\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\ - & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\ - $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\ -\hline 10 & 34 & 45 & 31 & 6 & Final Result \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Comba Multiplication Diagram} -\end{figure} - -At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler. -Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is -congruent to adding a leading zero digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Comba Fixup}. \\ -\textbf{Input}. Vector $\vec x$ of dimension $k$ \\ -\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\ -\hline \\ -1. for $n$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\ -\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\ -2. Return($\vec x$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Comba Fixup} -\end{figure} - -With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case -$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more -efficient than the baseline algorithm why not simply always use this algorithm? - -\subsubsection{Column Weight.} -At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output -independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix -the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of -three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then -an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is -min$(m, n)$ which is fairly obvious. - -The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall -from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these -two quantities we must not violate the following - -\begin{equation} -k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha} -\end{equation} - -Which reduces to - -\begin{equation} -k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha} -\end{equation} - -Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is -found. - -\begin{equation} -k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}} -\end{equation} - -The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration -the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since -$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} single precision digits named $W$ on the stack. \\ -1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\ -2. If step 1 failed return(\textit{MP\_MEM}).\\ -\\ -3. $pa \leftarrow \mbox{MIN}(digs, a.used + b.used)$ \\ -\\ -4. $\_ \hat W \leftarrow 0$ \\ -5. for $ix$ from 0 to $pa - 1$ do \\ -\hspace{3mm}5.1 $ty \leftarrow \mbox{MIN}(b.used - 1, ix)$ \\ -\hspace{3mm}5.2 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.3 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.4 for $iz$ from 0 to $iy - 1$ do \\ -\hspace{6mm}5.4.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx+iy}b_{ty-iy}$ \\ -\hspace{3mm}5.5 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$\\ -\hspace{3mm}5.6 $\_ \hat W \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow c.used$ \\ -7. $c.used \leftarrow digs$ \\ -8. for $ix$ from $0$ to $pa$ do \\ -\hspace{3mm}8.1 $c_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa + 1$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -\\ -10. Clamp $c$. \\ -11. Return MP\_OKAY. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_mul\_digs} -\label{fig:COMBAMULT} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_mul\_digs.} -This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. - -The outer loop of this algorithm is more complicated than that of the baseline multiplier. This is because on the inside of the -loop we want to produce one column per pass. This allows the accumulator $\_ \hat W$ to be placed in CPU registers and -reduce the memory bandwidth to two \textbf{mp\_digit} reads per iteration. - -The $ty$ variable is set to the minimum count of $ix$ or the number of digits in $b$. That way if $a$ has more digits than -$b$ this will be limited to $b.used - 1$. The $tx$ variable is set to the to the distance past $b.used$ the variable -$ix$ is. This is used for the immediately subsequent statement where we find $iy$. - -The variable $iy$ is the minimum digits we can read from either $a$ or $b$ before running out. Computing one column at a time -means we have to scan one integer upwards and the other downwards. $a$ starts at $tx$ and $b$ starts at $ty$. In each -pass we are producing the $ix$'th output column and we note that $tx + ty = ix$. As we move $tx$ upwards we have to -move $ty$ downards so the equality remains valid. The $iy$ variable is the number of iterations until -$tx \ge a.used$ or $ty < 0$ occurs. - -After every inner pass we store the lower half of the accumulator into $W_{ix}$ and then propagate the carry of the accumulator -into the next round by dividing $\_ \hat W$ by $\beta$. - -To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the -cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require -$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice, -the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply -and addition operations in the nested loop in parallel. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -As per the pseudo--code we first calculate $pa$ (line 48) as the number of digits to output. Next we begin the outer loop -to produce the individual columns of the product. We use the two aliases $tmpx$ and $tmpy$ (lines 62, 63) to point -inside the two multiplicands quickly. - -The inner loop (lines 71 to 74) of this implementation is where the tradeoff come into play. Originally this comba -implementation was ``row--major'' which means it adds to each of the columns in each pass. After the outer loop it would then fix -the carries. This was very fast except it had an annoying drawback. You had to read a mp\_word and two mp\_digits and write -one mp\_word per iteration. On processors such as the Athlon XP and P4 this did not matter much since the cache bandwidth -is very high and it can keep the ALU fed with data. It did, however, matter on older and embedded cpus where cache is often -slower and also often doesn't exist. This new algorithm only performs two reads per iteration under the assumption that the -compiler has aliased $\_ \hat W$ to a CPU register. - -After the inner loop we store the current accumulator in $W$ and shift $\_ \hat W$ (lines 77, 80) to forward it as -a carry for the next pass. After the outer loop we use the final carry (line 77) as the last digit of the product. - -\subsection{Polynomial Basis Multiplication} -To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms -the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and -$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree. - -The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will -directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients -requires $O(n^2)$ time and would in practice be slower than the Comba technique. - -However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown -coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with -Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in -effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$. - -The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since -$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the -fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required -by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs. - -When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term -is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product -$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather -simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication. -The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the -points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly. - -If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points} -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that -$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For -example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror. - -\begin{eqnarray} -\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\ -16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0) -\end{eqnarray} - -Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the -polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method. - -As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of -multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is -$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent} -summarizes the exponents for various values of $n$. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\ -\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\ -\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\ -\hline $4$ & $1.403677461$ &\\ -\hline $5$ & $1.365212389$ &\\ -\hline $10$ & $1.278753601$ &\\ -\hline $100$ & $1.149426538$ &\\ -\hline $1000$ & $1.100270931$ &\\ -\hline $10000$ & $1.075252070$ &\\ -\hline -\end{tabular} -\end{center} -\caption{Asymptotic Running Time of Polynomial Basis Multiplication} -\label{fig:exponent} -\end{figure} - -At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead -of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large -numbers. - -\subsubsection{Cutoff Point} -The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However, -the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the -polynomial basis approach more costly to use with small inputs. - -Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a -point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and -when $m > y$ the Comba methods are slower than the polynomial basis algorithms. - -The exact location of $y$ depends on several key architectural elements of the computer platform in question. - -\begin{enumerate} -\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example -on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower -the cutoff point $y$ will be. - -\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits -grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This -directly reflects on the ratio previous mentioned. - -\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an -influence over the cutoff point. - -\end{enumerate} - -A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point -is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when -a high resolution timer is available. - -\subsection{Karatsuba Multiplication} -Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for -general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with -light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent. - -\begin{equation} -f(x) \cdot g(x) = acx^2 + ((a + b)(c + d) - (ac + bd))x + bd -\end{equation} - -Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying -this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns -out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points -$\zeta_0$, $\zeta_{\infty}$ and $\zeta_{1}$. Consider the resultant system of equations. - -\begin{center} -\begin{tabular}{rcrcrcrc} -$\zeta_{0}$ & $=$ & & & & & $w_0$ \\ -$\zeta_{1}$ & $=$ & $w_2$ & $+$ & $w_1$ & $+$ & $w_0$ \\ -$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\ -\end{tabular} -\end{center} - -By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity -of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.} -making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\ -\hline \\ -1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\ -2. If step 2 failed then return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\ -3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\ -6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\ -7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\ -\\ -Calculate the three products. \\ -8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\ -9. $x1y1 \leftarrow x1 \cdot y1$ \\ -10. $t1 \leftarrow x1 + x0$ (\textit{mp\_add}) \\ -11. $x0 \leftarrow y1 + y0$ \\ -12. $t1 \leftarrow t1 \cdot x0$ \\ -\\ -Calculate the middle term. \\ -13. $x0 \leftarrow x0y0 + x1y1$ \\ -14. $t1 \leftarrow t1 - x0$ (\textit{s\_mp\_sub}) \\ -\\ -Calculate the final product. \\ -15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\ -16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\ -17. $t1 \leftarrow x0y0 + t1$ \\ -18. $c \leftarrow t1 + x1y1$ \\ -19. Clear all of the temporary variables. \\ -20. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_mul} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_mul.} -This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description -from Knuth \cite[pp. 294-295]{TAOCPV2}. - -\index{radix point} -In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must -be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the -smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5 -compute the lower halves. Step 6 and 7 computer the upper halves. - -After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products -$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 + x0$ has been computed. By using $x0$ instead -of an additional temporary variable, the algorithm can avoid an addition memory allocation operation. - -The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional -wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense -to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables -required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only -the temporaries that have been successfully allocated so far. - -The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the -additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective -number of digits for the next section of code. - -The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd -to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and -\textbf{sign} members are copied first. The first for loop on line 96 copies the lower halves. Since they are both the same magnitude it -is simpler to calculate both lower halves in a single loop. The for loop on lines 102 and 107 calculate the upper halves $x1$ and -$y1$ respectively. - -By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs. - -When line 151 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that -the same code that handles errors can be used to clear the temporary variables and return. - -\subsection{Toom-Cook $3$-Way Multiplication} -Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 2$ except that the points are -chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$, -$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients -of the $W(x)$. - -With the five relations that Toom-Cook specifies, the following system of equations is formed. - -\begin{center} -\begin{tabular}{rcrcrcrcrcr} -$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\ -$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\ -$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\ -$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\ -$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\ -\end{tabular} -\end{center} - -A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power -of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that -the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point -(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\ -1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\ -2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\ -\\ -Find the five equations for $w_0, w_1, ..., w_4$. \\ -8. $w_0 \leftarrow a_0 \cdot b_0$ \\ -9. $w_4 \leftarrow a_2 \cdot b_2$ \\ -10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\ -11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\ -13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\ -14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\ -15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\ -16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\ -17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\ -\\ -Continued on the next page.\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b $ \\ -\hline \\ -Now solve the system of equations. \\ -18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\ -19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\ -20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\ -21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\ -23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\ -24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\ -25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\ -\\ -Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\ -26. for $n$ from $1$ to $4$ do \\ -\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\ -27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\ -28. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toom\_mul (continued)} -\end{figure} - -\textbf{Algorithm mp\_toom\_mul.} -This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this -algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this -description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across -any given step. - -The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller -integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required. - -The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond -to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find -$f(y)$ and $g(y)$ which significantly speeds up the algorithm. - -After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients -$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of -the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates -that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$. - -Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer -result $a \cdot b$ is produced. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first obvious thing to note is that this algorithm is complicated. The complexity is worth it if you are multiplying very -large numbers. For example, a 10,000 digit multiplication takes approximaly 99,282,205 fewer single precision multiplications with -Toom--Cook than a Comba or baseline approach (this is a savings of more than 99$\%$). For most ``crypto'' sized numbers this -algorithm is not practical as Karatsuba has a much lower cutoff point. - -First we split $a$ and $b$ into three roughly equal portions. This has been accomplished (lines 41 to 70) with -combinations of mp\_rshd() and mp\_mod\_2d() function calls. At this point $a = a2 \cdot \beta^2 + a1 \cdot \beta + a0$ and similiarly -for $b$. - -Next we compute the five points $w0, w1, w2, w3$ and $w4$. Recall that $w0$ and $w4$ can be computed directly from the portions so -we get those out of the way first (lines 73 and 78). Next we compute $w1, w2$ and $w3$ using Horners method. - -After this point we solve for the actual values of $w1, w2$ and $w3$ by reducing the $5 \times 5$ system which is relatively -straight forward. - -\subsection{Signed Multiplication} -Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all -of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul}. \\ -\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\ -\textbf{Output}. $c \leftarrow a \cdot b$ \\ -\hline \\ -1. If $a.sign = b.sign$ then \\ -\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\ -2. else \\ -\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\ -3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\ -\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\ -4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\ -\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\ -5. else \\ -\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\ -\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\ -\hspace{3mm}5.3 else \\ -\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\ -6. $c.sign \leftarrow sign$ \\ -7. Return the result of the unsigned multiplication performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul} -\end{figure} - -\textbf{Algorithm mp\_mul.} -This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms -available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm -s\_mp\_mul\_digs will clear it. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation is rather simplistic and is not particularly noteworthy. Line 22 computes the sign of the result using the ``?'' -operator from the C programming language. Line 48 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$. - -\section{Squaring} -\label{sec:basesquare} - -Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization -available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications -performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider -the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$, -$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$ -and $3 \cdot 1 = 1 \cdot 3$. - -For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$ -required for multiplication. The following diagram gives an example of the operations required. - -\begin{figure}[here] -\begin{center} -\begin{tabular}{ccccc|c} -&&1&2&3&\\ -$\times$ &&1&2&3&\\ -\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\ - & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\ - $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\ -\end{tabular} -\end{center} -\caption{Squaring Optimization Diagram} -\end{figure} - -Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$ -represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it. - -The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will -appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double -products and at most one square (\textit{see the exercise section}). - -The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row, -occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero. -Column two of row one is a square and column three is the first unique column. - -\subsection{The Baseline Squaring Algorithm} -The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines -will not handle. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\ -2. If step 1 failed return(\textit{MP\_MEM}) \\ -3. $t.used \leftarrow 2 \cdot a.used + 1$ \\ -4. For $ix$ from 0 to $a.used - 1$ do \\ -\hspace{3mm}Calculate the square. \\ -\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\ -\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}Calculate the double products after the square. \\ -\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\ -\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\ -\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}Set the last carry. \\ -\hspace{3mm}4.5 While $u > 0$ do \\ -\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\ -\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\ -6. Exchange $b$ and $t$. \\ -7. Clear $t$ (\textit{mp\_clear}) \\ -8. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm s\_mp\_sqr.} -This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC -\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the -destination mp\_int to be the same as the source mp\_int. - -The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while -the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate -the carry and compute the double products. - -The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this -very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that -when it is multiplied by two, it can be properly represented by a mp\_word. - -Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial -results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Inside the outer loop (line 34) the square term is calculated on line 37. The carry (line 44) has been -extracted from the mp\_word accumulator using a right shift. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized -(lines 47 and 50) to simplify the inner loop. The doubling is performed using two -additions (line 59) since it is usually faster than shifting, if not at least as fast. - -The important observation is that the inner loop does not begin at $iy = 0$ like for multiplication. As such the inner loops -get progressively shorter as the algorithm proceeds. This is what leads to the savings compared to using a multiplication to -square a number. - -\subsection{Faster Squaring by the ``Comba'' Method} -A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional -drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these -performance hazards. - -The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry -propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact -that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example, -$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$. - -However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two -mp\_word arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and -carry propagation can be moved to a $O(n)$ work level outside the $O(n^2)$ level. In this case, we have an even simpler solution in mind. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_digits named $W$ on the stack. \\ -1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\ -2. If step 1 failed return(\textit{MP\_MEM}). \\ -\\ -3. $pa \leftarrow 2 \cdot a.used$ \\ -4. $\hat W1 \leftarrow 0$ \\ -5. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}5.1 $\_ \hat W \leftarrow 0$ \\ -\hspace{3mm}5.2 $ty \leftarrow \mbox{MIN}(a.used - 1, ix)$ \\ -\hspace{3mm}5.3 $tx \leftarrow ix - ty$ \\ -\hspace{3mm}5.4 $iy \leftarrow \mbox{MIN}(a.used - tx, ty + 1)$ \\ -\hspace{3mm}5.5 $iy \leftarrow \mbox{MIN}(iy, \lfloor \left (ty - tx + 1 \right )/2 \rfloor)$ \\ -\hspace{3mm}5.6 for $iz$ from $0$ to $iz - 1$ do \\ -\hspace{6mm}5.6.1 $\_ \hat W \leftarrow \_ \hat W + a_{tx + iz}a_{ty - iz}$ \\ -\hspace{3mm}5.7 $\_ \hat W \leftarrow 2 \cdot \_ \hat W + \hat W1$ \\ -\hspace{3mm}5.8 if $ix$ is even then \\ -\hspace{6mm}5.8.1 $\_ \hat W \leftarrow \_ \hat W + \left ( a_{\lfloor ix/2 \rfloor}\right )^2$ \\ -\hspace{3mm}5.9 $W_{ix} \leftarrow \_ \hat W (\mbox{mod }\beta)$ \\ -\hspace{3mm}5.10 $\hat W1 \leftarrow \lfloor \_ \hat W / \beta \rfloor$ \\ -\\ -6. $oldused \leftarrow b.used$ \\ -7. $b.used \leftarrow 2 \cdot a.used$ \\ -8. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}8.1 $b_{ix} \leftarrow W_{ix}$ \\ -9. for $ix$ from $pa$ to $oldused - 1$ do \\ -\hspace{3mm}9.1 $b_{ix} \leftarrow 0$ \\ -10. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_s\_mp\_sqr} -\end{figure} - -\textbf{Algorithm fast\_s\_mp\_sqr.} -This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm -s\_mp\_sqr when the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$. -This algorithm is very similar to the Comba multiplier except with a few key differences we shall make note of. - -First, we have an accumulator and carry variables $\_ \hat W$ and $\hat W1$ respectively. This is because the inner loop -products are to be doubled. If we had added the previous carry in we would be doubling too much. Next we perform an -addition MIN condition on $iy$ (step 5.5) to prevent overlapping digits. For example, $a_3 \cdot a_5$ is equal -$a_5 \cdot a_3$. Whereas in the multiplication case we would have $5 < a.used$ and $3 \ge 0$ is maintained since we double the sum -of the products just outside the inner loop we have to avoid doing this. This is also a good thing since we perform -fewer multiplications and the routine ends up being faster. - -Finally the last difference is the addition of the ``square'' term outside the inner loop (step 5.8). We add in the square -only to even outputs and it is the square of the term at the $\lfloor ix / 2 \rfloor$ position. