crypto/bn/bn_prime.c

   1 /*
   2  * Copyright 1995-2021 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
  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  */
  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  */
  94 static int bn_mr_min_checks(int bits)
  95 {
  96     if (bits > 2048)
  97         return 128;
  98     return 64;
  99 }
 100
 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
 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
 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
 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
 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 */
 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
 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  */
 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  */
 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  */
 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  */
 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 }