| /* |
| * Copyright 1995-2021 The OpenSSL Project Authors. All Rights Reserved. |
| * |
| * Licensed under the Apache License 2.0 (the "License"). You may not use |
| * this file except in compliance with the License. You can obtain a copy |
| * in the file LICENSE in the source distribution or at |
| * https://www.openssl.org/source/license.html |
| */ |
| |
| #include <stdio.h> |
| #include <time.h> |
| #include "internal/cryptlib.h" |
| #include "bn_local.h" |
| |
| /* |
| * The quick sieve algorithm approach to weeding out primes is Philip |
| * Zimmermann's, as implemented in PGP. I have had a read of his comments |
| * and implemented my own version. |
| */ |
| #include "bn_prime.h" |
| |
| static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods, |
| BN_CTX *ctx); |
| static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods, |
| const BIGNUM *add, const BIGNUM *rem, |
| BN_CTX *ctx); |
| static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx, |
| int do_trial_division, BN_GENCB *cb); |
| |
| #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x)) |
| |
| #if BN_BITS2 == 64 |
| # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo |
| #else |
| # define BN_DEF(lo, hi) lo, hi |
| #endif |
| |
| /* |
| * See SP800 89 5.3.3 (Step f) |
| * The product of the set of primes ranging from 3 to 751 |
| * Generated using process in test/bn_internal_test.c test_bn_small_factors(). |
| * This includes 751 (which is not currently included in SP 800-89). |
| */ |
| static const BN_ULONG small_prime_factors[] = { |
| BN_DEF(0x3ef4e3e1, 0xc4309333), BN_DEF(0xcd2d655f, 0x71161eb6), |
| BN_DEF(0x0bf94862, 0x95e2238c), BN_DEF(0x24f7912b, 0x3eb233d3), |
| BN_DEF(0xbf26c483, 0x6b55514b), BN_DEF(0x5a144871, 0x0a84d817), |
| BN_DEF(0x9b82210a, 0x77d12fee), BN_DEF(0x97f050b3, 0xdb5b93c2), |
| BN_DEF(0x4d6c026b, 0x4acad6b9), BN_DEF(0x54aec893, 0xeb7751f3), |
| BN_DEF(0x36bc85c4, 0xdba53368), BN_DEF(0x7f5ec78e, 0xd85a1b28), |
| BN_DEF(0x6b322244, 0x2eb072d8), BN_DEF(0x5e2b3aea, 0xbba51112), |
| BN_DEF(0x0e2486bf, 0x36ed1a6c), BN_DEF(0xec0c5727, 0x5f270460), |
| (BN_ULONG)0x000017b1 |
| }; |
| |
| #define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors) |
| static const BIGNUM _bignum_small_prime_factors = { |
| (BN_ULONG *)small_prime_factors, |
| BN_SMALL_PRIME_FACTORS_TOP, |
| BN_SMALL_PRIME_FACTORS_TOP, |
| 0, |
| BN_FLG_STATIC_DATA |
| }; |
| |
| const BIGNUM *ossl_bn_get0_small_factors(void) |
| { |
| return &_bignum_small_prime_factors; |
| } |
| |
| /* |
| * Calculate the number of trial divisions that gives the best speed in |
| * combination with Miller-Rabin prime test, based on the sized of the prime. |
| */ |
| static int calc_trial_divisions(int bits) |
| { |
| if (bits <= 512) |
| return 64; |
| else if (bits <= 1024) |
| return 128; |
| else if (bits <= 2048) |
| return 384; |
| else if (bits <= 4096) |
| return 1024; |
| return NUMPRIMES; |
| } |
| |
| /* |
| * Use a minimum of 64 rounds of Miller-Rabin, which should give a false |
| * positive rate of 2^-128. If the size of the prime is larger than 2048 |
| * the user probably wants a higher security level than 128, so switch |
| * to 128 rounds giving a false positive rate of 2^-256. |
| * Returns the number of rounds. |
| */ |
| static int bn_mr_min_checks(int bits) |
| { |
| if (bits > 2048) |
