/* Copyright (C) 1995-1998 Eric Young (eay@cryptsoft.com) * All rights reserved. * * This package is an SSL implementation written * by Eric Young (eay@cryptsoft.com). * The implementation was written so as to conform with Netscapes SSL. * * This library is free for commercial and non-commercial use as long as * the following conditions are aheared to. The following conditions * apply to all code found in this distribution, be it the RC4, RSA, * lhash, DES, etc., code; not just the SSL code. The SSL documentation * included with this distribution is covered by the same copyright terms * except that the holder is Tim Hudson (tjh@cryptsoft.com). * * Copyright remains Eric Young's, and as such any Copyright notices in * the code are not to be removed. * If this package is used in a product, Eric Young should be given attribution * as the author of the parts of the library used. * This can be in the form of a textual message at program startup or * in documentation (online or textual) provided with the package. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * 1. Redistributions of source code must retain the copyright * notice, this list of conditions and the following disclaimer. * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in the * documentation and/or other materials provided with the distribution. * 3. All advertising materials mentioning features or use of this software * must display the following acknowledgement: * "This product includes cryptographic software written by * Eric Young (eay@cryptsoft.com)" * The word 'cryptographic' can be left out if the rouines from the library * being used are not cryptographic related :-). * 4. If you include any Windows specific code (or a derivative thereof) from * the apps directory (application code) you must include an acknowledgement: * "This product includes software written by Tim Hudson (tjh@cryptsoft.com)" * * THIS SOFTWARE IS PROVIDED BY ERIC YOUNG ``AS IS'' AND * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF * SUCH DAMAGE. * * The licence and distribution terms for any publically available version or * derivative of this code cannot be changed. i.e. this code cannot simply be * copied and put under another distribution licence * [including the GNU Public Licence.] */ /* ==================================================================== * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * * 3. All advertising materials mentioning features or use of this * software must display the following acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" * * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to * endorse or promote products derived from this software without * prior written permission. For written permission, please contact * openssl-core@openssl.org. * * 5. Products derived from this software may not be called "OpenSSL" * nor may "OpenSSL" appear in their names without prior written * permission of the OpenSSL Project. * * 6. Redistributions of any form whatsoever must retain the following * acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit (http://www.openssl.org/)" * * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED * OF THE POSSIBILITY OF SUCH DAMAGE. * ==================================================================== * * This product includes cryptographic software written by Eric Young * (eay@cryptsoft.com). This product includes software written by Tim * Hudson (tjh@cryptsoft.com). */ #include <openssl_grpc/bn.h> #include <openssl_grpc/err.h> #include <openssl_grpc/mem.h> #include "internal.h" #include "../../internal.h" // kPrimes contains the first 1024 primes. static const uint16_t kPrimes[] = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, }; // BN_prime_checks_for_size returns the number of Miller-Rabin iterations // necessary for generating a 'bits'-bit candidate prime. // // // This table is generated using the algorithm of FIPS PUB 186-4 // Digital Signature Standard (DSS), section F.1, page 117. // (https://doi.org/10.6028/NIST.FIPS.186-4) // The following magma script was used to generate the output: // securitybits:=125; // k:=1024; // for t:=1 to 65 do // for M:=3 to Floor(2*Sqrt(k-1)-1) do // S:=0; // // Sum over m // for m:=3 to M do // s:=0; // // Sum over j // for j:=2 to m do // s+:=(RealField(32)!2)^-(j+(k-1)/j); // end for; // S+:=2^(m-(m-1)*t)*s; // end for; // A:=2^(k-2-M*t); // B:=8*(Pi(RealField(32))^2-6)/3*2^(k-2)*S; // pkt:=2.00743*Log(2)*k*2^-k*(A+B); // seclevel:=Floor(-Log(2,pkt)); // if seclevel ge securitybits then // printf "k: %5o, security: %o bits (t: %o, M: %o)\n",k,seclevel,t,M; // break; // end if; // end for; // if seclevel ge securitybits then break; end if; // end for; // // It can be run online at: http://magma.maths.usyd.edu.au/calc // And will output: // k: 1024, security: 129 bits (t: 6, M: 23) // k is the number of bits of the prime, securitybits is the level we want to // reach. // prime length | RSA key size | # MR tests | security level // -------------+--------------|------------+--------------- // (b) >= 6394 | >= 12788 | 3 | 256 bit // (b) >= 3747 | >= 7494 | 3 | 192 bit // (b) >= 1345 | >= 2690 | 4 | 128 bit // (b) >= 1080 | >= 2160 | 5 | 128 bit // (b) >= 852 | >= 1704 | 5 | 112 bit // (b) >= 476 | >= 952 | 5 | 80 bit // (b) >= 400 | >= 800 | 6 | 80 bit // (b) >= 347 | >= 694 | 7 | 80 bit // (b) >= 308 | >= 616 | 8 | 80 bit // (b) >= 55 | >= 110 | 27 | 64 bit // (b) >= 6 | >= 12 | 34 | 64 bit static int BN_prime_checks_for_size(int bits) { if (bits >= 3747) { return 3; } if (bits >= 1345) { return 4; } if (bits >= 476) { return 5; } if (bits >= 400) { return 6; } if (bits >= 347) { return 7; } if (bits >= 308) { return 8; } if (bits >= 55) { return 27; } return 34; } // num_trial_division_primes returns the number of primes to try with trial // division before using more expensive checks. For larger numbers, the value // of excluding a candidate with trial division is larger. static size_t num_trial_division_primes(const BIGNUM *n) { if (n->width * BN_BITS2 > 1024) { return OPENSSL_ARRAY_SIZE(kPrimes); } return OPENSSL_ARRAY_SIZE(kPrimes) / 2; } // BN_PRIME_CHECKS_BLINDED is the iteration count for blinding the constant-time // primality test. See |BN_primality_test| for details. This number is selected // so that, for a candidate N-bit RSA prime, picking |BN_PRIME_CHECKS_BLINDED| // random N-bit numbers will have at least |BN_prime_checks_for_size(N)| values // in range with high probability. // // The following Python script computes the blinding factor needed for the // corresponding iteration count. /* import math # We choose candidate RSA primes between sqrt(2)/2 * 2^N and 2^N and select # witnesses by generating random N-bit numbers. Thus the probability of # selecting one in range is at least sqrt(2)/2. p = math.sqrt(2) / 2 # Target around 2^-8 probability of the blinding being insufficient given that # key generation is a one-time, noisy operation. epsilon = 2**-8 def choose(a, b): r = 1 for i in xrange(b): r *= a - i r /= (i + 1) return r def failure_rate(min_uniform, iterations): """ Returns the probability that, for |iterations| candidate witnesses, fewer than |min_uniform| of them will be uniform. """ prob = 0.0 for i in xrange(min_uniform): prob += (choose(iterations, i) * p**i * (1-p)**(iterations - i)) return prob for min_uniform in (3, 4, 5, 6, 8, 13, 19, 28): # Find the smallest number of iterations under the target failure rate. iterations = min_uniform while True: prob = failure_rate(min_uniform, iterations) if prob < epsilon: print min_uniform, iterations, prob break iterations += 1 Output: 3 9 0.00368894873911 4 11 0.00363319494662 5 13 0.00336215573898 6 15 0.00300145783158 8 19 0.00225214119331 13 27 0.00385610026955 19 38 0.0021410539126 28 52 0.00325405801769 16 iterations suffices for 400-bit primes and larger (6 uniform samples needed), which is already well below the minimum acceptable key size for RSA. */ #define BN_PRIME_CHECKS_BLINDED 16 static int probable_prime(BIGNUM *rnd, int bits); static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx); static int probable_prime_dh_safe(BIGNUM *rnd, int bits, const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx); void BN_GENCB_set(BN_GENCB *callback, int (*f)(int event, int n, struct bn_gencb_st *), void *arg) { callback->callback = f; callback->arg = arg; } int BN_GENCB_call(BN_GENCB *callback, int