2 * Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
12 #include "internal/cryptlib.h"
16 * The quick sieve algorithm approach to weeding out primes is Philip
17 * Zimmermann's, as implemented in PGP. I have had a read of his comments
18 * and implemented my own version.
22 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
24 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
25 const BIGNUM *add, const BIGNUM *rem,
28 #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
31 # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
33 # define BN_DEF(lo, hi) lo, hi
37 * See SP800 89 5.3.3 (Step f)
38 * The product of the set of primes ranging from 3 to 751
39 * Generated using process in test/bn_internal_test.c test_bn_small_factors().
40 * This includes 751 (which is not currently included in SP 800-89).
42 static const BN_ULONG small_prime_factors[] = {
43 BN_DEF(0x3ef4e3e1, 0xc4309333), BN_DEF(0xcd2d655f, 0x71161eb6),
44 BN_DEF(0x0bf94862, 0x95e2238c), BN_DEF(0x24f7912b, 0x3eb233d3),
45 BN_DEF(0xbf26c483, 0x6b55514b), BN_DEF(0x5a144871, 0x0a84d817),
46 BN_DEF(0x9b82210a, 0x77d12fee), BN_DEF(0x97f050b3, 0xdb5b93c2),
47 BN_DEF(0x4d6c026b, 0x4acad6b9), BN_DEF(0x54aec893, 0xeb7751f3),
48 BN_DEF(0x36bc85c4, 0xdba53368), BN_DEF(0x7f5ec78e, 0xd85a1b28),
49 BN_DEF(0x6b322244, 0x2eb072d8), BN_DEF(0x5e2b3aea, 0xbba51112),
50 BN_DEF(0x0e2486bf, 0x36ed1a6c), BN_DEF(0xec0c5727, 0x5f270460),
54 #define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors)
55 static const BIGNUM _bignum_small_prime_factors = {
56 (BN_ULONG *)small_prime_factors,
57 BN_SMALL_PRIME_FACTORS_TOP,
58 BN_SMALL_PRIME_FACTORS_TOP,
63 const BIGNUM *bn_get0_small_factors(void)
65 return &_bignum_small_prime_factors;
68 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
70 /* No callback means continue */
75 /* Deprecated-style callbacks */
78 cb->cb.cb_1(a, b, cb->arg);
81 /* New-style callbacks */
82 return cb->cb.cb_2(a, b, cb);
86 /* Unrecognised callback type */
90 int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,
91 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,
98 int checks = BN_prime_checks_for_size(bits);
101 /* There are no prime numbers this small. */
102 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
104 } else if (add == NULL && safe && bits < 6 && bits != 3) {
106 * The smallest safe prime (7) is three bits.
107 * But the following two safe primes with less than 6 bits (11, 23)
108 * are unreachable for BN_rand with BN_RAND_TOP_TWO.
110 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
114 mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
123 /* make a random number and set the top and bottom bits */
125 if (!probable_prime(ret, bits, safe, mods, ctx))
128 if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
132 if (!BN_GENCB_call(cb, 0, c1++))
137 i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
144 * for "safe prime" generation, check that (p-1)/2 is prime. Since a
145 * prime is odd, We just need to divide by 2
147 if (!BN_rshift1(t, ret))
150 for (i = 0; i < checks; i++) {
151 j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
157 j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
163 if (!BN_GENCB_call(cb, 2, c1 - 1))
165 /* We have a safe prime test pass */
168 /* we have a prime :-) */
178 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
179 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
181 BN_CTX *ctx = BN_CTX_new();
187 retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
194 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
197 return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
200 /* See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test. */
201 int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
202 int do_trial_division, BN_GENCB *cb)
204 int i, status, ret = -1;
206 BN_CTX *ctxlocal = NULL;
213 /* w must be bigger than 1 */
214 if (BN_cmp(w, BN_value_one()) <= 0)
219 /* Take care of the really small prime 3 */
220 if (BN_is_word(w, 3))
223 /* 2 is the only even prime */
224 return BN_is_word(w, 2);
227 /* first look for small factors */
228 if (do_trial_division) {
229 for (i = 1; i < NUMPRIMES; i++) {
230 BN_ULONG mod = BN_mod_word(w, primes[i]);
231 if (mod == (BN_ULONG)-1)
234 return BN_is_word(w, primes[i]);
236 if (!BN_GENCB_call(cb, 1, -1))
240 if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
244 ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
247 ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
250 BN_CTX_free(ctxlocal);
256 * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.
257 * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).
258 * The Step numbers listed in the code refer to the enhanced case.
260 * if enhanced is set, then status returns one of the following:
261 * BN_PRIMETEST_PROBABLY_PRIME
262 * BN_PRIMETEST_COMPOSITE_WITH_FACTOR
263 * BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME
264 * if enhanced is zero, then status returns either
265 * BN_PRIMETEST_PROBABLY_PRIME or
266 * BN_PRIMETEST_COMPOSITE
268 * returns 0 if there was an error, otherwise it returns 1.
