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
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 *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 */
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        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
213int 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
230int 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
236int 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 */
244int 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
255int 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 */
266static 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);
316err:
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 */
338int 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;
433composite:
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;
448outer_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;
456err:
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 */
478static 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 */
536static 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