1 /*
2 * Copyright 1995-2022 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
22 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
23 BN_CTX *ctx);
24 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
25 const BIGNUM *add, const BIGNUM *rem,
26 BN_CTX *ctx);
27 static 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 */
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),
53 (BN_ULONG)0x000017b1
54 };
55
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,
61 0,
62 BN_FLG_STATIC_DATA
63 };
64
ossl_bn_get0_small_factors(void)65 const BIGNUM *ossl_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 */
calc_trial_divisions(int bits)74 static 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 */
bn_mr_min_checks(int bits)94 static int bn_mr_min_checks(int bits)
95 {
96 if (bits > 2048)
97 return 128;
98 return 64;
99 }
100
BN_GENCB_call(BN_GENCB * cb,int a,int b)101 int 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
BN_generate_prime_ex2(BIGNUM * ret,int bits,int safe,const BIGNUM * add,const BIGNUM * rem,BN_GENCB * cb,BN_CTX * ctx)123 int 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 ERR_raise(ERR_LIB_BN, 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 ERR_raise(ERR_LIB_BN, BN_R_BITS_TOO_SMALL);
144 return 0;
145 }
146
147 mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
148 if (mods == NULL) {
149 ERR_raise(ERR_LIB_BN, ERR_R_MALLOC_FAILURE);
150 return 0;
151 }
152
153 BN_CTX_start(ctx);
154 t = BN_CTX_get(ctx);
155 if (t == NULL)
156 goto err;
157 loop:
158 /* make a random number and set the top and bottom bits */
159 if (add == NULL) {
160 if (!probable_prime(ret, bits, safe, mods, ctx))
161 goto err;
162 } else {
163 if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
164 goto err;
165 }
166
167 if (!BN_GENCB_call(cb, 0, c1++))
168 /* aborted */
169 goto err;
170
171 if (!safe) {
172 i = bn_is_prime_int(ret, checks, ctx, 0, cb);
173 if (i == -1)
174 goto err;
175 if (i == 0)
176 goto loop;
177 } else {
178 /*
179 * for "safe prime" generation, check that (p-1)/2 is prime. Since a
180 * prime is odd, We just need to divide by 2
181 */
182 if (!BN_rshift1(t, ret))
183 goto err;
184
185 for (i = 0; i < checks; i++) {
186 j = bn_is_prime_int(ret, 1, ctx, 0, cb);
187 if (j == -1)
188 goto err;
189 if (j == 0)
190 goto loop;
191
192 j = bn_is_prime_int(t, 1, ctx, 0, cb);
193 if (j == -1)
194 goto err;
195 if (j == 0)
196 goto loop;
197
198 if (!BN_GENCB_call(cb, 2, c1 - 1))
199 goto err;
200 /* We have a safe prime test pass */
201 }
202 }
203 /* we have a prime :-) */
204 found = 1;
205 err:
206 OPENSSL_free(mods);
207 BN_CTX_end(ctx);
208 bn_check_top(ret);
209 return found;
210 }
211
212 #ifndef FIPS_MODULE
BN_generate_prime_ex(BIGNUM * ret,int bits,int safe,const BIGNUM * add,const BIGNUM * rem,BN_GENCB * cb)213 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
214 const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
215 {
216 BN_CTX *ctx = BN_CTX_new();
217 int retval;
218
219 if (ctx == NULL)
220 return 0;
221
222 retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
223
224 BN_CTX_free(ctx);
225 return retval;
226 }
227 #endif
228
229 #ifndef OPENSSL_NO_DEPRECATED_3_0
BN_is_prime_ex(const BIGNUM * a,int checks,BN_CTX * ctx_passed,BN_GENCB * cb)230 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
231 BN_GENCB *cb)
232 {
233 return ossl_bn_check_prime(a, checks, ctx_passed, 0, cb);
234 }
235
BN_is_prime_fasttest_ex(const BIGNUM * w,int checks,BN_CTX * ctx,int do_trial_division,BN_GENCB * cb)236 int BN_is_prime_fasttest_ex(const BIGNUM *w, int checks, BN_CTX *ctx,
237 int do_trial_division, BN_GENCB *cb)
238 {
239 return ossl_bn_check_prime(w, checks, ctx, do_trial_division, cb);
240 }
241 #endif
242
243 /* Wrapper around bn_is_prime_int that sets the minimum number of checks */
ossl_bn_check_prime(const BIGNUM * w,int checks,BN_CTX * ctx,int do_trial_division,BN_GENCB * cb)244 int ossl_bn_check_prime(const BIGNUM *w, int checks, BN_CTX *ctx,
245 int do_trial_division, BN_GENCB *cb)
246 {
247 int min_checks = bn_mr_min_checks(BN_num_bits(w));
248
249 if (checks < min_checks)
250 checks = min_checks;
251
252 return bn_is_prime_int(w, checks, ctx, do_trial_division, cb);
253 }
254
BN_check_prime(const BIGNUM * p,BN_CTX * ctx,BN_GENCB * cb)255 int BN_check_prime(const BIGNUM *p, BN_CTX *ctx, BN_GENCB *cb)
256 {
257 return ossl_bn_check_prime(p, 0, ctx, 1, cb);
258 }
259
260 /*
261 * Tests that |w| is probably prime
262 * See FIPS 186-4 C.3.1 Miller Rabin Probabilistic Primality Test.