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is essentially a copy of Comba multiplication with the appropriate changes added to make it faster for -the special case of squaring. - -\subsection{Polynomial Basis Squaring} -The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception -is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$ -multiplications to find the $\zeta$ relations, squaring operations are performed instead. - -\subsection{Karatsuba Squaring} -Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square. -Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a -number with the following equation. - -\begin{equation} -h(x) = a^2x^2 + \left ((a + b)^2 - (a^2 + b^2) \right )x + b^2 -\end{equation} - -Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a + b)^2$. As in -Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of -$O \left ( n^{lg(3)} \right )$. - -If the asymptotic times of Karatsuba squaring and multiplication are the same, why not simply use the multiplication algorithm -instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the -time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff -point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits. - -Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared. -The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication -were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\ -2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\ -\\ -Split the input. e.g. $a = x1\beta^B + x0$ \\ -3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\ -4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\ -\\ -Calculate the three squares. \\ -6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\ -7. $x1x1 \leftarrow x1^2$ \\ -8. $t1 \leftarrow x1 + x0$ (\textit{s\_mp\_add}) \\ -9. $t1 \leftarrow t1^2$ \\ -\\ -Compute the middle term. \\ -10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\ -11. $t1 \leftarrow t1 - t2$ \\ -\\ -Compute final product. \\ -12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\ -13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\ -14. $t1 \leftarrow t1 + x0x0$ \\ -15. $b \leftarrow t1 + x1x1$ \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_karatsuba\_sqr} -\end{figure} - -\textbf{Algorithm mp\_karatsuba\_sqr.} -This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based -multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings. - -The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is -placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$ -as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form. - -By expanding $\left (x1 + x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $(x0 - x1)^2 - (x1^2 + x0^2) = 2 \cdot x0 \cdot x1$. -Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then -this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality. - -Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or -machine clock cycles.}. - -\begin{equation} -5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2 -\end{equation} - -For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold. -\begin{center} -\begin{tabular}{rcl} -${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\ -${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\ -${13 \over 9}$ & $<$ & $n$ \\ -\end{tabular} -\end{center} - -This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors -where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On -the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a -ratio of 1:7. } than simpler operations such as addition. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and -shift the input into the two halves. The loop from line 54 to line 70 has been modified since only one input exists. The \textbf{used} -count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents -to the respective halves as if mp\_rshd and mp\_mod\_2d had been used. - -By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point -is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4 -it is actually below the Comba limit (\textit{at 110 digits}). - -This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are -redirected to the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and -mp\_clears are executed normally. - -\subsection{Toom-Cook Squaring} -The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used -instead of multiplication to find the five relations. The reader is encouraged to read the description of the latter algorithm and try to -derive their own Toom-Cook squaring algorithm. - -\subsection{High Level Squaring} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_sqr}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $b \leftarrow a^2$ \\ -\hline \\ -1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\ -\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\ -2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\ -\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\ -3. else \\ -\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\ -\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\ -\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\ -\hspace{3mm}3.3 else \\ -\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\ -4. $b.sign \leftarrow MP\_ZPOS$ \\ -5. Return the result of the unsigned squaring performed. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_sqr} -\end{figure} - -\textbf{Algorithm mp\_sqr.} -This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least -\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If -neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\ - & that have different number of digits in Karatsuba multiplication. \\ - & \\ -$\left [ 2 \right ] $ & In section 5.3 the fact that every column of a squaring is made up \\ - & of double products and at most one square is stated. Prove this statement. \\ - & \\ -$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\ - & \\ -$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\ - & \\ -$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\ - & required for equation $6.7$ to be true. \\ - & \\ -$\left [ 3 \right ] $ & Implement a threaded version of Comba multiplication (and squaring) where you \\ - & compute subsets of the columns in each thread. Determine a cutoff point where \\ - & it is effective and add the logic to mp\_mul() and mp\_sqr(). \\ - &\\ -$\left [ 4 \right ] $ & Same as the previous but also modify the Karatsuba and Toom-Cook. You must \\ - & increase the throughput of mp\_exptmod() for random odd moduli in the range \\ - & $512 \ldots 4096$ bits significantly ($> 2x$) to complete this challenge. \\ - & \\ -\end{tabular} - -\chapter{Modular Reduction} -\section{Basics of Modular Reduction} -\index{modular residue} -Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms, -such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be \textit{reduced} -modulo another number $b$ by finding the remainder of the division $a/b$. Full integer division with remainder is a topic to be covered -in~\ref{sec:division}. - -Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result -$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the -``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and -other forms of residues. - -Modular reductions are normally used to create either finite groups, rings or fields. The most common usage for performance driven modular reductions -is in modular exponentiation algorithms. That is to compute $d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. This operation is used in the -RSA and Diffie-Hellman public key algorithms, for example. Modular multiplication and squaring also appears as a fundamental operation in -elliptic curve cryptographic algorithms. As will be discussed in the subsequent chapter there exist fast algorithms for computing modular -exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial results in the -range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms. They have also been used to create redundancy check -algorithms known as CRCs, error correction codes such as Reed-Solomon and solve a variety of number theoeretic problems. - -\section{The Barrett Reduction} -The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate -division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to - -\begin{equation} -c = a - b \cdot \lfloor a/b \rfloor -\end{equation} - -Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP\footnote{It is worth noting that Barrett's paper -targeted the DSP56K processor.} intuition would indicate the next step would be to replace $a/b$ by a multiplication by the reciprocal. However, -DSP intuition on its own will not work as these numbers are considerably larger than the precision of common DSP floating point data types. -It would take another common optimization to optimize the algorithm. - -\subsection{Fixed Point Arithmetic} -The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed -point arithmetic would become very popular as it greatly optimize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were -fairly slow if not unavailable. The idea behind fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit -integer and a $q$-bit fraction part (\textit{where $p+q = k$}). - -In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the -value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized by -moving the implied decimal point back to where it should be. For example, with $q = 4$ to multiply the integers $9$ and $5$ they must be converted -to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$ represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the -fixed point representation of $5$. The product $ab$ is equal to $45(2^{2q})$ which when normalized by dividing by $2^q$ produces $45(2^q)$. - -This technique became popular since a normal integer multiplication and logical shift right are the only required operations to perform a multiplication -of two fixed point numbers. Using fixed point arithmetic, division can be easily approximated by multiplying by the reciprocal. If $2^q$ is -equivalent to one than $2^q/b$ is equivalent to the fixed point approximation of $1/b$ using real arithmetic. Using this fact dividing an integer -$a$ by another integer $b$ can be achieved with the following expression. - -\begin{equation} -\lfloor a / b \rfloor \mbox{ }\approx\mbox{ } \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with -modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations -are considerably faster than division on most processors. - -Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which -leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and -the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct. The value of $2^q$ must be close to or ideally -larger than the dividend. In effect if $a$ is the dividend then $q$ should allow $0 \le \lfloor a/2^q \rfloor \le 1$ in order for this approach -to work correctly. Plugging this form of divison into the original equation the following modular residue equation arises. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \lfloor 2^q / b \rfloor)/2^q \rfloor -\end{equation} - -Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$ -variable also helps re-inforce the idea that it is meant to be computed once and re-used. - -\begin{equation} -c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor -\end{equation} - -Provided that $2^q \ge a$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. In the context of Barrett -reduction the value of $a$ is bound by $0 \le a \le (b - 1)^2$ meaning that $2^q \ge b^2$ is sufficient to ensure the reciprocal will have enough -precision. - -Let $n$ represent the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and -another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to -reduce the number. - -For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing -$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$. -By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found. - -\subsection{Choosing a Radix Point} -Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best -that could be achieved a full division\footnote{A division requires approximately $O(2cn^2)$ single precision multiplications for a small value of $c$. -See~\ref{sec:division} for further details.} might as well be used in its place. The key to optimizing the reduction is to reduce the precision of -the initial multiplication that finds the quotient. - -Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent -the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$, which is generally true if -two $m$-digit numbers have been multiplied. Dividing $a$ by $b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the -$m - 1$'th digit of $a$ will contribute at most a value of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$. Another way to -express this is by re-writing $a$ as two parts. If $a' \equiv a \mbox{ (mod }b^m\mbox{)}$ and $a'' = a - a'$ then -${a \over b} \equiv {{a' + a''} \over b}$ which is equivalent to ${a' \over b} + {a'' \over b}$. Since $a'$ is bound to be less than $b$ the quotient -is bound by $0 \le {a' \over b} < 1$. - -Since the digits of $a'$ do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits -``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input -with the irrelevant digits trimmed. Now the modular reduction is trimmed to the almost equivalent equation - -\begin{equation} -c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor -\end{equation} - -Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$ where in this case $q$ is a multiple of $lg(\beta)$. Also note that the -exponent on the divisor when added to the amount $q_0$ was shifted by equals $2m$. If the optimization had not been performed the divisor -would have the exponent $2m$ so in the end the exponents do ``add up''. Using the above equation the quotient -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most two. The original fixed point quotient can be off -by as much as one (\textit{provided the radix point is chosen suitably}) and now that the lower irrelevent digits have been trimmed the quotient -can be off by an additional value of one for a total of at most two. This implies that -$0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then conditionally subtracting -$b$ once or twice the residue is found. - -The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single -precision multiplications, ignoring the subtractions required. In total $2m^2 + m$ single precision multiplications are required to find the residue. -This is considerably faster than the original attempt. - -For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$ -represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$. -With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then -$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$ -is found. - -\subsection{Trimming the Quotient} -So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As -it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for -optimization. - -After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower -half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision -multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly. -In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed. - -The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision -multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number -of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications. - -\subsection{Trimming the Residue} -After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small -multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the -result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are -implicitly zero. - -The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full -$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can -be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces -only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications. - -With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which -is considerably faster than the straightforward $3m^2$ method. - -\subsection{The Barrett Algorithm} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce}. \\ -\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor, m = \lceil lg_{\beta}(b) \rceil, (0 \le a < b^2, b > 1)$ \\ -\textbf{Output}. $a \mbox{ (mod }b\mbox{)}$ \\ -\hline \\ -Let $m$ represent the number of digits in $b$. \\ -1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\ -2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\ -\\ -Produce the quotient. \\ -3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\ -4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\ -\\ -Subtract the multiple of modulus from the input. \\ -5. $a \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\ -7. $a \leftarrow a - q$ (\textit{mp\_sub}) \\ -\\ -Add $\beta^{m+1}$ if a carry occured. \\ -8. If $a < 0$ then (\textit{mp\_cmp\_d}) \\ -\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\ -\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\ -\hspace{3mm}8.3 $a \leftarrow a + q$ \\ -\\ -Now subtract the modulus if the residue is too large (e.g. quotient too small). \\ -9. While $a \ge b$ do (\textit{mp\_cmp}) \\ -\hspace{3mm}9.1 $c \leftarrow a - b$ \\ -10. Clear $q$. \\ -11. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce} -\end{figure} - -\textbf{Algorithm mp\_reduce.} -This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC -\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must -be adhered to for the algorithm to work. - -First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting -a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order -for the quotient to have enough precision. If $a$ is the product of two numbers that were already reduced modulo $b$, this will not be a problem. -Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The value of $\mu$ is passed as an argument to this -algorithm and is assumed to be calculated and stored before the algorithm is used. - -Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called -$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. The algorithm is based on $s\_mp\_mul\_digs$ except that -instead of stopping at a given level of precision it starts at a given level of precision. This optimal algorithm can only be used if the number -of digits in $b$ is very much smaller than $\beta$. - -While it is known that -$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue, so an implied -``borrow'' from the higher digits might leave a negative result. After the multiple of the modulus has been subtracted from $a$ the residue must be -fixed up in case it is negative. The invariant $\beta^{m+1}$ must be added to the residue to make it positive again. - -The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is -performed at most twice, and on average once. However, if $a \ge b^2$ than it will iterate substantially more times than it should. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves -the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits -in the modulus. In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is -safe to do so. - -\subsection{The Barrett Setup Algorithm} -In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for -future use so that the Barrett algorithm can be used without delay. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_setup}. \\ -\textbf{Input}. mp\_int $a$ ($a > 1$) \\ -\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\ -\hline \\ -1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\ -2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_setup.} -This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which -is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable -which would received the remainder is passed as NULL. As will be discussed in~\ref{sec:division} the division routine allows both the quotient and the -remainder to be passed as NULL meaning to ignore the value. - -\section{The Montgomery Reduction} -Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting -form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a -residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient. - -Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of -$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input -is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established. - -\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way -to explain this is that $n$ is (\textit{or multiples of $n$ are}) congruent to zero modulo $n$. Adding zero will not change the value of the residue. - -\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually -this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to -multiplication by $k^{-1}$ modulo $n$. - -From these two simple facts the following simple algorithm can be derived. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction}. \\ -\textbf{Input}. Integer $x$, $n$ and $k$ \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If $x$ is odd then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\ -\hspace{3mm}1.2 $x \leftarrow x/2$ \\ -2. Return $x$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction} -\end{figure} - -The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is -added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since -$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the -final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to -$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\ -\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\ -\hline $2$ & $x/2 = 1453$ \\ -\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\ -\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\ -\hline $5$ & $x/2 = 278$ \\ -\hline $6$ & $x/2 = 139$ \\ -\hline $7$ & $x + n = 396$, $x/2 = 198$ \\ -\hline $8$ & $x/2 = 99$ \\ -\hline $9$ & $x + n = 356$, $x/2 = 178$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (I)} -\label{fig:MONT1} -\end{figure} - -Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 9$ (note $\beta^k = 512$ which is larger than $n$). The result of -the algorithm $r = 178$ is congruent to the value of $2^{-9} \cdot 5555 \mbox{ (mod }257\mbox{)}$. When $r$ is multiplied by $2^9$ modulo $257$ the correct residue -$r \equiv 158$ is produced. - -Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts -and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful. -Fortunately there exists an alternative representation of the algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($2^k > n$) \\ -\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $1$ to $k$ do \\ -\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\ -\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\ -2. Return $x/2^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified I)} -\end{figure} - -This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single -precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement. - -\begin{figure}[here] -\begin{small} -\begin{center} -\begin{tabular}{|c|l|r|} -\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} & \textbf{Result ($x$) in Binary} \\ -\hline -- & $5555$ & $1010110110011$ \\ -\hline $1$ & $x + 2^{0}n = 5812$ & $1011010110100$ \\ -\hline $2$ & $5812$ & $1011010110100$ \\ -\hline $3$ & $x + 2^{2}n = 6840$ & $1101010111000$ \\ -\hline $4$ & $x + 2^{3}n = 8896$ & $10001011000000$ \\ -\hline $5$ & $8896$ & $10001011000000$ \\ -\hline $6$ & $8896$ & $10001011000000$ \\ -\hline $7$ & $x + 2^{6}n = 25344$ & $110001100000000$ \\ -\hline $8$ & $25344$ & $110001100000000$ \\ -\hline $9$ & $x + 2^{7}n = 91136$ & $10110010000000000$ \\ -\hline -- & $x/2^k = 178$ & \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example of Montgomery Reduction (II)} -\label{fig:MONT2} -\end{figure} - -Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 5555$ modulo $n = 257$ with $k = 9$. -With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the -loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is -zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero. - -\subsection{Digit Based Montgomery Reduction} -Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the -previous algorithm re-written to compute the Montgomery reduction in this new fashion. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\ -\textbf{Input}. Integer $x$, $n$ and $k$ ($\beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. for $t$ from $0$ to $k - 1$ do \\ -\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\ -2. Return $x/\beta^k$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Montgomery Reduction (modified II)} -\end{figure} - -The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of -the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This -problem breaks down to solving the following congruency. - -\begin{center} -\begin{tabular}{rcl} -$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\ -$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\end{tabular} -\end{center} - -In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used -extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$. - -For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$ -represent the value to reduce. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\ -\hline -- & $33$ & --\\ -\hline $0$ & $33 + \mu n = 50$ & $1$ \\ -\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Montgomery Reduction} -\end{figure} - -The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$ -which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in -the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and -the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$. - -\subsection{Baseline Montgomery Reduction} -The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for -Montgomery reductions. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -1. $digs \leftarrow 2n.used + 1$ \\ -2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\ -\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\ -\\ -Setup $x$ for the reduction. \\ -3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\ -4. $x.used \leftarrow digs$ \\ -\\ -Eliminate the lower $k$ digits. \\ -5. For $ix$ from $0$ to $k - 1$ do \\ -\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}5.2 $u \leftarrow 0$ \\ -\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\ -\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\ -\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -\hspace{3mm}5.4 While $u > 0$ do \\ -\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\ -\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\ -\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\ -\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\ -\\ -Divide by $\beta^k$ and fix up as required. \\ -6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\ -7. If $x \ge n$ then \\ -\hspace{3mm}7.1 $x \leftarrow x - n$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_reduce.