368 lines
11 KiB
C
368 lines
11 KiB
C
#include "tommath_private.h"
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#ifdef BN_MP_PRIME_IS_PRIME_C
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/* LibTomMath, multiple-precision integer library -- Tom St Denis
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*
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* LibTomMath is a library that provides multiple-precision
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* integer arithmetic as well as number theoretic functionality.
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*
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* The library was designed directly after the MPI library by
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* Michael Fromberger but has been written from scratch with
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* additional optimizations in place.
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*
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* SPDX-License-Identifier: Unlicense
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*/
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/* portable integer log of two with small footprint */
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static unsigned int s_floor_ilog2(int value)
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{
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unsigned int r = 0;
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while ((value >>= 1) != 0) {
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r++;
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}
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return r;
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}
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int mp_prime_is_prime(const mp_int *a, int t, int *result)
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{
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mp_int b;
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int ix, err, res, p_max = 0, size_a, len;
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unsigned int fips_rand, mask;
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/* default to no */
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*result = MP_NO;
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/* valid value of t? */
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if (t > PRIME_SIZE) {
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return MP_VAL;
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}
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/* Some shortcuts */
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/* N > 3 */
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if (a->used == 1) {
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if ((a->dp[0] == 0u) || (a->dp[0] == 1u)) {
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*result = 0;
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return MP_OKAY;
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}
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if (a->dp[0] == 2u) {
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*result = 1;
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return MP_OKAY;
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}
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}
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/* N must be odd */
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if (mp_iseven(a) == MP_YES) {
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return MP_OKAY;
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}
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/* N is not a perfect square: floor(sqrt(N))^2 != N */
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if ((err = mp_is_square(a, &res)) != MP_OKAY) {
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return err;
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}
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if (res != 0) {
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return MP_OKAY;
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}
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/* is the input equal to one of the primes in the table? */
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for (ix = 0; ix < PRIME_SIZE; ix++) {
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if (mp_cmp_d(a, ltm_prime_tab[ix]) == MP_EQ) {
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*result = MP_YES;
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return MP_OKAY;
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}
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}
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#ifdef MP_8BIT
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/* The search in the loop above was exhaustive in this case */
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if (a->used == 1 && PRIME_SIZE >= 31) {
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return MP_OKAY;
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}
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#endif
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/* first perform trial division */
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if ((err = mp_prime_is_divisible(a, &res)) != MP_OKAY) {
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return err;
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}
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/* return if it was trivially divisible */
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if (res == MP_YES) {
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return MP_OKAY;
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}
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/*
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Run the Miller-Rabin test with base 2 for the BPSW test.
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*/
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if ((err = mp_init_set(&b, 2uL)) != MP_OKAY) {
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return err;
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}
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if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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/*
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Rumours have it that Mathematica does a second M-R test with base 3.
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Other rumours have it that their strong L-S test is slightly different.
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It does not hurt, though, beside a bit of extra runtime.
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*/
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b.dp[0]++;
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if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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/*
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* Both, the Frobenius-Underwood test and the the Lucas-Selfridge test are quite
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* slow so if speed is an issue, define LTM_USE_FIPS_ONLY to use M-R tests with
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* bases 2, 3 and t random bases.
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*/
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#ifndef LTM_USE_FIPS_ONLY
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if (t >= 0) {
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/*
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* Use a Frobenius-Underwood test instead of the Lucas-Selfridge test for
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* MP_8BIT (It is unknown if the Lucas-Selfridge test works with 16-bit
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* integers but the necesssary analysis is on the todo-list).
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*/
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#if defined (MP_8BIT) || defined (LTM_USE_FROBENIUS_TEST)
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err = mp_prime_frobenius_underwood(a, &res);
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if (err != MP_OKAY && err != MP_ITER) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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#else
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if ((err = mp_prime_strong_lucas_selfridge(a, &res)) != MP_OKAY) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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#endif
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}
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#endif
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/* run at least one Miller-Rabin test with a random base */
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if (t == 0) {
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t = 1;
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}
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/*
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abs(t) extra rounds of M-R to extend the range of primes it can find if t < 0.
