crypto/bn/bn_prime.c

   1 /*
   2  * Copyright 1995-2019 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_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 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, int safe, 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 #define square(x) ((BN_ULONG)(x) * (BN_ULONG)(x))
  31
  32 int BN_GENCB_call(BN_GENCB *cb, int a, int b)
  33 {
  34     /* No callback means continue */
  35     if (!cb)
  36         return 1;
  37     switch (cb->ver) {
  38     case 1:
  39         /* Deprecated-style callbacks */
  40         if (!cb->cb.cb_1)
  41             return 1;
  42         cb->cb.cb_1(a, b, cb->arg);
  43         return 1;
  44     case 2:
  45         /* New-style callbacks */
  46         return cb->cb.cb_2(a, b, cb);
  47     default:
  48         break;
  49     }
  50     /* Unrecognised callback type */
  51     return 0;
  52 }
  53
  54 int BN_generate_prime_ex(BIGNUM *ret, int bits, int safe,
  55                          const BIGNUM *add, const BIGNUM *rem, BN_GENCB *cb)
  56 {
  57     BIGNUM *t;
  58     int found = 0;
  59     int i, j, c1 = 0;
  60     BN_CTX *ctx = NULL;
  61     prime_t *mods = NULL;
  62     int checks = BN_prime_checks_for_size(bits);
  63
  64     if (bits < 2) {
  65         /* There are no prime numbers this small. */
  66         BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
  67         return 0;
  68     } else if (add == NULL && safe && bits < 6 && bits != 3) {
  69         /*
  70          * The smallest safe prime (7) is three bits.
  71          * But the following two safe primes with less than 6 bits (11, 23)
  72          * are unreachable for BN_rand with BN_RAND_TOP_TWO.
  73          */
  74         BNerr(BN_F_BN_GENERATE_PRIME_EX, BN_R_BITS_TOO_SMALL);
  75         return 0;
  76     }
  77
  78     mods = OPENSSL_zalloc(sizeof(*mods) * NUMPRIMES);
  79     if (mods == NULL)
  80         goto err;
  81
  82     ctx = BN_CTX_new();
  83     if (ctx == NULL)
  84         goto err;
  85     BN_CTX_start(ctx);
  86     t = BN_CTX_get(ctx);
  87     if (t == NULL)
  88         goto err;
  89  loop:
  90     /* make a random number and set the top and bottom bits */
  91     if (add == NULL) {
  92         if (!probable_prime(ret, bits, safe, mods))
  93             goto err;
  94     } else {
  95         if (safe) {
  96             if (!probable_prime_dh_safe(ret, bits, add, rem, ctx))
  97                 goto err;
  98         } else {
  99             if (!bn_probable_prime_dh(ret, bits, add, rem, ctx))
 100                 goto err;
 101         }
 102     }
 103
 104     if (!BN_GENCB_call(cb, 0, c1++))
 105         /* aborted */
 106         goto err;
 107
 108     if (!safe) {
 109         i = BN_is_prime_fasttest_ex(ret, checks, ctx, 0, cb);
 110         if (i == -1)
 111             goto err;
 112         if (i == 0)
 113             goto loop;
 114     } else {
 115         /*
 116          * for "safe prime" generation, check that (p-1)/2 is prime. Since a
 117          * prime is odd, We just need to divide by 2
 118          */
 119         if (!BN_rshift1(t, ret))
 120             goto err;
 121
 122         for (i = 0; i < checks; i++) {
 123             j = BN_is_prime_fasttest_ex(ret, 1, ctx, 0, cb);
 124             if (j == -1)
 125                 goto err;
 126             if (j == 0)
 127                 goto loop;
 128
 129             j = BN_is_prime_fasttest_ex(t, 1, ctx, 0, cb);
 130             if (j == -1)
 131                 goto err;
 132             if (j == 0)
 133                 goto loop;
 134
 135             if (!BN_GENCB_call(cb, 2, c1 - 1))
 136                 goto err;
 137             /* We have a safe prime test pass */
 138         }
 139     }
 140     /* we have a prime :-) */
 141     found = 1;
 142  err:
 143     OPENSSL_free(mods);
 144     BN_CTX_end(ctx);
 145     BN_CTX_free(ctx);
 146     bn_check_top(ret);
 147     return found;
 148 }
 149
 150 int BN_is_prime_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 151                    BN_GENCB *cb)
 152 {
 153     return BN_is_prime_fasttest_ex(a, checks, ctx_passed, 0, cb);
 154 }
 155
 156 int BN_is_prime_fasttest_ex(const BIGNUM *a, int checks, BN_CTX *ctx_passed,
 157                             int do_trial_division, BN_GENCB *cb)
 158 {
 159     int i, j, ret = -1;
 160     int k;
 161     BN_CTX *ctx = NULL;