263 *
264 * Returns 0 when composite, 1 when probable prime, -1 on error.
265 */
bn_is_prime_int(const BIGNUM * w,int checks,BN_CTX * ctx,int do_trial_division,BN_GENCB * cb)266 static int bn_is_prime_int(const BIGNUM *w, int checks, BN_CTX *ctx,
267 int do_trial_division, BN_GENCB *cb)
268 {
269 int i, status, ret = -1;
270 #ifndef FIPS_MODULE
271 BN_CTX *ctxlocal = NULL;
272 #else
273
274 if (ctx == NULL)
275 return -1;
276 #endif
277
278 /* w must be bigger than 1 */
279 if (BN_cmp(w, BN_value_one()) <= 0)
280 return 0;
281
282 /* w must be odd */
283 if (BN_is_odd(w)) {
284 /* Take care of the really small prime 3 */
285 if (BN_is_word(w, 3))
286 return 1;
287 } else {
288 /* 2 is the only even prime */
289 return BN_is_word(w, 2);
290 }
291
292 /* first look for small factors */
293 if (do_trial_division) {
294 int trial_divisions = calc_trial_divisions(BN_num_bits(w));
295
296 for (i = 1; i < trial_divisions; i++) {
297 BN_ULONG mod = BN_mod_word(w, primes[i]);
298 if (mod == (BN_ULONG)-1)
299 return -1;
300 if (mod == 0)
301 return BN_is_word(w, primes[i]);
302 }
303 if (!BN_GENCB_call(cb, 1, -1))
304 return -1;
305 }
306 #ifndef FIPS_MODULE
307 if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
308 goto err;
309 #endif
310
311 if (!ossl_bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status)) {
312 ret = -1;
313 goto err;
314 }
315 ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
316 err:
317 #ifndef FIPS_MODULE
318 BN_CTX_free(ctxlocal);
319 #endif
320 return ret;
321 }
322
323 /*
324 * Refer to FIPS 186-4 C.3.2 Enhanced Miller-Rabin Probabilistic Primality Test.
325 * OR C.3.1 Miller-Rabin Probabilistic Primality Test (if enhanced is zero).
326 * The Step numbers listed in the code refer to the enhanced case.
327 *
328 * if enhanced is set, then status returns one of the following:
329 * BN_PRIMETEST_PROBABLY_PRIME
330 * BN_PRIMETEST_COMPOSITE_WITH_FACTOR
331 * BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME
332 * if enhanced is zero, then status returns either
333 * BN_PRIMETEST_PROBABLY_PRIME or
334 * BN_PRIMETEST_COMPOSITE
335 *
336 * returns 0 if there was an error, otherwise it returns 1.
