crypto/bn/bn_prime.c

   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
  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
  28 #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
  29
  30 #if BN_BITS2 == 64
  31 # define BN_DEF(lo, hi) (BN_ULONG)hi<<32|lo
  32 #else
  33 # define BN_DEF(lo, hi) lo, hi
  34 #endif
  35
  36 /*
  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).
  41  */
  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),
  51     (BN_ULONG)0x000017b1
  52 };
  53
  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,
  59     0,
  60     BN_FLG_STATIC_DATA
  61 };
  62
  63 const BIGNUM *bn_get0_small_factors(void)
  64 {
  65     return &_bignum_small_prime_factors;
  66 }
  67
  68 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
  69 {
  70     /* No callback means continue */
  71     if (!cb)
  72         return 1;
  73     switch (cb->ver) {
  74     case 1:
  75         /* Deprecated-style callbacks */
  76         if (!cb->cb.cb_1)
  77             return 1;
  78         cb->cb.cb_1(a, b, cb->arg);
  79         return 1;
  80     case 2:
  81         /* New-style callbacks */
  82         return cb->cb.cb_2(a, b, cb);
  83     default:
  84         break;
  85     }
  86     /* Unrecognised callback type */
  87     return 0;
  88 }
  89
  90 int BN_generate_prime_ex2(BIGNUM *ret, int bits, int safe,
  91                           const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb,
  92                           BN_CTX *ctx)
  93 {
  94     BIGNUM *t;
  95     int found = 0;
  96     int i, j, c1 = 0;
  97     prime_t *mods = NULL;
  98     int checks = BN_prime_checks_for_size(bits);
  99
 100     if (bits < 2) {
 101         /* There are no prime numbers this small. */
 102         BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
 103         return 0;
 104     } else if (add == NULL && safe && bits < 6 && bits != 3) {
 105         /*
 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.
 109          */
 110         BNerr(BN_F_BN_GENERATE_PRIME_EX2, BN_R_BITS_TOO_SMALL);
 111         return 0;
 112     }
 113
 114     mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
 115     if (mods == NULL)
 116         goto err;
 117
 118     BN_CTX_start(ctx);
 119     t = BN_CTX_get(ctx);
 120     if (t == NULL)
 121         goto err;
 122  loop:
 123     /* make a random number and set the top and bottom bits */
 124     if (add == NULL) {
 125         if (!probable_prime(ret, bits, safe, mods, ctx))
 126             goto err;
 127     } else {
 128         if (!probable_prime_dh(ret, bits, safe, mods, add, rem, ctx))
 129             goto err;
 130     }
 131
 132     if (!BN_GENCB_call(cb, 0, c1++))
 133         /* aborted */
 134         goto err;
 135
 136     if (!safe) {
 137         i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
 138         if (i == -1)
 139             goto err;
 140         if (i == 0)
 141             goto loop;
 142     } else {
 143         /*
 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
 146          */
 147         if (!BN_rshift1(t, ret))
 148             goto err;
 149
 150         for (i = 0; i < checks; i++) {
 151             j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
 152             if (j == -1)
 153                 goto err;
 154             if (j == 0)
 155                 goto loop;
 156
 157             j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
 158             if (j == -1)
 159                 goto err;
 160             if (j == 0)
 161                 goto loop;
 162
 163             if (!BN_GENCB_call(cb, 2, c1 - 1))
 164                 goto err;
 165             /* We have a safe prime test pass */
 166         }
 167     }
 168     /* we have a prime :-) */
 169     found = 1;
 170  err:
 171     OPENSSL_free(mods);
 172     BN_CTX_end(ctx);
 173     bn_check_top(ret);
 174     return found;
 175 }
 176
 177 #ifndef FIPS_MODE
 178 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
 179                          const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
 180 {
 181     BN_CTX *ctx = BN_CTX_new();
 182     int retval;
 183
 184     if (ctx == NULL)
 185         return 0;
 186
 187     retval = BN_generate_prime_ex2(ret, bits, safe, add, rem, cb, ctx);
 188
 189     BN_CTX_free(ctx);
 190     return retval;
 191 }
 192 #endif
 193
 194 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 195                    BN_GENCB *cb)
 196 {
 197     return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
 198 }
 199
 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)
 203 {
 204     int i, status, ret = -1;
 205 #ifndef FIPS_MODE
 206     BN_CTX *ctxlocal = NULL;
 207 #else
 208
 209     if (ctx == NULL)
 210         return -1;
 211 #endif
 212
 213     /* w must be bigger than 1 */
 214     if (BN_cmp(w, BN_value_one()) <= 0)
 215         return 0;
 216
 217     /* w must be odd */
 218     if (BN_is_odd(w)) {
 219         /* Take care of the really small prime 3 */
 220         if (BN_is_word(w, 3))
 221             return 1;
 222     } else {
 223         /* 2 is the only even prime */
 224         return BN_is_word(w, 2);
 225     }
 226
 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)
 232                 return -1;
 233             if (mod == 0)
 234                 return BN_is_word(w, primes[i]);
 235         }
 236         if (!BN_GENCB_call(cb, 1, -1))
 237             return -1;
 238     }
 239 #ifndef FIPS_MODE
 240     if (ctx == NULL && (ctxlocal = ctx = BN_CTX_new()) == NULL)
 241         goto err;
 242 #endif
 243
 244     ret = bn_miller_rabin_is_prime(w, checks, ctx, cb, 0, &status);
 245     if (!ret)
 246         goto err;
 247     ret = (status == BN_PRIMETEST_PROBABLY_PRIME);
 248 err:
 249 #ifndef FIPS_MODE
 250     BN_CTX_free(ctxlocal);
 251 #endif
 252     return ret;
 253 }
 254
 255 /*
 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.
