2 * Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved.
4 * Licensed under the OpenSSL license (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 witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
23 const BIGNUM *a1_odd, int k, BN_CTX *ctx,
25 static int probable_prime(BIGNUM *rnd, int bits, prime_t *mods);
26 static int probable_prime_dh_safe(BIGNUM *rnd, int bits,
27 const BIGNUM *add, const BIGNUM *rem,
30 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
32 /* No callback means continue */
37 /* Deprecated-style callbacks */
40 cb->cb.cb_1(a, b, cb->arg);
43 /* New-style callbacks */
44 return cb->cb.cb_2(a, b, cb);
48 /* Unrecognised callback type */
52 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
53 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
60 int checks = BN_prime_checks_for_size(bits);
63 /* There are no prime numbers this small. */
64 BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
66 } else if (add == NULL && safe && bits < 6 && bits != 3) {
68 * The smallest safe prime (7) is three bits.
69 * But the following two safe primes with less than 6 bits (11, 23)
70 * are unreachable for BN_rand with BN_RAND_TOP_TWO.
72 BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
76 mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
88 /* make a random number and set the top and bottom bits */
90 if (!probable_prime(ret, bits, mods))
94 if (!probable_prime_dh_safe(ret, bits, add, rem, ctx))
97 if (!bn_probable_prime_dh(ret, bits, add, rem, ctx))
102 if (!BN_GENCB_call(cb, 0, c1++))
107 i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
114 * for "safe prime" generation, check that (p-1)/2 is prime. Since a
115 * prime is odd, We just need to divide by 2
117 if (!BN_rshift1(t, ret))
120 for (i = 0; i < checks; i++) {
121 j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
127 j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
133 if (!BN_GENCB_call(cb, 2, c1 - 1))
135 /* We have a safe prime test pass */
138 /* we have a prime :-) */
148 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
151 return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
154 int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
155 int do_trial_division, BN_GENCB *cb)
160 BIGNUM *A1, *A1_odd, *A3, *check; /* taken from ctx */
161 BN_MONT_CTX *mont = NULL;
163 /* Take care of the really small primes 2 & 3 */
164 if (BN_is_word(a, 2) || BN_is_word(a, 3))
167 /* Check odd and bigger than 1 */
168 if (!BN_is_odd(a) || BN_cmp(a, BN_value_one()) <= 0)
171 if (checks == BN_prime_checks)
172 checks = BN_prime_checks_for_size(BN_num_bits(a));
174 /* first look for small factors */
175 if (do_trial_division) {
176 for (i = 1; i < NUMPRIMES; i++) {
177 BN_ULONG mod = BN_mod_word(a, primes[i]);
178 if (mod == (BN_ULONG)-1)
181 return BN_is_word(a, primes[i]);
183 if (!BN_GENCB_call(cb, 1, -1))
187 if (ctx_passed != NULL)
189 else if ((ctx = BN_CTX_new()) == NULL)
193 A1 = BN_CTX_get(ctx);
194 A3 = BN_CTX_get(ctx);
195 A1_odd = BN_CTX_get(ctx);
196 check = BN_CTX_get(ctx);
200 /* compute A1 := a - 1 */
201 if (!BN_copy(A1, a) || !BN_sub_word(A1, 1))
203 /* compute A3 := a - 3 */
204 if (!BN_copy(A3, a) || !BN_sub_word(A3, 3))
207 /* write A1 as A1_odd * 2^k */
209 while (!BN_is_bit_set(A1, k))
211 if (!BN_rshift(A1_odd, A1, k))
214 /* Montgomery setup for computations mod a */
215 mont = BN_MONT_CTX_new();
218 if (!BN_MONT_CTX_set(mont, a, ctx))
221 for (i = 0; i < checks; i++) {
222 /* 1 < check < a-1 */
223 if (!BN_priv_rand_range(check, A3) || !BN_add_word(check, 2))
226 j = witness(check, a, A1, A1_odd, k, ctx, mont);
233 if (!BN_GENCB_call(cb, 1, i))
240 if (ctx_passed == NULL)
243 BN_MONT_CTX_free(mont);
248 static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
249 const BIGNUM *a1_odd, int k, BN_CTX *ctx,
252 if (!BN_mod_exp_mont(w, w, a1_odd, a, ctx, mont)) /* w := w^a1_odd mod a */
255 return 0; /* probably prime */
256 if (BN_cmp(w, a1) == 0)
257 return 0; /* w == -1 (mod a), 'a' is probably prime */
259 if (!BN_mod_mul(w, w, w, a, ctx)) /* w := w^2 mod a */
262 return 1; /* 'a' is composite, otherwise a previous 'w'
263 * would have been == -1 (mod 'a') */
264 if (BN_cmp(w, a1) == 0)
265 return 0; /* w == -1 (mod a), 'a' is probably prime */
268 * If we get here, 'w' is the (a-1)/2-th power of the original 'w', and
269 * it is neither -1 nor +1 -- so 'a' cannot be prime
275 static int probable_prime(BIGNUM *rnd, int bits, prime_t *mods)
279 BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
280 char is_single_word = bits <= BN_BITS2;
283 /* TODO: Not all primes are private */
284 if (!BN_priv_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD))
286 /* we now have a random number 'rnd' to test. */
287 for (i = 1; i < NUMPRIMES; i++) {
288 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
289 if (mod == (BN_ULONG)-1)
291 mods[i] = (prime_t) mod;
294 * If bits is so small that it fits into a single word then we
295 * additionally don't want to exceed that many bits.
