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,
27 static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
28 int do_trial_division, BN_GENCB *cb);
30 #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
33 # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
35 # define BN_DEF(lo, hi) lo, hi
39 * See SP800 89 5.3.3 (Step f)
40 * The product of the set of primes ranging from 3 to 751
41 * Generated using process in test/bn_internal_test.c test_bn_small_factors().
42 * This includes 751 (which is not currently included in SP 800-89).
44 static const BN_ULONG small_prime_factors[] = {
45 BN_DEF(0x3ef4e3e1, 0xc4309333), BN_DEF(0xcd2d655f, 0x71161eb6),
46 BN_DEF(0x0bf94862, 0x95e2238c), BN_DEF(0x24f7912b, 0x3eb233d3),
47 BN_DEF(0xbf26c483, 0x6b55514b), BN_DEF(0x5a144871, 0x0a84d817),
48 BN_DEF(0x9b82210a, 0x77d12fee), BN_DEF(0x97f050b3, 0xdb5b93c2),
49 BN_DEF(0x4d6c026b, 0x4acad6b9), BN_DEF(0x54aec893, 0xeb7751f3),
50 BN_DEF(0x36bc85c4, 0xdba53368), BN_DEF(0x7f5ec78e, 0xd85a1b28),
51 BN_DEF(0x6b322244, 0x2eb072d8), BN_DEF(0x5e2b3aea, 0xbba51112),
52 BN_DEF(0x0e2486bf, 0x36ed1a6c), BN_DEF(0xec0c5727, 0x5f270460),
56 #define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors)
57 static const BIGNUM _bignum_small_prime_factors = {
58 (BN_ULONG *)small_prime_factors,
59 BN_SMALL_PRIME_FACTORS_TOP,
60 BN_SMALL_PRIME_FACTORS_TOP,
65 const BIGNUM *bn_get0_small_factors(void)
67 return &_bignum_small_prime_factors;
71 * Calculate the number of trial divisions that gives the best speed in
72 * combination with Miller-Rabin prime test, based on the sized of the prime.
74 static int calc_trial_divisions(int bits)
78 else if (bits <= 1024)
80 else if (bits <= 2048)
82 else if (bits <= 4096)
88 * Use a minimum of 64 rounds of Miller-Rabin, which should give a false
89 * positive rate of 2^-128. If the size of the prime is larger than 2048
90 * the user probably wants a higher security level than 128, so switch
91 * to 128 rounds giving a false positive rate of 2^-256.
92 * Returns the number of rounds.
94 static int bn_mr_min_checks(int bits)
101 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
103 /* No callback means continue */
108 /* Deprecated-style callbacks */
111 cb->cb.cb_1(a, b, cb->arg);
114 /* New-style callbacks */
115 return cb->cb.cb_2(a, b, cb);
119 /* Unrecognised callback type */
123 int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,
124 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,
130 prime_t *mods = NULL;
131 int checks = bn_mr_min_checks(bits);
134 /* There are no prime numbers this small. */
135 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
137 } else if (add == NULL && safe && bits < 6 && bits != 3) {
139 * The smallest safe prime (7) is three bits.
140 * But the following two safe primes with less than 6 bits (11, 23)
141 * are unreachable for BN_rand with BN_RAND_TOP_TWO.