 259  *
 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
 267  *
 268  * returns 0 if there was an error, otherwise it returns 1.
 269  */
 270 int bn_miller_rabin_is_prime(const BIGNUM *w, int iterations, BN_CTX *ctx,
 271                              BN_GENCB *cb, int enhanced, int *status)
 272 {
 273     int i, j, a, ret = 0;
 274     BIGNUM *g, *w1, *w3, *x, *m, *z, *b;
 275     BN_MONT_CTX *mont = NULL;
 276
 277     /* w must be odd */
 278     if (!BN_is_odd(w))
 279         return 0;
 280
 281     BN_CTX_start(ctx);
 282     g = BN_CTX_get(ctx);
 283     w1 = BN_CTX_get(ctx);
 284     w3 = BN_CTX_get(ctx);
 285     x = BN_CTX_get(ctx);
 286     m = BN_CTX_get(ctx);
 287     z = BN_CTX_get(ctx);
 288     b = BN_CTX_get(ctx);
 289
 290     if (!(b != NULL
 291             /* w1 := w - 1 */
 292             && BN_copy(w1, w)
 293             && BN_sub_word(w1, 1)
 294             /* w3 := w - 3 */
 295             && BN_copy(w3, w)
 296             && BN_sub_word(w3, 3)))
 297         goto err;
 298
 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))
 301         goto err;
 302
 303     /* (Step 1) Calculate largest integer 'a' such that 2^a divides w-1 */
 304     a = 1;
 305     while (!BN_is_bit_set(w1, a))
 306         a++;
 307     /* (Step 2) m = (w-1) / 2^a */
 308     if (!BN_rshift(m, w1, a))
 309         goto err;
 310
 311     /* Montgomery setup for computations mod a */
 312     mont = BN_MONT_CTX_new();
 313     if (mont == NULL || !BN_MONT_CTX_set(mont, w, ctx))
 314         goto err;
 315
 316     if (iterations == BN_prime_checks)
 317         iterations = BN_prime_checks_for_size(BN_num_bits(w));
 318
 319     /* (Step 4) */
 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 */
 324             goto err;
 325
 326         if (enhanced) {
 327             /* (Step 4.3) */
 328             if (!BN_gcd(g, b, w, ctx))
 329                 goto err;
 330             /* (Step 4.4) */
 331             if (!BN_is_one(g)) {
 332                 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
 333                 ret = 1;
 334                 goto err;
 335             }
 336         }
 337         /* (Step 4.5) z = b^m mod w */
 338         if (!BN_mod_exp_mont(z, b, m, w, ctx, mont))
 339             goto err;
 340         /* (Step 4.6) if (z = 1 or z = w-1) */
 341         if (BN_is_one(z) || BN_cmp(z, w1) == 0)
 342             goto outer_loop;
 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))
 347                 goto err;
 348             /* (Step 4.7.3) */
 349             if (BN_cmp(z, w1) == 0)
 350                 goto outer_loop;
 351             /* (Step 4.7.4) */
 352             if (BN_is_one(z))
 353                 goto composite;
 354         }
 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))
 358             goto err;
 359         /* (Step 4.10) */
 360         if (BN_is_one(z))
 361             goto composite;
 362         /* (Step 4.11) x = b^(w-1) mod w */
 363         if (!BN_copy(x, z))
 364             goto err;
 365 composite:
 366         if (enhanced) {
 367             /* (Step 4.1.2) g = GCD(x-1, w) */
 368             if (!BN_sub_word(x, 1) || !BN_gcd(g, x, w, ctx))
 369                 goto err;
 370             /* (Steps 4.1.3 - 4.1.4) */
 371             if (BN_is_one(g))
 372                 *status = BN_PRIMETEST_COMPOSITE_NOT_POWER_OF_PRIME;
 373             else
 374                 *status = BN_PRIMETEST_COMPOSITE_WITH_FACTOR;
 375         } else {
 376             *status = BN_PRIMETEST_COMPOSITE;
 377         }
