crypto/bn/bn_prime.c

   1 /*
   2  * Copyright 1995-2017 The OpenSSL Project Authors. All Rights Reserved.
   3  *
   4  * Licensed under the OpenSSL license (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_lcl.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 witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
  23                    const BIGNUM *a1_odd, int k, BN_CTX *ctx,
  24                    BN_MONT_CTX *mont);
  25 static int probable_prime(BIGNUM *rnd, int bits, prime_t *mods);
  26 static int probable_prime_dh_safe(BIGNUM *rnd, int bits,
  27                                   const BIGNUM *add, const BIGNUM *rem,
  28                                   BN_CTX *ctx);
  29
  30 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
  31 {
  32     /* No callback means continue */
  33     if (!cb)
  34         return 1;
  35     switch (cb->ver) {
  36     case 1:
  37         /* Deprecated-style callbacks */
  38         if (!cb->cb.cb_1)
  39             return 1;
  40         cb->cb.cb_1(a, b, cb->arg);
  41         return 1;
  42     case 2:
  43         /* New-style callbacks */
  44         return cb->cb.cb_2(a, b, cb);
  45     default:
  46         break;
  47     }
  48     /* Unrecognised callback type */
  49     return 0;
  50 }
  51
  52 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
  53                          const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
  54 {
  55     BIGNUM *t;
  56     int found = 0;
  57     int i, j, c1 = 0;
  58     BN_CTX *ctx = NULL;
  59     prime_t *mods = NULL;
  60     int checks = BN_prime_checks_for_size(bits);
  61
  62     if (bits < 2) {
  63         /* There are no prime numbers this small. */
  64         BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
  65         return 0;
  66     } else if (bits == 2 && safe) {
  67         /* The smallest safe prime (7) is three bits. */
  68         BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
  69         return 0;
  70     }
  71
  72     mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
  73     if (mods == NULL)
  74         goto err;
  75
  76     ctx = BN_CTX_new();
  77     if (ctx == NULL)
  78         goto err;
  79     BN_CTX_start(ctx);
  80     t = BN_CTX_get(ctx);
  81     if (t == NULL)
  82         goto err;
  83  loop:
  84     /* make a random number and set the top and bottom bits */
  85     if (add == NULL) {
  86         if (!probable_prime(ret, bits, mods))
  87             goto err;
  88     } else {
  89         if (safe) {
  90             if (!probable_prime_dh_safe(ret, bits, add, rem, ctx))
  91                 goto err;
  92         } else {
  93             if (!bn_probable_prime_dh(ret, bits, add, rem, ctx))
  94                 goto err;
  95         }
  96     }
  97
  98     if (!BN_GENCB_call(cb, 0, c1++))
  99         /* aborted */
 100         goto err;
 101
 102     if (!safe) {
 103         i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
 104         if (i == -1)
 105             goto err;
 106         if (i == 0)
 107             goto loop;
 108     } else {
 109         /*
 110          * for "safe prime" generation, check that (p-1)/2 is prime. Since a
 111          * prime is odd, We just need to divide by 2
 112          */
 113         if (!BN_rshift1(t, ret))
 114             goto err;
 115
 116         for (i = 0; i < checks; i++) {
 117             j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
 118             if (j == -1)
 119                 goto err;
 120             if (j == 0)
 121                 goto loop;
 122
 123             j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
 124             if (j == -1)
 125                 goto err;
 126             if (j == 0)
 127                 goto loop;
 128
 129             if (!BN_GENCB_call(cb, 2, c1 - 1))
 130                 goto err;
 131             /* We have a safe prime test pass */
 132         }
 133     }
 134     /* we have a prime :-) */
 135     found = 1;
 136  err:
 137     OPENSSL_free(mods);
 138     if (ctx != NULL)
 139         BN_CTX_end(ctx);
 140     BN_CTX_free(ctx);
 141     bn_check_top(ret);
 142     return found;
 143 }
 144
 145 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 146                    BN_GENCB *cb)
 147 {
 148     return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
 149 }
 150
 151 int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 152                             int do_trial_division, BN_GENCB *cb)
 153 {
 154     int i, j, ret = -1;
 155     int k;
 156     BN_CTX *ctx = NULL;
 157     BIGNUM *A1, *A1_odd, *check; /* taken from ctx */
 158     BN_MONT_CTX *mont = NULL;
 159
 160     if (BN_cmp(a, BN_value_one()) <= 0)
 161         return 0;
 162
 163     if (checks == BN_prime_checks)
 164         checks = BN_prime_checks_for_size(BN_num_bits(a));
 165
 166     /* first look for small factors */
 167     if (!BN_is_odd(a))
 168         /* a is even => a is prime if and only if a == 2 */