 162     BIGNUM *A1, *A1_odd, *A3, *check; /* taken from ctx */
 163     BN_MONT_CTX *mont = NULL;
 164
 165     /* Take care of the really small primes 2 & 3 */
 166     if (BN_is_word(a, 2) || BN_is_word(a, 3))
 167         return 1;
 168
 169     /* Check odd and bigger than 1 */
 170     if (!BN_is_odd(a) || BN_cmp(a, BN_value_one()) <= 0)
 171         return 0;
 172
 173     if (checks == BN_prime_checks)
 174         checks = BN_prime_checks_for_size(BN_num_bits(a));
 175
 176     /* first look for small factors */
 177     if (do_trial_division) {
 178         for (i = 1; i < NUMPRIMES; i++) {
 179             BN_ULONG mod = BN_mod_word(a, primes[i]);
 180             if (mod == (BN_ULONG)-1)
 181                 goto err;
 182             if (mod == 0)
 183                 return BN_is_word(a, primes[i]);
 184         }
 185         if (!BN_GENCB_call(cb, 1, -1))
 186             goto err;
 187     }
 188
 189     if (ctx_passed != NULL)
 190         ctx = ctx_passed;
 191     else if ((ctx = BN_CTX_new()) == NULL)
 192         goto err;
 193     BN_CTX_start(ctx);
 194
 195     A1 = BN_CTX_get(ctx);
 196     A3 = BN_CTX_get(ctx);
 197     A1_odd = BN_CTX_get(ctx);
 198     check = BN_CTX_get(ctx);
 199     if (check == NULL)
 200         goto err;
 201
 202     /* compute A1 := a - 1 */
 203     if (!BN_copy(A1, a) || !BN_sub_word(A1, 1))
 204         goto err;
 205     /* compute A3 := a - 3 */
 206     if (!BN_copy(A3, a) || !BN_sub_word(A3, 3))
 207         goto err;
 208
 209     /* write  A1  as  A1_odd * 2^k */
 210     k = 1;
 211     while (!BN_is_bit_set(A1, k))
 212         k++;
 213     if (!BN_rshift(A1_odd, A1, k))
 214         goto err;
 215
 216     /* Montgomery setup for computations mod a */
 217     mont = BN_MONT_CTX_new();
 218     if (mont == NULL)
 219         goto err;
 220     if (!BN_MONT_CTX_set(mont, a, ctx))
 221         goto err;
 222
 223     for (i = 0; i < checks; i++) {
 224         /* 1 < check < a-1 */
 225         if (!BN_priv_rand_range(check, A3) || !BN_add_word(check, 2))
 226             goto err;
 227
 228         j = witness(check, a, A1, A1_odd, k, ctx, mont);
 229         if (j == -1)
 230             goto err;
 231         if (j) {
 232             ret = 0;
 233             goto err;
 234         }
 235         if (!BN_GENCB_call(cb, 1, i))
 236             goto err;
 237     }
 238     ret = 1;
 239  err:
 240     if (ctx != NULL) {
 241         BN_CTX_end(ctx);
 242         if (ctx_passed == NULL)
 243             BN_CTX_free(ctx);
 244     }
 245     BN_MONT_CTX_free(mont);
 246
 247     return ret;
 248 }
 249
 250 static int witness(BIGNUM *w, const BIGNUM *a, const BIGNUM *a1,
 251                    const BIGNUM *a1_odd, int k, BN_CTX *ctx,
 252                    BN_MONT_CTX *mont)
 253 {
 254     if (!BN_mod_exp_mont(w, w, a1_odd, a, ctx, mont)) /* w := w^a1_odd mod a */
 255         return -1;
 256     if (BN_is_one(w))
 257         return 0;               /* probably prime */
 258     if (BN_cmp(w, a1) == 0)
 259         return 0;               /* w == -1 (mod a), 'a' is probably prime */
 260     while (--k) {
 261         if (!BN_mod_mul(w, w, w, a, ctx)) /* w := w^2 mod a */
 262             return -1;
 263         if (BN_is_one(w))
 264             return 1;           /* 'a' is composite, otherwise a previous 'w'
 265                                  * would have been == -1 (mod 'a') */
 266         if (BN_cmp(w, a1) == 0)
 267             return 0;           /* w == -1 (mod a), 'a' is probably prime */
 268     }
 269     /*
 270      * If we get here, 'w' is the (a-1)/2-th power of the original 'w', and
 271      * it is neither -1 nor +1 -- so 'a' cannot be prime
 272      */
 273     bn_check_top(w);
 274     return 1;
 275 }
 276
 277 static int probable_prime(BIGNUM *rnd, int bits, int safe, prime_t *mods)
 278 {
 279     int i;
 280     BN_ULONG delta;
 281     BN_ULONG maxdelta = BN_MASK2 - primes[NUMPRIMES - 1];
 282
 283  again:
 284     /* TODO: Not all primes are private */
 285     if (!BN_priv_rand(rnd, bits, BN_RAND_TOP_TWO, BN_RAND_BOTTOM_ODD))
 286         return 0;
 287     if (safe && !BN_set_bit(rnd, 1))
 288         return 0;
 289     /* we now have a random number 'rnd' to test. */
 290     for (i = 1; i < NUMPRIMES; i++) {
 291         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 292         if (mod == (BN_ULONG)-1)