event, int n) { if (!callback) { return 1; } return callback->callback(event, n, callback); } int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe, const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb) { BIGNUM *t; int found = 0; int i, j, c1 = 0; BN_CTX *ctx; int checks = BN_prime_checks_for_size(bits); if (bits < 2) { // There are no prime numbers this small. OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL); return 0; } else if (bits == 2 && safe) { // The smallest safe prime (7) is three bits. OPENSSL_PUT_ERROR(BN, BN_R_BITS_TOO_SMALL); return 0; } ctx = BN_CTX_new(); if (ctx == NULL) { goto err; } BN_CTX_start(ctx); t = BN_CTX_get(ctx); if (!t) { goto err; } loop: // make a random number and set the top and bottom bits if (add == NULL) { if (!probable_prime(ret, bits)) { goto err; } } else { if (safe) { if (!probable_prime_dh_safe(ret, bits, add, rem, ctx)) { goto err; } } else { if (!probable_prime_dh(ret, bits, add, rem, ctx)) { goto err; } } } if (!BN_GENCB_call(cb, BN_GENCB_GENERATED, c1++)) { // aborted goto err; } if (!safe) { i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb); if (i == -1) { goto err; } else 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; } // Interleave |ret| and |t|'s primality tests to avoid paying the full // iteration count on |ret| only to quickly discover |t| is composite. // // TODO(davidben): This doesn't quite work because an iteration count of 1 // still runs the blinding mechanism. for (i = 0; i < checks; i++) { j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, NULL); if (j == -1) { goto err; } else if (j == 0) { goto loop; } j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, NULL); if (j == -1) { goto err; } else if (j == 0) { goto loop; } if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i)) { goto err; } // We have a safe prime test pass } } // we have a prime :-) found = 1; err: if (ctx != NULL) { BN_CTX_end(ctx); BN_CTX_free(ctx); } return found; } static int bn_trial_division(uint16_t *out, const BIGNUM *bn) { const size_t num_primes = num_trial_division_primes(bn); for (size_t i = 1; i < num_primes; i++) { if (bn_mod_u16_consttime(bn, kPrimes[i]) == 0) { *out = kPrimes[i]; return 1; } } return 0; } int bn_odd_number_is_obviously_composite(const BIGNUM *bn) { uint16_t prime; return bn_trial_division(&prime, bn) && !BN_is_word(bn, prime); } int bn_miller_rabin_init(BN_MILLER_RABIN *miller_rabin, const BN_MONT_CTX *mont, BN_CTX *ctx) { // This function corresponds to steps 1 through 3 of FIPS 186-4, C.3.1. const BIGNUM *w = &mont->N; // Note we do not call |BN_CTX_start| in this function. We intentionally // allocate values in the containing scope so they outlive this function. miller_rabin->w1 = BN_CTX_get(ctx); miller_rabin->m = BN_CTX_get(ctx); miller_rabin->one_mont = BN_CTX_get(ctx); miller_rabin->w1_mont = BN_CTX_get(ctx); if (miller_rabin->w1 == NULL || miller_rabin->m == NULL || miller_rabin->one_mont == NULL || miller_rabin->w1_mont == NULL) { return 0; } // See FIPS 186-4, C.3.1, steps 1 through 3. if (!bn_usub_consttime(miller_rabin->w1, w, BN_value_one())) { return 0; } miller_rabin->a = BN_count_low_zero_bits(miller_rabin->w1); if (!bn_rshift_secret_shift(miller_rabin->m, miller_rabin->w1, miller_rabin->a, ctx)) { return 0; } miller_rabin->w_bits = BN_num_bits(w); // Precompute some values in Montgomery form. if (!bn_one_to_montgomery(miller_rabin->one_mont, mont, ctx) || // w - 1 is -1 mod w, so we can compute it in the Montgomery domain, -R, // with a subtraction. (|one_mont| cannot be zero.) !bn_usub_consttime(miller_rabin->w1_mont, w, miller_rabin->one_mont)) { return 0; } return 1; } int bn_miller_rabin_iteration(const BN_MILLER_RABIN *miller_rabin, int *out_is_possibly_prime, const BIGNUM *b, const BN_MONT_CTX *mont, BN_CTX *ctx) { // This function corresponds to steps 4.3 through 4.5 of FIPS 186-4, C.3.1. int ret = 0; BN_CTX_start(ctx); // Step 4.3. We use Montgomery-encoding for better performance and to avoid // timing leaks. const BIGNUM *w = &mont->N; BIGNUM *z = BN_CTX_get(ctx); if (z == NULL || !BN_mod_exp_mont_consttime(z, b, miller_rabin->m, w, ctx, mont) || !BN_to_montgomery(z, z, mont, ctx)) { goto