 169         return BN_is_word(a, 2);
 170     if (do_trial_division) {
 171         for (i = 1; i < NUMPRIMES; i++) {
 172             BN_ULONG mod = BN_mod_word(a, primes[i]);
 173             if (mod == (BN_ULONG)-1)
 174                 goto err;
 175             if (mod == 0)
 176                 return BN_is_word(a, primes[i]);
 177         }
 178         if (!BN_GENCB_call(cb, 1, -1))
 179             goto err;
 180     }
 181
 182     if (ctx_passed != NULL)
 183         ctx = ctx_passed;
 184     else if ((ctx = BN_CTX_new()) == NULL)
 185         goto err;
 186     BN_CTX_start(ctx);
 187
 188     A1 = BN_CTX_get(ctx);
 189     A1_odd = BN_CTX_get(ctx);
 190     check = BN_CTX_get(ctx);
 191     if (check == NULL)
 192         goto err;
 193
 194     /* compute A1 := a - 1 */
 195     if (!BN_copy(A1, a))
 196         goto err;
 197     if (!BN_sub_word(A1, 1))
 198         goto err;
 199     if (BN_is_zero(A1)) {
 200         ret = 0;
 201         goto err;
 202     }
 203
 204     /* write  A1  as  A1_odd * 2^k */
 205     k = 1;
 206     while (!BN_is_bit_set(A1, k))
 207         k++;
 208     if (!BN_rshift(A1_odd, A1, k))
 209         goto err;
 210
 211     /* Montgomery setup for computations mod a */
 212     mont = BN_MONT_CTX_new();
 213     if (mont == NULL)
 214         goto err;
 215     if (!BN_MONT_CTX_set(mont, a, ctx))
 216         goto err;
 217
 218     for (i = 0; i < checks; i++) {
 219         if (!BN_priv_rand_range(check, A1))
 220             goto err;
 221         if (!BN_add_word(check, 1))
 222             goto err;
 223         /* now 1 <= check < a */
 224
 225         j = witness(check, a, A1, A1_odd, k, ctx, mont);
 226         if (j == -1)
 227             goto err;
 228         if (j) {
 229             ret = 0;
 230             goto err;
 231         }
 232         if (!BN_GENCB_call(cb, 1, i))
 233             goto err;
 234     }
 235     ret = 1;
 236  err:
 237     if (ctx != NULL) {
 238         BN_CTX_end(ctx);
 239         if (ctx_passed == NULL)
 240             BN_CTX_free(ctx);
 241     }
 242     BN_MONT_CTX_free(mont);
 243
 244     return (ret);
 245 }
 246
 247 static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
 248                    const BIGNUM *a1_odd, int k, BN_CTX *ctx,
 249                    BN_MONT_CTX *mont)
 250 {
 251     if (!BN_mod_exp_mont(w, w, a1_odd, a, ctx, mont)) /* w := w^a1_odd mod a */
 252         return -1;
 253     if (BN_is_one(w))
 254         return 0;               /* probably prime */
 255     if (BN_cmp(w, a1) == 0)
 256         return 0;               /* w == -1 (mod a), 'a' is probably prime */
 257     while (--k) {
 258         if (!BN_mod_mul(w, w, w, a, ctx)) /* w := w^2 mod a */
 259             return -1;
 260         if (BN_is_one(w))
 261             return 1;           /* 'a' is composite, otherwise a previous 'w'
 262                                  * would have been == -1 (mod 'a') */
 263         if (BN_cmp(w, a1) == 0)
 264             return 0;           /* w == -1 (mod a), 'a' is probably prime */
 265     }
 266     /*
 267      * If we get here, 'w' is the (a-1)/2-th power of the original 'w', and
 268      * it is neither -1 nor +1 -- so 'a' cannot be prime
 269      */
 270     bn_check_top(w);
 271     return 1;
 272 }
 273
 274 static int probable_prime(BIGNUM *rnd, int bits, prime_t *mods)
 275 {
 276     int i;
 277     BN_ULONG delta;
 278     BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
 279     char is_single_word = bits <= BN_BITS2;
 280
 281  again:
 282     if (!BN_priv_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD))
 283         return (0);
 284     /* we now have a random number 'rnd' to test. */
 285     for (i = 1; i < NUMPRIMES; i++) {
 286         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 287         if (mod == (BN_ULONG)-1)
 288             return 0;
 289         mods[i] = (prime_t) mod;
 290     }
 291     /*
 292      * If bits is so small that it fits into a single word then we
 293      * additionally don't want to exceed that many bits.
 294      */
 295     if (is_single_word) {
 296         BN_ULONG size_limit;
 297
 298         if (bits == BN_BITS2) {
 299             /*
 300              * Shifting by this much has undefined behaviour so we do it a
 301              * different way
 302              */
 303             size_limit = ~((BN_ULONG)0) - BN_get_word(rnd);
 304         } else {
 305             size_limit = (((BN_ULONG)1) << bits) - BN_get_word(rnd) - 1;
 306         }
 307         if (size_limit < maxdelta)
 308             maxdelta = size_limit;
 309     }
 310     delta = 0;
 311  loop:
 312     if (is_single_word) {
 313         BN_ULONG rnd_word = BN_get_word(rnd);
 314
 315         /*-
 316          * In the case that the candidate prime is a single word then
 317          * we check that:
 318          *   1) It's greater than primes[i] because we shouldn't reject
 319          *      3 as being a prime number because it's a multiple of
 320          *      three.
