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
))