1/*
2 * Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved.
3 *
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
8 */
9
10#include <stdio.h>
11#include <time.h>
12#include "internal/cryptlib.h"
13#include "bn_local.h"
14
15/*
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.
19 */
20#include "bn_prime.h"
21
22static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
23 BN_CTX *ctx);
24static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
25 const BIGNUM *add, const BIGNUM *rem,
26 BN_CTX *ctx);
27static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
28 int do_trial_division, BN_GENCB *cb);
29
30#define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
31
32#if BN_BITS2 == 64
33# define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
34#else
35# define BN_DEF(lo, hi) lo, hi
36#endif
37
38/*
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).
43 */
44static 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),
53 (BN_ULONG)0x000017b1
54};
55
56#define BN_SMALL_PRIME_FACTORS_TOP OSSL_NELEM(small_prime_factors)
57static const BIGNUM _bignum_small_prime_factors = {
58 (BN_ULONG *)small_prime_factors,
59 BN_SMALL_PRIME_FACTORS_TOP,
60 BN_SMALL_PRIME_FACTORS_TOP,
61 0,
62 BN_FLG_STATIC_DATA
63};
64
65const BIGNUM *bn_get0_small_factors(void)
66{
67 return &_bignum_small_prime_factors;
68}
69
70/*
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.
73 */
74static int calc_trial_divisions(int bits)
75{
76 if (bits <= 512)
77 return 64;
78 else if (bits <= 1024)
79 return 128;
80 else if (bits <= 2048)
81 return 384;
82 else if (bits <= 4096)
83 return 1024;
84 return NUMPRIMES;
85}
86
87/*
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.
93 */
94static int bn_mr_min_checks(int bits)
95{
96 if (bits > 2048)
97 return 128;
98 return 64;
99}
100
101int BN_GENCB_call(BN_GENCB *cb, int a, int b)
102{
103 /* No callback means continue */
104 if (!cb)
105 return 1;
106 switch (cb->ver) {
107 case 1:
108 /* Deprecated-style callbacks */
109 if (!cb->cb.cb_1)
110 return 1;
111 cb->cb.cb_1(a, b, cb->arg);
112 return 1;
113 case 2:
114 /* New-style callbacks */
115 return cb->cb.cb_2(a, b, cb);
116 default:
117 break;
118 }
119 /* Unrecognised callback type */
120 return 0;
121}
122
123int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,
124 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,
125 BN_CTX *ctx)
126{
127 BIGNUM *t;
128 int found = 0;
129 int i, j, c1 = 0;
130 prime_t *mods = NULL;
131 int checks = bn_mr_min_checks(bits);
132
133 if (bits < 2) {
134 /* There are no prime numbers this small. */
135 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
136 return 0;
137 } else if (add == NULL && safe && bits < 6 && bits != 3) {
138 /*
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.
142 */
143 BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
144 return 0;
145 }
146
147 mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
148 if (mods == NULL)
149 goto err;
150
151 BN_CTX_start(ctx);
152 t = BN_CTX_get(ctx);
153 if (t == NULL)
154 goto err;
155 loop:
156 /* make a random number and set the top and bottom bits */
157 if (add == NULL) {
158 if (!probable_prime(ret, bits, safe, mods, ctx))
159 goto err;
160 } else {
161 if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
162 goto err;
163 }
164
165 if (!BN_GENCB_call(cb, 0, c1++))
166 /* aborted */
167 goto err;
168
169 if (!safe) {
170 i = bn_is_prime_int(ret, checks, ctx, 0, cb);
171 if (i == -1)
172 goto err;
173 if (i == 0)
174 goto loop;
175 } else {
176 /*
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
179 */
180 if (!BN_rshift1(t, ret))
181 goto err;
182
183 for (i = 0; i < checks; i++) {
184 j = bn_is_prime_int(ret, 1, ctx, 0, cb);
185 if (j == -1)
186 goto err;
187 if (j == 0)
188 goto loop;
189
190 j = bn_is_prime_int(t, 1, ctx, 0, cb);
191 if (j == -1)
192 goto err;
193 if (j == 0)
194 goto loop;
195
196 if (!BN_GENCB_call(cb, 2, c1 - 1))
197 goto err;
198 /* We have a safe prime test pass */
199 }
200 }
201 /* we have a prime :-) */
202 found = 1;
203 err:
204 OPENSSL_free(mods);
205 BN_CTX_end(ctx);
206 bn_check_top(ret);
207 return found;
208}
209
210#ifndef FIPS_MODE
211int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
212 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
213{
214 BN_CTX *ctx = BN_CTX_new();
215 int retval;
216
217 if (ctx == NULL)
218 return 0;
219
220 retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
221
222 BN_CTX_free(ctx);
223 return retval;
224}
225#endif
226
227#ifndef OPENSSL_NO_DEPRECATED_3_0
228int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
229 BN_GENCB *cb)
230{
231 return bn_check_prime_int(a, checks, ctx_passed, 0, cb);
232}
233
234int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
235 int do_trial_division, BN_GENCB *cb)
236{
237 return bn_check_prime_int(w, checks, ctx, do_trial_division, cb);
238}
239#endif
240
241/* Wrapper around bn_is_prime_int that sets the minimum number of checks */
242int bn_check_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
243 int do_trial_division, BN_GENCB *cb)
244{
245 int min_checks = bn_mr_min_checks(BN_num_bits(w));
246
247 if (checks < min_checks)
248 checks = min_checks;
249
250 return bn_is_prime_int(w, checks, ctx, do_trial_division, cb);
251}
252
253int BN_check_prime(const BIGNUM *p, BN_CTX *ctx, BN_GENCB *cb)
254{
255 return bn_check_prime_int(p, 0, ctx, 1, cb);
256}
257
258/*
259 * Tests that |w| is probably prime
260 * See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test.
