/* * Copyright 1995-2020 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 #include #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 *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) goto err; 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 bn_check_prime_int(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 bn_check_prime_int(w, checks, ctx, do_trial_division, cb); } #endif /* Wrapper around bn_is_prime_int that sets the minimum number of checks */ int bn_check_prime_int(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 bn_check_prime_int(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 = 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 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, 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: /* TODO: Not all primes are private */ if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, 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, 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; }