297 if (is_single_word) {
300 if (bits == BN_BITS2) {
302 * Shifting by this much has undefined behaviour so we do it a
305 size_limit = ~((BN_ULONG)0) - BN_get_word(rnd);
307 size_limit = (((BN_ULONG)1) << bits) - BN_get_word(rnd) - 1;
309 if (size_limit < maxdelta)
310 maxdelta = size_limit;
314 if (is_single_word) {
315 BN_ULONG rnd_word = BN_get_word(rnd);
318 * In the case that the candidate prime is a single word then
320 * 1) It's greater than primes[i] because we shouldn't reject
321 * 3 as being a prime number because it's a multiple of
323 * 2) That it's not a multiple of a known prime. We don't
324 * check that rnd-1 is also coprime to all the known
325 * primes because there aren't many small primes where
328 for (i = 1; i < NUMPRIMES && primes[i] < rnd_word; i++) {
329 if ((mods[i] + delta) % primes[i] == 0) {
331 if (delta > maxdelta)
337 for (i = 1; i < NUMPRIMES; i++) {
339 * check that rnd is not a prime and also that gcd(rnd-1,primes)
340 * == 1 (except for 2)
342 if (((mods[i] + delta) % primes[i]) <= 1) {
344 if (delta > maxdelta)
350 if (!BN_add_word(rnd, delta))
352 if (BN_num_bits(rnd) != bits)
358 int bn_probable_prime_dh(BIGNUM *rnd, int bits,
359 const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx)
365 if ((t1 = BN_CTX_get(ctx)) == NULL)
368 if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
371 /* we need ((rnd-rem) % add) == 0 */
373 if (!BN_mod(t1, rnd, add, ctx))
375 if (!BN_sub(rnd, rnd, t1))
378 if (!BN_add_word(rnd, 1))
381 if (!BN_add(rnd, rnd, rem))
385 /* we now have a random number 'rand' to test. */
388 for (i = 1; i < NUMPRIMES; i++) {
389 /* check that rnd is a prime */
390 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
391 if (mod == (BN_ULONG)-1)
394 if (!BN_add(rnd, rnd, add))
407 static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
408 const BIGNUM *rem, BN_CTX *ctx)
411 BIGNUM *t1, *qadd, *q;
415 t1 = BN_CTX_get(ctx);
417 qadd = BN_CTX_get(ctx);
421 if (!BN_rshift1(qadd, padd))
424 if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
427 /* we need ((rnd-rem) % add) == 0 */
428 if (!BN_mod(t1, q, qadd, ctx))
430 if (!BN_sub(q, q, t1))
433 if (!BN_add_word(q, 1))
436 if (!BN_rshift1(t1, rem))
438 if (!BN_add(q, q, t1))
442 /* we now have a random number 'rand' to test. */
443 if (!BN_lshift1(p, q))
445 if (!BN_add_word(p, 1))
449 for (i = 1; i < NUMPRIMES; i++) {
450 /* check that p and q are prime */
452 * check that for p and q gcd(p-1,primes) == 1 (except for 2)
454 BN_ULONG pmod = BN_mod_word(p, (BN_ULONG)primes[i]);
455 BN_ULONG qmod = BN_mod_word(q, (BN_ULONG)primes[i]);
456 if (pmod == (BN_ULONG)-1 || qmod == (BN_ULONG)-1)
458 if (pmod == 0 || qmod == 0) {
459 if (!BN_add(p, p, padd))
461 if (!BN_add(q, q, qadd))