261 *
262 * Returns 0 when composite, 1 when probable prime, -1 on error.
263 */
264static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
265 int do_trial_division, BN_GENCB *cb)
266{
267 int i, status, ret = -1;
268#ifndef FIPS_MODE
269 BN_CTX *ctxlocal = NULL;
270#else
271
272 if (ctx == NULL)
273 return -1;
274#endif
275
276 /* w must be bigger than 1 */
277 if (BN_cmp(w, BN_value_one()) <= 0)
278 return 0;
279
280 /* w must be odd */
281 if (BN_is_odd(w)) {
282 /* Take care of the really small prime 3 */
283 if (BN_is_word(w, 3))
284 return 1;
285 } else {
286 /* 2 is the only even prime */
287 return BN_is_word(w, 2);
288 }
289
290 /* first look for small factors */
291 if (do_trial_division) {
292 int trial_divisions = calc_trial_divisions(BN_num_bits(w));
293
294 for (i = 1; i < trial_divisions; i++) {
295 BN_ULONG mod = BN_mod_word(w, primes[i]);
296 if (mod == (BN_ULONG)-1)
297 return -1;
298 if (mod == 0)
299 return BN_is_word(w, primes[i]);
300 }
301 if (!BN_GENCB_call(cb, 1, -1))
302 return -1;
303 }
304#ifndef FIPS_MODE
305 if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
306 goto err;
307#endif
308
309 ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
310 if (!ret)
311 goto err;
312 ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
313err:
314#ifndef FIPS_MODE
315 BN_CTX_free(ctxlocal);
316#endif
317 return ret;
318}
319
320/*
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.
324 *
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
332 *
333 * returns 0 if there was an error, otherwise it returns 1.
334 */
335int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
336 BN_GENCB *cb, int enhanced, int *status)
337{
338 int i, j, a, ret = 0;
339 BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
340 BN_MONT_CTX *mont = NULL;
341
342 /* w must be odd */
343 if (!BN_is_odd(w))
344 return 0;
345
346 BN_CTX_start(ctx);
347 g = BN_CTX_get(ctx);
348 w1 = BN_CTX_get(ctx);
349 w3 = BN_CTX_get(ctx);
350 x = BN_CTX_get(ctx);
351 m = BN_CTX_get(ctx);
352 z = BN_CTX_get(ctx);
353 b = BN_CTX_get(ctx);
354
355 if (!(b != NULL
356 /* w1 := w - 1 */
357 && BN_copy(w1, w)
358 && BN_sub_word(w1, 1)
359 /* w3 := w - 3 */
360 && BN_copy(w3, w)
361 && BN_sub_word(w3, 3)))
362 goto err;
363
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))
366 goto err;
367
368 /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
369 a = 1;
370 while (!BN_is_bit_set(w1, a))
371 a++;
372 /* (Step 2) m = (w-1) / 2^a */
373 if (!BN_rshift(m, w1, a))
374 goto err;
375
376 /* Montgomery setup for computations mod a */
377 mont = BN_MONT_CTX_new();
378 if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
379 goto err;
380
381 if (iterations == 0)
382 iterations = bn_mr_min_checks(BN_num_bits(w));
383
384 /* (Step 4) */
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 */
389 goto err;
390
391 if (enhanced) {
392 /* (Step 4.3) */
393 if (!BN_gcd(g, b, w, ctx))
394 goto err;
395 /* (Step 4.4) */
396 if (!BN_is_one(g)) {
397 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
398 ret = 1;
399 goto err;
400 }
401 }
402 /* (Step 4.5) z = b^m mod w */
403 if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
404 goto err;
405 /* (Step 4.6) if (z = 1 or z = w-1) */
406 if (BN_is_one(z) || BN_cmp(z, w1) == 0)
407 goto outer_loop;
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))
412 goto err;
413 /* (Step 4.7.3) */
414 if (BN_cmp(z, w1) == 0)
415 goto outer_loop;
416 /* (Step 4.7.4) */
417 if (BN_is_one(z))
418 goto composite;
419 }
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))
423 goto err;
424 /* (Step 4.10) */