err; } // is_possibly_prime is all ones if we have determined |b| is not a composite // witness for |w|. This is equivalent to going to step 4.7 in the original // algorithm. To avoid timing leaks, we run the algorithm to the end for prime // inputs. crypto_word_t is_possibly_prime = 0; // Step 4.4. If z = 1 or z = w-1, b is not a composite witness and w is still // possibly prime. is_possibly_prime = BN_equal_consttime(z, miller_rabin->one_mont) | BN_equal_consttime(z, miller_rabin->w1_mont); is_possibly_prime = 0 - is_possibly_prime; // Make it all zeros or all ones. // Step 4.5. // // To avoid leaking |a|, we run the loop to |w_bits| and mask off all // iterations once |j| = |a|. for (int j = 1; j < miller_rabin->w_bits; j++) { if (constant_time_eq_int(j, miller_rabin->a) & ~is_possibly_prime) { // If the loop is done and we haven't seen z = 1 or z = w-1 yet, the // value is composite and we can break in variable time. break; } // Step 4.5.1. if (!BN_mod_mul_montgomery(z, z, z, mont, ctx)) { goto err; } // Step 4.5.2. If z = w-1 and the loop is not done, this is not a composite // witness. crypto_word_t z_is_w1_mont = BN_equal_consttime(z, miller_rabin->w1_mont); z_is_w1_mont = 0 - z_is_w1_mont; // Make it all zeros or all ones. is_possibly_prime |= z_is_w1_mont; // Go to step 4.7 if |z_is_w1_mont|. // Step 4.5.3. If z = 1 and the loop is not done, the previous value of z // was not -1. There are no non-trivial square roots of 1 modulo a prime, so // w is composite and we may exit in variable time. if (BN_equal_consttime(z, miller_rabin->one_mont) & ~is_possibly_prime) { break; } } *out_is_possibly_prime = is_possibly_prime & 1; ret = 1; err: BN_CTX_end(ctx); return ret; } int BN_primality_test(int *out_is_probably_prime, const BIGNUM *w, int checks, BN_CTX *ctx, int do_trial_division, BN_GENCB *cb) { // This function's secrecy and performance requirements come from RSA key // generation. We generate RSA keys by selecting two large, secret primes with // rejection sampling. // // We thus treat |w| as secret if turns out to be a large prime. However, if // |w| is composite, we treat this and |w| itself as public. (Conversely, if // |w| is prime, that it is prime is public. Only the value is secret.) This // is fine for RSA key generation, but note it is important that we use // rejection sampling, with each candidate prime chosen independently. This // would not work for, e.g., an algorithm which looked for primes in // consecutive integers. These assumptions allow us to discard composites // quickly. We additionally treat |w| as public when it is a small prime to // simplify trial decryption and some edge cases. // // One RSA key generation will call this function on exactly two primes and // many more composites. The overall cost is a combination of several factors: // // 1. Checking if |w| is divisible by a small prime is much faster than // learning it is composite by Miller-Rabin (see below for details on that // cost). Trial division by p saves 1/p of Miller-Rabin calls, so this is // worthwhile until p exceeds the ratio of the two costs. // // 2. For a random (i.e. non-adversarial) candidate large prime and candidate // witness, the probability of false witness is very low. (This is why FIPS // 186-4 only requires a few iterations.) Thus composites not discarded by // trial decryption, in practice, cost one Miller-Rabin iteration. Only the // two actual primes cost the full iteration count. // // 3. A Miller-Rabin iteration is a modular exponentiation plus |a| additional // modular squares, where |a| is the number of factors of two in |w-1|. |a| // is likely small (the distribution falls exponentially), but it is also // potentially secret, so we loop up to its log(w) upper bound when |w| is // prime. When |w| is composite, we break early, so only two calls pay this // cost. (Note that all calls pay the modular exponentiation which is, // itself, log(w) modular multiplications and squares.) // // 4. While there are only two prime calls, they multiplicatively pay the full // costs of (2) and (3). // // 5. After the primes are chosen, RSA keys derive some values from the // primes, but this cost is negligible in