} -This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based -on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The -restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as -for the Barrett algorithm. Additionally if $n > 1$ and $n$ is odd there will exist a modular inverse $\rho$. $\rho$ must be calculated in -advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$. - -Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on -the size of the input. This algorithm is discussed in sub-section 6.3.3. - -Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop -calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and -multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop. - -Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications -in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision -multiplications. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This is the baseline implementation of the Montgomery reduction algorithm. Lines 31 to 36 determine if the Comba based -routine can be used instead. Line 47 computes the value of $\mu$ for that particular iteration of the outer loop. - -The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and -the alias $tmpn$ refers to the modulus $n$. - -\subsection{Faster ``Comba'' Montgomery Reduction} - -The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial -nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba -technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates -a $k \times 1$ product $k$ times. - -The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the -carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple. -Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry. - -With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases -the speed of the algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\ -\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\ -\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\ -\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\ -\hline \\ -Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\ -1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\ -Copy the digits of $x$ into the array $\hat W$ \\ -2. For $ix$ from $0$ to $x.used - 1$ do \\ -\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\ -3. For $ix$ from $x.used$ to $2n.used - 1$ do \\ -\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\ -Elimiate the lower $k$ digits. \\ -4. for $ix$ from $0$ to $n.used - 1$ do \\ -\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\ -\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\ -\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Propagate carries upwards. \\ -5. for $ix$ from $n.used$ to $2n.used + 1$ do \\ -\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\ -Shift right and reduce modulo $\beta$ simultaneously. \\ -6. for $ix$ from $0$ to $n.used + 1$ do \\ -\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\ -Zero excess digits and fixup $x$. \\ -7. if $x.used > n.used + 1$ then do \\ -\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\ -\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\ -8. $x.used \leftarrow n.used + 1$ \\ -9. Clamp excessive digits of $x$. \\ -10. If $x \ge n$ then \\ -\hspace{3mm}10.1 $x \leftarrow x - n$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm fast\_mp\_montgomery\_reduce} -\end{figure} - -\textbf{Algorithm fast\_mp\_montgomery\_reduce.} -This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly -faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions -on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the -the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo -a modulus of at most $3,556$ bits in length. - -As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the -contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step -4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such -as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing -a single precision multiplication instead half the amount of time is spent. - -Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step -4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note -how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no -point. - -Step 5 will propagate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are -stored in the destination $x$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The $\hat W$ array is first filled with digits of $x$ on line 48 then the rest of the digits are zeroed on line 55. Both loops share -the same alias variables to make the code easier to read. - -The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This -forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 110 fixes the carry -for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$. - -The for loop on line 109 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns -modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th -digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$. - -\subsection{Montgomery Setup} -To calculate the variable $\rho$ a relatively simple algorithm will be required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_montgomery\_setup}. \\ -\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\ -\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\ -\hline \\ -1. $b \leftarrow n_0$ \\ -2. If $b$ is even return(\textit{MP\_VAL}) \\ -3. $x \leftarrow (((b + 2) \mbox{ AND } 4) << 1) + b$ \\ -4. for $k$ from 0 to $\lceil lg(lg(\beta)) \rceil - 2$ do \\ -\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\ -5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_montgomery\_setup} -\end{figure} - -\textbf{Algorithm mp\_montgomery\_setup.} -This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick -to calculate $1/n_0$ when $\beta$ is a power of two. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess -multiplications when $\beta$ is not the default 28-bits. - -\section{The Diminished Radix Algorithm} -The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett -or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence. - -\begin{equation} -(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)} -\end{equation} - -This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that -then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof -of the above equation is very simple. First write $x$ in the product form. - -\begin{equation} -x = qn + r -\end{equation} - -Now reduce both sides modulo $(n - k)$. - -\begin{equation} -x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)} -\end{equation} - -The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$ -into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Diminished Radix Reduction}. \\ -\textbf{Input}. Integer $x$, $n$, $k$ \\ -\textbf{Output}. $x \mbox{ mod } (n - k)$ \\ -\hline \\ -1. $q \leftarrow \lfloor x / n \rfloor$ \\ -2. $q \leftarrow k \cdot q$ \\ -3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\ -4. $x \leftarrow x + q$ \\ -5. If $x \ge (n - k)$ then \\ -\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\ -\hspace{3mm}5.2 Goto step 1. \\ -6. Return $x$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Diminished Radix Reduction} -\label{fig:DR} -\end{figure} - -This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always -once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial. - -\begin{equation} -0 \le x < n^2 + k^2 - 2nk -\end{equation} - -The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following. - -\begin{equation} -q < n - 2k - k^2/n -\end{equation} - -Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as -$0 \le x < n$. By step four the sum $x + q$ is bounded by - -\begin{equation} -0 \le q + x < (k + 1)n - 2k^2 - 1 -\end{equation} - -With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the -sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the -range $0 \le x < (n - k - 1)^2$. - -\begin{figure} -\begin{small} -\begin{center} -\begin{tabular}{|l|} -\hline -$x = 123456789, n = 256, k = 3$ \\ -\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\ -$q \leftarrow q*k = 1446759$ \\ -$x \leftarrow x \mbox{ mod } n = 21$ \\ -$x \leftarrow x + q = 1446780$ \\ -$x \leftarrow x - (n - k) = 1446527$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 5650$ \\ -$q \leftarrow q*k = 16950$ \\ -$x \leftarrow x \mbox{ mod } n = 127$ \\ -$x \leftarrow x + q = 17077$ \\ -$x \leftarrow x - (n - k) = 16824$ \\ -\hline -$q \leftarrow \lfloor x/n \rfloor = 65$ \\ -$q \leftarrow q*k = 195$ \\ -$x \leftarrow x \mbox{ mod } n = 184$ \\ -$x \leftarrow x + q = 379$ \\ -$x \leftarrow x - (n - k) = 126$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Example Diminished Radix Reduction} -\label{fig:EXDR} -\end{figure} - -Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$ -is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only -three passes were required to find the residue $x \equiv 126$. - - -\subsection{Choice of Moduli} -On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other -modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate modulus is chosen. - -Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used. -Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division -by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$ -which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits. - -However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be -performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$. -Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ merely requires zeroing the digits above the $p-1$'th digit of $x$. - -Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ whereas the term ``unrestricted -modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the -$2^p$ logic except $p$ must be a multiple of $lg(\beta)$. - -\subsection{Choice of $k$} -Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$ -in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might -as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be. - -\subsection{Restricted Diminished Radix Reduction} -The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce -an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation -of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition -of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular -exponentiations are performed. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_reduce}. \\ -\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\ -\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\ -\textbf{Output}. $x \mbox{ mod } n$ \\ -\hline \\ -1. $m \leftarrow n.used$ \\ -2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\ -3. $\mu \leftarrow 0$ \\ -4. for $i$ from $0$ to $m - 1$ do \\ -\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\ -\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -5. $x_{m} \leftarrow \mu$ \\ -6. for $i$ from $m + 1$ to $x.used - 1$ do \\ -\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\ -7. Clamp excess digits of $x$. \\ -8. If $x \ge n$ then \\ -\hspace{3mm}8.1 $x \leftarrow x - n$ \\ -\hspace{3mm}8.2 Goto step 3. \\ -9. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_reduce} -\end{figure} - -\textbf{Algorithm mp\_dr\_reduce.} -This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction -with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$. - -This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$ -and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing -the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th -digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to -$x$ before the addition of the multiple of the upper half. - -At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes -at step 3. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 52 is where -the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of -the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time. - -The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits -a division by $\beta^m$ can be simulated virtually for free. The loop on line 64 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11}) -in this algorithm. - -By line 67 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 74 the -same pointer will point to the $m+1$'th digit where the zeroes will be placed. - -Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required. -With the same logic at line 81 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used -as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code -does not need to be checked. - -\subsubsection{Setup} -To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for -completeness. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = \beta - n_0$ \\ -\hline \\ -1. $k \leftarrow \beta - n_0$ \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_setup} -\end{figure} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Modulus Detection} -Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be -of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\ -\hline -1. If $n.used < 2$ then return($0$). \\ -2. for $ix$ from $1$ to $n.used - 1$ do \\ -\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\ -3. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_dr\_is\_modulus} -\end{figure} - -\textbf{Algorithm mp\_dr\_is\_modulus.} -This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are -in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to -step 3 then $n$ must be of Diminished Radix form. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{Unrestricted Diminished Radix Reduction} -The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm -is a straightforward adaptation of algorithm~\ref{fig:DR}. - -In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new -algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k}. \\ -\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\ -\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\ -\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. While $a \ge n$ do \\ -\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\ -\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\ -\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\ -\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\ -\hspace{3mm}2.5 If $a \ge n$ then do \\ -\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k.} -This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right -shift which makes the algorithm fairly inexpensive to use. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d -on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size -is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without -any multiplications. - -The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are -positive. By using the unsigned versions the overhead is kept to a minimum. - -\subsubsection{Unrestricted Setup} -To setup this reduction algorithm the value of $k = 2^p - n$ is required. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $k = 2^p - n$ \\ -\hline -1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\ -3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\ -4. $k \leftarrow x_0$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_2k\_setup} -\end{figure} - -\textbf{Algorithm mp\_reduce\_2k\_setup.} -This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction -is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Unrestricted Detection} -An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true. - -\begin{enumerate} -\item The number has only one digit. -\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one. -\end{enumerate} - -If either condition is true than there is a power of two $2^p$ such that $0 < 2^p - n < \beta$. If the input is only -one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact -that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most -significant bit. The resulting sum will be a power of two. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\ -\textbf{Input}. mp\_int $n$ \\ -\textbf{Output}. $1$ if of proper form, $0$ otherwise \\ -\hline -1. If $n.used = 0$ then return($0$). \\ -2. If $n.used = 1$ then return($1$). \\ -3. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\ -4. for $x$ from $lg(\beta)$ to $p$ do \\ -\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\ -5. Return($1$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_reduce\_is\_2k} -\end{figure} - -\textbf{Algorithm mp\_reduce\_is\_2k.} -This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - - - -\section{Algorithm Comparison} -So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses -that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since -all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table. - -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\ -\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\ -\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\ -\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\ -\hline -\end{tabular} -\end{small} -\end{center} - -In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery -reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of -calling the half precision multipliers, addition and division by $\beta$ algorithms. - -For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly -shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms -primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in -modular exponentiation to greatly speed up the operation. - - - -\section*{Exercises} -\begin{tabular}{cl} -$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\ - & calculates the correct value of $\rho$. \\ - & \\ -$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\ - & \\ -$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\ - & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\ - & terminate within $1 \le k \le 10$ iterations. \\ - & \\ -\end{tabular} - - -\chapter{Exponentiation} -Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed -in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key -cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any -such cryptosystem and many methods have been sought to speed it up. - -\section{Exponentiation Basics} -A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size -the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature -with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long. - -Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which -are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least -significant bit. If $b$ is a $k$-bit integer than the following equation is true. - -\begin{equation} -a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i} -\end{equation} - -By taking the base $a$ logarithm of both sides of the equation the following equation is the result. - -\begin{equation} -b = \sum_{i=0}^{k-1}2^i \cdot b_i -\end{equation} - -The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to -$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average -$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times. - -While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to -be computed in an auxilary variable. Consider the following equivalent algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Left to Right Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$ and $k$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $k - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Left to Right Exponentiation} -\label{fig:LTOR} -\end{figure} - -This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is -multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the -product. - -For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm. - -\newpage\begin{figure} -\begin{center} -\begin{tabular}{|c|c|} -\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\ -\hline - & $1$ \\ -\hline $5$ & $a$ \\ -\hline $4$ & $a^2$ \\ -\hline $3$ & $a^4 \cdot a$ \\ -\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\ -\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\ -\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Left to Right Exponentiation} -\end{figure} - -When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is -called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature. - -\subsection{Single Digit Exponentiation} -The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended -to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of -$b$ that are greater than three. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_expt\_d}. \\ -\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\ -2. $c \leftarrow 1$ (\textit{mp\_set}) \\ -3. for $x$ from 1 to $lg(\beta)$ do \\ -\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\ -\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\ -\hspace{3mm}3.3 $b \leftarrow b << 1$ \\ -4. Clear $g$. \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_expt\_d} -\end{figure} - -\textbf{Algorithm mp\_expt\_d.} -This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to -quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the -exponent is a fixed width. - -A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of -$1$ in the subsequent step. - -Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared -on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value -of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each -iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Line 29 sets the initial value of the result to $1$. Next the loop on line 31 steps through each bit of the exponent starting from -the most significant down towards the least significant. The invariant squaring operation placed on line 33 is performed first. After -the squaring the result $c$ is multiplied by the base $g$ if and only if the most significant bit of the exponent is set. The shift on line -47 moves all of the bits of the exponent upwards towards the most significant location. - -\section{$k$-ary Exponentiation} -When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor -slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to -the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY} -computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a -portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{$k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\ -\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\ -\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{$k$-ary Exponentiation} -\label{fig:KARY} -\end{figure} - -The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been -precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and -$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$. -However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}. - -Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The -original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings -has increased slightly but the number of multiplications has nearly halved. - -\subsection{Optimal Values of $k$} -An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest -approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$ -for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\ -\hline $16$ & $2$ & $27$ & $24$ \\ -\hline $32$ & $3$ & $49$ & $48$ \\ -\hline $64$ & $3$ & $92$ & $96$ \\ -\hline $128$ & $4$ & $175$ & $192$ \\ -\hline $256$ & $4$ & $335$ & $384$ \\ -\hline $512$ & $5$ & $645$ & $768$ \\ -\hline $1024$ & $6$ & $1257$ & $1536$ \\ -\hline $2048$ & $6$ & $2452$ & $3072$ \\ -\hline $4096$ & $7$ & $4808$ & $6144$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for $k$-ary Exponentiation} -\label{fig:OPTK} -\end{figure} - -\subsection{Sliding-Window Exponentiation} -A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially -this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the -algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided. - -Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}. - -\begin{figure}[here] -\begin{center} -\begin{small} -\begin{tabular}{|c|c|c|c|c|c|} -\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\ -\hline $16$ & $3$ & $24$ & $27$ \\ -\hline $32$ & $3$ & $45$ & $49$ \\ -\hline $64$ & $4$ & $87$ & $92$ \\ -\hline $128$ & $4$ & $167$ & $175$ \\ -\hline $256$ & $5$ & $322$ & $335$ \\ -\hline $512$ & $6$ & $628$ & $645$ \\ -\hline $1024$ & $6$ & $1225$ & $1257$ \\ -\hline $2048$ & $7$ & $2403$ & $2452$ \\ -\hline $4096$ & $8$ & $4735$ & $4808$ \\ -\hline -\end{tabular} -\end{small} -\end{center} -\caption{Optimal Values of $k$ for Sliding Window Exponentiation} -\label{fig:OPTK2} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\ -\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\ -\textbf{Output}. $c = a^b$ \\ -\hline \\ -1. $c \leftarrow 1$ \\ -2. for $i$ from $t - 1$ to $0$ do \\ -\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\ -\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\ -\hspace{3mm}2.2 else do \\ -\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\ -\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\ -\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\ -\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\ -3. Return $c$. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Sliding Window $k$-ary Exponentiation} -\end{figure} - -Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this -algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half -the size as the previous table. - -Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as -the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the -exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where -a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$ -squarings. The second method requires $8$ multiplications and $18$ squarings. - -In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster. - -\section{Modular Exponentiation} - -Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing -$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it -modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation. - -This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using -one of the algorithms presented in chapter six. - -Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm -will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The -value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see \ref{sec;modinv}}). If no inverse exists the algorithm -terminates with an error. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. If $b.sign = MP\_NEG$ then \\ -\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\ -\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\ -\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\ -3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\ -\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\ -4. else \\ -\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_exptmod} -\end{figure} - -\textbf{Algorithm mp\_exptmod.