143 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
147 mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
156 /* make a random number and set the top and bottom bits */
158 if (!probable_prime(ret, bits, safe, mods, ctx))
161 if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
165 if (!BN_GENCB_call(cb, 0, c1++))
170 i = bn_is_prime_int(ret, checks, ctx, 0, cb);
177 * for "safe prime" generation, check that (p-1)/2 is prime. Since a
178 * prime is odd, We just need to divide by 2
180 if (!BN_rshift1(t, ret))
183 for (i = 0; i < checks; i++) {
184 j = bn_is_prime_int(ret, 1, ctx, 0, cb);
190 j = bn_is_prime_int(t, 1, ctx, 0, cb);
196 if (!BN_GENCB_call(cb, 2, c1 - 1))
198 /* We have a safe prime test pass */
201 /* we have a prime :-) */
211 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
212 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
214 BN_CTX *ctx = BN_CTX_new();
220 retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
227 #ifndef OPENSSL_NO_DEPRECATED_3_0
228 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
231 return bn_check_prime_int(a, checks, ctx_passed, 0, cb);
234 int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
235 int do_trial_division, BN_GENCB *cb)
237 return bn_check_prime_int(w, checks, ctx, do_trial_division, cb);
241 /* Wrapper around bn_is_prime_int that sets the minimum number of checks */
242 int bn_check_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
243 int do_trial_division, BN_GENCB *cb)
245 int min_checks = bn_mr_min_checks(BN_num_bits(w));
247 if (checks < min_checks)
250 return bn_is_prime_int(w, checks, ctx, do_trial_division, cb);
253 int BN_check_prime(const BIGNUM *p, BN_CTX *ctx, BN_GENCB *cb)
255 return bn_check_prime_int(p, 0, ctx, 1, cb);
259 * Tests that |w| is probably prime
260 * See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test.
262 * Returns 0 when composite, 1 when probable prime, -1 on error.
264 static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
265 int do_trial_division, BN_GENCB *cb)
267 int i, status, ret = -1;
269 BN_CTX *ctxlocal = NULL;
276 /* w must be bigger than 1 */
277 if (BN_cmp(w, BN_value_one()) <= 0)
282 /* Take care of the really small prime 3 */
283 if (BN_is_word(w, 3))
286 /* 2 is the only even prime */
287 return BN_is_word(w, 2);
290 /* first look for small factors */
291 if (do_trial_division) {
292 int trial_divisions = calc_trial_divisions(BN_num_bits(w));
294 for (i = 1; i < trial_divisions; i++) {
295 BN_ULONG mod = BN_mod_word(w, primes[i]);
296 if (mod == (BN_ULONG)-1)
299 return BN_is_word(w, primes[i]);
301 if (!BN_GENCB_call(cb, 1, -1))
305 if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
309 ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
312 ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
315 BN_CTX_free(ctxlocal);
321 * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.
322 * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).
323 * The Step numbers listed in the code refer to the enhanced case.
325 * if enhanced is set, then status returns one of the following:
326 * BN_PRIMETEST_PROBABLY_PRIME
327 * BN_PRIMETEST_COMPOSITE_WITH_FACTOR
328 * BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME
329 * if enhanced is zero, then status returns either
330 * BN_PRIMETEST_PROBABLY_PRIME or
331 * BN_PRIMETEST_COMPOSITE
333 * returns 0 if there was an error, otherwise it returns 1.
335 int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
336 BN_GENCB *cb, int enhanced, int *status)
338 int i, j, a, ret = 0;
339 BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
340 BN_MONT_CTX *mont = NULL;
348 w1 = BN_CTX_get(ctx);
349 w3 = BN_CTX_get(ctx);
358 && BN_sub_word(w1, 1)
361 && BN_sub_word(w3, 3)))
364 /* check w is larger than 3, otherwise the random b will be too small */
365 if (BN_is_zero(w3) || BN_is_negative(w3))
368 /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
370 while (!BN_is_bit_set(w1, a))
372 /* (Step 2) m = (w-1) / 2^a */
373 if (!BN_rshift(m, w1, a))
376 /* Montgomery setup for computations mod a */
377 mont = BN_MONT_CTX_new();
378 if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
382 iterations = bn_mr_min_checks(BN_num_bits(w));
385 for (i = 0; i < iterations; ++i) {
386 /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */
387 if (!BN_priv_rand_range_ex(b, w3, ctx)
388 || !BN_add_word(b, 2)) /* 1 < b < w-1 */
393 if (!BN_gcd(g, b, w, ctx))
397 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
402 /* (Step 4.5) z = b^m mod w */
403 if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
405 /* (Step 4.6) if (z = 1 or z = w-1) */
406 if (BN_is_one(z) || BN_cmp(z, w1) == 0)
408 /* (Step 4.7) for j = 1 to a-1 */
409 for (j = 1; j < a ; ++j) {
410 /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */
411 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
414 if (BN_cmp(z, w1) == 0)
420 /* At this point z = b^((w-1)/2) mod w */
421 /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */
422 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
427 /* (Step 4.11) x = b^(w-1) mod w */
432 /* (Step 4.1.2) g = GCD(x-1, w) */
433 if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
435 /* (Steps 4.1.3 - 4.1.4) */
437 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
439 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
441 *status = BN_PRIMETEST_COMPOSITE;
447 if (!BN_GENCB_call(cb, 1, i))
451 *status = BN_PRIMETEST_PROBABLY_PRIME;
462 BN_MONT_CTX_free(mont);
467 * Generate a random number of |bits| bits that is probably prime by sieving.