337 */
ossl_bn_miller_rabin_is_prime(const BIGNUM * w,int iterations,BN_CTX * ctx,BN_GENCB * cb,int enhanced,int * status)338 int ossl_bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
339 BN_GENCB *cb, int enhanced, int *status)
340 {
341 int i, j, a, ret = 0;
342 BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
343 BN_MONT_CTX *mont = NULL;
344
345 /* w must be odd */
346 if (!BN_is_odd(w))
347 return 0;
348
349 BN_CTX_start(ctx);
350 g = BN_CTX_get(ctx);
351 w1 = BN_CTX_get(ctx);
352 w3 = BN_CTX_get(ctx);
353 x = BN_CTX_get(ctx);
354 m = BN_CTX_get(ctx);
355 z = BN_CTX_get(ctx);
356 b = BN_CTX_get(ctx);
357
358 if (!(b != NULL
359 /* w1 := w - 1 */
360 && BN_copy(w1, w)
361 && BN_sub_word(w1, 1)
362 /* w3 := w - 3 */
363 && BN_copy(w3, w)
364 && BN_sub_word(w3, 3)))
365 goto err;
366
367 /* check w is larger than 3, otherwise the random b will be too small */
368 if (BN_is_zero(w3) || BN_is_negative(w3))
369 goto err;
370
371 /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
372 a = 1;
373 while (!BN_is_bit_set(w1, a))
374 a++;
375 /* (Step 2) m = (w-1) / 2^a */
376 if (!BN_rshift(m, w1, a))
377 goto err;
378
379 /* Montgomery setup for computations mod a */
380 mont = BN_MONT_CTX_new();
381 if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
382 goto err;
383
384 if (iterations == 0)
385 iterations = bn_mr_min_checks(BN_num_bits(w));
386
387 /* (Step 4) */
388 for (i = 0; i < iterations; ++i) {
389 /* (Step 4.1) obtain a Random string of bits b where 1 < b < w-1 */
390 if (!BN_priv_rand_range_ex(b, w3, 0, ctx)
391 || !BN_add_word(b, 2)) /* 1 < b < w-1 */
392 goto err;
393
394 if (enhanced) {
395 /* (Step 4.3) */
396 if (!BN_gcd(g, b, w, ctx))
397 goto err;
398 /* (Step 4.4) */
399 if (!BN_is_one(g)) {
400 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
401 ret = 1;
402 goto err;
403 }
404 }
405 /* (Step 4.5) z = b^m mod w */
406 if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
407 goto err;
408 /* (Step 4.6) if (z = 1 or z = w-1) */
409 if (BN_is_one(z) || BN_cmp(z, w1) == 0)
410 goto outer_loop;
411 /* (Step 4.7) for j = 1 to a-1 */
412 for (j = 1; j < a ; ++j) {
413 /* (Step 4.7.1 - 4.7.2) x = z. z = x^2 mod w */
414 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
415 goto err;
416 /* (Step 4.7.3) */
417 if (BN_cmp(z, w1) == 0)
418 goto outer_loop;
419 /* (Step 4.7.4) */
420 if (BN_is_one(z))
421 goto composite;
422 }
423 /* At this point z = b^((w-1)/2) mod w */
424 /* (Steps 4.8 - 4.9) x = z, z = x^2 mod w */
425 if (!BN_copy(x, z) || !BN_mod_mul(z, x, x, w, ctx))
426 goto err;
427 /* (Step 4.10) */
428 if (BN_is_one(z))
429 goto composite;
430 /* (Step 4.11) x = b^(w-1) mod w */
431 if (!BN_copy(x, z))
432 goto err;
433 composite:
434 if (enhanced) {
435 /* (Step 4.1.2) g = GCD(x-1, w) */
436 if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
437 goto err;
438 /* (Steps 4.1.3 - 4.1.4) */
439 if (BN_is_one(g))
440 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
441 else
442 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
443 } else {
444 *status = BN_PRIMETEST_COMPOSITE;
445 }
446 ret = 1;
447 goto err;
448 outer_loop: ;
449 /* (Step 4.1.5) */
450 if (!BN_GENCB_call(cb, 1, i))
451 goto err;
452 }
453 /* (Step 5) */
454 *status = BN_PRIMETEST_PROBABLY_PRIME;
455 ret = 1;
456 err:
457 BN_clear(g);
458 BN_clear(w1);
459 BN_clear(w3);
460 BN_clear(x);
461 BN_clear(m);
462 BN_clear(z);
463 BN_clear(b);
464 BN_CTX_end(ctx);
465 BN_MONT_CTX_free(mont);
466 return ret;
467 }
468
469 /*
470 * Generate a random number of |bits| bits that is probably prime by sieving.
471 * If |safe| != 0, it generates a safe prime.
472 * |mods| is a preallocated array that gets reused when called again.
473 *
474 * The probably prime is saved in |rnd|.
475 *
476 * Returns 1 on success and 0 on error.