} -The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm -which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation -except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation -algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}). - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input. If the exponent is -negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned -the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive -exponent. - -If the exponent is positive the algorithm resumes the exponentiation. Line 77 determines if the modulus is of the restricted Diminished Radix -form. If it is not line 70 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one -of three values. - -\begin{enumerate} -\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form. -\item $dr = 1$ means that the modulus is of restricted Diminished Radix form. -\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form. -\end{enumerate} - -Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise, -the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction. - -\subsection{Barrett Modular Exponentiation} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod}. \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -1. $k \leftarrow lg(x)$ \\ -2. $winsize \leftarrow \left \lbrace \begin{array}{ll} - 2 & \mbox{if }k \le 7 \\ - 3 & \mbox{if }7 < k \le 36 \\ - 4 & \mbox{if }36 < k \le 140 \\ - 5 & \mbox{if }140 < k \le 450 \\ - 6 & \mbox{if }450 < k \le 1303 \\ - 7 & \mbox{if }1303 < k \le 3529 \\ - 8 & \mbox{if }3529 < k \\ - \end{array} \right .$ \\ -3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\ -4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\ -5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\ -\\ -Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\ -6. $k \leftarrow 2^{winsize - 1}$ \\ -7. $M_{k} \leftarrow M_1$ \\ -8. for $ix$ from 0 to $winsize - 2$ do \\ -\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\ -\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\ -\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\ -\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\ -10. $res \leftarrow 1$ \\ -\\ -Start Sliding Window. \\ -11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\ -12. Loop \\ -\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\ -\hspace{3mm}12.2 If $bitcnt = 0$ then do \\ -\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\ -\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\ -\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\ -\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\ -Continued on next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\ -\textbf{Input}. mp\_int $a$, $b$ and $c$ \\ -\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\ -\hline \\ -\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\ -\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\ -\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\ -\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\ -\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\ -\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.6.3 Goto step 12. \\ -\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\ -\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\ -\hspace{3mm}12.9 $mode \leftarrow 2$ \\ -\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\ -\hspace{6mm}Window is full so perform the squarings and single multiplication. \\ -\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\ -\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\ -\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\ -\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}Reset the window. \\ -\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\ -\\ -No more windows left. Check for residual bits of exponent. \\ -13. If $mode = 2$ and $bitcpy > 0$ then do \\ -\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\ -\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\ -\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\ -\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\ -\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\ -\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\ -14. $y \leftarrow res$ \\ -15. Clear $res$, $mu$ and the $M$ array. \\ -16. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm s\_mp\_exptmod (continued)} -\end{figure} - -\textbf{Algorithm s\_mp\_exptmod.} -This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction -algorithm to keep the product small throughout the algorithm. - -The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the -larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This -table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$. - -After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make -the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$ -times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$. - -Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window. -\begin{enumerate} -\item The variable $mode$ dictates how the bits of the exponent are interpreted. -\begin{enumerate} - \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply - $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found. - \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits - are read and a single squaring is performed. If a non-zero bit is read a new window is created. - \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit - downwards. -\end{enumerate} -\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit - is fetched from the exponent. -\item The variable $buf$ holds the currently read digit of the exponent. -\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit. -\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and - the appropriate operations performed. -\item The variable $bitbuf$ holds the current bits of the window being formed. -\end{enumerate} - -All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step -inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is -read and if there are no digits left than the loop terminates. - -After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit -upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to -trailing edges the entire exponent is read from most significant bit to least significant bit. - -At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the -algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle -the two cases of $mode = 1$ and $mode = 2$ respectively. - -\begin{center} -\begin{figure}[here] -\includegraphics{pics/expt_state.ps} -\caption{Sliding Window State Diagram} -\label{pic:expt_state} -\end{figure} -\end{center} - -By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then -a Left-to-Right algorithm is used to process the remaining few bits. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Lines 32 through 46 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted -from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement -on line 38 the value of $x$ is already known to be greater than $140$. - -The conditional piece of code beginning on line 48 allows the window size to be restricted to five bits. This logic is used to ensure -the table of precomputed powers of $G$ remains relatively small. - -The for loop on line 61 initializes the $M$ array while lines 72 and 77 through 86 initialize the reduction -function that will be used for this modulus. - --- More later. - -\section{Quick Power of Two} -Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is -equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_2expt}. \\ -\textbf{Input}. integer $b$ \\ -\textbf{Output}. $a \leftarrow 2^b$ \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\ -3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\ -4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\ -5. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_2expt} -\end{figure} - -\textbf{Algorithm mp\_2expt.} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\chapter{Higher Level Algorithms} - -This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These -routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important. - -The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic -for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations. -These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate -various representations of integers. For example, converting from an mp\_int to a string of character. - -\section{Integer Division with Remainder} -\label{sec:division} - -Integer division aside from modular exponentiation is the most intensive algorithm to compute. Like addition, subtraction and multiplication -the basis of this algorithm is the long-hand division algorithm taught to school children. Throughout this discussion several common variables -will be used. Let $x$ represent the divisor and $y$ represent the dividend. Let $q$ represent the integer quotient $\lfloor y / x \rfloor$ and -let $r$ represent the remainder $r = y - x \lfloor y / x \rfloor$. The following simple algorithm will be used to start the discussion. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Radix-$\beta$ Integer Division}. \\ -\textbf{Input}. integer $x$ and $y$ \\ -\textbf{Output}. $q = \lfloor y/x\rfloor, r = y - xq$ \\ -\hline \\ -1. $q \leftarrow 0$ \\ -2. $n \leftarrow \vert \vert y \vert \vert - \vert \vert x \vert \vert$ \\ -3. for $t$ from $n$ down to $0$ do \\ -\hspace{3mm}3.1 Maximize $k$ such that $kx\beta^t$ is less than or equal to $y$ and $(k + 1)x\beta^t$ is greater. \\ -\hspace{3mm}3.2 $q \leftarrow q + k\beta^t$ \\ -\hspace{3mm}3.3 $y \leftarrow y - kx\beta^t$ \\ -4. $r \leftarrow y$ \\ -5. Return($q, r$) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Radix-$\beta$ Integer Division} -\label{fig:raddiv} -\end{figure} - -As children we are taught this very simple algorithm for the case of $\beta = 10$. Almost instinctively several optimizations are taught for which -their reason of existing are never explained. For this example let $y = 5471$ represent the dividend and $x = 23$ represent the divisor. - -To find the first digit of the quotient the value of $k$ must be maximized such that $kx\beta^t$ is less than or equal to $y$ and -simultaneously $(k + 1)x\beta^t$ is greater than $y$. Implicitly $k$ is the maximum value the $t$'th digit of the quotient may have. The habitual method -used to find the maximum is to ``eyeball'' the two numbers, typically only the leading digits and quickly estimate a quotient. By only using leading -digits a much simpler division may be used to form an educated guess at what the value must be. In this case $k = \lfloor 54/23\rfloor = 2$ quickly -arises as a possible solution. Indeed $2x\beta^2 = 4600$ is less than $y = 5471$ and simultaneously $(k + 1)x\beta^2 = 6900$ is larger than $y$. -As a result $k\beta^2$ is added to the quotient which now equals $q = 200$ and $4600$ is subtracted from $y$ to give a remainder of $y = 841$. - -Again this process is repeated to produce the quotient digit $k = 3$ which makes the quotient $q = 200 + 3\beta = 230$ and the remainder -$y = 841 - 3x\beta = 181$. Finally the last iteration of the loop produces $k = 7$ which leads to the quotient $q = 230 + 7 = 237$ and the -remainder $y = 181 - 7x = 20$. The final quotient and remainder found are $q = 237$ and $r = y = 20$ which are indeed correct since -$237 \cdot 23 + 20 = 5471$ is true. - -\subsection{Quotient Estimation} -\label{sec:divest} -As alluded to earlier the quotient digit $k$ can be estimated from only the leading digits of both the divisor and dividend. When $p$ leading -digits are used from both the divisor and dividend to form an estimation the accuracy of the estimation rises as $p$ grows. Technically -speaking the estimation is based on assuming the lower $\vert \vert y \vert \vert - p$ and $\vert \vert x \vert \vert - p$ lower digits of the -dividend and divisor are zero. - -The value of the estimation may off by a few values in either direction and in general is fairly correct. A simplification \cite[pp. 271]{TAOCPV2} -of the estimation technique is to use $t + 1$ digits of the dividend and $t$ digits of the divisor, in particularly when $t = 1$. The estimate -using this technique is never too small. For the following proof let $t = \vert \vert y \vert \vert - 1$ and $s = \vert \vert x \vert \vert - 1$ -represent the most significant digits of the dividend and divisor respectively. - -\textbf{Proof.}\textit{ The quotient $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ is greater than or equal to -$k = \lfloor y / (x \cdot \beta^{\vert \vert y \vert \vert - \vert \vert x \vert \vert - 1}) \rfloor$. } -The first obvious case is when $\hat k = \beta - 1$ in which case the proof is concluded since the real quotient cannot be larger. For all other -cases $\hat k = \lfloor (y_t\beta + y_{t-1}) / x_s \rfloor$ and $\hat k x_s \ge y_t\beta + y_{t-1} - x_s + 1$. The latter portion of the inequalility -$-x_s + 1$ arises from the fact that a truncated integer division will give the same quotient for at most $x_s - 1$ values. Next a series of -inequalities will prove the hypothesis. - -\begin{equation} -y - \hat k x \le y - \hat k x_s\beta^s -\end{equation} - -This is trivially true since $x \ge x_s\beta^s$. Next we replace $\hat kx_s\beta^s$ by the previous inequality for $\hat kx_s$. - -\begin{equation} -y - \hat k x \le y_t\beta^t + \ldots + y_0 - (y_t\beta^t + y_{t-1}\beta^{t-1} - x_s\beta^t + \beta^s) -\end{equation} - -By simplifying the previous inequality the following inequality is formed. - -\begin{equation} -y - \hat k x \le y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s -\end{equation} - -Subsequently, - -\begin{equation} -y_{t-2}\beta^{t-2} + \ldots + y_0 + x_s\beta^s - \beta^s < x_s\beta^s \le x -\end{equation} - -Which proves that $y - \hat kx \le x$ and by consequence $\hat k \ge k$ which concludes the proof. \textbf{QED} - - -\subsection{Normalized Integers} -For the purposes of division a normalized input is when the divisors leading digit $x_n$ is greater than or equal to $\beta / 2$. By multiplying both -$x$ and $y$ by $j = \lfloor (\beta / 2) / x_n \rfloor$ the quotient remains unchanged and the remainder is simply $j$ times the original -remainder. The purpose of normalization is to ensure the leading digit of the divisor is sufficiently large such that the estimated quotient will -lie in the domain of a single digit. Consider the maximum dividend $(\beta - 1) \cdot \beta + (\beta - 1)$ and the minimum divisor $\beta / 2$. - -\begin{equation} -{{\beta^2 - 1} \over { \beta / 2}} \le 2\beta - {2 \over \beta} -\end{equation} - -At most the quotient approaches $2\beta$, however, in practice this will not occur since that would imply the previous quotient digit was too small. - -\subsection{Radix-$\beta$ Division with Remainder} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div}. \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -1. If $b = 0$ return(\textit{MP\_VAL}). \\ -2. If $\vert a \vert < \vert b \vert$ then do \\ -\hspace{3mm}2.1 $d \leftarrow a$ \\ -\hspace{3mm}2.2 $c \leftarrow 0$ \\ -\hspace{3mm}2.3 Return(\textit{MP\_OKAY}). \\ -\\ -Setup the quotient to receive the digits. \\ -3. Grow $q$ to $a.used + 2$ digits. \\ -4. $q \leftarrow 0$ \\ -5. $x \leftarrow \vert a \vert , y \leftarrow \vert b \vert$ \\ -6. $sign \leftarrow \left \lbrace \begin{array}{ll} - MP\_ZPOS & \mbox{if }a.sign = b.sign \\ - MP\_NEG & \mbox{otherwise} \\ - \end{array} \right .$ \\ -\\ -Normalize the inputs such that the leading digit of $y$ is greater than or equal to $\beta / 2$. \\ -7. $norm \leftarrow (lg(\beta) - 1) - (\lceil lg(y) \rceil \mbox{ (mod }lg(\beta)\mbox{)})$ \\ -8. $x \leftarrow x \cdot 2^{norm}, y \leftarrow y \cdot 2^{norm}$ \\ -\\ -Find the leading digit of the quotient. \\ -9. $n \leftarrow x.used - 1, t \leftarrow y.used - 1$ \\ -10. $y \leftarrow y \cdot \beta^{n - t}$ \\ -11. While ($x \ge y$) do \\ -\hspace{3mm}11.1 $q_{n - t} \leftarrow q_{n - t} + 1$ \\ -\hspace{3mm}11.2 $x \leftarrow x - y$ \\ -12. $y \leftarrow \lfloor y / \beta^{n-t} \rfloor$ \\ -\\ -Continued on the next page. \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div} (continued). \\ -\textbf{Input}. mp\_int $a, b$ \\ -\textbf{Output}. $c = \lfloor a/b \rfloor$, $d = a - bc$ \\ -\hline \\ -Now find the remainder fo the digits. \\ -13. for $i$ from $n$ down to $(t + 1)$ do \\ -\hspace{3mm}13.1 If $i > x.used$ then jump to the next iteration of this loop. \\ -\hspace{3mm}13.2 If $x_{i} = y_{t}$ then \\ -\hspace{6mm}13.2.1 $q_{i - t - 1} \leftarrow \beta - 1$ \\ -\hspace{3mm}13.3 else \\ -\hspace{6mm}13.3.1 $\hat r \leftarrow x_{i} \cdot \beta + x_{i - 1}$ \\ -\hspace{6mm}13.3.2 $\hat r \leftarrow \lfloor \hat r / y_{t} \rfloor$ \\ -\hspace{6mm}13.3.3 $q_{i - t - 1} \leftarrow \hat r$ \\ -\hspace{3mm}13.4 $q_{i - t - 1} \leftarrow q_{i - t - 1} + 1$ \\ -\\ -Fixup quotient estimation. \\ -\hspace{3mm}13.5 Loop \\ -\hspace{6mm}13.5.1 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\hspace{6mm}13.5.2 t$1 \leftarrow 0$ \\ -\hspace{6mm}13.5.3 t$1_0 \leftarrow y_{t - 1}, $ t$1_1 \leftarrow y_t,$ t$1.used \leftarrow 2$ \\ -\hspace{6mm}13.5.4 $t1 \leftarrow t1 \cdot q_{i - t - 1}$ \\ -\hspace{6mm}13.5.5 t$2_0 \leftarrow x_{i - 2}, $ t$2_1 \leftarrow x_{i - 1}, $ t$2_2 \leftarrow x_i, $ t$2.used \leftarrow 3$ \\ -\hspace{6mm}13.5.6 If $\vert t1 \vert > \vert t2 \vert$ then goto step 13.5. \\ -\hspace{3mm}13.6 t$1 \leftarrow y \cdot q_{i - t - 1}$ \\ -\hspace{3mm}13.7 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{3mm}13.8 $x \leftarrow x - $ t$1$ \\ -\hspace{3mm}13.9 If $x.sign = MP\_NEG$ then \\ -\hspace{6mm}13.10 t$1 \leftarrow y$ \\ -\hspace{6mm}13.11 t$1 \leftarrow $ t$1 \cdot \beta^{i - t - 1}$ \\ -\hspace{6mm}13.12 $x \leftarrow x + $ t$1$ \\ -\hspace{6mm}13.13 $q_{i - t - 1} \leftarrow q_{i - t - 1} - 1$ \\ -\\ -Finalize the result. \\ -14. Clamp excess digits of $q$ \\ -15. $c \leftarrow q, c.sign \leftarrow sign$ \\ -16. $x.sign \leftarrow a.sign$ \\ -17. $d \leftarrow \lfloor x / 2^{norm} \rfloor$ \\ -18. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div (continued)} -\end{figure} -\textbf{Algorithm mp\_div.} -This algorithm will calculate quotient and remainder from an integer division given a dividend and divisor. The algorithm is a signed -division and will produce a fully qualified quotient and remainder. - -First the divisor $b$ must be non-zero which is enforced in step one. If the divisor is larger than the dividend than the quotient is implicitly -zero and the remainder is the dividend. - -After the first two trivial cases of inputs are handled the variable $q$ is setup to receive the digits of the quotient. Two unsigned copies of the -divisor $y$ and dividend $x$ are made as well. The core of the division algorithm is an unsigned division and will only work if the values are -positive. Now the two values $x$ and $y$ must be normalized such that the leading digit of $y$ is greater than or equal to $\beta / 2$. -This is performed by shifting both to the left by enough bits to get the desired normalization. - -At this point the division algorithm can begin producing digits of the quotient. Recall that maximum value of the estimation used is -$2\beta - {2 \over \beta}$ which means that a digit of the quotient must be first produced by another means. In this case $y$ is shifted -to the left (\textit{step ten}) so that it has the same number of digits as $x$. The loop on step eleven will subtract multiples of the -shifted copy of $y$ until $x$ is smaller. Since the leading digit of $y$ is greater than or equal to $\beta/2$ this loop will iterate at most two -times to produce the desired leading digit of the quotient. - -Now the remainder of the digits can be produced. The equation $\hat q = \lfloor {{x_i \beta + x_{i-1}}\over y_t} \rfloor$ is used to fairly -accurately approximate the true quotient digit. The estimation can in theory produce an estimation as high as $2\beta - {2 \over \beta}$ but by -induction the upper quotient digit is correct (\textit{as established on step eleven}) and the estimate must be less than $\beta$. - -Recall from section~\ref{sec:divest} that the estimation is never too low but may be too high. The next step of the estimation process is -to refine the estimation. The loop on step 13.5 uses $x_i\beta^2 + x_{i-1}\beta + x_{i-2}$ and $q_{i - t - 1}(y_t\beta + y_{t-1})$ as a higher -order approximation to adjust the quotient digit. - -After both phases of estimation the quotient digit may still be off by a value of one\footnote{This is similar to the error introduced -by optimizing Barrett reduction.}. Steps 13.6 and 13.7 subtract the multiple of the divisor from the dividend (\textit{Similar to step 3.3 of -algorithm~\ref{fig:raddiv}} and then subsequently add a multiple of the divisor if the quotient was too large. - -Now that the quotient has been determine finializing the result is a matter of clamping the quotient, fixing the sizes and de-normalizing the -remainder. An important aspect of this algorithm seemingly overlooked in other descriptions such as that of Algorithm 14.20 HAC \cite[pp. 598]{HAC} -is that when the estimations are being made (\textit{inside the loop on step 13.5}) that the digits $y_{t-1}$, $x_{i-2}$ and $x_{i-1}$ may lie -outside their respective boundaries. For example, if $t = 0$ or $i \le 1$ then the digits would be undefined. In those cases the digits should -respectively be replaced with a zero. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The implementation of this algorithm differs slightly from the pseudo code presented previously. In this algorithm either of the quotient $c$ or -remainder $d$ may be passed as a \textbf{NULL} pointer which indicates their value is not desired. For example, the C code to call the division -algorithm with only the quotient is - -\begin{verbatim} -mp_div(&a, &b, &c, NULL); /* c = [a/b] */ -\end{verbatim} - -Lines 109 and 113 handle the two trivial cases of inputs which are division by zero and dividend smaller than the divisor -respectively. After the two trivial cases all of the temporary variables are initialized. Line 148 determines the sign of -the quotient and line 148 ensures that both $x$ and $y$ are positive. - -The number of bits in the leading digit is calculated on line 151. Implictly an mp\_int with $r$ digits will require $lg(\beta)(r-1) + k$ bits -of precision which when reduced modulo $lg(\beta)$ produces the value of $k$. In this case $k$ is the number of bits in the leading digit which is -exactly what is required. For the algorithm to operate $k$ must equal $lg(\beta) - 1$ and when it does not the inputs must be normalized by shifting -them to the left by $lg(\beta) - 1 - k$ bits. - -Throughout the variables $n$ and $t$ will represent the highest digit of $x$ and $y$ respectively. These are first used to produce the -leading digit of the quotient. The loop beginning on line 184 will produce the remainder of the quotient digits. - -The conditional ``continue'' on line 187 is used to prevent the algorithm from reading past the leading edge of $x$ which can occur when the -algorithm eliminates multiple non-zero digits in a single iteration. This ensures that $x_i$ is always non-zero since by definition the digits -above the $i$'th position $x$ must be zero in order for the quotient to be precise\footnote{Precise as far as integer division is concerned.}. - -Lines 214, 216 and 223 through 225 manually construct the high accuracy estimations by setting the digits of the two mp\_int -variables directly. - -\section{Single Digit Helpers} - -This section briefly describes a series of single digit helper algorithms which come in handy when working with small constants. All of -the helper functions assume the single digit input is positive and will treat them as such. - -\subsection{Single Digit Addition and Subtraction} - -Both addition and subtraction are performed by ``cheating'' and using mp\_set followed by the higher level addition or subtraction -algorithms. As a result these algorithms are subtantially simpler with a slight cost in performance. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_add\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = a + b$ \\ -\hline \\ -1. $t \leftarrow b$ (\textit{mp\_set}) \\ -2. $c \leftarrow a + t$ \\ -3. Return(\textit{MP\_OKAY}) \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_add\_d} -\end{figure} - -\textbf{Algorithm mp\_add\_d.} -This algorithm initiates a temporary mp\_int with the value of the single digit and uses algorithm mp\_add to add the two values together. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_add\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Clever use of the letter 't'. - -\subsubsection{Subtraction} -The single digit subtraction algorithm mp\_sub\_d is essentially the same except it uses mp\_sub to subtract the digit from the mp\_int. - -\subsection{Single Digit Multiplication} -Single digit multiplication arises enough in division and radix conversion that it ought to be implement as a special case of the baseline -multiplication algorithm. Essentially this algorithm is a modified version of algorithm s\_mp\_mul\_digs where one of the multiplicands -only has one digit. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_mul\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = ab$ \\ -\hline \\ -1. $pa \leftarrow a.used$ \\ -2. Grow $c$ to at least $pa + 1$ digits. \\ -3. $oldused \leftarrow c.used$ \\ -4. $c.used \leftarrow pa + 1$ \\ -5. $c.sign \leftarrow a.sign$ \\ -6. $\mu \leftarrow 0$ \\ -7. for $ix$ from $0$ to $pa - 1$ do \\ -\hspace{3mm}7.1 $\hat r \leftarrow \mu + a_{ix}b$ \\ -\hspace{3mm}7.2 $c_{ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\ -\hspace{3mm}7.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\ -8. $c_{pa} \leftarrow \mu$ \\ -9. for $ix$ from $pa + 1$ to $oldused$ do \\ -\hspace{3mm}9.1 $c_{ix} \leftarrow 0$ \\ -10. Clamp excess digits of $c$. \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_mul\_d} -\end{figure} -\textbf{Algorithm mp\_mul\_d.} -This algorithm quickly multiplies an mp\_int by a small single digit value. It is specially tailored to the job and has a minimal of overhead. -Unlike the full multiplication algorithms this algorithm does not require any significnat temporary storage or memory allocations. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -In this implementation the destination $c$ may point to the same mp\_int as the source $a$ since the result is written after the digit is -read from the source. This function uses pointer aliases $tmpa$ and $tmpc$ for the digits of $a$ and $c$ respectively. - -\subsection{Single Digit Division} -Like the single digit multiplication algorithm, single digit division is also a fairly common algorithm used in radix conversion. Since the -divisor is only a single digit a specialized variant of the division algorithm can be used to compute the quotient. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_div\_d}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c = \lfloor a / b \rfloor, d = a - cb$ \\ -\hline \\ -1. If $b = 0$ then return(\textit{MP\_VAL}).\\ -2. If $b = 3$ then use algorithm mp\_div\_3 instead. \\ -3. Init $q$ to $a.used$ digits. \\ -4. $q.used \leftarrow a.used$ \\ -5. $q.sign \leftarrow a.sign$ \\ -6. $\hat w \leftarrow 0$ \\ -7. for $ix$ from $a.used - 1$ down to $0$ do \\ -\hspace{3mm}7.1 $\hat w \leftarrow \hat w \beta + a_{ix}$ \\ -\hspace{3mm}7.2 If $\hat w \ge b$ then \\ -\hspace{6mm}7.2.1 $t \leftarrow \lfloor \hat w / b \rfloor$ \\ -\hspace{6mm}7.2.2 $\hat w \leftarrow \hat w \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}7.3 else\\ -\hspace{6mm}7.3.1 $t \leftarrow 0$ \\ -\hspace{3mm}7.4 $q_{ix} \leftarrow t$ \\ -8. $d \leftarrow \hat w$ \\ -9. Clamp excess digits of $q$. \\ -10. $c \leftarrow q$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_div\_d} -\end{figure} -\textbf{Algorithm mp\_div\_d.