425 if (BN_is_one(z))
426 goto composite;
427 /* (Step 4.11) x = b^(w-1) mod w */
428 if (!BN_copy(x, z))
429 goto err;
430composite:
431 if (enhanced) {
432 /* (Step 4.1.2) g = GCD(x-1, w) */
433 if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
434 goto err;
435 /* (Steps 4.1.3 - 4.1.4) */
436 if (BN_is_one(g))
437 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
438 else
439 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
440 } else {
441 *status = BN_PRIMETEST_COMPOSITE;
442 }
443 ret = 1;
444 goto err;
445outer_loop: ;
446 /* (Step 4.1.5) */
447 if (!BN_GENCB_call(cb, 1, i))
448 goto err;
449 }
450 /* (Step 5) */
451 *status = BN_PRIMETEST_PROBABLY_PRIME;
452 ret = 1;
453err:
454 BN_clear(g);
455 BN_clear(w1);
456 BN_clear(w3);
457 BN_clear(x);
458 BN_clear(m);
459 BN_clear(z);
460 BN_clear(b);
461 BN_CTX_end(ctx);
462 BN_MONT_CTX_free(mont);
463 return ret;
464}
465
466/*
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.
470 *
471 * The probably prime is saved in |rnd|.
472 *
473 * Returns 1 on success and 0 on error.
474 */
475static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
476 BN_CTX *ctx)
477{
478 int i;
479 BN_ULONG delta;
480 int trial_divisions = calc_trial_divisions(bits);
481 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
482
483 again:
484 /* TODO: Not all primes are private */
485 if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
486 return 0;
487 if (safe && !BN_set_bit(rnd, 1))
488 return 0;
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)
493 return 0;
494 mods[i] = (prime_t) mod;
495 }
496 delta = 0;
497 loop:
498 for (i = 1; i < trial_divisions; i++) {
499 /*
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)
505 */
506 if (bits <= 31 && delta <= 0x7fffffff
507 && square(primes[i]) > BN_get_word(rnd) + delta)
508 break;
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)
513 goto again;
514 goto loop;
515 }
516 }
517 if (!BN_add_word(rnd, delta))
518 return 0;
519 if (BN_num_bits(rnd) != bits)
520 goto again;
521 bn_check_top(rnd);
522 return 1;
523}
524
525/*
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.
530 *
531 * Returns 1 on success and 0 on error.
532 */
533static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
534 const BIGNUM *add, const BIGNUM *rem,
535 BN_CTX *ctx)
536{
537 int i, ret = 0;
538 BIGNUM *t1;
539 BN_ULONG delta;
540 int trial_divisions = calc_trial_divisions(bits);
541 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
542
543 BN_CTX_start(ctx);
544 if ((t1 = BN_CTX_get(ctx)) == NULL)
545 goto err;
546
547 if (maxdelta > BN_MASK2 - BN_get_word(add))
548 maxdelta = BN_MASK2 - BN_get_word(add);
549
550 again:
551 if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
552 goto err;
553
554 /* we need ((rnd-rem) % add) == 0 */
555
556 if (!BN_mod(t1, rnd, add, ctx))
557 goto err;
558 if (!BN_sub(rnd, rnd, t1))
559 goto err;
560 if (rem == NULL) {
561 if (!BN_add_word(rnd, safe ? 3u : 1u))
562 goto err;
563 } else {
564 if (!BN_add(rnd, rnd, rem))
565 goto err;
566 }
567
568 if (BN_num_bits(rnd) < bits
569 || BN_get_word(rnd) < (safe ? 5u : 3u)) {
570 if (!BN_add(rnd, rnd, add))
571 goto err;
572 }
573
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)
578 goto err;
579 mods[i] = (prime_t) mod;
580 }
581 delta = 0;
582 loop:
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)
587 break;
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)
593 goto again;
594 goto loop;
595 }
596 }
597 if (!BN_add_word(rnd, delta))
598 goto err;
599 ret = 1;
600
601 err:
602 BN_CTX_end(ctx);
603 bn_check_top(rnd);
604 return ret;
605}
606