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Only recommended if the input range is known to be < 3317044064679887385961981
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It uses the bases for a deterministic M-R test if input < 3317044064679887385961981
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The caller has to check the size.
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Not for cryptographic use because with known bases strong M-R pseudoprimes can
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be constructed. Use at least one M-R test with a random base (t >= 1).
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The 1119 bit large number
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80383745745363949125707961434194210813883768828755814583748891752229742737653\
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33652186502336163960045457915042023603208766569966760987284043965408232928738\
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79185086916685732826776177102938969773947016708230428687109997439976544144845\
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34115587245063340927902227529622941498423068816854043264575340183297861112989\
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60644845216191652872597534901
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has been constructed by F. Arnault (F. Arnault, "Rabin-Miller primality test:
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composite numbers which pass it.", Mathematics of Computation, 1995, 64. Jg.,
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Nr. 209, S. 355-361), is a semiprime with the two factors
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40095821663949960541830645208454685300518816604113250877450620473800321707011\
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96242716223191597219733582163165085358166969145233813917169287527980445796800\
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452592031836601
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20047910831974980270915322604227342650259408302056625438725310236900160853505\
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98121358111595798609866791081582542679083484572616906958584643763990222898400\
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226296015918301
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and it is a strong pseudoprime to all forty-six prime M-R bases up to 200
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It does not fail the strong Bailley-PSP test as implemented here, it is just
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given as an example, if not the reason to use the BPSW-test instead of M-R-tests
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with a sequence of primes 2...n.
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*/
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if (t < 0) {
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t = -t;
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/*
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Sorenson, Jonathan; Webster, Jonathan (2015).
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"Strong Pseudoprimes to Twelve Prime Bases".
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*/
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/* 0x437ae92817f9fc85b7e5 = 318665857834031151167461 */
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if ((err = mp_read_radix(&b, "437ae92817f9fc85b7e5", 16)) != MP_OKAY) {
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goto LBL_B;
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}
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if (mp_cmp(a, &b) == MP_LT) {
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p_max = 12;
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} else {
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/* 0x2be6951adc5b22410a5fd = 3317044064679887385961981 */
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if ((err = mp_read_radix(&b, "2be6951adc5b22410a5fd", 16)) != MP_OKAY) {
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goto LBL_B;
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}
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if (mp_cmp(a, &b) == MP_LT) {
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p_max = 13;
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} else {
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err = MP_VAL;
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goto LBL_B;
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}
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}
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/* for compatibility with the current API (well, compatible within a sign's width) */
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if (p_max < t) {
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p_max = t;
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}
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if (p_max > PRIME_SIZE) {
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err = MP_VAL;
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goto LBL_B;
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}
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/* we did bases 2 and 3 already, skip them */
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for (ix = 2; ix < p_max; ix++) {
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mp_set(&b, ltm_prime_tab[ix]);
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if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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}
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}
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/*
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Do "t" M-R tests with random bases between 3 and "a".
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See Fips 186.4 p. 126ff
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*/
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else if (t > 0) {
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/*
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* The mp_digit's have a defined bit-size but the size of the
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* array a.dp is a simple 'int' and this library can not assume full
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* compliance to the current C-standard (ISO/IEC 9899:2011) because
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* it gets used for small embeded processors, too. Some of those MCUs
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* have compilers that one cannot call standard compliant by any means.
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* Hence the ugly type-fiddling in the following code.
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*/
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size_a = mp_count_bits(a);
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mask = (1u << s_floor_ilog2(size_a)) - 1u;
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/*
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Assuming the General Rieman hypothesis (never thought to write that in a
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comment) the upper bound can be lowered to 2*(log a)^2.