| return 128; |
| return 64; |
| } |
| |
| int BN_GENCB_call(BN_GENCB *cb, int a, int b) |
| { |
| /* No callback means continue */ |
| if (!cb) |
| return 1; |
| switch (cb->ver) { |
| case 1: |
| /* Deprecated-style callbacks */ |
| if (!cb->cb.cb_1) |
| return 1; |
| cb->cb.cb_1(a, b, cb->arg); |
| return 1; |
| case 2: |
| /* New-style callbacks */ |
| return cb->cb.cb_2(a, b, cb); |
| default: |
| break; |
| } |
| /* Unrecognised callback type */ |
| return 0; |
| } |
| |
| int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe, |
| const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb, |
| BN_CTX *ctx) |
| { |
| BIGNUM *t; |
| int found = 0; |
| int i, j, c1 = 0; |
| prime_t *mods = NULL; |
| int checks = bn_mr_min_checks(bits); |
| |
| if (bits < 2) { |
| /* There are no prime numbers this small. */ |
| ERR_raise(ERR_LIB_BN, BN_R_BITS_TOO_SMALL); |
| return 0; |
| } else if (add == NULL && safe && bits < 6 && bits != 3) { |
| /* |
| * The smallest safe prime (7) is three bits. |
| * But the following two safe primes with less than 6 bits (11, 23) |
| * are unreachable for BN_rand with BN_RAND_TOP_TWO. |
| */ |
| ERR_raise(ERR_LIB_BN, BN_R_BITS_TOO_SMALL); |
| return 0; |
| } |
| |
| mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES); |
| if (mods == NULL) { |
| ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE); |
| return 0; |
| } |
| |
| BN_CTX_start(ctx); |
| t = BN_CTX_get(ctx); |
| if (t == NULL) |
| goto err; |
| loop: |
| /* make a random number and set the top and bottom bits */ |
| if (add == NULL) { |
| if (!probable_prime(ret, bits, safe, mods, ctx)) |
| goto err; |
| } else { |
| if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx)) |
| goto err; |
| } |
| |
| if (!BN_GENCB_call(cb, 0, c1++)) |
| /* aborted */ |
| goto err; |
| |
| if (!safe) { |
| i = bn_is_prime_int(ret, checks, ctx, 0, cb); |
| if (i == -1) |
| goto err; |
| if (i == 0) |
| goto loop; |
| } else { |
| /* |
| * for "safe prime" generation, check that (p-1)/2 is prime. Since a |
| * prime is odd, We just need to divide by 2 |
| */ |
| if (!BN_rshift1(t, ret)) |
| goto err; |
| |
| for (i = 0; i < checks; i++) { |
| j = bn_is_prime_int(ret, 1, ctx, 0, cb); |
| if (j == -1) |
| goto err; |
| if (j == 0) |
| goto loop; |
| |
| j = bn_is_prime_int(t, 1, ctx, 0, cb); |
| if (j == -1) |
| goto err; |
| if (j == 0) |
| goto loop; |
| |
| if (!BN_GENCB_call(cb, 2, c1 - 1)) |
| goto err; |
| /* We have a safe prime test pass */ |
| } |
| } |
| /* we have a prime :-) */ |
| found = 1; |
| err: |
| OPENSSL_free(mods); |
| BN_CTX_end(ctx); |
| bn_check_top(ret); |
| return found; |
| } |
| |
| #ifndef FIPS_MODULE |
| int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, |
| const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb) |
| { |
| BN_CTX *ctx = BN_CTX_new(); |
| int retval; |
| |
| if (ctx == NULL) |
| return 0; |
| |
| retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx); |
| |
| BN_CTX_free(ctx); |
| return retval; |
| } |
| #endif |
| |
| #ifndef OPENSSL_NO_DEPRECATED_3_0 |
| int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed, |
| BN_GENCB *cb) |
| { |
| return ossl_bn_check_prime(a, checks, ctx_passed, 0, cb); |
| } |
| |
| int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx, |
| int do_trial_division, BN_GENCB *cb) |
| { |