comparison. *out_is_probably_prime = 0; if (BN_cmp(w, BN_value_one()) <= 0) { return 1; } if (!BN_is_odd(w)) { // The only even prime is two. *out_is_probably_prime = BN_is_word(w, 2); return 1; } // Miller-Rabin does not work for three. if (BN_is_word(w, 3)) { *out_is_probably_prime = 1; return 1; } if (do_trial_division) { // Perform additional trial division checks to discard small primes. uint16_t prime; if (bn_trial_division(&prime, w)) { *out_is_probably_prime = BN_is_word(w, prime); return 1; } if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, -1)) { return 0; } } if (checks == BN_prime_checks_for_generation) { checks = BN_prime_checks_for_size(BN_num_bits(w)); } BN_CTX *new_ctx = NULL; if (ctx == NULL) { new_ctx = BN_CTX_new(); if (new_ctx == NULL) { return 0; } ctx = new_ctx; } // See C.3.1 from FIPS 186-4. int ret = 0; BN_CTX_start(ctx); BIGNUM *b = BN_CTX_get(ctx); BN_MONT_CTX *mont = BN_MONT_CTX_new_consttime(w, ctx); BN_MILLER_RABIN miller_rabin; if (b == NULL || mont == NULL || // Steps 1-3. !bn_miller_rabin_init(&miller_rabin, mont, ctx)) { goto err; } // The following loop performs in inner iteration of the Miller-Rabin // Primality test (Step 4). // // The algorithm as specified in FIPS 186-4 leaks information on |w|, the RSA // private key. Instead, we run through each iteration unconditionally, // performing modular multiplications, masking off any effects to behave // equivalently to the specified algorithm. // // We also blind the number of values of |b| we try. Steps 4.1–4.2 say to // discard out-of-range values. To avoid leaking information on |w|, we use // |bn_rand_secret_range| which, rather than discarding bad values, adjusts // them to be in range. Though not uniformly selected, these adjusted values // are still usable as Miller-Rabin checks. // // Miller-Rabin is already probabilistic, so we could reach the desired // confidence levels by just suitably increasing the iteration count. However, // to align with FIPS 186-4, we use a more pessimal analysis: we do not count // the non-uniform values towards the iteration count. As a result, this // function is more complex and has more timing risk than necessary. // // We count both total iterations and uniform ones and iterate until we've // reached at least |BN_PRIME_CHECKS_BLINDED| and |iterations|, respectively. // If the latter is large enough, it will be the limiting factor with high // probability and we won't leak information. // // Note this blinding does not impact most calls when picking primes because // composites are rejected early. Only the two secret primes see extra work. crypto_word_t uniform_iterations = 0; // Using |constant_time_lt_w| seems to prevent the compiler from optimizing // this into two jumps. for (int i = 1; (i <= BN_PRIME_CHECKS_BLINDED) | constant_time_lt_w(uniform_iterations, checks); i++) { // Step 4.1-4.2 int is_uniform; if (!bn_rand_secret_range(b, &is_uniform, 2, miller_rabin.w1)) { goto err; } uniform_iterations += is_uniform; // Steps 4.3-4.5 int is_possibly_prime = 0; if (!bn_miller_rabin_iteration(&miller_rabin, &is_possibly_prime, b, mont, ctx)) { goto err; } if (!is_possibly_prime) { // Step 4.6. We did not see z = w-1 before z = 1, so w must be composite. *out_is_probably_prime = 0; ret = 1; goto err; } // Step 4.7 if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) { goto err; } } assert(uniform_iterations >= (crypto_word_t)checks); *out_is_probably_prime = 1; ret = 1; err: BN_MONT_CTX_free(mont); BN_CTX_end(ctx); BN_CTX_free(new_ctx); return ret; } int BN_is_prime_ex(const BIGNUM *candidate, int checks, BN_CTX *ctx, BN_GENCB *cb) { return BN_is_prime_fasttest_ex(candidate, checks, ctx, 0, cb); } int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx, int do_trial_division, BN_GENCB *cb) { int is_probably_prime; if (!BN_primality_test(&is_probably_prime, a, checks, ctx, do_trial_division, cb)) { return -1; } return is_probably_prime; } int BN_enhanced_miller_rabin_primality_test( enum bn_primality_result_t *out_result, const BIGNUM *w, int checks, BN_CTX *ctx, BN_GENCB *cb) { // Enhanced Miller-Rabin is only valid on odd integers greater than 3. if (!BN_is_odd(w) || BN_cmp_word(w, 3) <= 