477 */
probable_prime(BIGNUM * rnd,int bits,int safe,prime_t * mods,BN_CTX * ctx)478 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
479 BN_CTX *ctx)
480 {
481 int i;
482 BN_ULONG delta;
483 int trial_divisions = calc_trial_divisions(bits);
484 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
485
486 again:
487 if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, 0,
488 ctx))
489 return 0;
490 if (safe && !BN_set_bit(rnd, 1))
491 return 0;
492 /* we now have a random number 'rnd' to test. */
493 for (i = 1; i < trial_divisions; i++) {
494 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
495 if (mod == (BN_ULONG)-1)
496 return 0;
497 mods[i] = (prime_t) mod;
498 }
499 delta = 0;
500 loop:
501 for (i = 1; i < trial_divisions; i++) {
502 /*
503 * check that rnd is a prime and also that
504 * gcd(rnd-1,primes) == 1 (except for 2)
505 * do the second check only if we are interested in safe primes
506 * in the case that the candidate prime is a single word then
507 * we check only the primes up to sqrt(rnd)
508 */
509 if (bits <= 31 && delta <= 0x7fffffff
510 && square(primes[i]) > BN_get_word(rnd) + delta)
511 break;
512 if (safe ? (mods[i] + delta) % primes[i] <= 1
513 : (mods[i] + delta) % primes[i] == 0) {
514 delta += safe ? 4 : 2;
515 if (delta > maxdelta)
516 goto again;
517 goto loop;
518 }
519 }
520 if (!BN_add_word(rnd, delta))
521 return 0;
522 if (BN_num_bits(rnd) != bits)
523 goto again;
524 bn_check_top(rnd);
525 return 1;
526 }
527
528 /*
529 * Generate a random number |rnd| of |bits| bits that is probably prime
530 * and satisfies |rnd| % |add| == |rem| by sieving.
531 * If |safe| != 0, it generates a safe prime.
532 * |mods| is a preallocated array that gets reused when called again.
533 *
534 * Returns 1 on success and 0 on error.
535 */
probable_prime_dh(BIGNUM * rnd,int bits,int safe,prime_t * mods,const BIGNUM * add,const BIGNUM * rem,BN_CTX * ctx)536 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
537 const BIGNUM *add, const BIGNUM *rem,
538 BN_CTX *ctx)
539 {
540 int i, ret = 0;
541 BIGNUM *t1;
542 BN_ULONG delta;
543 int trial_divisions = calc_trial_divisions(bits);
544 BN_ULONG maxdelta = BN_MASK2 - primes[trial_divisions - 1];
545
546 BN_CTX_start(ctx);
547 if ((t1 = BN_CTX_get(ctx)) == NULL)
548 goto err;
549
550 if (maxdelta > BN_MASK2 - BN_get_word(add))
551 maxdelta = BN_MASK2 - BN_get_word(add);
552
553 again:
554 if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, 0, ctx))
555 goto err;
556
557 /* we need ((rnd-rem) % add) == 0 */
558
559 if (!BN_mod(t1, rnd, add, ctx))
560 goto err;
561 if (!BN_sub(rnd, rnd, t1))
562 goto err;
563 if (rem == NULL) {
564 if (!BN_add_word(rnd, safe ? 3u : 1u))
565 goto err;
566 } else {
567 if (!BN_add(rnd, rnd, rem))
568 goto err;
569 }
570
571 if (BN_num_bits(rnd) < bits
572 || BN_get_word(rnd) < (safe ? 5u : 3u)) {
573 if (!BN_add(rnd, rnd, add))
574 goto err;
575 }
576
577 /* we now have a random number 'rnd' to test. */
578 for (i = 1; i < trial_divisions; i++) {
579 BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
580 if (mod == (BN_ULONG)-1)
581 goto err;
582 mods[i] = (prime_t) mod;
583 }
584 delta = 0;
585 loop:
586 for (i = 1; i < trial_divisions; i++) {
587 /* check that rnd is a prime */
588 if (bits <= 31 && delta <= 0x7fffffff
589 && square(primes[i]) > BN_get_word(rnd) + delta)
590 break;
591 /* rnd mod p == 1 implies q = (rnd-1)/2 is divisible by p */
592 if (safe ? (mods[i] + delta) % primes[i] <= 1
593 : (mods[i] + delta) % primes[i] == 0) {
594 delta += BN_get_word(add);
595 if (delta > maxdelta)
596 goto again;
597 goto loop;
598 }
599 }
600 if (!BN_add_word(rnd, delta))
601 goto err;
602 ret = 1;
603
604 err:
605 BN_CTX_end(ctx);
606 bn_check_top(rnd);
607 return ret;
608 }
609