270 int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
271 BN_GENCB *cb, int enhanced, int *status)
273 int i, j, a, ret = 0;
274 BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
275 BN_MONT_CTX *mont = NULL;
283 w1 = BN_CTX_get(ctx);
284 w3 = BN_CTX_get(ctx);
293 && BN_sub_word(w1, 1)
296 && BN_sub_word(w3, 3)))
299 /* check w is larger than 3, otherwise the random b will be too small */
300 if (BN_is_zero(w3) || BN_is_negative(w3))
303 /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
305 while (!BN_is_bit_set(w1, a))
307 /* (Step 2) m = (w-1) / 2^a */
308 if (!BN_rshift(m, w1, a))
311 /* Montgomery setup for computations mod a */
312 mont = BN_MONT_CTX_new();
313 if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
316 if (iterations == BN_prime_checks)
317 iterations = BN_prime_checks_for_size(BN_num_bits(w));
320 for (i = 0; i < iterations; ++i) {
321 /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */
322 if (!BN_priv_rand_range_ex(b, w3, ctx)
323 || !BN_add_word(b, 2)) /* 1 < b < w-1 */
328 if (!BN_gcd(g, b, w, ctx))
332 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
337 /* (Step 4.5) z = b^m mod w */
338 if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
340 /* (Step 4.6) if (z = 1 or z = w-1) */
341 if (BN_is_one(z) || BN_cmp(z, w1) == 0)
343 /* (Step 4.7) for j = 1 to a-1 */
344 for (j = 1; j < a ; ++j) {
345 /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */
346 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
349 if (BN_cmp(z, w1) == 0)
355 /* At this point z = b^((w-1)/2) mod w */
356 /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */
357 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
362 /* (Step 4.11) x = b^(w-1) mod w */
367 /* (Step 4.1.2) g = GCD(x-1, w) */
368 if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
370 /* (Steps 4.1.3 - 4.1.4) */
372 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
374 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
376 *status = BN_PRIMETEST_COMPOSITE;
382 if (!BN_GENCB_call(cb, 1, i))
386 *status = BN_PRIMETEST_PROBABLY_PRIME;
397 BN_MONT_CTX_free(mont);
401 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
406 BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
409 /* TODO: Not all primes are private */
410 if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
412 if (safe && !BN_set_bit(rnd, 1))
414 /* we now have a random number 'rnd' to test. */
415 for (i = 1; i < NUMPRIMES; i++) {
416 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
417 if (mod == (BN_ULONG)-1)
419 mods[i] = (prime_t) mod;
423 for (i = 1; i < NUMPRIMES; i++) {
425 * check that rnd is a prime and also that
426 * gcd(rnd-1,primes) == 1 (except for 2)
427 * do the second check only if we are interested in safe primes
428 * in the case that the candidate prime is a single word then
429 * we check only the primes up to sqrt(rnd)
431 if (bits <= 31 && delta <= 0x7fffffff
432 && square(primes[i]) > BN_get_word(rnd) + delta)
434 if (safe ? (mods[i] + delta) % primes[i] <= 1
435 : (mods[i] + delta) % primes[i] == 0) {
436 delta += safe ? 4 : 2;
437 if (delta > maxdelta)
442 if (!BN_add_word(rnd, delta))
444 if (BN_num_bits(rnd) != bits)
450 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
451 const BIGNUM *add, const BIGNUM *rem,
457 BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
460 if ((t1 = BN_CTX_get(ctx)) == NULL)
463 if (maxdelta > BN_MASK2 - BN_get_word(add))
464 maxdelta = BN_MASK2 - BN_get_word(add);
467 if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
470 /* we need ((rnd-rem) % add) == 0 */
472 if (!BN_mod(t1, rnd, add, ctx))
474 if (!BN_sub(rnd, rnd, t1))
477 if (!BN_add_word(rnd, safe ? 3u : 1u))
480 if (!BN_add(rnd, rnd, rem))
484 if (BN_num_bits(rnd) < bits
485 || BN_get_word(rnd) < (safe ? 5u : 3u)) {
486 if (!BN_add(rnd, rnd, add))
490 /* we now have a random number 'rnd' to test. */
491 for (i = 1; i < NUMPRIMES; i++) {
492 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
493 if (mod == (BN_ULONG)-1)
495 mods[i] = (prime_t) mod;
499 for (i = 1; i < NUMPRIMES; i++) {
500 /* check that rnd is a prime */
501 if (bits <= 31 && delta <= 0x7fffffff
502 && square(primes[i]) > BN_get_word(rnd) + delta)
504 /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */
505 if (safe ? (mods[i] + delta) % primes[i] <= 1
506 : (mods[i] + delta) % primes[i] == 0) {
507 delta += BN_get_word(add);
508 if (delta > maxdelta)
513 if (!BN_add_word(rnd, delta))