 321          *   2) That it's not a multiple of a known prime. We don't
 322          *      check that rnd-1 is also coprime to all the known
 323          *      primes because there aren't many small primes where
 324          *      that's true.
 325          */
 326         for (i = 1; i < NUMPRIMES && primes[i] < rnd_word; i++) {
 327             if ((mods[i] + delta) % primes[i] == 0) {
 328                 delta += 2;
 329                 if (delta > maxdelta)
 330                     goto again;
 331                 goto loop;
 332             }
 333         }
 334     } else {
 335         for (i = 1; i < NUMPRIMES; i++) {
 336             /*
 337              * check that rnd is not a prime and also that gcd(rnd-1,primes)
 338              * == 1 (except for 2)
 339              */
 340             if (((mods[i] + delta) % primes[i]) <= 1) {
 341                 delta += 2;
 342                 if (delta > maxdelta)
 343                     goto again;
 344                 goto loop;
 345             }
 346         }
 347     }
 348     if (!BN_add_word(rnd, delta))
 349         return (0);
 350     if (BN_num_bits(rnd) != bits)
 351         goto again;
 352     bn_check_top(rnd);
 353     return (1);
 354 }
 355
 356 int bn_probable_prime_dh(BIGNUM *rnd, int bits,
 357                          const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx)
 358 {
 359     int i, ret = 0;
 360     BIGNUM *t1;
 361
 362     BN_CTX_start(ctx);
 363     if ((t1 = BN_CTX_get(ctx)) == NULL)
 364         goto err;
 365
 366     if (!BN_priv_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
 367         goto err;
 368
 369     /* we need ((rnd-rem) % add) == 0 */
 370
 371     if (!BN_mod(t1, rnd, add, ctx))
 372         goto err;
 373     if (!BN_sub(rnd, rnd, t1))
 374         goto err;
 375     if (rem == NULL) {
 376         if (!BN_add_word(rnd, 1))
 377             goto err;
 378     } else {
 379         if (!BN_add(rnd, rnd, rem))
 380             goto err;
 381     }
 382
 383     /* we now have a random number 'rand' to test. */
 384
 385  loop:
 386     for (i = 1; i < NUMPRIMES; i++) {
 387         /* check that rnd is a prime */
 388         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 389         if (mod == (BN_ULONG)-1)
 390             goto err;
 391         if (mod <= 1) {
 392             if (!BN_add(rnd, rnd, add))
 393                 goto err;
 394             goto loop;
 395         }
 396     }
 397     ret = 1;
 398
 399  err:
 400     BN_CTX_end(ctx);
 401     bn_check_top(rnd);
 402     return (ret);
 403 }
 404
 405 static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
 406                                   const BIGNUM *rem, BN_CTX *ctx)
 407 {
 408     int i, ret = 0;
 409     BIGNUM *t1, *qadd, *q;
 410
 411     bits--;
 412     BN_CTX_start(ctx);
 413     t1 = BN_CTX_get(ctx);
 414     q = BN_CTX_get(ctx);
 415     qadd = BN_CTX_get(ctx);
 416     if (qadd == NULL)
 417         goto err;
 418
 419     if (!BN_rshift1(qadd, padd))
 420         goto err;
 421
 422     if (!BN_priv_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
 423         goto err;
 424
 425     /* we need ((rnd-rem) % add) == 0 */
 426     if (!BN_mod(t1, q, qadd, ctx))
 427         goto err;
 428     if (!BN_sub(q, q, t1))
 429         goto err;
 430     if (rem == NULL) {
 431         if (!BN_add_word(q, 1))
 432             goto err;
 433     } else {
 434         if (!BN_rshift1(t1, rem))
 435             goto err;
 436         if (!BN_add(q, q, t1))
 437             goto err;
 438     }
 439
 440     /* we now have a random number 'rand' to test. */
 441     if (!BN_lshift1(p, q))
 442         goto err;
 443     if (!BN_add_word(p, 1))
 444         goto err;
 445
 446  loop:
 447     for (i = 1; i < NUMPRIMES; i++) {
 448         /* check that p and q are prime */
 449         /*
 450          * check that for p and q gcd(p-1,primes) == 1 (except for 2)
 451          */
 452         BN_ULONG pmod = BN_mod_word(p, (BN_ULONG)primes[i]);
 453         BN_ULONG qmod = BN_mod_word(q, (BN_ULONG)primes[i]);
 454         if (pmod == (BN_ULONG)-1 || qmod == (BN_ULONG)-1)
 455             goto err;
 456         if (pmod == 0 || qmod == 0) {
 457             if (!BN_add(p, p, padd))
 458                 goto err;
 459             if (!BN_add(q, q, qadd))
 460                 goto err;
 461             goto loop;
 462         }
 463     }
 464     ret = 1;
 465
 466  err:
 467     BN_CTX_end(ctx);
 468     bn_check_top(p);
 469     return (ret);
 470 }