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E. Bach, "Explicit bounds for primality testing and related problems,"
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Math. Comp. 55 (1990), 355-380.
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size_a = (size_a/10) * 7;
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len = 2 * (size_a * size_a);
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E.g.: a number of size 2^2048 would be reduced to the upper limit
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floor(2048/10)*7 = 1428
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2 * 1428^2 = 4078368
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(would have been ~4030331.9962 with floats and natural log instead)
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That number is smaller than 2^28, the default bit-size of mp_digit.
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*/
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/*
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How many tests, you might ask? Dana Jacobsen of Math::Prime::Util fame
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does exactly 1. In words: one. Look at the end of _GMP_is_prime() in
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Math-Prime-Util-GMP-0.50/primality.c if you do not believe it.
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The function mp_rand() goes to some length to use a cryptographically
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good PRNG. That also means that the chance to always get the same base
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in the loop is non-zero, although very low.
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If the BPSW test and/or the addtional Frobenious test have been
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performed instead of just the Miller-Rabin test with the bases 2 and 3,
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a single extra test should suffice, so such a very unlikely event
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will not do much harm.
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To preemptivly answer the dangling question: no, a witness does not
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need to be prime.
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*/
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for (ix = 0; ix < t; ix++) {
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/* mp_rand() guarantees the first digit to be non-zero */
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if ((err = mp_rand(&b, 1)) != MP_OKAY) {
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goto LBL_B;
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}
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/*
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* Reduce digit before casting because mp_digit might be bigger than
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* an unsigned int and "mask" on the other side is most probably not.
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*/
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fips_rand = (unsigned int)(b.dp[0] & (mp_digit) mask);
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#ifdef MP_8BIT
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/*
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* One 8-bit digit is too small, so concatenate two if the size of
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* unsigned int allows for it.
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*/
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if ((sizeof(unsigned int) * CHAR_BIT)/2 >= (sizeof(mp_digit) * CHAR_BIT)) {
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if ((err = mp_rand(&b, 1)) != MP_OKAY) {
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goto LBL_B;
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}
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fips_rand <<= sizeof(mp_digit) * CHAR_BIT;
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fips_rand |= (unsigned int) b.dp[0];
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fips_rand &= mask;
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}
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#endif
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/* Ceil, because small numbers have a right to live, too, */
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len = (((int)fips_rand + DIGIT_BIT) / DIGIT_BIT);
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/* Unlikely. */
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if (len < 0) {
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ix--;
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continue;
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}
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/*
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* As mentioned above, one 8-bit digit is too small and
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* although it can only happen in the unlikely case that
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* an "unsigned int" is smaller than 16 bit a simple test
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* is cheap and the correction even cheaper.
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*/
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#ifdef MP_8BIT
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/* All "a" < 2^8 have been caught before */
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if (len == 1) {
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len++;
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}
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#endif
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if ((err = mp_rand(&b, len)) != MP_OKAY) {
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goto LBL_B;
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}
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/*
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* That number might got too big and the witness has to be
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* smaller than or equal to "a"
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*/
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len = mp_count_bits(&b);
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if (len > size_a) {
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len = len - size_a;
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if ((err = mp_div_2d(&b, len, &b, NULL)) != MP_OKAY) {
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goto LBL_B;
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}
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}
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/* Although the chance for b <= 3 is miniscule, try again. */
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if (mp_cmp_d(&b, 3uL) != MP_GT) {
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ix--;
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continue;
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}
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if ((err = mp_prime_miller_rabin(a, &b, &res)) != MP_OKAY) {
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goto LBL_B;
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}
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if (res == MP_NO) {
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goto LBL_B;
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}
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}
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}
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/* passed the test */
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*result = MP_YES;
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LBL_B:
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mp_clear(&b);
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return err;
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}
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#endif
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/* ref: $Format:%D$ */
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/* git commit: $Format:%H$ */
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/* commit time: $Format:%ai$ */
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