 293             return 0;
 294         mods[i] = (prime_t) mod;
 295     }
 296     delta = 0;
 297  loop:
 298     for (i = 1; i < NUMPRIMES; i++) {
 299         /*
 300          * check that rnd is a prime and also that
 301          * gcd(rnd-1,primes) == 1 (except for 2)
 302          * do the second check only if we are interested in safe primes
 303          * in the case that the candidate prime is a single word then
 304          * we check only the primes up to sqrt(rnd)
 305          */
 306         if (bits <= 31 && delta <= 0x7fffffff
 307                 && square(primes[i]) > BN_get_word(rnd) + delta)
 308             break;
 309         if (safe ? (mods[i] + delta) % primes[i] <= 1
 310                  : (mods[i] + delta) % primes[i] == 0) {
 311             delta += safe ? 4 : 2;
 312             if (delta > maxdelta)
 313                 goto again;
 314             goto loop;
 315         }
 316     }
 317     if (!BN_add_word(rnd, delta))
 318         return 0;
 319     if (BN_num_bits(rnd) != bits)
 320         goto again;
 321     bn_check_top(rnd);
 322     return 1;
 323 }
 324
 325 int bn_probable_prime_dh(BIGNUM *rnd, int bits,
 326                          const BIGNUM *add, const BIGNUM *rem, BN_CTX *ctx)
 327 {
 328     int i, ret = 0;
 329     BIGNUM *t1;
 330
 331     BN_CTX_start(ctx);
 332     if ((t1 = BN_CTX_get(ctx)) == NULL)
 333         goto err;
 334
 335     if (!BN_rand(rnd, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
 336         goto err;
 337
 338     /* we need ((rnd-rem) % add) == 0 */
 339
 340     if (!BN_mod(t1, rnd, add, ctx))
 341         goto err;
 342     if (!BN_sub(rnd, rnd, t1))
 343         goto err;
 344     if (rem == NULL) {
 345         if (!BN_add_word(rnd, 1))
 346             goto err;
 347     } else {
 348         if (!BN_add(rnd, rnd, rem))
 349             goto err;
 350     }
 351
 352     /* we now have a random number 'rand' to test. */
 353
 354  loop:
 355     for (i = 1; i < NUMPRIMES; i++) {
 356         /* check that rnd is a prime */
 357         BN_ULONG mod = BN_mod_word(rnd, (BN_ULONG)primes[i]);
 358         if (mod == (BN_ULONG)-1)
 359             goto err;
 360         if (mod <= 1) {
 361             if (!BN_add(rnd, rnd, add))
 362                 goto err;
 363             goto loop;
 364         }
 365     }
 366     ret = 1;
 367
 368  err:
 369     BN_CTX_end(ctx);
 370     bn_check_top(rnd);
 371     return ret;
 372 }
 373
 374 static int probable_prime_dh_safe(BIGNUM *p, int bits, const BIGNUM *padd,
 375                                   const BIGNUM *rem, BN_CTX *ctx)
 376 {
 377     int i, ret = 0;
 378     BIGNUM *t1, *qadd, *q;
 379
 380     bits--;
 381     BN_CTX_start(ctx);
 382     t1 = BN_CTX_get(ctx);
 383     q = BN_CTX_get(ctx);
 384     qadd = BN_CTX_get(ctx);
 385     if (qadd == NULL)
 386         goto err;
 387
 388     if (!BN_rshift1(qadd, padd))
 389         goto err;
 390
 391     if (!BN_rand(q, bits, BN_RAND_TOP_ONE, BN_RAND_BOTTOM_ODD))
 392         goto err;
 393
 394     /* we need ((rnd-rem) % add) == 0 */
 395     if (!BN_mod(t1, q, qadd, ctx))
 396         goto err;
 397     if (!BN_sub(q, q, t1))
 398         goto err;
 399     if (rem == NULL) {
 400         if (!BN_add_word(q, 1))
 401             goto err;
 402     } else {
 403         if (!BN_rshift1(t1, rem))
 404             goto err;
 405         if (!BN_add(q, q, t1))
 406             goto err;
 407     }
 408
 409     /* we now have a random number 'rand' to test. */
 410     if (!BN_lshift1(p, q))
 411         goto err;
 412     if (!BN_add_word(p, 1))
 413         goto err;
 414
 415  loop:
 416     for (i = 1; i < NUMPRIMES; i++) {
 417         /* check that p and q are prime */
 418         /*
 419          * check that for p and q gcd(p-1,primes) == 1 (except for 2)
 420          */
 421         BN_ULONG pmod = BN_mod_word(p, (BN_ULONG)primes[i]);
 422         BN_ULONG qmod = BN_mod_word(q, (BN_ULONG)primes[i]);
 423         if (pmod == (BN_ULONG)-1 || qmod == (BN_ULONG)-1)
 424             goto err;
 425         if (pmod == 0 || qmod == 0) {
 426             if (!BN_add(p, p, padd))
 427                 goto err;
 428             if (!BN_add(q, q, qadd))
 429                 goto err;
 430             goto loop;
 431         }
 432     }
 433     ret = 1;
 434
 435  err:
 436     BN_CTX_end(ctx);
 437     bn_check_top(p);
 438     return ret;
 439 }