| return ossl_bn_check_prime(w, checks, ctx, do_trial_division, cb); |
| } |
| #endif |
| |
| /* Wrapper around bn_is_prime_int that sets the minimum number of checks */ |
| int ossl_bn_check_prime(const BIGNUM *w, int checks, BN_CTX *ctx, |
| int do_trial_division, BN_GENCB *cb) |
| { |
| int min_checks = bn_mr_min_checks(BN_num_bits(w)); |
| |
| if (checks < min_checks) |
| checks = min_checks; |
| |
| return bn_is_prime_int(w, checks, ctx, do_trial_division, cb); |
| } |
| |
| int BN_check_prime(const BIGNUM *p, BN_CTX *ctx, BN_GENCB *cb) |
| { |
| return ossl_bn_check_prime(p, 0, ctx, 1, cb); |
| } |
| |
| /* |
| * Tests that |w| is probably prime |
| * See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test. |
| * |
| * Returns 0 when composite, 1 when probable prime, -1 on error. |
| */ |
| static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx, |
| int do_trial_division, BN_GENCB *cb) |
| { |
| int i, status, ret = -1; |
| #ifndef FIPS_MODULE |
| BN_CTX *ctxlocal = NULL; |
| #else |
| |
| if (ctx == NULL) |
| return -1; |
| #endif |
| |
| /* w must be bigger than 1 */ |
| if (BN_cmp(w, BN_value_one()) <= 0) |
| return 0; |
| |
| /* w must be odd */ |
| if (BN_is_odd(w)) { |
| /* Take care of the really small prime 3 */ |
| if (BN_is_word(w, 3)) |
| return 1; |
| } else { |
| /* 2 is the only even prime */ |
| return BN_is_word(w, 2); |
| } |
| |
| /* first look for small factors */ |
| if (do_trial_division) { |
| int trial_divisions = calc_trial_divisions(BN_num_bits(w)); |
| |
| for (i = 1; i < trial_divisions; i++) { |
| BN_ULONG mod = BN_mod_word(w, primes[i]); |
| if (mod == (BN_ULONG)-1) |
| return -1; |
| if (mod == 0) |
| return BN_is_word(w, primes[i]); |
| } |
| if (!BN_GENCB_call(cb, 1, -1)) |
| return -1; |
| } |
| #ifndef FIPS_MODULE |
| if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL) |
| goto err; |
| #endif |
| |
| ret = ossl_bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status); |
| if (!ret) |
| goto err; |
| ret = (status == BN_PRIMETEST_PROBABLY_PRIME); |
| err: |
| #ifndef FIPS_MODULE |
| BN_CTX_free(ctxlocal); |
| #endif |
| return ret; |
| } |
| |
| /* |
| * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test. |
| * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero). |
| * The Step numbers listed in the code refer to the enhanced case. |
| * |
| * if enhanced is set, then status returns one of the following: |
| * BN_PRIMETEST_PROBABLY_PRIME |
| * BN_PRIMETEST_COMPOSITE_WITH_FACTOR |
| * BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME |
| * if enhanced is zero, then status returns either |
| * BN_PRIMETEST_PROBABLY_PRIME or |
| * BN_PRIMETEST_COMPOSITE |
| * |
| * returns 0 if there was an error, otherwise it returns 1. |
| */ |
| int ossl_bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx, |
| BN_GENCB *cb, int enhanced, int *status) |
| { |
| int i, j, a, ret = 0; |
| BIGNUM *g, *w1, *w3, *x, *m, *z, *b; |
| BN_MONT_CTX *mont = NULL; |
| |
| /* w must be odd */ |
| if (!BN_is_odd(w)) |
| return 0; |
| |
| BN_CTX_start(ctx); |
| g = BN_CTX_get(ctx); |
| w1 = BN_CTX_get(ctx); |
| w3 = BN_CTX_get(ctx); |
| x = BN_CTX_get(ctx); |
| m = BN_CTX_get(ctx); |
| z = BN_CTX_get(ctx); |
| b = BN_CTX_get(ctx); |
| |
| if (!(b != NULL |
| /* w1 := w - 1 */ |
| && BN_copy(w1, w) |
| && BN_sub_word(w1, 1) |
| /* w3 := w - 3 */ |