} -This algorithm divides the mp\_int $a$ by the single mp\_digit $b$ using an optimized approach. Essentially in every iteration of the -algorithm another digit of the dividend is reduced and another digit of quotient produced. Provided $b < \beta$ the value of $\hat w$ -after step 7.1 will be limited such that $0 \le \lfloor \hat w / b \rfloor < \beta$. - -If the divisor $b$ is equal to three a variant of this algorithm is used which is called mp\_div\_3. It replaces the division by three with -a multiplication by $\lfloor \beta / 3 \rfloor$ and the appropriate shift and residual fixup. In essence it is much like the Barrett reduction -from chapter seven. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_d.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Like the implementation of algorithm mp\_div this algorithm allows either of the quotient or remainder to be passed as a \textbf{NULL} pointer to -indicate the respective value is not required. This allows a trivial single digit modular reduction algorithm, mp\_mod\_d to be created. - -The division and remainder on lines 44 and @45,%@ can be replaced often by a single division on most processors. For example, the 32-bit x86 based -processors can divide a 64-bit quantity by a 32-bit quantity and produce the quotient and remainder simultaneously. Unfortunately the GCC -compiler does not recognize that optimization and will actually produce two function calls to find the quotient and remainder respectively. - -\subsection{Single Digit Root Extraction} - -Finding the $n$'th root of an integer is fairly easy as far as numerical analysis is concerned. Algorithms such as the Newton-Raphson approximation -(\ref{eqn:newton}) series will converge very quickly to a root for any continuous function $f(x)$. - -\begin{equation} -x_{i+1} = x_i - {f(x_i) \over f'(x_i)} -\label{eqn:newton} -\end{equation} - -In this case the $n$'th root is desired and $f(x) = x^n - a$ where $a$ is the integer of which the root is desired. The derivative of $f(x)$ is -simply $f'(x) = nx^{n - 1}$. Of particular importance is that this algorithm will be used over the integers not over the a more continuous domain -such as the real numbers. As a result the root found can be above the true root by few and must be manually adjusted. Ideally at the end of the -algorithm the $n$'th root $b$ of an integer $a$ is desired such that $b^n \le a$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_n\_root}. \\ -\textbf{Input}. mp\_int $a$ and a mp\_digit $b$ \\ -\textbf{Output}. $c^b \le a$ \\ -\hline \\ -1. If $b$ is even and $a.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\ -2. $sign \leftarrow a.sign$ \\ -3. $a.sign \leftarrow MP\_ZPOS$ \\ -4. t$2 \leftarrow 2$ \\ -5. Loop \\ -\hspace{3mm}5.1 t$1 \leftarrow $ t$2$ \\ -\hspace{3mm}5.2 t$3 \leftarrow $ t$1^{b - 1}$ \\ -\hspace{3mm}5.3 t$2 \leftarrow $ t$3 $ $\cdot$ t$1$ \\ -\hspace{3mm}5.4 t$2 \leftarrow $ t$2 - a$ \\ -\hspace{3mm}5.5 t$3 \leftarrow $ t$3 \cdot b$ \\ -\hspace{3mm}5.6 t$3 \leftarrow \lfloor $t$2 / $t$3 \rfloor$ \\ -\hspace{3mm}5.7 t$2 \leftarrow $ t$1 - $ t$3$ \\ -\hspace{3mm}5.8 If t$1 \ne $ t$2$ then goto step 5. \\ -6. Loop \\ -\hspace{3mm}6.1 t$2 \leftarrow $ t$1^b$ \\ -\hspace{3mm}6.2 If t$2 > a$ then \\ -\hspace{6mm}6.2.1 t$1 \leftarrow $ t$1 - 1$ \\ -\hspace{6mm}6.2.2 Goto step 6. \\ -7. $a.sign \leftarrow sign$ \\ -8. $c \leftarrow $ t$1$ \\ -9. $c.sign \leftarrow sign$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_n\_root} -\end{figure} -\textbf{Algorithm mp\_n\_root.} -This algorithm finds the integer $n$'th root of an input using the Newton-Raphson approach. It is partially optimized based on the observation -that the numerator of ${f(x) \over f'(x)}$ can be derived from a partial denominator. That is at first the denominator is calculated by finding -$x^{b - 1}$. This value can then be multiplied by $x$ and have $a$ subtracted from it to find the numerator. This saves a total of $b - 1$ -multiplications by t$1$ inside the loop. - -The initial value of the approximation is t$2 = 2$ which allows the algorithm to start with very small values and quickly converge on the -root. Ideally this algorithm is meant to find the $n$'th root of an input where $n$ is bounded by $2 \le n \le 5$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_n\_root.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Random Number Generation} - -Random numbers come up in a variety of activities from public key cryptography to simple simulations and various randomized algorithms. Pollard-Rho -factoring for example, can make use of random values as starting points to find factors of a composite integer. In this case the algorithm presented -is solely for simulations and not intended for cryptographic use. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_rand}. \\ -\textbf{Input}. An integer $b$ \\ -\textbf{Output}. A pseudo-random number of $b$ digits \\ -\hline \\ -1. $a \leftarrow 0$ \\ -2. If $b \le 0$ return(\textit{MP\_OKAY}) \\ -3. Pick a non-zero random digit $d$. \\ -4. $a \leftarrow a + d$ \\ -5. for $ix$ from 1 to $d - 1$ do \\ -\hspace{3mm}5.1 $a \leftarrow a \cdot \beta$ \\ -\hspace{3mm}5.2 Pick a random digit $d$. \\ -\hspace{3mm}5.3 $a \leftarrow a + d$ \\ -6. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_rand} -\end{figure} -\textbf{Algorithm mp\_rand.} -This algorithm produces a pseudo-random integer of $b$ digits. By ensuring that the first digit is non-zero the algorithm also guarantees that the -final result has at least $b$ digits. It relies heavily on a third-part random number generator which should ideally generate uniformly all of -the integers from $0$ to $\beta - 1$. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_rand.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Formatted Representations} -The ability to emit a radix-$n$ textual representation of an integer is useful for interacting with human parties. For example, the ability to -be given a string of characters such as ``114585'' and turn it into the radix-$\beta$ equivalent would make it easier to enter numbers -into a program. - -\subsection{Reading Radix-n Input} -For the purposes of this text we will assume that a simple lower ASCII map (\ref{fig:ASC}) is used for the values of from $0$ to $63$ to -printable characters. For example, when the character ``N'' is read it represents the integer $23$. The first $16$ characters of the -map are for the common representations up to hexadecimal. After that they match the ``base64'' encoding scheme which are suitable chosen -such that they are printable. While outputting as base64 may not be too helpful for human operators it does allow communication via non binary -mediums. - -\newpage\begin{figure}[here] -\begin{center} -\begin{tabular}{cc|cc|cc|cc} -\hline \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} & \textbf{Value} & \textbf{Char} \\ -\hline -0 & 0 & 1 & 1 & 2 & 2 & 3 & 3 \\ -4 & 4 & 5 & 5 & 6 & 6 & 7 & 7 \\ -8 & 8 & 9 & 9 & 10 & A & 11 & B \\ -12 & C & 13 & D & 14 & E & 15 & F \\ -16 & G & 17 & H & 18 & I & 19 & J \\ -20 & K & 21 & L & 22 & M & 23 & N \\ -24 & O & 25 & P & 26 & Q & 27 & R \\ -28 & S & 29 & T & 30 & U & 31 & V \\ -32 & W & 33 & X & 34 & Y & 35 & Z \\ -36 & a & 37 & b & 38 & c & 39 & d \\ -40 & e & 41 & f & 42 & g & 43 & h \\ -44 & i & 45 & j & 46 & k & 47 & l \\ -48 & m & 49 & n & 50 & o & 51 & p \\ -52 & q & 53 & r & 54 & s & 55 & t \\ -56 & u & 57 & v & 58 & w & 59 & x \\ -60 & y & 61 & z & 62 & $+$ & 63 & $/$ \\ -\hline -\end{tabular} -\end{center} -\caption{Lower ASCII Map} -\label{fig:ASC} -\end{figure} - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_read\_radix}. \\ -\textbf{Input}. A string $str$ of length $sn$ and radix $r$. \\ -\textbf{Output}. The radix-$\beta$ equivalent mp\_int. \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. $ix \leftarrow 0$ \\ -3. If $str_0 =$ ``-'' then do \\ -\hspace{3mm}3.1 $ix \leftarrow ix + 1$ \\ -\hspace{3mm}3.2 $sign \leftarrow MP\_NEG$ \\ -4. else \\ -\hspace{3mm}4.1 $sign \leftarrow MP\_ZPOS$ \\ -5. $a \leftarrow 0$ \\ -6. for $iy$ from $ix$ to $sn - 1$ do \\ -\hspace{3mm}6.1 Let $y$ denote the position in the map of $str_{iy}$. \\ -\hspace{3mm}6.2 If $str_{iy}$ is not in the map or $y \ge r$ then goto step 7. \\ -\hspace{3mm}6.3 $a \leftarrow a \cdot r$ \\ -\hspace{3mm}6.4 $a \leftarrow a + y$ \\ -7. If $a \ne 0$ then $a.sign \leftarrow sign$ \\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_read\_radix} -\end{figure} -\textbf{Algorithm mp\_read\_radix.} -This algorithm will read an ASCII string and produce the radix-$\beta$ mp\_int representation of the same integer. A minus symbol ``-'' may precede the -string to indicate the value is negative, otherwise it is assumed to be positive. The algorithm will read up to $sn$ characters from the input -and will stop when it reads a character it cannot map the algorithm stops reading characters from the string. This allows numbers to be embedded -as part of larger input without any significant problem. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_read\_radix.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{Generating Radix-$n$ Output} -Generating radix-$n$ output is fairly trivial with a division and remainder algorithm. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_toradix}. \\ -\textbf{Input}. A mp\_int $a$ and an integer $r$\\ -\textbf{Output}. The radix-$r$ representation of $a$ \\ -\hline \\ -1. If $r < 2$ or $r > 64$ return(\textit{MP\_VAL}). \\ -2. If $a = 0$ then $str = $ ``$0$'' and return(\textit{MP\_OKAY}). \\ -3. $t \leftarrow a$ \\ -4. $str \leftarrow$ ``'' \\ -5. if $t.sign = MP\_NEG$ then \\ -\hspace{3mm}5.1 $str \leftarrow str + $ ``-'' \\ -\hspace{3mm}5.2 $t.sign = MP\_ZPOS$ \\ -6. While ($t \ne 0$) do \\ -\hspace{3mm}6.1 $d \leftarrow t \mbox{ (mod }r\mbox{)}$ \\ -\hspace{3mm}6.2 $t \leftarrow \lfloor t / r \rfloor$ \\ -\hspace{3mm}6.3 Look up $d$ in the map and store the equivalent character in $y$. \\ -\hspace{3mm}6.4 $str \leftarrow str + y$ \\ -7. If $str_0 = $``$-$'' then \\ -\hspace{3mm}7.1 Reverse the digits $str_1, str_2, \ldots str_n$. \\ -8. Otherwise \\ -\hspace{3mm}8.1 Reverse the digits $str_0, str_1, \ldots str_n$. \\ -9. Return(\textit{MP\_OKAY}).\\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_toradix} -\end{figure} -\textbf{Algorithm mp\_toradix.} -This algorithm computes the radix-$r$ representation of an mp\_int $a$. The ``digits'' of the representation are extracted by reducing -successive powers of $\lfloor a / r^k \rfloor$ the input modulo $r$ until $r^k > a$. Note that instead of actually dividing by $r^k$ in -each iteration the quotient $\lfloor a / r \rfloor$ is saved for the next iteration. As a result a series of trivial $n \times 1$ divisions -are required instead of a series of $n \times k$ divisions. One design flaw of this approach is that the digits are produced in the reverse order -(see~\ref{fig:mpradix}). To remedy this flaw the digits must be swapped or simply ``reversed''. - -\begin{figure} -\begin{center} -\begin{tabular}{|c|c|c|} -\hline \textbf{Value of $a$} & \textbf{Value of $d$} & \textbf{Value of $str$} \\ -\hline $1234$ & -- & -- \\ -\hline $123$ & $4$ & ``4'' \\ -\hline $12$ & $3$ & ``43'' \\ -\hline $1$ & $2$ & ``432'' \\ -\hline $0$ & $1$ & ``4321'' \\ -\hline -\end{tabular} -\end{center} -\caption{Example of Algorithm mp\_toradix.} -\label{fig:mpradix} -\end{figure} - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_toradix.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\chapter{Number Theoretic Algorithms} -This chapter discusses several fundamental number theoretic algorithms such as the greatest common divisor, least common multiple and Jacobi -symbol computation. These algorithms arise as essential components in several key cryptographic algorithms such as the RSA public key algorithm and -various Sieve based factoring algorithms. - -\section{Greatest Common Divisor} -The greatest common divisor of two integers $a$ and $b$, often denoted as $(a, b)$ is the largest integer $k$ that is a proper divisor of -both $a$ and $b$. That is, $k$ is the largest integer such that $0 \equiv a \mbox{ (mod }k\mbox{)}$ and $0 \equiv b \mbox{ (mod }k\mbox{)}$ occur -simultaneously. - -The most common approach (cite) is to reduce one input modulo another. That is if $a$ and $b$ are divisible by some integer $k$ and if $qa + r = b$ then -$r$ is also divisible by $k$. The reduction pattern follows $\left < a , b \right > \rightarrow \left < b, a \mbox{ mod } b \right >$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (I)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 $r \leftarrow a \mbox{ (mod }b\mbox{)}$ \\ -\hspace{3mm}1.2 $a \leftarrow b$ \\ -\hspace{3mm}1.3 $b \leftarrow r$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (I)} -\label{fig:gcd1} -\end{figure} - -This algorithm will quickly converge on the greatest common divisor since the residue $r$ tends diminish rapidly. However, divisions are -relatively expensive operations to perform and should ideally be avoided. There is another approach based on a similar relationship of -greatest common divisors. The faster approach is based on the observation that if $k$ divides both $a$ and $b$ it will also divide $a - b$. -In particular, we would like $a - b$ to decrease in magnitude which implies that $b \ge a$. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (II)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. While ($b > 0$) do \\ -\hspace{3mm}1.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}1.2 $b \leftarrow b - a$ \\ -2. Return($a$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (II)} -\label{fig:gcd2} -\end{figure} - -\textbf{Proof} \textit{Algorithm~\ref{fig:gcd2} will return the greatest common divisor of $a$ and $b$.} -The algorithm in figure~\ref{fig:gcd2} will eventually terminate since $b \ge a$ the subtraction in step 1.2 will be a value less than $b$. In other -words in every iteration that tuple $\left < a, b \right >$ decrease in magnitude until eventually $a = b$. Since both $a$ and $b$ are always -divisible by the greatest common divisor (\textit{until the last iteration}) and in the last iteration of the algorithm $b = 0$, therefore, in the -second to last iteration of the algorithm $b = a$ and clearly $(a, a) = a$ which concludes the proof. \textbf{QED}. - -As a matter of practicality algorithm \ref{fig:gcd1} decreases far too slowly to be useful. Specially if $b$ is much larger than $a$ such that -$b - a$ is still very much larger than $a$. A simple addition to the algorithm is to divide $b - a$ by a power of some integer $p$ which does -not divide the greatest common divisor but will divide $b - a$. In this case ${b - a} \over p$ is also an integer and still divisible by -the greatest common divisor. - -However, instead of factoring $b - a$ to find a suitable value of $p$ the powers of $p$ can be removed from $a$ and $b$ that are in common first. -Then inside the loop whenever $b - a$ is divisible by some power of $p$ it can be safely removed. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{Greatest Common Divisor (III)}. \\ -\textbf{Input}. Two positive integers $a$ and $b$ greater than zero. \\ -\textbf{Output}. The greatest common divisor $(a, b)$. \\ -\hline \\ -1. $k \leftarrow 0$ \\ -2. While $a$ and $b$ are both divisible by $p$ do \\ -\hspace{3mm}2.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -\hspace{3mm}2.2 $b \leftarrow \lfloor b / p \rfloor$ \\ -\hspace{3mm}2.3 $k \leftarrow k + 1$ \\ -3. While $a$ is divisible by $p$ do \\ -\hspace{3mm}3.1 $a \leftarrow \lfloor a / p \rfloor$ \\ -4. While $b$ is divisible by $p$ do \\ -\hspace{3mm}4.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -5. While ($b > 0$) do \\ -\hspace{3mm}5.1 Swap $a$ and $b$ such that $a$ is the smallest of the two. \\ -\hspace{3mm}5.2 $b \leftarrow b - a$ \\ -\hspace{3mm}5.3 While $b$ is divisible by $p$ do \\ -\hspace{6mm}5.3.1 $b \leftarrow \lfloor b / p \rfloor$ \\ -6. Return($a \cdot p^k$). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm Greatest Common Divisor (III)} -\label{fig:gcd3} -\end{figure} - -This algorithm is based on the first except it removes powers of $p$ first and inside the main loop to ensure the tuple $\left < a, b \right >$ -decreases more rapidly. The first loop on step two removes powers of $p$ that are in common. A count, $k$, is kept which will present a common -divisor of $p^k$. After step two the remaining common divisor of $a$ and $b$ cannot be divisible by $p$. This means that $p$ can be safely -divided out of the difference $b - a$ so long as the division leaves no remainder. - -In particular the value of $p$ should be chosen such that the division on step 5.3.1 occur often. It also helps that division by $p$ be easy -to compute. The ideal choice of $p$ is two since division by two amounts to a right logical shift. Another important observation is that by -step five both $a$ and $b$ are odd. Therefore, the diffrence $b - a$ must be even which means that each iteration removes one bit from the -largest of the pair. - -\subsection{Complete Greatest Common Divisor} -The algorithms presented so far cannot handle inputs which are zero or negative. The following algorithm can handle all input cases properly -and will produce the greatest common divisor. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_gcd}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The greatest common divisor $c = (a, b)$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow \vert b \vert $ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $b = 0$ then \\ -\hspace{3mm}2.1 $c \leftarrow \vert a \vert $ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $u \leftarrow \vert a \vert, v \leftarrow \vert b \vert$ \\ -4. $k \leftarrow 0$ \\ -5. While $u.used > 0$ and $v.used > 0$ and $u_0 \equiv v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}5.3 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -8. While $v.used > 0$ \\ -\hspace{3mm}8.1 If $\vert u \vert > \vert v \vert$ then \\ -\hspace{6mm}8.1.1 Swap $u$ and $v$. \\ -\hspace{3mm}8.2 $v \leftarrow \vert v \vert - \vert u \vert$ \\ -\hspace{3mm}8.3 While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{6mm}8.3.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -9. $c \leftarrow u \cdot 2^k$ \\ -10. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_gcd} -\end{figure} -\textbf{Algorithm mp\_gcd.} -This algorithm will produce the greatest common divisor of two mp\_ints $a$ and $b$. The algorithm was originally based on Algorithm B of -Knuth \cite[pp. 338]{TAOCPV2} but has been modified to be simpler to explain. In theory it achieves the same asymptotic working time as -Algorithm B and in practice this appears to be true. - -The first two steps handle the cases where either one of or both inputs are zero. If either input is zero the greatest common divisor is the -largest input or zero if they are both zero. If the inputs are not trivial than $u$ and $v$ are assigned the absolute values of -$a$ and $b$ respectively and the algorithm will proceed to reduce the pair. - -Step five will divide out any common factors of two and keep track of the count in the variable $k$. After this step, two is no longer a -factor of the remaining greatest common divisor between $u$ and $v$ and can be safely evenly divided out of either whenever they are even. Step -six and seven ensure that the $u$ and $v$ respectively have no more factors of two. At most only one of the while--loops will iterate since -they cannot both be even. - -By step eight both of $u$ and $v$ are odd which is required for the inner logic. First the pair are swapped such that $v$ is equal to -or greater than $u$. This ensures that the subtraction on step 8.2 will always produce a positive and even result. Step 8.3 removes any -factors of two from the difference $u$ to ensure that in the next iteration of the loop both are once again odd. - -After $v = 0$ occurs the variable $u$ has the greatest common divisor of the pair $\left < u, v \right >$ just after step six. The result -must be adjusted by multiplying by the common factors of two ($2^k$) removed earlier. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_gcd.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -This function makes use of the macros mp\_iszero and mp\_iseven. The former evaluates to $1$ if the input mp\_int is equivalent to the -integer zero otherwise it evaluates to $0$. The latter evaluates to $1$ if the input mp\_int represents a non-zero even integer otherwise -it evaluates to $0$. Note that just because mp\_iseven may evaluate to $0$ does not mean the input is odd, it could also be zero. The three -trivial cases of inputs are handled on lines 24 through 30. After those lines the inputs are assumed to be non-zero. - -Lines 32 and 37 make local copies $u$ and $v$ of the inputs $a$ and $b$ respectively. At this point the common factors of two -must be divided out of the two inputs. The block starting at line 44 removes common factors of two by first counting the number of trailing -zero bits in both. The local integer $k$ is used to keep track of how many factors of $2$ are pulled out of both values. It is assumed that -the number of factors will not exceed the maximum value of a C ``int'' data type\footnote{Strictly speaking no array in C may have more than -entries than are accessible by an ``int'' so this is not a limitation.}. - -At this point there are no more common factors of two in the two values. The divisions by a power of two on lines 62 and 68 remove -any independent factors of two such that both $u$ and $v$ are guaranteed to be an odd integer before hitting the main body of the algorithm. The while loop -on line 73 performs the reduction of the pair until $v$ is equal to zero. The unsigned comparison and subtraction algorithms are used in -place of the full signed routines since both values are guaranteed to be positive and the result of the subtraction is guaranteed to be non-negative. - -\section{Least Common Multiple} -The least common multiple of a pair of integers is their product divided by their greatest common divisor. For two integers $a$ and $b$ the -least common multiple is normally denoted as $[ a, b ]$ and numerically equivalent to ${ab} \over {(a, b)}$. For example, if $a = 2 \cdot 2 \cdot 3 = 12$ -and $b = 2 \cdot 3 \cdot 3 \cdot 7 = 126$ the least common multiple is ${126 \over {(12, 126)}} = {126 \over 6} = 21$. - -The least common multiple arises often in coding theory as well as number theory. If two functions have periods of $a$ and $b$ respectively they will -collide, that is be in synchronous states, after only $[ a, b ]$ iterations. This is why, for example, random number generators based on -Linear Feedback Shift Registers (LFSR) tend to use registers with periods which are co-prime (\textit{e.g. the greatest common divisor is one.}). -Similarly in number theory if a composite $n$ has two prime factors $p$ and $q$ then maximal order of any unit of $\Z/n\Z$ will be $[ p - 1, q - 1] $. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_lcm}. \\ -\textbf{Input}. mp\_int $a$ and $b$ \\ -\textbf{Output}. The least common multiple $c = [a, b]$. \\ -\hline \\ -1. $c \leftarrow (a, b)$ \\ -2. $t \leftarrow a \cdot b$ \\ -3. $c \leftarrow \lfloor t / c \rfloor$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_lcm} -\end{figure} -\textbf{Algorithm mp\_lcm.} -This algorithm computes the least common multiple of two mp\_int inputs $a$ and $b$. It computes the least common multiple directly by -dividing the product of the two inputs by their greatest common divisor. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_lcm.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\section{Jacobi Symbol Computation} -To explain the Jacobi Symbol we shall first discuss the Legendre function\footnote{Arrg. What is the name of this?} off which the Jacobi symbol is -defined. The Legendre function computes whether or not an integer $a$ is a quadratic residue modulo an odd prime $p$. Numerically it is -equivalent to equation \ref{eqn:legendre}. - -\textit{-- Tom, don't be an ass, cite your source here...!} - -\begin{equation} -a^{(p-1)/2} \equiv \begin{array}{rl} - -1 & \mbox{if }a\mbox{ is a quadratic non-residue.} \\ - 0 & \mbox{if }a\mbox{ divides }p\mbox{.} \\ - 1 & \mbox{if }a\mbox{ is a quadratic residue}. - \end{array} \mbox{ (mod }p\mbox{)} -\label{eqn:legendre} -\end{equation} - -\textbf{Proof.} \textit{Equation \ref{eqn:legendre} correctly identifies the residue status of an integer $a$ modulo a prime $p$.} -An integer $a$ is a quadratic residue if the following equation has a solution. - -\begin{equation} -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\label{eqn:root} -\end{equation} - -Consider the following equation. - -\begin{equation} -0 \equiv x^{p-1} - 1 \equiv \left \lbrace \left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \right \rbrace + \left ( a^{(p-1)/2} - 1 \right ) \mbox{ (mod }p\mbox{)} -\label{eqn:rooti} -\end{equation} - -Whether equation \ref{eqn:root} has a solution or not equation \ref{eqn:rooti} is always true. If $a^{(p-1)/2} - 1 \equiv 0 \mbox{ (mod }p\mbox{)}$ -then the quantity in the braces must be zero. By reduction, - -\begin{eqnarray} -\left (x^2 \right )^{(p-1)/2} - a^{(p-1)/2} \equiv 0 \nonumber \\ -\left (x^2 \right )^{(p-1)/2} \equiv a^{(p-1)/2} \nonumber \\ -x^2 \equiv a \mbox{ (mod }p\mbox{)} -\end{eqnarray} - -As a result there must be a solution to the quadratic equation and in turn $a$ must be a quadratic residue. If $a$ does not divide $p$ and $a$ -is not a quadratic residue then the only other value $a^{(p-1)/2}$ may be congruent to is $-1$ since -\begin{equation} -0 \equiv a^{p - 1} - 1 \equiv (a^{(p-1)/2} + 1)(a^{(p-1)/2} - 1) \mbox{ (mod }p\mbox{)} -\end{equation} -One of the terms on the right hand side must be zero. \textbf{QED} - -\subsection{Jacobi Symbol} -The Jacobi symbol is a generalization of the Legendre function for any odd non prime moduli $p$ greater than 2. If $p = \prod_{i=0}^n p_i$ then -the Jacobi symbol $\left ( { a \over p } \right )$ is equal to the following equation. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { a \over p_0} \right ) \left ( { a \over p_1} \right ) \ldots \left ( { a \over p_n} \right ) -\end{equation} - -By inspection if $p$ is prime the Jacobi symbol is equivalent to the Legendre function. The following facts\footnote{See HAC \cite[pp. 72-74]{HAC} for -further details.