468 * If |safe| != 0, it generates a safe prime.
469 * |mods| is a preallocated array that gets reused when called again.
471 * The probably prime is saved in |rnd|.
473 * Returns 1 on success and 0 on error.
475 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
480 int trial_divisions = calc_trial_divisions(bits);
481 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
484 /* TODO: Not all primes are private */
485 if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
487 if (safe && !BN_set_bit(rnd, 1))
489 /* we now have a random number 'rnd' to test. */
490 for (i = 1; i < trial_divisions; i++) {
491 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
492 if (mod == (BN_ULONG)-1)
494 mods[i] = (prime_t) mod;
498 for (i = 1; i < trial_divisions; i++) {
500 * check that rnd is a prime and also that
501 * gcd(rnd-1,primes) == 1 (except for 2)
502 * do the second check only if we are interested in safe primes
503 * in the case that the candidate prime is a single word then
504 * we check only the primes up to sqrt(rnd)
506 if (bits <= 31 && delta <= 0x7fffffff
507 && square(primes[i]) > BN_get_word(rnd) + delta)
509 if (safe ? (mods[i] + delta) % primes[i] <= 1
510 : (mods[i] + delta) % primes[i] == 0) {
511 delta += safe ? 4 : 2;
512 if (delta > maxdelta)
517 if (!BN_add_word(rnd, delta))
519 if (BN_num_bits(rnd) != bits)
526 * Generate a random number |rnd| of |bits| bits that is probably prime
527 * and satisfies |rnd| % |add| == |rem| by sieving.
528 * If |safe| != 0, it generates a safe prime.
529 * |mods| is a preallocated array that gets reused when called again.
531 * Returns 1 on success and 0 on error.
533 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
534 const BIGNUM *add, const BIGNUM *rem,
540 int trial_divisions = calc_trial_divisions(bits);
541 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
544 if ((t1 = BN_CTX_get(ctx)) == NULL)
547 if (maxdelta > BN_MASK2 - BN_get_word(add))
548 maxdelta = BN_MASK2 - BN_get_word(add);
551 if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
554 /* we need ((rnd-rem) % add) == 0 */
556 if (!BN_mod(t1, rnd, add, ctx))
558 if (!BN_sub(rnd, rnd, t1))
561 if (!BN_add_word(rnd, safe ? 3u : 1u))
564 if (!BN_add(rnd, rnd, rem))
568 if (BN_num_bits(rnd) < bits
569 || BN_get_word(rnd) < (safe ? 5u : 3u)) {
570 if (!BN_add(rnd, rnd, add))
574 /* we now have a random number 'rnd' to test. */
575 for (i = 1; i < trial_divisions; i++) {
576 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
577 if (mod == (BN_ULONG)-1)
579 mods[i] = (prime_t) mod;
583 for (i = 1; i < trial_divisions; i++) {
584 /* check that rnd is a prime */
585 if (bits <= 31 && delta <= 0x7fffffff
586 && square(primes[i]) > BN_get_word(rnd) + delta)
588 /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */
589 if (safe ? (mods[i] + delta) % primes[i] <= 1
590 : (mods[i] + delta) % primes[i] == 0) {
591 delta += BN_get_word(add);
592 if (delta > maxdelta)
597 if (!BN_add_word(rnd, delta))