| && BN_copy(w3, w) |
| && BN_sub_word(w3, 3))) |
| goto err; |
| |
| /* check w is larger than 3, otherwise the random b will be too small */ |
| if (BN_is_zero(w3) || BN_is_negative(w3)) |
| goto err; |
| |
| /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */ |
| a = 1; |
| while (!BN_is_bit_set(w1, a)) |
| a++; |
| /* (Step 2) m = (w-1) / 2^a */ |
| if (!BN_rshift(m, w1, a)) |
| goto err; |
| |
| /* Montgomery setup for computations mod a */ |
| mont = BN_MONT_CTX_new(); |
| if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx)) |
| goto err; |
| |
| if (iterations == 0) |
| iterations = bn_mr_min_checks(BN_num_bits(w)); |
| |
| /* (Step 4) */ |
| for (i = 0; i < iterations; ++i) { |
| /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */ |
| if (!BN_priv_rand_range_ex(b, w3, 0, ctx) |
| || !BN_add_word(b, 2)) /* 1 < b < w-1 */ |
| goto err; |
| |
| if (enhanced) { |
| /* (Step 4.3) */ |
| if (!BN_gcd(g, b, w, ctx)) |
| goto err; |
| /* (Step 4.4) */ |
| if (!BN_is_one(g)) { |
| *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR; |
| ret = 1; |
| goto err; |
| } |
| } |
| /* (Step 4.5) z = b^m mod w */ |
| if (!BN_mod_exp_mont(z, b, m, w, ctx, mont)) |
| goto err; |
| /* (Step 4.6) if (z = 1 or z = w-1) */ |
| if (BN_is_one(z) || BN_cmp(z, w1) == 0) |
| goto outer_loop; |
| /* (Step 4.7) for j = 1 to a-1 */ |
| for (j = 1; j < a ; ++j) { |
| /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */ |
| if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) |
| goto err; |
| /* (Step 4.7.3) */ |
| if (BN_cmp(z, w1) == 0) |
| goto outer_loop; |
| /* (Step 4.7.4) */ |
| if (BN_is_one(z)) |
| goto composite; |
| } |
| /* At this point z = b^((w-1)/2) mod w */ |
| /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */ |
| if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) |
| goto err; |
| /* (Step 4.10) */ |
| if (BN_is_one(z)) |
| goto composite; |
| /* (Step 4.11) x = b^(w-1) mod w */ |
| if (!BN_copy(x, z)) |
| goto err; |
| composite: |
| if (enhanced) { |
| /* (Step 4.1.2) g = GCD(x-1, w) */ |
| if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx)) |
| goto err; |
| /* (Steps 4.1.3 - 4.1.4) */ |
| if (BN_is_one(g)) |
| *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME; |
| else |
| *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR; |
| } else { |
| *status = BN_PRIMETEST_COMPOSITE; |
| } |
| ret = 1; |
| goto err; |
| outer_loop: ; |
| /* (Step 4.1.5) */ |
| if (!BN_GENCB_call(cb, 1, i)) |
| goto err; |
| } |
| /* (Step 5) */ |
| *status = BN_PRIMETEST_PROBABLY_PRIME; |
| ret = 1; |
| err: |
| BN_clear(g); |
| BN_clear(w1); |
| BN_clear(w3); |
| BN_clear(x); |
| BN_clear(m); |
| BN_clear(z); |
| BN_clear(b); |
| BN_CTX_end(ctx); |
| BN_MONT_CTX_free(mont); |
| return ret; |
| } |
| |
| /* |
| * Generate a random number of |bits| bits that is probably prime by sieving. |
| * If |safe| != 0, it generates a safe prime. |
| * |mods| is a preallocated array that gets reused when called again. |
| * |
| * The probably prime is saved in |rnd|. |
| * |
| * Returns 1 on success and 0 on error. |
| */ |
| static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods, |
| BN_CTX *ctx) |
| { |
| int i; |
| BN_ULONG delta; |
| int trial_divisions = calc_trial_divisions(bits); |
| BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1]; |
| |