} will be used to derive an efficient Jacobi symbol algorithm. Where $p$ is an odd integer greater than two and $a, b \in \Z$ the -following are true. - -\begin{enumerate} -\item $\left ( { a \over p} \right )$ equals $-1$, $0$ or $1$. -\item $\left ( { ab \over p} \right ) = \left ( { a \over p} \right )\left ( { b \over p} \right )$. -\item If $a \equiv b$ then $\left ( { a \over p} \right ) = \left ( { b \over p} \right )$. -\item $\left ( { 2 \over p} \right )$ equals $1$ if $p \equiv 1$ or $7 \mbox{ (mod }8\mbox{)}$. Otherwise, it equals $-1$. -\item $\left ( { a \over p} \right ) \equiv \left ( { p \over a} \right ) \cdot (-1)^{(p-1)(a-1)/4}$. More specifically -$\left ( { a \over p} \right ) = \left ( { p \over a} \right )$ if $p \equiv a \equiv 1 \mbox{ (mod }4\mbox{)}$. -\end{enumerate} - -Using these facts if $a = 2^k \cdot a'$ then - -\begin{eqnarray} -\left ( { a \over p } \right ) = \left ( {{2^k} \over p } \right ) \left ( {a' \over p} \right ) \nonumber \\ - = \left ( {2 \over p } \right )^k \left ( {a' \over p} \right ) -\label{eqn:jacobi} -\end{eqnarray} - -By fact five, - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { p \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -Subsequently by fact three since $p \equiv (p \mbox{ mod }a) \mbox{ (mod }a\mbox{)}$ then - -\begin{equation} -\left ( { a \over p } \right ) = \left ( { {p \mbox{ mod } a} \over a } \right ) \cdot (-1)^{(p-1)(a-1)/4} -\end{equation} - -By putting both observations into equation \ref{eqn:jacobi} the following simplified equation is formed. - -\begin{equation} -\left ( { a \over p } \right ) = \left ( {2 \over p } \right )^k \left ( {{p\mbox{ mod }a'} \over a'} \right ) \cdot (-1)^{(p-1)(a'-1)/4} -\end{equation} - -The value of $\left ( {{p \mbox{ mod }a'} \over a'} \right )$ can be found by using the same equation recursively. The value of -$\left ( {2 \over p } \right )^k$ equals $1$ if $k$ is even otherwise it equals $\left ( {2 \over p } \right )$. Using this approach the -factors of $p$ do not have to be known. Furthermore, if $(a, p) = 1$ then the algorithm will terminate when the recursion requests the -Jacobi symbol computation of $\left ( {1 \over a'} \right )$ which is simply $1$. - -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_jacobi}. \\ -\textbf{Input}. mp\_int $a$ and $p$, $a \ge 0$, $p \ge 3$, $p \equiv 1 \mbox{ (mod }2\mbox{)}$ \\ -\textbf{Output}. The Jacobi symbol $c = \left ( {a \over p } \right )$. \\ -\hline \\ -1. If $a = 0$ then \\ -\hspace{3mm}1.1 $c \leftarrow 0$ \\ -\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\ -2. If $a = 1$ then \\ -\hspace{3mm}2.1 $c \leftarrow 1$ \\ -\hspace{3mm}2.2 Return(\textit{MP\_OKAY}). \\ -3. $a' \leftarrow a$ \\ -4. $k \leftarrow 0$ \\ -5. While $a'.used > 0$ and $a'_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}5.1 $k \leftarrow k + 1$ \\ -\hspace{3mm}5.2 $a' \leftarrow \lfloor a' / 2 \rfloor$ \\ -6. If $k \equiv 0 \mbox{ (mod }2\mbox{)}$ then \\ -\hspace{3mm}6.1 $s \leftarrow 1$ \\ -7. else \\ -\hspace{3mm}7.1 $r \leftarrow p_0 \mbox{ (mod }8\mbox{)}$ \\ -\hspace{3mm}7.2 If $r = 1$ or $r = 7$ then \\ -\hspace{6mm}7.2.1 $s \leftarrow 1$ \\ -\hspace{3mm}7.3 else \\ -\hspace{6mm}7.3.1 $s \leftarrow -1$ \\ -8. If $p_0 \equiv a'_0 \equiv 3 \mbox{ (mod }4\mbox{)}$ then \\ -\hspace{3mm}8.1 $s \leftarrow -s$ \\ -9. If $a' \ne 1$ then \\ -\hspace{3mm}9.1 $p' \leftarrow p \mbox{ (mod }a'\mbox{)}$ \\ -\hspace{3mm}9.2 $s \leftarrow s \cdot \mbox{mp\_jacobi}(p', a')$ \\ -10. $c \leftarrow s$ \\ -11. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_jacobi} -\end{figure} -\textbf{Algorithm mp\_jacobi.} -This algorithm computes the Jacobi symbol for an arbitrary positive integer $a$ with respect to an odd integer $p$ greater than three. The algorithm -is based on algorithm 2.149 of HAC \cite[pp. 73]{HAC}. - -Step numbers one and two handle the trivial cases of $a = 0$ and $a = 1$ respectively. Step five determines the number of two factors in the -input $a$. If $k$ is even than the term $\left ( { 2 \over p } \right )^k$ must always evaluate to one. If $k$ is odd than the term evaluates to one -if $p_0$ is congruent to one or seven modulo eight, otherwise it evaluates to $-1$. After the the $\left ( { 2 \over p } \right )^k$ term is handled -the $(-1)^{(p-1)(a'-1)/4}$ is computed and multiplied against the current product $s$. The latter term evaluates to one if both $p$ and $a'$ -are congruent to one modulo four, otherwise it evaluates to negative one. - -By step nine if $a'$ does not equal one a recursion is required. Step 9.1 computes $p' \equiv p \mbox{ (mod }a'\mbox{)}$ and will recurse to compute -$\left ( {p' \over a'} \right )$ which is multiplied against the current Jacobi product. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_jacobi.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -As a matter of practicality the variable $a'$ as per the pseudo-code is reprensented by the variable $a1$ since the $'$ symbol is not valid for a C -variable name character. - -The two simple cases of $a = 0$ and $a = 1$ are handled at the very beginning to simplify the algorithm. If the input is non-trivial the algorithm -has to proceed compute the Jacobi. The variable $s$ is used to hold the current Jacobi product. Note that $s$ is merely a C ``int'' data type since -the values it may obtain are merely $-1$, $0$ and $1$. - -After a local copy of $a$ is made all of the factors of two are divided out and the total stored in $k$. Technically only the least significant -bit of $k$ is required, however, it makes the algorithm simpler to follow to perform an addition. In practice an exclusive-or and addition have the same -processor requirements and neither is faster than the other. - -Line 58 through 71 determines the value of $\left ( { 2 \over p } \right )^k$. If the least significant bit of $k$ is zero than -$k$ is even and the value is one. Otherwise, the value of $s$ depends on which residue class $p$ belongs to modulo eight. The value of -$(-1)^{(p-1)(a'-1)/4}$ is compute and multiplied against $s$ on lines 71 through 74. - -Finally, if $a1$ does not equal one the algorithm must recurse and compute $\left ( {p' \over a'} \right )$. - -\textit{-- Comment about default $s$ and such...} - -\section{Modular Inverse} -\label{sec:modinv} -The modular inverse of a number actually refers to the modular multiplicative inverse. Essentially for any integer $a$ such that $(a, p) = 1$ there -exist another integer $b$ such that $ab \equiv 1 \mbox{ (mod }p\mbox{)}$. The integer $b$ is called the multiplicative inverse of $a$ which is -denoted as $b = a^{-1}$. Technically speaking modular inversion is a well defined operation for any finite ring or field not just for rings and -fields of integers. However, the former will be the matter of discussion. - -The simplest approach is to compute the algebraic inverse of the input. That is to compute $b \equiv a^{\Phi(p) - 1}$. If $\Phi(p)$ is the -order of the multiplicative subgroup modulo $p$ then $b$ must be the multiplicative inverse of $a$. The proof of which is trivial. - -\begin{equation} -ab \equiv a \left (a^{\Phi(p) - 1} \right ) \equiv a^{\Phi(p)} \equiv a^0 \equiv 1 \mbox{ (mod }p\mbox{)} -\end{equation} - -However, as simple as this approach may be it has two serious flaws. It requires that the value of $\Phi(p)$ be known which if $p$ is composite -requires all of the prime factors. This approach also is very slow as the size of $p$ grows. - -A simpler approach is based on the observation that solving for the multiplicative inverse is equivalent to solving the linear -Diophantine\footnote{See LeVeque \cite[pp. 40-43]{LeVeque} for more information.} equation. - -\begin{equation} -ab + pq = 1 -\end{equation} - -Where $a$, $b$, $p$ and $q$ are all integers. If such a pair of integers $ \left < b, q \right >$ exist than $b$ is the multiplicative inverse of -$a$ modulo $p$. The extended Euclidean algorithm (Knuth \cite[pp. 342]{TAOCPV2}) can be used to solve such equations provided $(a, p) = 1$. -However, instead of using that algorithm directly a variant known as the binary Extended Euclidean algorithm will be used in its place. The -binary approach is very similar to the binary greatest common divisor algorithm except it will produce a full solution to the Diophantine -equation. - -\subsection{General Case} -\newpage\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_invmod}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $(a, b) = 1$, $p \ge 2$, $0 < a < p$. \\ -\textbf{Output}. The modular inverse $c \equiv a^{-1} \mbox{ (mod }b\mbox{)}$. \\ -\hline \\ -1. If $b \le 0$ then return(\textit{MP\_VAL}). \\ -2. If $b_0 \equiv 1 \mbox{ (mod }2\mbox{)}$ then use algorithm fast\_mp\_invmod. \\ -3. $x \leftarrow \vert a \vert, y \leftarrow b$ \\ -4. If $x_0 \equiv y_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ then return(\textit{MP\_VAL}). \\ -5. $B \leftarrow 0, C \leftarrow 0, A \leftarrow 1, D \leftarrow 1$ \\ -6. While $u.used > 0$ and $u_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}6.1 $u \leftarrow \lfloor u / 2 \rfloor$ \\ -\hspace{3mm}6.2 If ($A.used > 0$ and $A_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($B.used > 0$ and $B_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}6.2.1 $A \leftarrow A + y$ \\ -\hspace{6mm}6.2.2 $B \leftarrow B - x$ \\ -\hspace{3mm}6.3 $A \leftarrow \lfloor A / 2 \rfloor$ \\ -\hspace{3mm}6.4 $B \leftarrow \lfloor B / 2 \rfloor$ \\ -7. While $v.used > 0$ and $v_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}7.1 $v \leftarrow \lfloor v / 2 \rfloor$ \\ -\hspace{3mm}7.2 If ($C.used > 0$ and $C_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) or ($D.used > 0$ and $D_0 \equiv 1 \mbox{ (mod }2\mbox{)}$) then \\ -\hspace{6mm}7.2.1 $C \leftarrow C + y$ \\ -\hspace{6mm}7.2.2 $D \leftarrow D - x$ \\ -\hspace{3mm}7.3 $C \leftarrow \lfloor C / 2 \rfloor$ \\ -\hspace{3mm}7.4 $D \leftarrow \lfloor D / 2 \rfloor$ \\ -8. If $u \ge v$ then \\ -\hspace{3mm}8.1 $u \leftarrow u - v$ \\ -\hspace{3mm}8.2 $A \leftarrow A - C$ \\ -\hspace{3mm}8.3 $B \leftarrow B - D$ \\ -9. else \\ -\hspace{3mm}9.1 $v \leftarrow v - u$ \\ -\hspace{3mm}9.2 $C \leftarrow C - A$ \\ -\hspace{3mm}9.3 $D \leftarrow D - B$ \\ -10. If $u \ne 0$ goto step 6. \\ -11. If $v \ne 1$ return(\textit{MP\_VAL}). \\ -12. While $C \le 0$ do \\ -\hspace{3mm}12.1 $C \leftarrow C + b$ \\ -13. While $C \ge b$ do \\ -\hspace{3mm}13.1 $C \leftarrow C - b$ \\ -14. $c \leftarrow C$ \\ -15. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\end{figure} -\textbf{Algorithm mp\_invmod.} -This algorithm computes the modular multiplicative inverse of an integer $a$ modulo an integer $b$. This algorithm is a variation of the -extended binary Euclidean algorithm from HAC \cite[pp. 608]{HAC}. It has been modified to only compute the modular inverse and not a complete -Diophantine solution. - -If $b \le 0$ than the modulus is invalid and MP\_VAL is returned. Similarly if both $a$ and $b$ are even then there cannot be a multiplicative -inverse for $a$ and the error is reported. - -The astute reader will observe that steps seven through nine are very similar to the binary greatest common divisor algorithm mp\_gcd. In this case -the other variables to the Diophantine equation are solved. The algorithm terminates when $u = 0$ in which case the solution is - -\begin{equation} -Ca + Db = v -\end{equation} - -If $v$, the greatest common divisor of $a$ and $b$ is not equal to one then the algorithm will report an error as no inverse exists. Otherwise, $C$ -is the modular inverse of $a$. The actual value of $C$ is congruent to, but not necessarily equal to, the ideal modular inverse which should lie -within $1 \le a^{-1} < b$. Step numbers twelve and thirteen adjust the inverse until it is in range. If the original input $a$ is within $0 < a < p$ -then only a couple of additions or subtractions will be required to adjust the inverse. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_invmod.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsubsection{Odd Moduli} - -When the modulus $b$ is odd the variables $A$ and $C$ are fixed and are not required to compute the inverse. In particular by attempting to solve -the Diophantine $Cb + Da = 1$ only $B$ and $D$ are required to find the inverse of $a$. - -The algorithm fast\_mp\_invmod is a direct adaptation of algorithm mp\_invmod with all all steps involving either $A$ or $C$ removed. This -optimization will halve the time required to compute the modular inverse. - -\section{Primality Tests} - -A non-zero integer $a$ is said to be prime if it is not divisible by any other integer excluding one and itself. For example, $a = 7$ is prime -since the integers $2 \ldots 6$ do not evenly divide $a$. By contrast, $a = 6$ is not prime since $a = 6 = 2 \cdot 3$. - -Prime numbers arise in cryptography considerably as they allow finite fields to be formed. The ability to determine whether an integer is prime or -not quickly has been a viable subject in cryptography and number theory for considerable time. The algorithms that will be presented are all -probablistic algorithms in that when they report an integer is composite it must be composite. However, when the algorithms report an integer is -prime the algorithm may be incorrect. - -As will be discussed it is possible to limit the probability of error so well that for practical purposes the probablity of error might as -well be zero. For the purposes of these discussions let $n$ represent the candidate integer of which the primality is in question. - -\subsection{Trial Division} - -Trial division means to attempt to evenly divide a candidate integer by small prime integers. If the candidate can be evenly divided it obviously -cannot be prime. By dividing by all primes $1 < p \le \sqrt{n}$ this test can actually prove whether an integer is prime. However, such a test -would require a prohibitive amount of time as $n$ grows. - -Instead of dividing by every prime, a smaller, more mangeable set of primes may be used instead. By performing trial division with only a subset -of the primes less than $\sqrt{n} + 1$ the algorithm cannot prove if a candidate is prime. However, often it can prove a candidate is not prime. - -The benefit of this test is that trial division by small values is fairly efficient. Specially compared to the other algorithms that will be -discussed shortly. The probability that this approach correctly identifies a composite candidate when tested with all primes upto $q$ is given by -$1 - {1.12 \over ln(q)}$. The graph (\ref{pic:primality}, will be added later) demonstrates the probability of success for the range -$3 \le q \le 100$. - -At approximately $q = 30$ the gain of performing further tests diminishes fairly quickly. At $q = 90$ further testing is generally not going to -be of any practical use. In the case of LibTomMath the default limit $q = 256$ was chosen since it is not too high and will eliminate -approximately $80\%$ of all candidate integers. The constant \textbf{PRIME\_SIZE} is equal to the number of primes in the test base. The -array \_\_prime\_tab is an array of the first \textbf{PRIME\_SIZE} prime numbers. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_is\_divisible}. \\ -\textbf{Input}. mp\_int $a$ \\ -\textbf{Output}. $c = 1$ if $n$ is divisible by a small prime, otherwise $c = 0$. \\ -\hline \\ -1. for $ix$ from $0$ to $PRIME\_SIZE$ do \\ -\hspace{3mm}1.1 $d \leftarrow n \mbox{ (mod }\_\_prime\_tab_{ix}\mbox{)}$ \\ -\hspace{3mm}1.2 If $d = 0$ then \\ -\hspace{6mm}1.2.1 $c \leftarrow 1$ \\ -\hspace{6mm}1.2.2 Return(\textit{MP\_OKAY}). \\ -2. $c \leftarrow 0$ \\ -3. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_is\_divisible} -\end{figure} -\textbf{Algorithm mp\_prime\_is\_divisible.} -This algorithm attempts to determine if a candidate integer $n$ is composite by performing trial divisions. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_is\_divisible.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -The algorithm defaults to a return of $0$ in case an error occurs. The values in the prime table are all specified to be in the range of a -mp\_digit. The table \_\_prime\_tab is defined in the following file. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_prime\_tab.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -Note that there are two possible tables. When an mp\_digit is 7-bits long only the primes upto $127$ may be included, otherwise the primes -upto $1619$ are used. Note that the value of \textbf{PRIME\_SIZE} is a constant dependent on the size of a mp\_digit. - -\subsection{The Fermat Test} -The Fermat test is probably one the oldest tests to have a non-trivial probability of success. It is based on the fact that if $n$ is in -fact prime then $a^{n} \equiv a \mbox{ (mod }n\mbox{)}$ for all $0 < a < n$. The reason being that if $n$ is prime than the order of -the multiplicative sub group is $n - 1$. Any base $a$ must have an order which divides $n - 1$ and as such $a^n$ is equivalent to -$a^1 = a$. - -If $n$ is composite then any given base $a$ does not have to have a period which divides $n - 1$. In which case -it is possible that $a^n \nequiv a \mbox{ (mod }n\mbox{)}$. However, this test is not absolute as it is possible that the order -of a base will divide $n - 1$ which would then be reported as prime. Such a base yields what is known as a Fermat pseudo-prime. Several -integers known as Carmichael numbers will be a pseudo-prime to all valid bases. Fortunately such numbers are extremely rare as $n$ grows -in size. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_fermat}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $b^a \equiv b \mbox{ (mod }a\mbox{)}$, otherwise $c = 0$. \\ -\hline \\ -1. $t \leftarrow b^a \mbox{ (mod }a\mbox{)}$ \\ -2. If $t = b$ then \\ -\hspace{3mm}2.1 $c = 1$ \\ -3. else \\ -\hspace{3mm}3.1 $c = 0$ \\ -4. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_fermat} -\end{figure} -\textbf{Algorithm mp\_prime\_fermat.} -This algorithm determines whether an mp\_int $a$ is a Fermat prime to the base $b$ or not. It uses a single modular exponentiation to -determine the result. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_fermat.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - -\subsection{The Miller-Rabin Test} -The Miller-Rabin (citation) test is another primality test which has tighter error bounds than the Fermat test specifically with sequentially chosen -candidate integers. The algorithm is based on the observation that if $n - 1 = 2^kr$ and if $b^r \nequiv \pm 1$ then after upto $k - 1$ squarings the -value must be equal to $-1$. The squarings are stopped as soon as $-1$ is observed. If the value of $1$ is observed first it means that -some value not congruent to $\pm 1$ when squared equals one which cannot occur if $n$ is prime. - -\begin{figure}[!here] -\begin{small} -\begin{center} -\begin{tabular}{l} -\hline Algorithm \textbf{mp\_prime\_miller\_rabin}. \\ -\textbf{Input}. mp\_int $a$ and $b$, $a \ge 2$, $0 < b < a$. \\ -\textbf{Output}. $c = 1$ if $a$ is a Miller-Rabin prime to the base $a$, otherwise $c = 0$. \\ -\hline -1. $a' \leftarrow a - 1$ \\ -2. $r \leftarrow n1$ \\ -3. $c \leftarrow 0, s \leftarrow 0$ \\ -4. While $r.used > 0$ and $r_0 \equiv 0 \mbox{ (mod }2\mbox{)}$ \\ -\hspace{3mm}4.1 $s \leftarrow s + 1$ \\ -\hspace{3mm}4.2 $r \leftarrow \lfloor r / 2 \rfloor$ \\ -5. $y \leftarrow b^r \mbox{ (mod }a\mbox{)}$ \\ -6. If $y \nequiv \pm 1$ then \\ -\hspace{3mm}6.1 $j \leftarrow 1$ \\ -\hspace{3mm}6.2 While $j \le (s - 1)$ and $y \nequiv a'$ \\ -\hspace{6mm}6.2.1 $y \leftarrow y^2 \mbox{ (mod }a\mbox{)}$ \\ -\hspace{6mm}6.2.2 If $y = 1$ then goto step 8. \\ -\hspace{6mm}6.2.3 $j \leftarrow j + 1$ \\ -\hspace{3mm}6.3 If $y \nequiv a'$ goto step 8. \\ -7. $c \leftarrow 1$\\ -8. Return(\textit{MP\_OKAY}). \\ -\hline -\end{tabular} -\end{center} -\end{small} -\caption{Algorithm mp\_prime\_miller\_rabin} -\end{figure} -\textbf{Algorithm mp\_prime\_miller\_rabin.} -This algorithm performs one trial round of the Miller-Rabin algorithm to the base $b$. It will set $c = 1$ if the algorithm cannot determine -if $b$ is composite or $c = 0$ if $b$ is provably composite. The values of $s$ and $r$ are computed such that $a' = a - 1 = 2^sr$. - -If the value $y \equiv b^r$ is congruent to $\pm 1$ then the algorithm cannot prove if $a$ is composite or not. Otherwise, the algorithm will -square $y$ upto $s - 1$ times stopping only when $y \equiv -1$. If $y^2 \equiv 1$ and $y \nequiv \pm 1$ then the algorithm can report that $a$ -is provably composite. If the algorithm performs $s - 1$ squarings and $y \nequiv -1$ then $a$ is provably composite. If $a$ is not provably -composite then it is \textit{probably} prime. - -\vspace{+3mm}\begin{small} -\hspace{-5.1mm}{\bf File}: bn\_mp\_prime\_miller\_rabin.c -\vspace{-3mm} -\begin{alltt} -\end{alltt} -\end{small} - - - - -\backmatter -\appendix -\begin{thebibliography}{ABCDEF} -\bibitem[1]{TAOCPV2} -Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998 - -\bibitem[2]{HAC} -A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996 - -\bibitem[3]{ROSE} -Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999 - -\bibitem[4]{COMBA} -Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990) - -\bibitem[5]{KARA} -A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294 - -\bibitem[6]{KARAP} -Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002 - -\bibitem[7]{BARRETT} -Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag. - -\bibitem[8]{MONT} -P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985. - -\bibitem[9]{DRMET} -Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories - -\bibitem[10]{MMB} -J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89 - -\bibitem[11]{RSAREF} -R.L. Rivest, A. Shamir, L. Adleman, \textit{A Method for Obtaining Digital Signatures and Public-Key Cryptosystems} - -\bibitem[12]{DHREF} -Whitfield Diffie, Martin E. Hellman, \textit{New Directions in Cryptography}, IEEE Transactions on Information Theory, 1976 - -\bibitem[13]{IEEE} -IEEE Standard for Binary Floating-Point Arithmetic (ANSI/IEEE Std 754-1985) - -\bibitem[14]{GMP} -GNU Multiple Precision (GMP), \url{http://www.swox.com/gmp/} - -\bibitem[15]{MPI} -Multiple Precision Integer Library (MPI), Michael Fromberger, \url{http://thayer.dartmouth.edu/~sting/mpi/} - -\bibitem[16]{OPENSSL} -OpenSSL Cryptographic Toolkit, \url{http://openssl.org} - -\bibitem[17]{LIP} -Large Integer Package, \url{http://home.hetnet.nl/~ecstr/LIP.zip} - -\bibitem[18]{ISOC} -JTC1/SC22/WG14, ISO/IEC 9899:1999, ``A draft rationale for the C99 standard.'' - -\bibitem[19]{JAVA} -The Sun Java Website, \url{http://java.sun.com/} - -\end{thebibliography} - -\input{tommath.ind} - -\end{document} diff --git a/lib/hcrypto/libtommath/tommath_class.h b/lib/hcrypto/libtommath/tommath_class.h index 8fd0f52f4..52ba585ec 100644 --- a/lib/hcrypto/libtommath/tommath_class.h +++ b/lib/hcrypto/libtommath/tommath_class.h @@ -1,214 +1,296 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + #if !(defined(LTM1) && defined(LTM2) && defined(LTM3)) +#define LTM_INSIDE #if defined(LTM2) -#define LTM3 +# define LTM3 #endif #if defined(LTM1) -#define LTM2 +# define LTM2 #endif #define LTM1 - #if defined(LTM_ALL) -#define BN_ERROR_C -#define BN_FAST_MP_INVMOD_C -#define BN_FAST_MP_MONTGOMERY_REDUCE_C -#define BN_FAST_S_MP_MUL_DIGS_C -#define BN_FAST_S_MP_MUL_HIGH_DIGS_C -#define BN_FAST_S_MP_SQR_C -#define BN_MP_2EXPT_C -#define BN_MP_ABS_C -#define BN_MP_ADD_C -#define BN_MP_ADD_D_C -#define BN_MP_ADDMOD_C -#define BN_MP_AND_C -#define BN_MP_CLAMP_C -#define BN_MP_CLEAR_C -#define BN_MP_CLEAR_MULTI_C -#define BN_MP_CMP_C -#define BN_MP_CMP_D_C -#define BN_MP_CMP_MAG_C -#define BN_MP_CNT_LSB_C -#define BN_MP_COPY_C -#define BN_MP_COUNT_BITS_C -#define BN_MP_DIV_C -#define BN_MP_DIV_2_C -#define BN_MP_DIV_2D_C -#define BN_MP_DIV_3_C -#define BN_MP_DIV_D_C -#define BN_MP_DR_IS_MODULUS_C -#define BN_MP_DR_REDUCE_C -#define BN_MP_DR_SETUP_C -#define BN_MP_EXCH_C -#define BN_MP_EXPT_D_C -#define BN_MP_EXPTMOD_C -#define BN_MP_EXPTMOD_FAST_C -#define BN_MP_EXTEUCLID_C -#define BN_MP_FIND_PRIME_C -#define BN_MP_FREAD_C -#define BN_MP_FWRITE_C -#define BN_MP_GCD_C -#define BN_MP_GET_INT_C -#define BN_MP_GROW_C -#define BN_MP_INIT_C -#define BN_MP_INIT_COPY_C -#define BN_MP_INIT_MULTI_C -#define BN_MP_INIT_SET_C -#define BN_MP_INIT_SET_INT_C -#define BN_MP_INIT_SIZE_C -#define BN_MP_INVMOD_C -#define BN_MP_INVMOD_SLOW_C -#define BN_MP_ISPRIME_C -#define BN_MP_IS_SQUARE_C -#define BN_MP_JACOBI_C -#define BN_MP_KARATSUBA_MUL_C -#define BN_MP_KARATSUBA_SQR_C -#define BN_MP_LCM_C -#define BN_MP_LSHD_C -#define BN_MP_MOD_C -#define BN_MP_MOD_2D_C -#define BN_MP_MOD_D_C -#define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C -#define BN_MP_MONTGOMERY_REDUCE_C -#define BN_MP_MONTGOMERY_SETUP_C -#define BN_MP_MUL_C -#define BN_MP_MUL_2_C -#define BN_MP_MUL_2D_C -#define BN_MP_MUL_D_C -#define BN_MP_MULMOD_C -#define BN_MP_N_ROOT_C -#define BN_MP_NEG_C -#define BN_MP_OR_C -#define BN_MP_PRIME_FERMAT_C -#define BN_MP_PRIME_IS_DIVISIBLE_C -#define BN_MP_PRIME_IS_PRIME_C -#define BN_MP_PRIME_MILLER_RABIN_C -#define BN_MP_PRIME_NEXT_PRIME_C -#define BN_MP_PRIME_RABIN_MILLER_TRIALS_C -#define BN_MP_PRIME_RANDOM_EX_C -#define BN_MP_RADIX_SIZE_C -#define BN_MP_RADIX_SMAP_C -#define BN_MP_RAND_C -#define BN_MP_READ_RADIX_C -#define BN_MP_READ_SIGNED_BIN_C -#define BN_MP_READ_UNSIGNED_BIN_C -#define BN_MP_REDUCE_C -#define BN_MP_REDUCE_2K_C -#define BN_MP_REDUCE_2K_L_C -#define BN_MP_REDUCE_2K_SETUP_C -#define BN_MP_REDUCE_2K_SETUP_L_C -#define BN_MP_REDUCE_IS_2K_C -#define BN_MP_REDUCE_IS_2K_L_C -#define BN_MP_REDUCE_SETUP_C -#define BN_MP_RSHD_C -#define BN_MP_SET_C -#define BN_MP_SET_INT_C -#define BN_MP_SHRINK_C -#define BN_MP_SIGNED_BIN_SIZE_C -#define BN_MP_SQR_C -#define BN_MP_SQRMOD_C -#define BN_MP_SQRT_C -#define BN_MP_SUB_C -#define BN_MP_SUB_D_C -#define BN_MP_SUBMOD_C -#define BN_MP_TO_SIGNED_BIN_C -#define BN_MP_TO_SIGNED_BIN_N_C -#define BN_MP_TO_UNSIGNED_BIN_C -#define BN_MP_TO_UNSIGNED_BIN_N_C -#define BN_MP_TOOM_MUL_C -#define BN_MP_TOOM_SQR_C -#define BN_MP_TORADIX_C -#define BN_MP_TORADIX_N_C -#define BN_MP_UNSIGNED_BIN_SIZE_C -#define BN_MP_XOR_C -#define BN_MP_ZERO_C -#define BN_MP_ZERO_MULTI_C -#define BN_PRIME_TAB_C -#define BN_REVERSE_C -#define BN_S_MP_ADD_C -#define BN_S_MP_EXPTMOD_C -#define BN_S_MP_MUL_DIGS_C -#define BN_S_MP_MUL_HIGH_DIGS_C -#define BN_S_MP_SQR_C -#define BN_S_MP_SUB_C -#define BNCORE_C +# define BN_CUTOFFS_C +# define BN_DEPRECATED_C +# define BN_MP_2EXPT_C +# define BN_MP_ABS_C +# define BN_MP_ADD_C +# define BN_MP_ADD_D_C +# define BN_MP_ADDMOD_C +# define BN_MP_AND_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_CMP_MAG_C +# define BN_MP_CNT_LSB_C +# define BN_MP_COMPLEMENT_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DECR_C +# define BN_MP_DIV_C +# define BN_MP_DIV_2_C +# define BN_MP_DIV_2D_C +# define BN_MP_DIV_3_C +# define BN_MP_DIV_D_C +# define BN_MP_DR_IS_MODULUS_C +# define BN_MP_DR_REDUCE_C +# define BN_MP_DR_SETUP_C +# define BN_MP_ERROR_TO_STRING_C +# define BN_MP_EXCH_C +# define BN_MP_EXPT_U32_C +# define BN_MP_EXPTMOD_C +# define BN_MP_EXTEUCLID_C +# define BN_MP_FREAD_C +# define BN_MP_FROM_SBIN_C +# define BN_MP_FROM_UBIN_C +# define BN_MP_FWRITE_C +# define BN_MP_GCD_C +# define BN_MP_GET_DOUBLE_C +# define BN_MP_GET_I32_C +# define BN_MP_GET_I64_C +# define BN_MP_GET_L_C +# define BN_MP_GET_LL_C +# define BN_MP_GET_MAG_U32_C +# define BN_MP_GET_MAG_U64_C +# define BN_MP_GET_MAG_UL_C +# define BN_MP_GET_MAG_ULL_C +# define BN_MP_GROW_C +# define BN_MP_INCR_C +# define BN_MP_INIT_C +# define BN_MP_INIT_COPY_C +# define BN_MP_INIT_I32_C +# define BN_MP_INIT_I64_C +# define BN_MP_INIT_L_C +# define