 378         ret = 1;
 379         goto err;
 380 outer_loop: ;
 381         /* (Step 4.1.5) */
 382         if (!BN_GENCB_call(cb, 1, i))
 383             goto err;
 384     }
 385     /* (Step 5) */
 386     *status = BN_PRIMETEST_PROBABLY_PRIME;
 387     ret = 1;
 388 err:
 389     BN_clear(g);
 390     BN_clear(w1);
 391     BN_clear(w3);
 392     BN_clear(x);
 393     BN_clear(m);
 394     BN_clear(z);
 395     BN_clear(b);
 396     BN_CTX_end(ctx);
 397     BN_MONT_CTX_free(mont);
 398     return ret;
 399 }
 400
 401 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods,
 402                           BN_CTX *ctx)
 403 {
 404     int i;
 405     BN_ULONG delta;
 406     BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
 407
 408  again:
 409     /* TODO: Not all primes are private */
 410     if (!BN_priv_rand_ex(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD, ctx))
 411         return 0;
 412     if (safe && !BN_set_bit(rnd, 1))
 413         return 0;
 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)
 418             return 0;
 419         mods[i] = (prime_t) mod;
 420     }
 421     delta = 0;
 422  loop:
 423     for (i = 1; i < NUMPRIMES; i++) {
 424         /*
 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)
 430          */
 431         if (bits <= 31 && delta <= 0x7fffffff
 432                 && square(primes[i]) > BN_get_word(rnd) + delta)
 433             break;
 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)
 438                 goto again;
 439             goto loop;
 440         }
 441     }
 442     if (!BN_add_word(rnd, delta))
 443         return 0;
 444     if (BN_num_bits(rnd) != bits)
 445         goto again;
 446     bn_check_top(rnd);
 447     return 1;
 448 }
 449
 450 static int probable_prime_dh(BIGNUM *rnd, int bits, int safe, prime_t *mods,
 451                              const BIGNUM *add, const BIGNUM *rem,
 452                              BN_CTX *ctx)
 453 {
 454     int i, ret = 0;
 455     BIGNUM *t1;
 456     BN_ULONG delta;
 457     BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
 458
 459     BN_CTX_start(ctx);
 460     if ((t1 = BN_CTX_get(ctx)) == NULL)
 461         goto err;
 462
 463     if (maxdelta > BN_MASK2 - BN_get_word(add))
 464         maxdelta = BN_MASK2 - BN_get_word(add);
 465
 466  again:
 467     if (!BN_rand_ex(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD, ctx))
 468         goto err;
 469
 470     /* we need ((rnd-rem) % add) == 0 */
 471
 472     if (!BN_mod(t1, rnd, add, ctx))
 473         goto err;
 474     if (!BN_sub(rnd, rnd, t1))
 475         goto err;
 476     if (rem == NULL) {
 477         if (!BN_add_word(rnd, safe ? 3u : 1u))
 478             goto err;
 479     } else {
 480         if (!BN_add(rnd, rnd, rem))
 481             goto err;
 482     }
 483
 484     if (BN_num_bits(rnd) < bits
 485             || BN_get_word(rnd) < (safe ? 5u : 3u)) {
 486         if (!BN_add(rnd, rnd, add))
 487             goto err;
 488     }
 489
 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)
 494             goto err;
 495         mods[i] = (prime_t) mod;
 496     }
 497     delta = 0;
 498  loop:
 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)
 503             break;
 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)
 509                 goto again;
 510             goto loop;
 511         }
 512     }
 513     if (!BN_add_word(rnd, delta))
 514         goto err;
 515     ret = 1;
 516
 517  err:
 518     BN_CTX_end(ctx);
 519     bn_check_top(rnd);
 520     return ret;
 521 }