| again: |
| if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, 0, |
| ctx)) |
| return 0; |
| if (safe && !BN_set_bit(rnd, 1)) |
| return 0; |
| /* we now have a random number 'rnd' to test. */ |
| for (i = 1; i < trial_divisions; i++) { |
| BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]); |
| if (mod == (BN_ULONG)-1) |
| return 0; |
| mods[i] = (prime_t) mod; |
| } |
| delta = 0; |
| loop: |
| for (i = 1; i < trial_divisions; i++) { |
| /* |
| * check that rnd is a prime and also that |
| * gcd(rnd-1,primes) == 1 (except for 2) |
| * do the second check only if we are interested in safe primes |
| * in the case that the candidate prime is a single word then |
| * we check only the primes up to sqrt(rnd) |
| */ |
| if (bits <= 31 && delta <= 0x7fffffff |
| && square(primes[i]) > BN_get_word(rnd) + delta) |
| break; |
| if (safe ? (mods[i] + delta) % primes[i] <= 1 |
| : (mods[i] + delta) % primes[i] == 0) { |
| delta += safe ? 4 : 2; |
| if (delta > maxdelta) |
| goto again; |
| goto loop; |
| } |
| } |
| if (!BN_add_word(rnd, delta)) |
| return 0; |
| if (BN_num_bits(rnd) != bits) |
| goto again; |
| bn_check_top(rnd); |
| return 1; |
| } |
| |
| /* |
| * Generate a random number |rnd| of |bits| bits that is probably prime |
| * and satisfies |rnd| % |add| == |rem| by sieving. |
| * If |safe| != 0, it generates a safe prime. |
| * |mods| is a preallocated array that gets reused when called again. |
| * |
| * Returns 1 on success and 0 on error. |
| */ |
| static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods, |
| const BIGNUM *add, const BIGNUM *rem, |
| BN_CTX *ctx) |
| { |
| int i, ret = 0; |
| BIGNUM *t1; |
| BN_ULONG delta; |
| int trial_divisions = calc_trial_divisions(bits); |
| BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1]; |
| |
| BN_CTX_start(ctx); |
| if ((t1 = BN_CTX_get(ctx)) == NULL) |
| goto err; |
| |
| if (maxdelta > BN_MASK2 - BN_get_word(add)) |
| maxdelta = BN_MASK2 - BN_get_word(add); |
| |
| again: |
| if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, 0, ctx)) |
| goto err; |
| |
| /* we need ((rnd-rem) % add) == 0 */ |
| |
| if (!BN_mod(t1, rnd, add, ctx)) |
| goto err; |
| if (!BN_sub(rnd, rnd, t1)) |
| goto err; |
| if (rem == NULL) { |
| if (!BN_add_word(rnd, safe ? 3u : 1u)) |
| goto err; |
| } else { |
| if (!BN_add(rnd, rnd, rem)) |
| goto err; |
| } |
| |
| if (BN_num_bits(rnd) < bits |
| || BN_get_word(rnd) < (safe ? 5u : 3u)) { |
| if (!BN_add(rnd, rnd, add)) |
| goto err; |
| } |
| |
| /* we now have a random number 'rnd' to test. */ |
| for (i = 1; i < trial_divisions; i++) { |
| BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]); |
| if (mod == (BN_ULONG)-1) |
| goto err; |
| mods[i] = (prime_t) mod; |
| } |
| delta = 0; |
| loop: |
| for (i = 1; i < trial_divisions; i++) { |
| /* check that rnd is a prime */ |
| if (bits <= 31 && delta <= 0x7fffffff |
| && square(primes[i]) > BN_get_word(rnd) + delta) |
| break; |
| /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */ |
| if (safe ? (mods[i] + delta) % primes[i] <= 1 |
| : (mods[i] + delta) % primes[i] == 0) { |
| delta += BN_get_word(add); |
| if (delta > maxdelta) |
| goto again; |
| goto loop; |
| } |
| } |
| if (!BN_add_word(rnd, delta)) |
| goto err; |
| ret = 1; |
| |
| err: |
| BN_CTX_end(ctx); |
| bn_check_top(rnd); |
| return ret; |
| } |