BN_MP_INIT_LL_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_INIT_SET_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_INIT_U32_C +# define BN_MP_INIT_U64_C +# define BN_MP_INIT_UL_C +# define BN_MP_INIT_ULL_C +# define BN_MP_INVMOD_C +# define BN_MP_IS_SQUARE_C +# define BN_MP_ISEVEN_C +# define BN_MP_ISODD_C +# define BN_MP_KRONECKER_C +# define BN_MP_LCM_C +# define BN_MP_LOG_U32_C +# define BN_MP_LSHD_C +# define BN_MP_MOD_C +# define BN_MP_MOD_2D_C +# define BN_MP_MOD_D_C +# define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +# define BN_MP_MONTGOMERY_REDUCE_C +# define BN_MP_MONTGOMERY_SETUP_C +# define BN_MP_MUL_C +# define BN_MP_MUL_2_C +# define BN_MP_MUL_2D_C +# define BN_MP_MUL_D_C +# define BN_MP_MULMOD_C +# define BN_MP_NEG_C +# define BN_MP_OR_C +# define BN_MP_PACK_C +# define BN_MP_PACK_COUNT_C +# define BN_MP_PRIME_FERMAT_C +# define BN_MP_PRIME_FROBENIUS_UNDERWOOD_C +# define BN_MP_PRIME_IS_PRIME_C +# define BN_MP_PRIME_MILLER_RABIN_C +# define BN_MP_PRIME_NEXT_PRIME_C +# define BN_MP_PRIME_RABIN_MILLER_TRIALS_C +# define BN_MP_PRIME_RAND_C +# define BN_MP_PRIME_STRONG_LUCAS_SELFRIDGE_C +# define BN_MP_RADIX_SIZE_C +# define BN_MP_RADIX_SMAP_C +# define BN_MP_RAND_C +# define BN_MP_READ_RADIX_C +# define BN_MP_REDUCE_C +# define BN_MP_REDUCE_2K_C +# define BN_MP_REDUCE_2K_L_C +# define BN_MP_REDUCE_2K_SETUP_C +# define BN_MP_REDUCE_2K_SETUP_L_C +# define BN_MP_REDUCE_IS_2K_C +# define BN_MP_REDUCE_IS_2K_L_C +# define BN_MP_REDUCE_SETUP_C +# define BN_MP_ROOT_U32_C +# define BN_MP_RSHD_C +# define BN_MP_SBIN_SIZE_C +# define BN_MP_SET_C +# define BN_MP_SET_DOUBLE_C +# define BN_MP_SET_I32_C +# define BN_MP_SET_I64_C +# define BN_MP_SET_L_C +# define BN_MP_SET_LL_C +# define BN_MP_SET_U32_C +# define BN_MP_SET_U64_C +# define BN_MP_SET_UL_C +# define BN_MP_SET_ULL_C +# define BN_MP_SHRINK_C +# define BN_MP_SIGNED_RSH_C +# define BN_MP_SQR_C +# define BN_MP_SQRMOD_C +# define BN_MP_SQRT_C +# define BN_MP_SQRTMOD_PRIME_C +# define BN_MP_SUB_C +# define BN_MP_SUB_D_C +# define BN_MP_SUBMOD_C +# define BN_MP_TO_RADIX_C +# define BN_MP_TO_SBIN_C +# define BN_MP_TO_UBIN_C +# define BN_MP_UBIN_SIZE_C +# define BN_MP_UNPACK_C +# define BN_MP_XOR_C +# define BN_MP_ZERO_C +# define BN_PRIME_TAB_C +# define BN_S_MP_ADD_C +# define BN_S_MP_BALANCE_MUL_C +# define BN_S_MP_EXPTMOD_C +# define BN_S_MP_EXPTMOD_FAST_C +# define BN_S_MP_GET_BIT_C +# define BN_S_MP_INVMOD_FAST_C +# define BN_S_MP_INVMOD_SLOW_C +# define BN_S_MP_KARATSUBA_MUL_C +# define BN_S_MP_KARATSUBA_SQR_C +# define BN_S_MP_MONTGOMERY_REDUCE_FAST_C +# define BN_S_MP_MUL_DIGS_C +# define BN_S_MP_MUL_DIGS_FAST_C +# define BN_S_MP_MUL_HIGH_DIGS_C +# define BN_S_MP_MUL_HIGH_DIGS_FAST_C +# define BN_S_MP_PRIME_IS_DIVISIBLE_C +# define BN_S_MP_RAND_JENKINS_C +# define BN_S_MP_RAND_PLATFORM_C +# define BN_S_MP_REVERSE_C +# define BN_S_MP_SQR_C +# define BN_S_MP_SQR_FAST_C +# define BN_S_MP_SUB_C +# define BN_S_MP_TOOM_MUL_C +# define BN_S_MP_TOOM_SQR_C +#endif +#endif +#if defined(BN_CUTOFFS_C) #endif -#if defined(BN_ERROR_C) - #define BN_MP_ERROR_TO_STRING_C -#endif - -#if defined(BN_FAST_MP_INVMOD_C) - #define BN_MP_ISEVEN_C - #define BN_MP_INIT_MULTI_C - #define BN_MP_COPY_C - #define BN_MP_MOD_C - #define BN_MP_SET_C - #define BN_MP_DIV_2_C - #define BN_MP_ISODD_C - #define BN_MP_SUB_C - #define BN_MP_CMP_C - #define BN_MP_ISZERO_C - #define BN_MP_CMP_D_C - #define BN_MP_ADD_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_MULTI_C -#endif - -#if defined(BN_FAST_MP_MONTGOMERY_REDUCE_C) - #define BN_MP_GROW_C - #define BN_MP_RSHD_C - #define BN_MP_CLAMP_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C -#endif - -#if defined(BN_FAST_S_MP_MUL_DIGS_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C -#endif - -#if defined(BN_FAST_S_MP_MUL_HIGH_DIGS_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C -#endif - -#if defined(BN_FAST_S_MP_SQR_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C +#if defined(BN_DEPRECATED_C) +# define BN_FAST_MP_INVMOD_C +# define BN_FAST_MP_MONTGOMERY_REDUCE_C +# define BN_FAST_S_MP_MUL_DIGS_C +# define BN_FAST_S_MP_MUL_HIGH_DIGS_C +# define BN_FAST_S_MP_SQR_C +# define BN_MP_AND_C +# define BN_MP_BALANCE_MUL_C +# define BN_MP_CMP_D_C +# define BN_MP_EXPORT_C +# define BN_MP_EXPTMOD_FAST_C +# define BN_MP_EXPT_D_C +# define BN_MP_EXPT_D_EX_C +# define BN_MP_EXPT_U32_C +# define BN_MP_FROM_SBIN_C +# define BN_MP_FROM_UBIN_C +# define BN_MP_GET_BIT_C +# define BN_MP_GET_INT_C +# define BN_MP_GET_LONG_C +# define BN_MP_GET_LONG_LONG_C +# define BN_MP_GET_MAG_U32_C +# define BN_MP_GET_MAG_ULL_C +# define BN_MP_GET_MAG_UL_C +# define BN_MP_IMPORT_C +# define BN_MP_INIT_SET_INT_C +# define BN_MP_INIT_U32_C +# define BN_MP_INVMOD_SLOW_C +# define BN_MP_JACOBI_C +# define BN_MP_KARATSUBA_MUL_C +# define BN_MP_KARATSUBA_SQR_C +# define BN_MP_KRONECKER_C +# define BN_MP_N_ROOT_C +# define BN_MP_N_ROOT_EX_C +# define BN_MP_OR_C +# define BN_MP_PACK_C +# define BN_MP_PRIME_IS_DIVISIBLE_C +# define BN_MP_PRIME_RANDOM_EX_C +# define BN_MP_RAND_DIGIT_C +# define BN_MP_READ_SIGNED_BIN_C +# define BN_MP_READ_UNSIGNED_BIN_C +# define BN_MP_ROOT_U32_C +# define BN_MP_SBIN_SIZE_C +# define BN_MP_SET_INT_C +# define BN_MP_SET_LONG_C +# define BN_MP_SET_LONG_LONG_C +# define BN_MP_SET_U32_C +# define BN_MP_SET_U64_C +# define BN_MP_SIGNED_BIN_SIZE_C +# define BN_MP_SIGNED_RSH_C +# define BN_MP_TC_AND_C +# define BN_MP_TC_DIV_2D_C +# define BN_MP_TC_OR_C +# define BN_MP_TC_XOR_C +# define BN_MP_TOOM_MUL_C +# define BN_MP_TOOM_SQR_C +# define BN_MP_TORADIX_C +# define BN_MP_TORADIX_N_C +# define BN_MP_TO_RADIX_C +# define BN_MP_TO_SBIN_C +# define BN_MP_TO_SIGNED_BIN_C +# define BN_MP_TO_SIGNED_BIN_N_C +# define BN_MP_TO_UBIN_C +# define BN_MP_TO_UNSIGNED_BIN_C +# define BN_MP_TO_UNSIGNED_BIN_N_C +# define BN_MP_UBIN_SIZE_C +# define BN_MP_UNPACK_C +# define BN_MP_UNSIGNED_BIN_SIZE_C +# define BN_MP_XOR_C +# define BN_S_MP_BALANCE_MUL_C +# define BN_S_MP_EXPTMOD_FAST_C +# define BN_S_MP_GET_BIT_C +# define BN_S_MP_INVMOD_FAST_C +# define BN_S_MP_INVMOD_SLOW_C +# define BN_S_MP_KARATSUBA_MUL_C +# define BN_S_MP_KARATSUBA_SQR_C +# define BN_S_MP_MONTGOMERY_REDUCE_FAST_C +# define BN_S_MP_MUL_DIGS_FAST_C +# define BN_S_MP_MUL_HIGH_DIGS_FAST_C +# define BN_S_MP_PRIME_IS_DIVISIBLE_C +# define BN_S_MP_PRIME_RANDOM_EX_C +# define BN_S_MP_RAND_SOURCE_C +# define BN_S_MP_REVERSE_C +# define BN_S_MP_SQR_FAST_C +# define BN_S_MP_TOOM_MUL_C +# define BN_S_MP_TOOM_SQR_C #endif #if defined(BN_MP_2EXPT_C) - #define BN_MP_ZERO_C - #define BN_MP_GROW_C +# define BN_MP_GROW_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_ABS_C) - #define BN_MP_COPY_C +# define BN_MP_COPY_C #endif #if defined(BN_MP_ADD_C) - #define BN_S_MP_ADD_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C +# define BN_MP_CMP_MAG_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_ADD_D_C) - #define BN_MP_GROW_C - #define BN_MP_SUB_D_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_ADDMOD_C) - #define BN_MP_INIT_C - #define BN_MP_ADD_C - #define BN_MP_CLEAR_C - #define BN_MP_MOD_C +# define BN_MP_ADD_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_C +# define BN_MP_MOD_C #endif #if defined(BN_MP_AND_C) - #define BN_MP_INIT_COPY_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_CLAMP_C) @@ -218,11 +300,11 @@ #endif #if defined(BN_MP_CLEAR_MULTI_C) - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C #endif #if defined(BN_MP_CMP_C) - #define BN_MP_CMP_MAG_C +# define BN_MP_CMP_MAG_C #endif #if defined(BN_MP_CMP_D_C) @@ -232,699 +314,788 @@ #endif #if defined(BN_MP_CNT_LSB_C) - #define BN_MP_ISZERO_C +#endif + +#if defined(BN_MP_COMPLEMENT_C) +# define BN_MP_NEG_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_COPY_C) - #define BN_MP_GROW_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_COUNT_BITS_C) #endif +#if defined(BN_MP_DECR_C) +# define BN_MP_INCR_C +# define BN_MP_SET_C +# define BN_MP_SUB_D_C +# define BN_MP_ZERO_C +#endif + #if defined(BN_MP_DIV_C) - #define BN_MP_ISZERO_C - #define BN_MP_CMP_MAG_C - #define BN_MP_COPY_C - #define BN_MP_ZERO_C - #define BN_MP_INIT_MULTI_C - #define BN_MP_SET_C - #define BN_MP_COUNT_BITS_C - #define BN_MP_ABS_C - #define BN_MP_MUL_2D_C - #define BN_MP_CMP_C - #define BN_MP_SUB_C - #define BN_MP_ADD_C - #define BN_MP_DIV_2D_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_MULTI_C - #define BN_MP_INIT_SIZE_C - #define BN_MP_INIT_C - #define BN_MP_INIT_COPY_C - #define BN_MP_LSHD_C - #define BN_MP_RSHD_C - #define BN_MP_MUL_D_C - #define BN_MP_CLAMP_C - #define BN_MP_CLEAR_C +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_2D_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_C +# define BN_MP_INIT_COPY_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_MUL_2D_C +# define BN_MP_MUL_D_C +# define BN_MP_RSHD_C +# define BN_MP_SUB_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_DIV_2_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_DIV_2D_C) - #define BN_MP_COPY_C - #define BN_MP_ZERO_C - #define BN_MP_INIT_C - #define BN_MP_MOD_2D_C - #define BN_MP_CLEAR_C - #define BN_MP_RSHD_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C +# define BN_MP_CLAMP_C +# define BN_MP_COPY_C +# define BN_MP_MOD_2D_C +# define BN_MP_RSHD_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_DIV_3_C) - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C #endif #if defined(BN_MP_DIV_D_C) - #define BN_MP_ISZERO_C - #define BN_MP_COPY_C - #define BN_MP_DIV_2D_C - #define BN_MP_DIV_3_C - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_COPY_C +# define BN_MP_DIV_2D_C +# define BN_MP_DIV_3_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C #endif #if defined(BN_MP_DR_IS_MODULUS_C) #endif #if defined(BN_MP_DR_REDUCE_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C +# define BN_MP_CLAMP_C +# define BN_MP_CMP_MAG_C +# define BN_MP_GROW_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_DR_SETUP_C) #endif +#if defined(BN_MP_ERROR_TO_STRING_C) +#endif + #if defined(BN_MP_EXCH_C) #endif -#if defined(BN_MP_EXPT_D_C) - #define BN_MP_INIT_COPY_C - #define BN_MP_SET_C - #define BN_MP_SQR_C - #define BN_MP_CLEAR_C - #define BN_MP_MUL_C +#if defined(BN_MP_EXPT_U32_C) +# define BN_MP_CLEAR_C +# define BN_MP_INIT_COPY_C +# define BN_MP_MUL_C +# define BN_MP_SET_C +# define BN_MP_SQR_C #endif #if defined(BN_MP_EXPTMOD_C) - #define BN_MP_INIT_C - #define BN_MP_INVMOD_C - #define BN_MP_CLEAR_C - #define BN_MP_ABS_C - #define BN_MP_CLEAR_MULTI_C - #define BN_MP_REDUCE_IS_2K_L_C - #define BN_S_MP_EXPTMOD_C - #define BN_MP_DR_IS_MODULUS_C - #define BN_MP_REDUCE_IS_2K_C - #define BN_MP_ISODD_C - #define BN_MP_EXPTMOD_FAST_C -#endif - -#if defined(BN_MP_EXPTMOD_FAST_C) - #define BN_MP_COUNT_BITS_C - #define BN_MP_INIT_C - #define BN_MP_CLEAR_C - #define BN_MP_MONTGOMERY_SETUP_C - #define BN_FAST_MP_MONTGOMERY_REDUCE_C - #define BN_MP_MONTGOMERY_REDUCE_C - #define BN_MP_DR_SETUP_C - #define BN_MP_DR_REDUCE_C - #define BN_MP_REDUCE_2K_SETUP_C - #define BN_MP_REDUCE_2K_C - #define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C - #define BN_MP_MULMOD_C - #define BN_MP_SET_C - #define BN_MP_MOD_C - #define BN_MP_COPY_C - #define BN_MP_SQR_C - #define BN_MP_MUL_C - #define BN_MP_EXCH_C +# define BN_MP_ABS_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_DR_IS_MODULUS_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_INVMOD_C +# define BN_MP_REDUCE_IS_2K_C +# define BN_MP_REDUCE_IS_2K_L_C +# define BN_S_MP_EXPTMOD_C +# define BN_S_MP_EXPTMOD_FAST_C #endif #if defined(BN_MP_EXTEUCLID_C) - #define BN_MP_INIT_MULTI_C - #define BN_MP_SET_C - #define BN_MP_COPY_C - #define BN_MP_ISZERO_C - #define BN_MP_DIV_C - #define BN_MP_MUL_C - #define BN_MP_SUB_C - #define BN_MP_NEG_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_MULTI_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_COPY_C +# define BN_MP_DIV_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MUL_C +# define BN_MP_NEG_C +# define BN_MP_SET_C +# define BN_MP_SUB_C #endif #if defined(BN_MP_FREAD_C) - #define BN_MP_ZERO_C - #define BN_MP_S_RMAP_C - #define BN_MP_MUL_D_C - #define BN_MP_ADD_D_C - #define BN_MP_CMP_D_C +# define BN_MP_ADD_D_C +# define BN_MP_MUL_D_C +# define BN_MP_ZERO_C +#endif + +#if defined(BN_MP_FROM_SBIN_C) +# define BN_MP_FROM_UBIN_C +#endif + +#if defined(BN_MP_FROM_UBIN_C) +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C +# define BN_MP_MUL_2D_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_FWRITE_C) - #define BN_MP_RADIX_SIZE_C - #define BN_MP_TORADIX_C +# define BN_MP_RADIX_SIZE_C +# define BN_MP_TO_RADIX_C #endif #if defined(BN_MP_GCD_C) - #define BN_MP_ISZERO_C - #define BN_MP_ABS_C - #define BN_MP_ZERO_C - #define BN_MP_INIT_COPY_C - #define BN_MP_CNT_LSB_C - #define BN_MP_DIV_2D_C - #define BN_MP_CMP_MAG_C - #define BN_MP_EXCH_C - #define BN_S_MP_SUB_C - #define BN_MP_MUL_2D_C - #define BN_MP_CLEAR_C +# define BN_MP_ABS_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_MAG_C +# define BN_MP_CNT_LSB_C +# define BN_MP_DIV_2D_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_COPY_C +# define BN_MP_MUL_2D_C +# define BN_S_MP_SUB_C #endif -#if defined(BN_MP_GET_INT_C) +#if defined(BN_MP_GET_DOUBLE_C) +#endif + +#if defined(BN_MP_GET_I32_C) +# define BN_MP_GET_MAG_U32_C +#endif + +#if defined(BN_MP_GET_I64_C) +# define BN_MP_GET_MAG_U64_C +#endif + +#if defined(BN_MP_GET_L_C) +# define BN_MP_GET_MAG_UL_C +#endif + +#if defined(BN_MP_GET_LL_C) +# define BN_MP_GET_MAG_ULL_C +#endif + +#if defined(BN_MP_GET_MAG_U32_C) +#endif + +#if defined(BN_MP_GET_MAG_U64_C) +#endif + +#if defined(BN_MP_GET_MAG_UL_C) +#endif + +#if defined(BN_MP_GET_MAG_ULL_C) #endif #if defined(BN_MP_GROW_C) #endif +#if defined(BN_MP_INCR_C) +# define BN_MP_ADD_D_C +# define BN_MP_DECR_C +# define BN_MP_SET_C +#endif + #if defined(BN_MP_INIT_C) #endif #if defined(BN_MP_INIT_COPY_C) - #define BN_MP_COPY_C +# define BN_MP_CLEAR_C +# define BN_MP_COPY_C +# define BN_MP_INIT_SIZE_C +#endif + +#if defined(BN_MP_INIT_I32_C) +# define BN_MP_INIT_C +# define BN_MP_SET_I32_C +#endif + +#if defined(BN_MP_INIT_I64_C) +# define BN_MP_INIT_C +# define BN_MP_SET_I64_C +#endif + +#if defined(BN_MP_INIT_L_C) +# define BN_MP_INIT_C +# define BN_MP_SET_L_C +#endif + +#if defined(BN_MP_INIT_LL_C) +# define BN_MP_INIT_C +# define BN_MP_SET_LL_C #endif #if defined(BN_MP_INIT_MULTI_C) - #define BN_MP_ERR_C - #define BN_MP_INIT_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_C #endif #if defined(BN_MP_INIT_SET_C) - #define BN_MP_INIT_C - #define BN_MP_SET_C -#endif - -#if defined(BN_MP_INIT_SET_INT_C) - #define BN_MP_INIT_C - #define BN_MP_SET_INT_C +# define BN_MP_INIT_C +# define BN_MP_SET_C #endif #if defined(BN_MP_INIT_SIZE_C) - #define BN_MP_INIT_C +#endif + +#if defined(BN_MP_INIT_U32_C) +# define BN_MP_INIT_C +# define BN_MP_SET_U32_C +#endif + +#if defined(BN_MP_INIT_U64_C) +# define BN_MP_INIT_C +# define BN_MP_SET_U64_C +#endif + +#if defined(BN_MP_INIT_UL_C) +# define BN_MP_INIT_C +# define BN_MP_SET_UL_C +#endif + +#if defined(BN_MP_INIT_ULL_C) +# define BN_MP_INIT_C +# define BN_MP_SET_ULL_C #endif #if defined(BN_MP_INVMOD_C) - #define BN_MP_ISZERO_C - #define BN_MP_ISODD_C - #define BN_FAST_MP_INVMOD_C - #define BN_MP_INVMOD_SLOW_C -#endif - -#if defined(BN_MP_INVMOD_SLOW_C) - #define BN_MP_ISZERO_C - #define BN_MP_INIT_MULTI_C - #define BN_MP_MOD_C - #define BN_MP_COPY_C - #define BN_MP_ISEVEN_C - #define BN_MP_SET_C - #define BN_MP_DIV_2_C - #define BN_MP_ISODD_C - #define BN_MP_ADD_C - #define BN_MP_SUB_C - #define BN_MP_CMP_C - #define BN_MP_CMP_D_C - #define BN_MP_CMP_MAG_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_D_C +# define BN_S_MP_INVMOD_FAST_C +# define BN_S_MP_INVMOD_SLOW_C #endif #if defined(BN_MP_IS_SQUARE_C) - #define BN_MP_MOD_D_C - #define BN_MP_INIT_SET_INT_C - #define BN_MP_MOD_C - #define BN_MP_GET_INT_C - #define BN_MP_SQRT_C - #define BN_MP_SQR_C - #define BN_MP_CMP_MAG_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_MAG_C +# define BN_MP_GET_I32_C +# define BN_MP_INIT_U32_C +# define BN_MP_MOD_C +# define BN_MP_MOD_D_C +# define BN_MP_SQRT_C +# define BN_MP_SQR_C #endif -#if defined(BN_MP_JACOBI_C) - #define BN_MP_CMP_D_C - #define BN_MP_ISZERO_C - #define BN_MP_INIT_COPY_C - #define BN_MP_CNT_LSB_C - #define BN_MP_DIV_2D_C - #define BN_MP_MOD_C - #define BN_MP_CLEAR_C +#if defined(BN_MP_ISEVEN_C) #endif -#if defined(BN_MP_KARATSUBA_MUL_C) - #define BN_MP_MUL_C - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_SUB_C - #define BN_MP_ADD_C - #define BN_MP_LSHD_C - #define BN_MP_CLEAR_C +#if defined(BN_MP_ISODD_C) #endif -#if defined(BN_MP_KARATSUBA_SQR_C) - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_SQR_C - #define BN_MP_SUB_C - #define BN_S_MP_ADD_C - #define BN_MP_LSHD_C - #define BN_MP_ADD_C - #define BN_MP_CLEAR_C +#if defined(BN_MP_KRONECKER_C) +# define BN_MP_CLEAR_C +# define BN_MP_CMP_D_C +# define BN_MP_CNT_LSB_C +# define BN_MP_COPY_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_C +# define BN_MP_INIT_COPY_C +# define BN_MP_MOD_C #endif #if defined(BN_MP_LCM_C) - #define BN_MP_INIT_MULTI_C - #define BN_MP_GCD_C - #define BN_MP_CMP_MAG_C - #define BN_MP_DIV_C - #define BN_MP_MUL_C - #define BN_MP_CLEAR_MULTI_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_MAG_C +# define BN_MP_DIV_C +# define BN_MP_GCD_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MUL_C +#endif + +#if defined(BN_MP_LOG_U32_C) +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_EXCH_C +# define BN_MP_EXPT_U32_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MUL_C +# define BN_MP_SET_C +# define BN_MP_SQR_C #endif #if defined(BN_MP_LSHD_C) - #define BN_MP_GROW_C - #define BN_MP_RSHD_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_MOD_C) - #define BN_MP_INIT_C - #define BN_MP_DIV_C - #define BN_MP_CLEAR_C - #define BN_MP_ADD_C - #define BN_MP_EXCH_C +# define BN_MP_ADD_C +# define BN_MP_CLEAR_C +# define BN_MP_DIV_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C #endif #if defined(BN_MP_MOD_2D_C) - #define BN_MP_ZERO_C - #define BN_MP_COPY_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_COPY_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_MOD_D_C) - #define BN_MP_DIV_D_C +# define BN_MP_DIV_D_C #endif #if defined(BN_MP_MONTGOMERY_CALC_NORMALIZATION_C) - #define BN_MP_COUNT_BITS_C - #define BN_MP_2EXPT_C - #define BN_MP_SET_C - #define BN_MP_MUL_2_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C +# define BN_MP_2EXPT_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_MUL_2_C +# define BN_MP_SET_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_MONTGOMERY_REDUCE_C) - #define BN_FAST_MP_MONTGOMERY_REDUCE_C - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C - #define BN_MP_RSHD_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C +# define BN_MP_CLAMP_C +# define BN_MP_CMP_MAG_C +# define BN_MP_GROW_C +# define BN_MP_RSHD_C +# define BN_S_MP_MONTGOMERY_REDUCE_FAST_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_MONTGOMERY_SETUP_C) #endif #if defined(BN_MP_MUL_C) - #define BN_MP_TOOM_MUL_C - #define BN_MP_KARATSUBA_MUL_C - #define BN_FAST_S_MP_MUL_DIGS_C - #define BN_S_MP_MUL_C - #define BN_S_MP_MUL_DIGS_C +# define BN_S_MP_BALANCE_MUL_C +# define BN_S_MP_KARATSUBA_MUL_C +# define BN_S_MP_MUL_DIGS_C +# define BN_S_MP_MUL_DIGS_FAST_C +# define BN_S_MP_TOOM_MUL_C #endif #if defined(BN_MP_MUL_2_C) - #define BN_MP_GROW_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_MUL_2D_C) - #define BN_MP_COPY_C - #define BN_MP_GROW_C - #define BN_MP_LSHD_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_COPY_C +# define BN_MP_GROW_C +# define BN_MP_LSHD_C #endif #if defined(BN_MP_MUL_D_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_MULMOD_C) - #define BN_MP_INIT_C - #define BN_MP_MUL_C - #define BN_MP_CLEAR_C - #define BN_MP_MOD_C -#endif - -#if defined(BN_MP_N_ROOT_C) - #define BN_MP_INIT_C - #define BN_MP_SET_C - #define BN_MP_COPY_C - #define BN_MP_EXPT_D_C - #define BN_MP_MUL_C - #define BN_MP_SUB_C - #define BN_MP_MUL_D_C - #define BN_MP_DIV_C - #define BN_MP_CMP_C - #define BN_MP_SUB_D_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_MOD_C +# define BN_MP_MUL_C #endif #if defined(BN_MP_NEG_C) - #define BN_MP_COPY_C - #define BN_MP_ISZERO_C +# define BN_MP_COPY_C #endif #if defined(BN_MP_OR_C) - #define BN_MP_INIT_COPY_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C +#endif + +#if defined(BN_MP_PACK_C) +# define BN_MP_CLEAR_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_COPY_C +# define BN_MP_PACK_COUNT_C +#endif + +#if defined(BN_MP_PACK_COUNT_C) +# define BN_MP_COUNT_BITS_C #endif #if defined(BN_MP_PRIME_FERMAT_C) - #define BN_MP_CMP_D_C - #define BN_MP_INIT_C - #define BN_MP_EXPTMOD_C - #define BN_MP_CMP_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_EXPTMOD_C +# define BN_MP_INIT_C #endif -#if defined(BN_MP_PRIME_IS_DIVISIBLE_C) - #define BN_MP_MOD_D_C +#if defined(BN_MP_PRIME_FROBENIUS_UNDERWOOD_C) +# define BN_MP_ADD_C +# define BN_MP_ADD_D_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_EXCH_C +# define BN_MP_GCD_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_KRONECKER_C +# define BN_MP_MOD_C +# define BN_MP_MUL_2_C +# define BN_MP_MUL_C +# define BN_MP_MUL_D_C +# define BN_MP_SET_C +# define BN_MP_SET_U32_C +# define BN_MP_SQR_C +# define BN_MP_SUB_C +# define BN_MP_SUB_D_C +# define BN_S_MP_GET_BIT_C #endif #if defined(BN_MP_PRIME_IS_PRIME_C) - #define BN_MP_CMP_D_C - #define BN_MP_PRIME_IS_DIVISIBLE_C - #define BN_MP_INIT_C - #define BN_MP_SET_C - #define BN_MP_PRIME_MILLER_RABIN_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_SET_C +# define BN_MP_IS_SQUARE_C +# define BN_MP_PRIME_MILLER_RABIN_C +# define BN_MP_PRIME_STRONG_LUCAS_SELFRIDGE_C +# define BN_MP_RAND_C +# define BN_MP_READ_RADIX_C +# define BN_MP_SET_C +# define BN_S_MP_PRIME_IS_DIVISIBLE_C #endif #if defined(BN_MP_PRIME_MILLER_RABIN_C) - #define BN_MP_CMP_D_C - #define BN_MP_INIT_COPY_C - #define BN_MP_SUB_D_C - #define BN_MP_CNT_LSB_C - #define BN_MP_DIV_2D_C - #define BN_MP_EXPTMOD_C - #define BN_MP_CMP_C - #define BN_MP_SQRMOD_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_CNT_LSB_C +# define BN_MP_DIV_2D_C +# define BN_MP_EXPTMOD_C +# define BN_MP_INIT_C +# define BN_MP_INIT_COPY_C +# define BN_MP_SQRMOD_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_PRIME_NEXT_PRIME_C) - #define BN_MP_CMP_D_C - #define BN_MP_SET_C - #define BN_MP_SUB_D_C - #define BN_MP_ISEVEN_C - #define BN_MP_MOD_D_C - #define BN_MP_INIT_C - #define BN_MP_ADD_D_C - #define BN_MP_PRIME_MILLER_RABIN_C - #define BN_MP_CLEAR_C +# define BN_MP_ADD_D_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_D_C +# define BN_MP_INIT_C +# define BN_MP_MOD_D_C +# define BN_MP_PRIME_IS_PRIME_C +# define BN_MP_SET_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_PRIME_RABIN_MILLER_TRIALS_C) #endif -#if defined(BN_MP_PRIME_RANDOM_EX_C) - #define BN_MP_READ_UNSIGNED_BIN_C - #define BN_MP_PRIME_IS_PRIME_C - #define BN_MP_SUB_D_C - #define BN_MP_DIV_2_C - #define BN_MP_MUL_2_C - #define BN_MP_ADD_D_C +#if defined(BN_MP_PRIME_RAND_C) +# define BN_MP_ADD_D_C +# define BN_MP_DIV_2_C +# define BN_MP_FROM_UBIN_C +# define BN_MP_MUL_2_C +# define BN_MP_PRIME_IS_PRIME_C +# define BN_MP_SUB_D_C +# define BN_S_MP_PRIME_RANDOM_EX_C +# define BN_S_MP_RAND_CB_C +# define BN_S_MP_RAND_SOURCE_C +#endif + +#if defined(BN_MP_PRIME_STRONG_LUCAS_SELFRIDGE_C) +# define BN_MP_ADD_C +# define BN_MP_ADD_D_C +# define BN_MP_CLEAR_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_CNT_LSB_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_2D_C +# define BN_MP_DIV_2_C +# define BN_MP_GCD_C +# define BN_MP_INIT_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_KRONECKER_C +# define BN_MP_MOD_C +# define BN_MP_MUL_2_C +# define BN_MP_MUL_C +# define BN_MP_SET_C +# define BN_MP_SET_I32_C +# define BN_MP_SET_U32_C +# define BN_MP_SQR_C +# define BN_MP_SUB_C +# define BN_MP_SUB_D_C +# define BN_S_MP_GET_BIT_C +# define BN_S_MP_MUL_SI_C #endif #if defined(BN_MP_RADIX_SIZE_C) - #define BN_MP_COUNT_BITS_C - #define BN_MP_INIT_COPY_C - #define BN_MP_ISZERO_C - #define BN_MP_DIV_D_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_D_C +# define BN_MP_INIT_COPY_C #endif #if defined(BN_MP_RADIX_SMAP_C) - #define BN_MP_S_RMAP_C #endif #if defined(BN_MP_RAND_C) - #define BN_MP_ZERO_C - #define BN_MP_ADD_D_C - #define BN_MP_LSHD_C +# define BN_MP_GROW_C +# define BN_MP_RAND_SOURCE_C +# define BN_MP_ZERO_C +# define BN_S_MP_RAND_PLATFORM_C +# define BN_S_MP_RAND_SOURCE_C #endif #if defined(BN_MP_READ_RADIX_C) - #define BN_MP_ZERO_C - #define BN_MP_S_RMAP_C - #define BN_MP_RADIX_SMAP_C - #define BN_MP_MUL_D_C - #define BN_MP_ADD_D_C - #define BN_MP_ISZERO_C -#endif - -#if defined(BN_MP_READ_SIGNED_BIN_C) - #define BN_MP_READ_UNSIGNED_BIN_C -#endif - -#if defined(BN_MP_READ_UNSIGNED_BIN_C) - #define BN_MP_GROW_C - #define BN_MP_ZERO_C - #define BN_MP_MUL_2D_C - #define BN_MP_CLAMP_C +# define BN_MP_ADD_D_C +# define BN_MP_MUL_D_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_REDUCE_C) - #define BN_MP_REDUCE_SETUP_C - #define BN_MP_INIT_COPY_C - #define BN_MP_RSHD_C - #define BN_MP_MUL_C - #define BN_S_MP_MUL_HIGH_DIGS_C - #define BN_FAST_S_MP_MUL_HIGH_DIGS_C - #define BN_MP_MOD_2D_C - #define BN_S_MP_MUL_DIGS_C - #define BN_MP_SUB_C - #define BN_MP_CMP_D_C - #define BN_MP_SET_C - #define BN_MP_LSHD_C - #define BN_MP_ADD_C - #define BN_MP_CMP_C - #define BN_S_MP_SUB_C - #define BN_MP_CLEAR_C +# define BN_MP_ADD_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_INIT_COPY_C +# define BN_MP_LSHD_C +# define BN_MP_MOD_2D_C +# define BN_MP_MUL_C +# define BN_MP_RSHD_C +# define BN_MP_SET_C +# define BN_MP_SUB_C +# define BN_S_MP_MUL_DIGS_C +# define BN_S_MP_MUL_HIGH_DIGS_C +# define BN_S_MP_MUL_HIGH_DIGS_FAST_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_REDUCE_2K_C) - #define BN_MP_INIT_C - #define BN_MP_COUNT_BITS_C - #define BN_MP_DIV_2D_C - #define BN_MP_MUL_D_C - #define BN_S_MP_ADD_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_C +# define BN_MP_MUL_D_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_REDUCE_2K_L_C) - #define BN_MP_INIT_C - #define BN_MP_COUNT_BITS_C - #define BN_MP_DIV_2D_C - #define BN_MP_MUL_C - #define BN_S_MP_ADD_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C - #define BN_MP_CLEAR_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_C +# define BN_MP_MUL_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_REDUCE_2K_SETUP_C) - #define BN_MP_INIT_C - #define BN_MP_COUNT_BITS_C - #define BN_MP_2EXPT_C - #define BN_MP_CLEAR_C - #define BN_S_MP_SUB_C +# define BN_MP_2EXPT_C +# define BN_MP_CLEAR_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_INIT_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_REDUCE_2K_SETUP_L_C) - #define BN_MP_INIT_C - #define BN_MP_2EXPT_C - #define BN_MP_COUNT_BITS_C - #define BN_S_MP_SUB_C - #define BN_MP_CLEAR_C +# define BN_MP_2EXPT_C +# define BN_MP_CLEAR_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_INIT_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_REDUCE_IS_2K_C) - #define BN_MP_REDUCE_2K_C - #define BN_MP_COUNT_BITS_C +# define BN_MP_COUNT_BITS_C #endif #if defined(BN_MP_REDUCE_IS_2K_L_C) #endif #if defined(BN_MP_REDUCE_SETUP_C) - #define BN_MP_2EXPT_C - #define BN_MP_DIV_C +# define BN_MP_2EXPT_C +# define BN_MP_DIV_C +#endif + +#if defined(BN_MP_ROOT_U32_C) +# define BN_MP_2EXPT_C +# define BN_MP_ADD_D_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DIV_C +# define BN_MP_EXCH_C +# define BN_MP_EXPT_U32_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MUL_C +# define BN_MP_MUL_D_C +# define BN_MP_SET_C +# define BN_MP_SUB_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_RSHD_C) - #define BN_MP_ZERO_C +# define BN_MP_ZERO_C +#endif + +#if defined(BN_MP_SBIN_SIZE_C) +# define BN_MP_UBIN_SIZE_C #endif #if defined(BN_MP_SET_C) - #define BN_MP_ZERO_C #endif -#if defined(BN_MP_SET_INT_C) - #define BN_MP_ZERO_C - #define BN_MP_MUL_2D_C - #define BN_MP_CLAMP_C +#if defined(BN_MP_SET_DOUBLE_C) +# define BN_MP_DIV_2D_C +# define BN_MP_MUL_2D_C +# define BN_MP_SET_U64_C +#endif + +#if defined(BN_MP_SET_I32_C) +# define BN_MP_SET_U32_C +#endif + +#if defined(BN_MP_SET_I64_C) +# define BN_MP_SET_U64_C +#endif + +#if defined(BN_MP_SET_L_C) +# define BN_MP_SET_UL_C +#endif + +#if defined(BN_MP_SET_LL_C) +# define BN_MP_SET_ULL_C +#endif + +#if defined(BN_MP_SET_U32_C) +#endif + +#if defined(BN_MP_SET_U64_C) +#endif + +#if defined(BN_MP_SET_UL_C) +#endif + +#if defined(BN_MP_SET_ULL_C) #endif #if defined(BN_MP_SHRINK_C) #endif -#if defined(BN_MP_SIGNED_BIN_SIZE_C) - #define BN_MP_UNSIGNED_BIN_SIZE_C +#if defined(BN_MP_SIGNED_RSH_C) +# define BN_MP_ADD_D_C +# define BN_MP_DIV_2D_C +# define BN_MP_SUB_D_C #endif #if defined(BN_MP_SQR_C) - #define BN_MP_TOOM_SQR_C - #define BN_MP_KARATSUBA_SQR_C - #define BN_FAST_S_MP_SQR_C - #define BN_S_MP_SQR_C +# define BN_S_MP_KARATSUBA_SQR_C +# define BN_S_MP_SQR_C +# define BN_S_MP_SQR_FAST_C +# define BN_S_MP_TOOM_SQR_C #endif #if defined(BN_MP_SQRMOD_C) - #define BN_MP_INIT_C - #define BN_MP_SQR_C - #define BN_MP_CLEAR_C - #define BN_MP_MOD_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_C +# define BN_MP_MOD_C +# define BN_MP_SQR_C #endif #if defined(BN_MP_SQRT_C) - #define BN_MP_N_ROOT_C - #define BN_MP_ISZERO_C - #define BN_MP_ZERO_C - #define BN_MP_INIT_COPY_C - #define BN_MP_RSHD_C - #define BN_MP_DIV_C - #define BN_MP_ADD_C - #define BN_MP_DIV_2_C - #define BN_MP_CMP_MAG_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_ADD_C +# define BN_MP_CLEAR_C +# define BN_MP_CMP_MAG_C +# define BN_MP_DIV_2_C +# define BN_MP_DIV_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_C +# define BN_MP_INIT_COPY_C +# define BN_MP_RSHD_C +# define BN_MP_ZERO_C +#endif + +#if defined(BN_MP_SQRTMOD_PRIME_C) +# define BN_MP_ADD_D_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_D_C +# define BN_MP_COPY_C +# define BN_MP_DIV_2_C +# define BN_MP_EXPTMOD_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_KRONECKER_C +# define BN_MP_MOD_D_C +# define BN_MP_MULMOD_C +# define BN_MP_SET_C +# define BN_MP_SET_U32_C +# define BN_MP_SQRMOD_C +# define BN_MP_SUB_D_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_SUB_C) - #define BN_S_MP_ADD_C - #define BN_MP_CMP_MAG_C - #define BN_S_MP_SUB_C +# define BN_MP_CMP_MAG_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C #endif #if defined(BN_MP_SUB_D_C) - #define BN_MP_GROW_C - #define BN_MP_ADD_D_C - #define BN_MP_CLAMP_C +# define BN_MP_ADD_D_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_SUBMOD_C) - #define BN_MP_INIT_C - #define BN_MP_SUB_C - #define BN_MP_CLEAR_C - #define BN_MP_MOD_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_C +# define BN_MP_MOD_C +# define BN_MP_SUB_C #endif -#if defined(BN_MP_TO_SIGNED_BIN_C) - #define BN_MP_TO_UNSIGNED_BIN_C +#if defined(BN_MP_TO_RADIX_C) +# define BN_MP_CLEAR_C +# define BN_MP_DIV_D_C +# define BN_MP_INIT_COPY_C +# define BN_S_MP_REVERSE_C #endif -#if defined(BN_MP_TO_SIGNED_BIN_N_C) - #define BN_MP_SIGNED_BIN_SIZE_C - #define BN_MP_TO_SIGNED_BIN_C +#if defined(BN_MP_TO_SBIN_C) +# define BN_MP_TO_UBIN_C #endif -#if defined(BN_MP_TO_UNSIGNED_BIN_C) - #define BN_MP_INIT_COPY_C - #define BN_MP_ISZERO_C - #define BN_MP_DIV_2D_C - #define BN_MP_CLEAR_C +#if defined(BN_MP_TO_UBIN_C) +# define BN_MP_CLEAR_C +# define BN_MP_DIV_2D_C +# define BN_MP_INIT_COPY_C +# define BN_MP_UBIN_SIZE_C #endif -#if defined(BN_MP_TO_UNSIGNED_BIN_N_C) - #define BN_MP_UNSIGNED_BIN_SIZE_C - #define BN_MP_TO_UNSIGNED_BIN_C +#if defined(BN_MP_UBIN_SIZE_C) +# define BN_MP_COUNT_BITS_C #endif -#if defined(BN_MP_TOOM_MUL_C) - #define BN_MP_INIT_MULTI_C - #define BN_MP_MOD_2D_C - #define BN_MP_COPY_C - #define BN_MP_RSHD_C - #define BN_MP_MUL_C - #define BN_MP_MUL_2_C - #define BN_MP_ADD_C - #define BN_MP_SUB_C - #define BN_MP_DIV_2_C - #define BN_MP_MUL_2D_C - #define BN_MP_MUL_D_C - #define BN_MP_DIV_3_C - #define BN_MP_LSHD_C - #define BN_MP_CLEAR_MULTI_C -#endif - -#if defined(BN_MP_TOOM_SQR_C) - #define BN_MP_INIT_MULTI_C - #define BN_MP_MOD_2D_C - #define BN_MP_COPY_C - #define BN_MP_RSHD_C - #define BN_MP_SQR_C - #define BN_MP_MUL_2_C - #define BN_MP_ADD_C - #define BN_MP_SUB_C - #define BN_MP_DIV_2_C - #define BN_MP_MUL_2D_C - #define BN_MP_MUL_D_C - #define BN_MP_DIV_3_C - #define BN_MP_LSHD_C - #define BN_MP_CLEAR_MULTI_C -#endif - -#if defined(BN_MP_TORADIX_C) - #define BN_MP_ISZERO_C - #define BN_MP_INIT_COPY_C - #define BN_MP_DIV_D_C - #define BN_MP_CLEAR_C - #define BN_MP_S_RMAP_C -#endif - -#if defined(BN_MP_TORADIX_N_C) - #define BN_MP_ISZERO_C - #define BN_MP_INIT_COPY_C - #define BN_MP_DIV_D_C - #define BN_MP_CLEAR_C - #define BN_MP_S_RMAP_C -#endif - -#if defined(BN_MP_UNSIGNED_BIN_SIZE_C) - #define BN_MP_COUNT_BITS_C +#if defined(BN_MP_UNPACK_C) +# define BN_MP_CLAMP_C +# define BN_MP_MUL_2D_C +# define BN_MP_ZERO_C #endif #if defined(BN_MP_XOR_C) - #define BN_MP_INIT_COPY_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_MP_ZERO_C) @@ -933,70 +1104,216 @@ #if defined(BN_PRIME_TAB_C) #endif -#if defined(BN_REVERSE_C) +#if defined(BN_S_MP_ADD_C) +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif -#if defined(BN_S_MP_ADD_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C +#if defined(BN_S_MP_BALANCE_MUL_C) +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_MUL_C #endif #if defined(BN_S_MP_EXPTMOD_C) - #define BN_MP_COUNT_BITS_C - #define BN_MP_INIT_C - #define BN_MP_CLEAR_C - #define BN_MP_REDUCE_SETUP_C - #define BN_MP_REDUCE_C - #define BN_MP_REDUCE_2K_SETUP_L_C - #define BN_MP_REDUCE_2K_L_C - #define BN_MP_MOD_C - #define BN_MP_COPY_C - #define BN_MP_SQR_C - #define BN_MP_MUL_C - #define BN_MP_SET_C - #define BN_MP_EXCH_C +# define BN_MP_CLEAR_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_C +# define BN_MP_MOD_C +# define BN_MP_MUL_C +# define BN_MP_REDUCE_2K_L_C +# define BN_MP_REDUCE_2K_SETUP_L_C +# define BN_MP_REDUCE_C +# define BN_MP_REDUCE_SETUP_C +# define BN_MP_SET_C +# define BN_MP_SQR_C +#endif + +#if defined(BN_S_MP_EXPTMOD_FAST_C) +# define BN_MP_CLEAR_C +# define BN_MP_COPY_C +# define BN_MP_COUNT_BITS_C +# define BN_MP_DR_REDUCE_C +# define BN_MP_DR_SETUP_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_MOD_C +# define BN_MP_MONTGOMERY_CALC_NORMALIZATION_C +# define BN_MP_MONTGOMERY_REDUCE_C +# define BN_MP_MONTGOMERY_SETUP_C +# define BN_MP_MULMOD_C +# define BN_MP_MUL_C +# define BN_MP_REDUCE_2K_C +# define BN_MP_REDUCE_2K_SETUP_C +# define BN_MP_SET_C +# define BN_MP_SQR_C +# define BN_S_MP_MONTGOMERY_REDUCE_FAST_C +#endif + +#if defined(BN_S_MP_GET_BIT_C) +#endif + +#if defined(BN_S_MP_INVMOD_FAST_C) +# define BN_MP_ADD_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COPY_C +# define BN_MP_DIV_2_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MOD_C +# define BN_MP_SET_C +# define BN_MP_SUB_C +#endif + +#if defined(BN_S_MP_INVMOD_SLOW_C) +# define BN_MP_ADD_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_CMP_C +# define BN_MP_CMP_D_C +# define BN_MP_CMP_MAG_C +# define BN_MP_COPY_C +# define BN_MP_DIV_2_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_MOD_C +# define BN_MP_SET_C +# define BN_MP_SUB_C +#endif + +#if defined(BN_S_MP_KARATSUBA_MUL_C) +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_MUL_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C +#endif + +#if defined(BN_S_MP_KARATSUBA_SQR_C) +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_SQR_C +# define BN_S_MP_ADD_C +# define BN_S_MP_SUB_C +#endif + +#if defined(BN_S_MP_MONTGOMERY_REDUCE_FAST_C) +# define BN_MP_CLAMP_C +# define BN_MP_CMP_MAG_C +# define BN_MP_GROW_C +# define BN_S_MP_SUB_C #endif #if defined(BN_S_MP_MUL_DIGS_C) - #define BN_FAST_S_MP_MUL_DIGS_C - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C +# define BN_S_MP_MUL_DIGS_FAST_C +#endif + +#if defined(BN_S_MP_MUL_DIGS_FAST_C) +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_S_MP_MUL_HIGH_DIGS_C) - #define BN_FAST_S_MP_MUL_HIGH_DIGS_C - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C +# define BN_S_MP_MUL_HIGH_DIGS_FAST_C +#endif + +#if defined(BN_S_MP_MUL_HIGH_DIGS_FAST_C) +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C +#endif + +#if defined(BN_S_MP_PRIME_IS_DIVISIBLE_C) +# define BN_MP_MOD_D_C +#endif + +#if defined(BN_S_MP_RAND_JENKINS_C) +# define BN_S_MP_RAND_JENKINS_INIT_C +#endif + +#if defined(BN_S_MP_RAND_PLATFORM_C) +#endif + +#if defined(BN_S_MP_REVERSE_C) #endif #if defined(BN_S_MP_SQR_C) - #define BN_MP_INIT_SIZE_C - #define BN_MP_CLAMP_C - #define BN_MP_EXCH_C - #define BN_MP_CLEAR_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_EXCH_C +# define BN_MP_INIT_SIZE_C +#endif + +#if defined(BN_S_MP_SQR_FAST_C) +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif #if defined(BN_S_MP_SUB_C) - #define BN_MP_GROW_C - #define BN_MP_CLAMP_C +# define BN_MP_CLAMP_C +# define BN_MP_GROW_C #endif -#if defined(BNCORE_C) +#if defined(BN_S_MP_TOOM_MUL_C) +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_DIV_2_C +# define BN_MP_DIV_3_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_MUL_2_C +# define BN_MP_MUL_C +# define BN_MP_SUB_C #endif +#if defined(BN_S_MP_TOOM_SQR_C) +# define BN_MP_ADD_C +# define BN_MP_CLAMP_C +# define BN_MP_CLEAR_C +# define BN_MP_DIV_2_C +# define BN_MP_INIT_C +# define BN_MP_INIT_SIZE_C +# define BN_MP_LSHD_C +# define BN_MP_MUL_2_C +# define BN_MP_MUL_C +# define BN_MP_SQR_C +# define BN_MP_SUB_C +#endif + +#ifdef LTM_INSIDE +#undef LTM_INSIDE #ifdef LTM3 -#define LTM_LAST -#endif -#include -#include -#else -#define LTM_LAST +# define LTM_LAST #endif -/* $Source: /cvs/libtom/libtommath/tommath_class.h,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2005/07/28 11:59:32 $ */ +#include "tommath_superclass.h" +#include "tommath_class.h" +#else +# define LTM_LAST +#endif diff --git a/lib/hcrypto/libtommath/tommath_cutoffs.h b/lib/hcrypto/libtommath/tommath_cutoffs.h new file mode 100644 index 000000000..a65a9b3e3 --- /dev/null +++ b/lib/hcrypto/libtommath/tommath_cutoffs.h @@ -0,0 +1,13 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ +/* + Current values evaluated on an AMD A8-6600K (64-bit). + Type "make tune" to optimize them for your machine but + be aware that it may take a long time. It took 2:30 minutes + on the aforementioned machine for example. + */ + +#define MP_DEFAULT_KARATSUBA_MUL_CUTOFF 80 +#define MP_DEFAULT_KARATSUBA_SQR_CUTOFF 120 +#define MP_DEFAULT_TOOM_MUL_CUTOFF 350 +#define MP_DEFAULT_TOOM_SQR_CUTOFF 400 diff --git a/lib/hcrypto/libtommath/tommath_private.h b/lib/hcrypto/libtommath/tommath_private.h new file mode 100644 index 000000000..1a0096f85 --- /dev/null +++ b/lib/hcrypto/libtommath/tommath_private.h @@ -0,0 +1,303 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + +#ifndef TOMMATH_PRIV_H_ +#define TOMMATH_PRIV_H_ + +#include "tommath.h" +#include "tommath_class.h" + +/* + * Private symbols + * --------------- + * + * On Unix symbols can be marked as hidden if libtommath is compiled + * as a shared object. By default, symbols are visible. + * As of now, this feature is opt-in via the MP_PRIVATE_SYMBOLS define. + * + * On Win32 a .def file must be used to specify the exported symbols. + */ +#if defined (MP_PRIVATE_SYMBOLS) && defined(__GNUC__) && __GNUC__ >= 4 +# define MP_PRIVATE __attribute__ ((visibility ("hidden"))) +#else +# define MP_PRIVATE +#endif + +/* Hardening libtommath + * -------------------- + * + * By default memory is zeroed before calling + * MP_FREE to avoid leaking data. This is good + * practice in cryptographical applications. + * + * Note however that memory allocators used + * in cryptographical applications can often + * be configured by itself to clear memory, + * rendering the clearing in tommath unnecessary. + * See for example https://github.com/GrapheneOS/hardened_malloc + * and the option CONFIG_ZERO_ON_FREE. + * + * Furthermore there are applications which + * value performance more and want this + * feature to be disabled. For such applications + * define MP_NO_ZERO_ON_FREE during compilation. + */ +#ifdef MP_NO_ZERO_ON_FREE +# define MP_FREE_BUFFER(mem, size) MP_FREE((mem), (size)) +# define MP_FREE_DIGITS(mem, digits) MP_FREE((mem), sizeof (mp_digit) * (size_t)(digits)) +#else +# define MP_FREE_BUFFER(mem, size) \ +do { \ + size_t fs_ = (size); \ + void* fm_ = (mem); \ + if (fm_ != NULL) { \ + MP_ZERO_BUFFER(fm_, fs_); \ + MP_FREE(fm_, fs_); \ + } \ +} while (0) +# define MP_FREE_DIGITS(mem, digits) \ +do { \ + int fd_ = (digits); \ + void* fm_ = (mem); \ + if (fm_ != NULL) { \ + size_t fs_ = sizeof (mp_digit) * (size_t)fd_; \ + MP_ZERO_BUFFER(fm_, fs_); \ + MP_FREE(fm_, fs_); \ + } \ +} while (0) +#endif + +#ifdef MP_USE_MEMSET +# include +# define MP_ZERO_BUFFER(mem, size) memset((mem), 0, (size)) +# define MP_ZERO_DIGITS(mem, digits) \ +do { \ + int zd_ = (digits); \ + if (zd_ > 0) { \ + memset((mem), 0, sizeof(mp_digit) * (size_t)zd_); \ + } \ +} while (0) +#else +# define MP_ZERO_BUFFER(mem, size) \ +do { \ + size_t zs_ = (size); \ + char* zm_ = (char*)(mem); \ + while (zs_-- > 0u) { \ + *zm_++ = '\0'; \ + } \ +} while (0) +# define MP_ZERO_DIGITS(mem, digits) \ +do { \ + int zd_ = (digits); \ + mp_digit* zm_ = (mem); \ + while (zd_-- > 0) { \ + *zm_++ = 0; \ + } \ +} while (0) +#endif + +/* Tunable cutoffs + * --------------- + * + * - In the default settings, a cutoff X can be modified at runtime + * by adjusting the corresponding X_CUTOFF variable. + * + * - Tunability of the library can be disabled at compile time + * by defining the MP_FIXED_CUTOFFS macro. + * + * - There is an additional file tommath_cutoffs.h, which defines + * the default cutoffs. These can be adjusted manually or by the + * autotuner. + * + */ + +#ifdef MP_FIXED_CUTOFFS +# include "tommath_cutoffs.h" +# define MP_KARATSUBA_MUL_CUTOFF MP_DEFAULT_KARATSUBA_MUL_CUTOFF +# define MP_KARATSUBA_SQR_CUTOFF MP_DEFAULT_KARATSUBA_SQR_CUTOFF +# define MP_TOOM_MUL_CUTOFF MP_DEFAULT_TOOM_MUL_CUTOFF +# define MP_TOOM_SQR_CUTOFF MP_DEFAULT_TOOM_SQR_CUTOFF +#else +# define MP_KARATSUBA_MUL_CUTOFF KARATSUBA_MUL_CUTOFF +# define MP_KARATSUBA_SQR_CUTOFF KARATSUBA_SQR_CUTOFF +# define MP_TOOM_MUL_CUTOFF TOOM_MUL_CUTOFF +# define MP_TOOM_SQR_CUTOFF TOOM_SQR_CUTOFF +#endif + +/* define heap macros */ +#ifndef MP_MALLOC +/* default to libc stuff */ +# include +# define MP_MALLOC(size) malloc(size) +# define MP_REALLOC(mem, oldsize, newsize) realloc((mem), (newsize)) +# define MP_CALLOC(nmemb, size) calloc((nmemb), (size)) +# define MP_FREE(mem, size) free(mem) +#else +/* prototypes for our heap functions */ +extern void *MP_MALLOC(size_t size); +extern void *MP_REALLOC(void *mem, size_t oldsize, size_t newsize); +extern void *MP_CALLOC(size_t nmemb, size_t size); +extern void MP_FREE(void *mem, size_t size); +#endif + +/* feature detection macro */ +#ifdef _MSC_VER +/* Prevent false positive: not enough arguments for function-like macro invocation */ +#pragma warning(disable: 4003) +#endif +#define MP_STRINGIZE(x) MP__STRINGIZE(x) +#define MP__STRINGIZE(x) ""#x"" +#define MP_HAS(x) (sizeof(MP_STRINGIZE(BN_##x##_C)) == 1u) + +/* TODO: Remove private_mp_word as soon as deprecated mp_word is removed from tommath. */ +#undef mp_word +typedef private_mp_word mp_word; + +#define MP_MIN(x, y) (((x) < (y)) ? (x) : (y)) +#define MP_MAX(x, y) (((x) > (y)) ? (x) : (y)) + +/* Static assertion */ +#define MP_STATIC_ASSERT(msg, cond) typedef char mp_static_assert_##msg[(cond) ? 1 : -1]; + +/* ---> Basic Manipulations <--- */ +#define MP_IS_ZERO(a) ((a)->used == 0) +#define MP_IS_EVEN(a) (((a)->used == 0) || (((a)->dp[0] & 1u) == 0u)) +#define MP_IS_ODD(a) (((a)->used > 0) && (((a)->dp[0] & 1u) == 1u)) + +#define MP_SIZEOF_BITS(type) ((size_t)CHAR_BIT * sizeof(type)) +#define MP_MAXFAST (int)(1uL << (MP_SIZEOF_BITS(mp_word) - (2u * (size_t)MP_DIGIT_BIT))) + +/* TODO: Remove PRIVATE_MP_WARRAY as soon as deprecated MP_WARRAY is removed from tommath.h */ +#undef MP_WARRAY +#define MP_WARRAY PRIVATE_MP_WARRAY + +/* TODO: Remove PRIVATE_MP_PREC as soon as deprecated MP_PREC is removed from tommath.h */ +#ifdef PRIVATE_MP_PREC +# undef MP_PREC +# define MP_PREC PRIVATE_MP_PREC +#endif + +/* Minimum number of available digits in mp_int, MP_PREC >= MP_MIN_PREC */ +#define MP_MIN_PREC ((((int)MP_SIZEOF_BITS(long long) + MP_DIGIT_BIT) - 1) / MP_DIGIT_BIT) + +MP_STATIC_ASSERT(prec_geq_min_prec, MP_PREC >= MP_MIN_PREC) + +/* random number source */ +extern MP_PRIVATE mp_err(*s_mp_rand_source)(void *out, size_t size); + +/* lowlevel functions, do not call! */ +MP_PRIVATE mp_bool s_mp_get_bit(const mp_int *a, unsigned int b); +MP_PRIVATE mp_err s_mp_add(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_sub(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_mul_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) MP_WUR; +MP_PRIVATE mp_err s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) MP_WUR; +MP_PRIVATE mp_err s_mp_mul_high_digs_fast(const mp_int *a, const mp_int *b, mp_int *c, int digs) MP_WUR; +MP_PRIVATE mp_err s_mp_mul_high_digs(const mp_int *a, const mp_int *b, mp_int *c, int digs) MP_WUR; +MP_PRIVATE mp_err s_mp_sqr_fast(const mp_int *a, mp_int *b) MP_WUR; +MP_PRIVATE mp_err s_mp_sqr(const mp_int *a, mp_int *b) MP_WUR; +MP_PRIVATE mp_err s_mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_karatsuba_sqr(const mp_int *a, mp_int *b) MP_WUR; +MP_PRIVATE mp_err s_mp_toom_sqr(const mp_int *a, mp_int *b) MP_WUR; +MP_PRIVATE mp_err s_mp_invmod_fast(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c) MP_WUR; +MP_PRIVATE mp_err s_mp_montgomery_reduce_fast(mp_int *x, const mp_int *n, mp_digit rho) MP_WUR; +MP_PRIVATE mp_err s_mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) MP_WUR; +MP_PRIVATE mp_err s_mp_exptmod(const mp_int *G, const mp_int *X, const mp_int *P, mp_int *Y, int redmode) MP_WUR; +MP_PRIVATE mp_err s_mp_rand_platform(void *p, size_t n) MP_WUR; +MP_PRIVATE mp_err s_mp_prime_random_ex(mp_int *a, int t, int size, int flags, private_mp_prime_callback cb, void *dat); +MP_PRIVATE void s_mp_reverse(unsigned char *s, size_t len); +MP_PRIVATE mp_err s_mp_prime_is_divisible(const mp_int *a, mp_bool *result); + +/* TODO: jenkins prng is not thread safe as of now */ +MP_PRIVATE mp_err s_mp_rand_jenkins(void *p, size_t n) MP_WUR; +MP_PRIVATE void s_mp_rand_jenkins_init(uint64_t seed); + +extern MP_PRIVATE const char *const mp_s_rmap; +extern MP_PRIVATE const uint8_t mp_s_rmap_reverse[]; +extern MP_PRIVATE const size_t mp_s_rmap_reverse_sz; +extern MP_PRIVATE const mp_digit *s_mp_prime_tab; + +/* deprecated functions */ +MP_DEPRECATED(s_mp_invmod_fast) mp_err fast_mp_invmod(const mp_int *a, const mp_int *b, mp_int *c); +MP_DEPRECATED(s_mp_montgomery_reduce_fast) mp_err fast_mp_montgomery_reduce(mp_int *x, const mp_int *n, + mp_digit rho); +MP_DEPRECATED(s_mp_mul_digs_fast) mp_err fast_s_mp_mul_digs(const mp_int *a, const mp_int *b, mp_int *c, + int digs); +MP_DEPRECATED(s_mp_mul_high_digs_fast) mp_err fast_s_mp_mul_high_digs(const mp_int *a, const mp_int *b, + mp_int *c, + int digs); +MP_DEPRECATED(s_mp_sqr_fast) mp_err fast_s_mp_sqr(const mp_int *a, mp_int *b); +MP_DEPRECATED(s_mp_balance_mul) mp_err mp_balance_mul(const mp_int *a, const mp_int *b, mp_int *c); +MP_DEPRECATED(s_mp_exptmod_fast) mp_err mp_exptmod_fast(const mp_int *G, const mp_int *X, const mp_int *P, + mp_int *Y, + int redmode); +MP_DEPRECATED(s_mp_invmod_slow) mp_err mp_invmod_slow(const mp_int *a, const mp_int *b, mp_int *c); +MP_DEPRECATED(s_mp_karatsuba_mul) mp_err mp_karatsuba_mul(const mp_int *a, const mp_int *b, mp_int *c); +MP_DEPRECATED(s_mp_karatsuba_sqr) mp_err mp_karatsuba_sqr(const mp_int *a, mp_int *b); +MP_DEPRECATED(s_mp_toom_mul) mp_err mp_toom_mul(const mp_int *a, const mp_int *b, mp_int *c); +MP_DEPRECATED(s_mp_toom_sqr) mp_err mp_toom_sqr(const mp_int *a, mp_int *b); +MP_DEPRECATED(s_mp_reverse) void bn_reverse(unsigned char *s, int len); + +#define MP_GET_ENDIANNESS(x) \ + do{\ + int16_t n = 0x1; \ + char *p = (char *)&n; \ + x = (p[0] == '\x01') ? MP_LITTLE_ENDIAN : MP_BIG_ENDIAN; \ + } while (0) + +/* code-generating macros */ +#define MP_SET_UNSIGNED(name, type) \ + void name(mp_int * a, type b) \ + { \ + int i = 0; \ + while (b != 0u) { \ + a->dp[i++] = ((mp_digit)b & MP_MASK); \ + if (MP_SIZEOF_BITS(type) <= MP_DIGIT_BIT) { break; } \ + b >>= ((MP_SIZEOF_BITS(type) <= MP_DIGIT_BIT) ? 0 : MP_DIGIT_BIT); \ + } \ + a->used = i; \ + a->sign = MP_ZPOS; \ + MP_ZERO_DIGITS(a->dp + a->used, a->alloc - a->used); \ + } + +#define MP_SET_SIGNED(name, uname, type, utype) \ + void name(mp_int * a, type b) \ + { \ + uname(a, (b < 0) ? -(utype)b : (utype)b); \ + if (b < 0) { a->sign = MP_NEG; } \ + } + +#define MP_INIT_INT(name , set, type) \ + mp_err name(mp_int * a, type b) \ + { \ + mp_err err; \ + if ((err = mp_init(a)) != MP_OKAY) { \ + return err; \ + } \ + set(a, b); \ + return MP_OKAY; \ + } + +#define MP_GET_MAG(name, type) \ + type name(const mp_int* a) \ + { \ + unsigned i = MP_MIN((unsigned)a->used, (unsigned)((MP_SIZEOF_BITS(type) + MP_DIGIT_BIT - 1) / MP_DIGIT_BIT)); \ + type res = 0u; \ + while (i --> 0u) { \ + res <<= ((MP_SIZEOF_BITS(type) <= MP_DIGIT_BIT) ? 0 : MP_DIGIT_BIT); \ + res |= (type)a->dp[i]; \ + if (MP_SIZEOF_BITS(type) <= MP_DIGIT_BIT) { break; } \ + } \ + return res; \ + } + +#define MP_GET_SIGNED(name, mag, type, utype) \ + type name(const mp_int* a) \ + { \ + utype res = mag(a); \ + return (a->sign == MP_NEG) ? (type)-res : (type)res; \ + } + +#endif diff --git a/lib/hcrypto/libtommath/tommath_superclass.h b/lib/hcrypto/libtommath/tommath_superclass.h index a96c36feb..d88bce9c7 100644 --- a/lib/hcrypto/libtommath/tommath_superclass.h +++ b/lib/hcrypto/libtommath/tommath_superclass.h @@ -1,10 +1,16 @@ +/* LibTomMath, multiple-precision integer library -- Tom St Denis */ +/* SPDX-License-Identifier: Unlicense */ + /* super class file for PK algos */ /* default ... include all MPI */ +#ifndef LTM_NOTHING #define LTM_ALL +#endif /* RSA only (does not support DH/DSA/ECC) */ /* #define SC_RSA_1 */ +/* #define SC_RSA_1_WITH_TESTS */ /* For reference.... On an Athlon64 optimizing for speed... @@ -12,65 +18,93 @@ */ +#ifdef SC_RSA_1_WITH_TESTS +# define BN_MP_ERROR_TO_STRING_C +# define BN_MP_FREAD_C +# define BN_MP_FWRITE_C +# define BN_MP_INCR_C +# define BN_MP_ISEVEN_C +# define BN_MP_ISODD_C +# define BN_MP_NEG_C +# define BN_MP_PRIME_FROBENIUS_UNDERWOOD_C +# define BN_MP_RADIX_SIZE_C +# define BN_MP_RAND_C +# define BN_MP_REDUCE_C +# define BN_MP_REDUCE_2K_L_C +# define BN_MP_FROM_SBIN_C +# define BN_MP_ROOT_U32_C +# define BN_MP_SET_L_C +# define BN_MP_SET_UL_C +# define BN_MP_SBIN_SIZE_C +# define BN_MP_TO_RADIX_C +# define BN_MP_TO_SBIN_C +# define BN_S_MP_RAND_JENKINS_C +# define BN_S_MP_RAND_PLATFORM_C +#endif + /* Works for RSA only, mpi.o is 68KiB */ -#ifdef SC_RSA_1 - #define BN_MP_SHRINK_C - #define BN_MP_LCM_C - #define BN_MP_PRIME_RANDOM_EX_C - #define BN_MP_INVMOD_C - #define BN_MP_GCD_C - #define BN_MP_MOD_C - #define BN_MP_MULMOD_C - #define BN_MP_ADDMOD_C - #define BN_MP_EXPTMOD_C - #define BN_MP_SET_INT_C - #define BN_MP_INIT_MULTI_C - #define BN_MP_CLEAR_MULTI_C - #define BN_MP_UNSIGNED_BIN_SIZE_C - #define BN_MP_TO_UNSIGNED_BIN_C - #define BN_MP_MOD_D_C - #define BN_MP_PRIME_RABIN_MILLER_TRIALS_C - #define BN_REVERSE_C - #define BN_PRIME_TAB_C +#if defined(SC_RSA_1) || defined (SC_RSA_1_WITH_TESTS) +# define BN_CUTOFFS_C +# define BN_MP_ADDMOD_C +# define BN_MP_CLEAR_MULTI_C +# define BN_MP_EXPTMOD_C +# define BN_MP_GCD_C +# define BN_MP_INIT_MULTI_C +# define BN_MP_INVMOD_C +# define BN_MP_LCM_C +# define BN_MP_MOD_C +# define BN_MP_MOD_D_C +# define BN_MP_MULMOD_C +# define BN_MP_PRIME_IS_PRIME_C +# define BN_MP_PRIME_RABIN_MILLER_TRIALS_C +# define BN_MP_PRIME_RAND_C +# define BN_MP_RADIX_SMAP_C +# define BN_MP_SET_INT_C +# define BN_MP_SHRINK_C +# define BN_MP_TO_UNSIGNED_BIN_C +# define BN_MP_UNSIGNED_BIN_SIZE_C +# define BN_PRIME_TAB_C +# define BN_S_MP_REVERSE_C - /* other modifiers */ - #define BN_MP_DIV_SMALL /* Slower division, not critical */ +/* other modifiers */ +# define BN_MP_DIV_SMALL /* Slower division, not critical */ - /* here we are on the last pass so we turn things off. The functions classes are still there - * but we remove them specifically from the build. This also invokes tweaks in functions - * like removing support for even moduli, etc... - */ -#ifdef LTM_LAST - #undef BN_MP_TOOM_MUL_C - #undef BN_MP_TOOM_SQR_C - #undef BN_MP_KARATSUBA_MUL_C - #undef BN_MP_KARATSUBA_SQR_C - #undef BN_MP_REDUCE_C - #undef BN_MP_REDUCE_SETUP_C - #undef BN_MP_DR_IS_MODULUS_C - #undef BN_MP_DR_SETUP_C - #undef BN_MP_DR_REDUCE_C - #undef BN_MP_REDUCE_IS_2K_C - #undef BN_MP_REDUCE_2K_SETUP_C - #undef BN_MP_REDUCE_2K_C - #undef BN_S_MP_EXPTMOD_C - #undef BN_MP_DIV_3_C - #undef BN_S_MP_MUL_HIGH_DIGS_C - #undef BN_FAST_S_MP_MUL_HIGH_DIGS_C - #undef BN_FAST_MP_INVMOD_C - /* To safely undefine these you have to make sure your RSA key won't exceed the Comba threshold - * which is roughly 255 digits [7140 bits for 32-bit machines, 15300 bits for 64-bit machines] - * which means roughly speaking you can handle upto 2536-bit RSA keys with these defined without - * trouble. - */ - #undef BN_S_MP_MUL_DIGS_C - #undef BN_S_MP_SQR_C - #undef BN_MP_MONTGOMERY_REDUCE_C -#endif +/* here we are on the last pass so we turn things off. The functions classes are still there + * but we remove them specifically from the build. This also invokes tweaks in functions + * like removing support for even moduli, etc... + */ +# ifdef LTM_LAST +# undef BN_MP_DR_IS_MODULUS_C +# undef BN_MP_DR_SETUP_C +# undef BN_MP_DR_REDUCE_C +# undef BN_MP_DIV_3_C +# undef BN_MP_REDUCE_2K_SETUP_C +# undef BN_MP_REDUCE_2K_C +# undef BN_MP_REDUCE_IS_2K_C +# undef BN_MP_REDUCE_SETUP_C +# undef BN_S_MP_BALANCE_MUL_C +# undef BN_S_MP_EXPTMOD_C +# undef BN_S_MP_INVMOD_FAST_C +# undef BN_S_MP_KARATSUBA_MUL_C +# undef BN_S_MP_KARATSUBA_SQR_C +# undef BN_S_MP_MUL_HIGH_DIGS_C +# undef BN_S_MP_MUL_HIGH_DIGS_FAST_C +# undef BN_S_MP_TOOM_MUL_C +# undef BN_S_MP_TOOM_SQR_C + +# ifndef SC_RSA_1_WITH_TESTS +# undef BN_MP_REDUCE_C +# endif + +/* To safely undefine these you have to make sure your RSA key won't exceed the Comba threshold + * which is roughly 255 digits [7140 bits for 32-bit machines, 15300 bits for 64-bit machines] + * which means roughly speaking you can handle upto 2536-bit RSA keys with these defined without + * trouble. + */ +# undef BN_MP_MONTGOMERY_REDUCE_C +# undef BN_S_MP_MUL_DIGS_C +# undef BN_S_MP_SQR_C +# endif #endif - -/* $Source: /cvs/libtom/libtommath/tommath_superclass.h,v $ */ -/* $Revision: 1.3 $ */ -/* $Date: 2005/05/14 13:29:17 $ */