0) { OPENSSL_PUT_ERROR(BN, BN_R_INVALID_INPUT); return 0; } if (checks == BN_prime_checks_for_generation) { checks = BN_prime_checks_for_size(BN_num_bits(w)); } int ret = 0; BN_MONT_CTX *mont = NULL; BN_CTX_start(ctx); BIGNUM *w1 = BN_CTX_get(ctx); if (w1 == NULL || !BN_copy(w1, w) || !BN_sub_word(w1, 1)) { goto err; } // Write w1 as m*2^a (Steps 1 and 2). int a = 0; while (!BN_is_bit_set(w1, a)) { a++; } BIGNUM *m = BN_CTX_get(ctx); if (m == NULL || !BN_rshift(m, w1, a)) { goto err; } BIGNUM *b = BN_CTX_get(ctx); BIGNUM *g = BN_CTX_get(ctx); BIGNUM *z = BN_CTX_get(ctx); BIGNUM *x = BN_CTX_get(ctx); BIGNUM *x1 = BN_CTX_get(ctx); if (b == NULL || g == NULL || z == NULL || x == NULL || x1 == NULL) { goto err; } // Montgomery setup for computations mod w mont = BN_MONT_CTX_new_for_modulus(w, ctx); if (mont == NULL) { goto err; } // The following loop performs in inner iteration of the Enhanced Miller-Rabin // Primality test (Step 4). for (int i = 1; i <= checks; i++) { // Step 4.1-4.2 if (!BN_rand_range_ex(b, 2, w1)) { goto err; } // Step 4.3-4.4 if (!BN_gcd(g, b, w, ctx)) { goto err; } if (BN_cmp_word(g, 1) > 0) { *out_result = bn_composite; ret = 1; goto err; } // Step 4.5 if (!BN_mod_exp_mont(z, b, m, w, ctx, mont)) { goto err; } // Step 4.6 if (BN_is_one(z) || BN_cmp(z, w1) == 0) { goto loop; } // Step 4.7 for (int j = 1; j < a; j++) { if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) { goto err; } if (BN_cmp(z, w1) == 0) { goto loop; } if (BN_is_one(z)) { goto composite; } } // Step 4.8-4.9 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx)) { goto err; } // Step 4.10-4.11 if (!BN_is_one(z) && !BN_copy(x, z)) { goto err; } composite: // Step 4.12-4.14 if (!BN_copy(x1, x) || !BN_sub_word(x1, 1) || !BN_gcd(g, x1, w, ctx)) { goto err; } if (BN_cmp_word(g, 1) > 0) { *out_result = bn_composite; } else { *out_result = bn_non_prime_power_composite; } ret = 1; goto err; loop: // Step 4.15 if (!BN_GENCB_call(cb, BN_GENCB_PRIME_TEST, i - 1)) { goto err; } } *out_result = bn_probably_prime; ret = 1; err: BN_MONT_CTX_free(mont); BN_CTX_end(ctx); return ret; } static int probable_prime(BIGNUM *rnd, int bits) { do { if (!BN_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD)) { return 0; } } while (bn_odd_number_is_obviously_composite(rnd)); return 1; } static int probable_prime_dh(BIGNUM *rnd, int bits, const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx) { int ret = 0; BIGNUM *t1; BN_CTX_start(ctx); if ((t1 = BN_CTX_get(ctx)) == NULL) { goto err; } if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) { 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, 1)) { goto err; } } else { if (!BN_add(rnd, rnd, rem)) { goto err; } } // we now have a random number 'rand' to test. const size_t num_primes = num_trial_division_primes(rnd); loop: for (size_t i = 1; i < num_primes; i++) { // check that rnd is a prime if (bn_mod_u16_consttime(rnd, kPrimes[i]) <= 1) { if (!BN_add(rnd, rnd, add)) { goto err; } goto loop; } } ret = 1; err: BN_CTX_end(ctx); return ret; } static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd, const BIGNUM *rem, BN_CTX *ctx) { int ret = 0; BIGNUM *t1, *qadd, *q; bits--; BN_CTX_start(ctx); t1 = BN_CTX_get(ctx); q = BN_CTX_get(ctx); qadd = BN_CTX_get(ctx); if (qadd == NULL) { goto err; } if (!BN_rshift1(qadd, padd)) { goto err; } if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD)) { goto err; } // we need ((rnd-rem) % add) == 0 if (!BN_mod(t1, q, qadd, ctx)) { goto err; } if (!BN_sub(q, q, t1)) { goto err; } if (rem == NULL) { if (!BN_add_word(q, 1)) { goto err; } } else { if (!BN_rshift1(t1, rem)) { goto err; } if (!BN_add(q, q, t1)) { goto err; } } // we now have a random number 'rand' to test. if (!BN_lshift1(p, q)) { goto err; } if (!BN_add_word(p, 1)) { goto err; } const size_t num_primes = num_trial_division_primes(p); loop: for (size_t i = 1; i < num_primes; i++) { // check that p and q are prime // check that for p and q // gcd(p-1,primes) == 1 (except for 2) if (bn_mod_u16_consttime(p, kPrimes[i]) == 0 || bn_mod_u16_consttime(q, kPrimes[i]) == 0) { if (!BN_add(p, p, padd)) { goto err; } if (!BN_add(q, q, qadd)) { goto err; } goto loop; } } ret = 1; err: BN_CTX_end(ctx); return ret; }