/* primegen.c - prime number generator * Copyright (C) 1998, 2000, 2001, 2002 Free Software Foundation, Inc. * * This file is part of Libgcrypt. * * Libgcrypt is free software; you can redistribute it and/or modify * it under the terms of the GNU Lesser general Public License as * published by the Free Software Foundation; either version 2.1 of * the License, or (at your option) any later version. * * Libgcrypt is distributed in the hope that it will be useful, * but WITHOUT ANY WARRANTY; without even the implied warranty of * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * GNU Lesser General Public License for more details. * * You should have received a copy of the GNU Lesser General Public * License along with this program; if not, write to the Free Software * Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA * * *********************************************************************** * The algorithm used to generate practically save primes is due to * Lim and Lee as described in the CRYPTO '97 proceedings (ISBN3540633847) * page 260. */ #include #include #include #include #include #include "g10lib.h" #include "mpi.h" #include "cipher.h" static int no_of_small_prime_numbers; static MPI gen_prime( unsigned nbits, int mode, int randomlevel ); static int check_prime( MPI prime, MPI val_2 ); static int is_prime( MPI n, int steps, int *count ); static void m_out_of_n( char *array, int m, int n ); static void (*progress_cb) ( void *, int ); static void *progress_cb_data; /* Note: 2 is not included because it can be tested more easily by looking at bit 0. The last entry in this list is marked by a zero */ static ushort small_prime_numbers[] = { 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 0 }; void _gcry_register_primegen_progress ( void (*cb)(void *,const char*,int,int,int), void *cb_data ) { progress_cb = cb; progress_cb_data = cb_data; } static void progress( int c ) { if ( progress_cb ) progress_cb ( progress_cb_data, "primegen", c, 0, 0 ); } /**************** * Generate a prime number (stored in secure memory) */ MPI _gcry_generate_secret_prime( unsigned nbits ) { MPI prime; prime = gen_prime( nbits, 1, 2 ); progress('\n'); return prime; } MPI _gcry_generate_public_prime( unsigned nbits ) { MPI prime; prime = gen_prime( nbits, 0, 2 ); progress('\n'); return prime; } /**************** * We do not need to use the strongest RNG because we gain no extra * security from it - The prime number is public and we could also * offer the factors for those who are willing to check that it is * indeed a strong prime. * * mode 0: Standard * 1: Make sure that at least one factor is of size qbits. */ MPI _gcry_generate_elg_prime( int mode, unsigned pbits, unsigned qbits, MPI g, MPI **ret_factors ) { int n; /* number of factors */ int m; /* number of primes in pool */ unsigned fbits; /* length of prime factors */ MPI *factors; /* current factors */ MPI *pool; /* pool of primes */ MPI q; /* first prime factor (variable)*/ MPI prime; /* prime test value */ MPI q_factor; /* used for mode 1 */ byte *perms = NULL; int i, j; int count1, count2; unsigned nprime; unsigned req_qbits = qbits; /* the requested q bits size */ MPI val_2 = mpi_alloc_set_ui( 2 ); /* find number of needed prime factors */ for(n=1; (pbits - qbits - 1) / n >= qbits; n++ ) ; n--; if( !n || (mode==1 && n < 2) ) log_fatal("can't gen prime with pbits=%u qbits=%u\n", pbits, qbits ); if( mode == 1 ) { n--; fbits = (pbits - 2*req_qbits -1) / n; qbits = pbits - req_qbits - n*fbits; } else { fbits = (pbits - req_qbits -1) / n; qbits = pbits - n*fbits; } if( DBG_CIPHER ) log_debug("gen prime: pbits=%u qbits=%u fbits=%u/%u n=%d\n", pbits, req_qbits, qbits, fbits, n ); prime = gcry_mpi_new ( pbits ); q = gen_prime( qbits, 0, 0 ); q_factor = mode==1? gen_prime( req_qbits, 0, 0 ) : NULL; /* allocate an array to hold the factors + 2 for later usage */ factors = gcry_xcalloc( n+2, sizeof *factors ); /* make a pool of 3n+5 primes (this is an arbitrary value) */ m = n*3+5; if( mode == 1 ) m += 5; /* need some more for DSA */ if( m < 25 ) m = 25; pool = gcry_xcalloc( m , sizeof *pool ); /* permutate over the pool of primes */ count1=count2=0; do { next_try: if( !perms ) { /* allocate new primes */ for(i=0; i < m; i++ ) { mpi_free(pool[i]); pool[i] = NULL; } /* init m_out_of_n() */ perms = gcry_xcalloc( 1, m ); for(i=0; i < n; i++ ) { perms[i] = 1; pool[i] = gen_prime( fbits, 0, 1 ); factors[i] = pool[i]; } } else { m_out_of_n( perms, n, m ); for(i=j=0; i < m && j < n ; i++ ) if( perms[i] ) { if( !pool[i] ) pool[i] = gen_prime( fbits, 0, 1 ); factors[j++] = pool[i]; } if( i == n ) { gcry_free(perms); perms = NULL; progress('!'); goto next_try; /* allocate new primes */ } } mpi_set( prime, q ); mpi_mul_ui( prime, prime, 2 ); if( mode == 1 ) mpi_mul( prime, prime, q_factor ); for(i=0; i < n; i++ ) mpi_mul( prime, prime, factors[i] ); mpi_add_ui( prime, prime, 1 ); nprime = mpi_get_nbits(prime); if( nprime < pbits ) { if( ++count1 > 20 ) { count1 = 0; qbits++; progress('>'); mpi_free (q); q = gen_prime( qbits, 0, 0 ); goto next_try; } } else count1 = 0; if( nprime > pbits ) { if( ++count2 > 20 ) { count2 = 0; qbits--; progress('<'); mpi_free (q); q = gen_prime( qbits, 0, 0 ); goto next_try; } } else count2 = 0; } while( !(nprime == pbits && check_prime( prime, val_2 )) ); if( DBG_CIPHER ) { progress('\n'); log_mpidump( "prime : ", prime ); log_mpidump( "factor q: ", q ); if( mode == 1 ) log_mpidump( "factor q0: ", q_factor ); for(i=0; i < n; i++ ) log_mpidump( "factor pi: ", factors[i] ); log_debug("bit sizes: prime=%u, q=%u", mpi_get_nbits(prime), mpi_get_nbits(q) ); if( mode == 1 ) fprintf(stderr, ", q0=%u", mpi_get_nbits(q_factor) ); for(i=0; i < n; i++ ) fprintf(stderr, ", p%d=%u", i, mpi_get_nbits(factors[i]) ); progress('\n'); } if( ret_factors ) { /* caller wants the factors */ *ret_factors = gcry_xcalloc( n+2 , sizeof **ret_factors); i = 0; if( mode == 1 ) { (*ret_factors)[i++] = mpi_copy( q_factor ); for(; i <= n; i++ ) (*ret_factors)[i] = mpi_copy( factors[i] ); } else { for(; i < n; i++ ) (*ret_factors)[i] = mpi_copy( factors[i] ); } } if( g ) { /* create a generator (start with 3)*/ MPI tmp = mpi_alloc( mpi_get_nlimbs(prime) ); MPI b = mpi_alloc( mpi_get_nlimbs(prime) ); MPI pmin1 = mpi_alloc( mpi_get_nlimbs(prime) ); if( mode == 1 ) BUG(); /* not yet implemented */ factors[n] = q; factors[n+1] = mpi_alloc_set_ui(2); mpi_sub_ui( pmin1, prime, 1 ); mpi_set_ui(g,2); do { mpi_add_ui(g, g, 1); if( DBG_CIPHER ) { log_debug("checking g: "); /*mpi_print( stderr, g, 1 );*/ #if __GNUC__ >= 2 # warning we need an internal mpi_print for debugging #endif } else progress('^'); for(i=0; i < n+2; i++ ) { /*fputc('~', stderr);*/ mpi_fdiv_q(tmp, pmin1, factors[i] ); /* (no mpi_pow(), but it is okay to use this with mod prime) */ gcry_mpi_powm(b, g, tmp, prime ); if( !mpi_cmp_ui(b, 1) ) break; } if( DBG_CIPHER ) progress('\n'); } while( i < n+2 ); mpi_free(factors[n+1]); mpi_free(tmp); mpi_free(b); mpi_free(pmin1); } if( !DBG_CIPHER ) progress('\n'); gcry_free( factors ); /* (factors are shallow copies) */ for(i=0; i < m; i++ ) mpi_free( pool[i] ); gcry_free( pool ); gcry_free(perms); mpi_free(val_2); mpi_free (q); return prime; } static MPI gen_prime( unsigned nbits, int secret, int randomlevel ) { MPI prime, ptest, pminus1, val_2, val_3, result; int i; unsigned x, step; unsigned count1, count2; int *mods; if( 0 && DBG_CIPHER ) log_debug("generate a prime of %u bits ", nbits ); if( !no_of_small_prime_numbers ) { for(i=0; small_prime_numbers[i]; i++ ) no_of_small_prime_numbers++; } mods = gcry_xmalloc( no_of_small_prime_numbers * sizeof *mods ); /* make nbits fit into MPI implementation */ val_2 = mpi_alloc_set_ui( 2 ); val_3 = mpi_alloc_set_ui( 3); prime = secret? gcry_mpi_snew ( nbits ): gcry_mpi_new ( nbits ); result = mpi_alloc_like( prime ); pminus1= mpi_alloc_like( prime ); ptest = mpi_alloc_like( prime ); count1 = count2 = 0; for(;;) { /* try forvever */ int dotcount=0; /* generate a random number */ gcry_mpi_randomize( prime, nbits, randomlevel ); /* Set high order bit to 1, set low order bit to 0. If we are generating a secret prime we are most probably doing that for RSA, to make sure that the modulus does have the requested keysize we set the 2 high order bits */ mpi_set_highbit (prime, nbits-1); if (secret) mpi_set_bit (prime, nbits-2); mpi_set_bit(prime, 0); /* calculate all remainders */ for(i=0; (x = small_prime_numbers[i]); i++ ) mods[i] = mpi_fdiv_r_ui(NULL, prime, x); /* now try some primes starting with prime */ for(step=0; step < 20000; step += 2 ) { /* check against all the small primes we have in mods */ count1++; for(i=0; (x = small_prime_numbers[i]); i++ ) { while( mods[i] + step >= x ) mods[i] -= x; if( !(mods[i] + step) ) break; } if( x ) continue; /* found a multiple of an already known prime */ mpi_add_ui( ptest, prime, step ); /* do a faster Fermat test */ count2++; mpi_sub_ui( pminus1, ptest, 1); gcry_mpi_powm( result, val_2, pminus1, ptest ); if( !mpi_cmp_ui( result, 1 ) ) { /* not composite */ /* perform stronger tests */ if( is_prime(ptest, 5, &count2 ) ) { if( !mpi_test_bit( ptest, nbits-1-secret ) ) { progress('\n'); log_debug("overflow in prime generation\n"); break; /* step loop, continue with a new prime */ } mpi_free(val_2); mpi_free(val_3); mpi_free(result); mpi_free(pminus1); mpi_free(prime); gcry_free(mods); return ptest; } } if( ++dotcount == 10 ) { progress('.'); dotcount = 0; } } progress(':'); /* restart with a new random value */ } } /**************** * Returns: true if this may be a prime */ static int check_prime( MPI prime, MPI val_2 ) { int i; unsigned x; int count=0; /* check against small primes */ for(i=0; (x = small_prime_numbers[i]); i++ ) { if( mpi_divisible_ui( prime, x ) ) return 0; } /* a quick fermat test */ { MPI result = mpi_alloc_like( prime ); MPI pminus1 = mpi_alloc_like( prime ); mpi_sub_ui( pminus1, prime, 1); gcry_mpi_powm( result, val_2, pminus1, prime ); mpi_free( pminus1 ); if( mpi_cmp_ui( result, 1 ) ) { /* if composite */ mpi_free( result ); progress('.'); return 0; } mpi_free( result ); } /* perform stronger tests */ if( is_prime(prime, 5, &count ) ) return 1; /* is probably a prime */ progress('.'); return 0; } /**************** * Return true if n is probably a prime */ static int is_prime( MPI n, int steps, int *count ) { MPI x = mpi_alloc( mpi_get_nlimbs( n ) ); MPI y = mpi_alloc( mpi_get_nlimbs( n ) ); MPI z = mpi_alloc( mpi_get_nlimbs( n ) ); MPI nminus1 = mpi_alloc( mpi_get_nlimbs( n ) ); MPI a2 = mpi_alloc_set_ui( 2 ); MPI q; unsigned i, j, k; int rc = 0; unsigned nbits = mpi_get_nbits( n ); mpi_sub_ui( nminus1, n, 1 ); /* find q and k, so that n = 1 + 2^k * q */ q = mpi_copy( nminus1 ); k = mpi_trailing_zeros( q ); mpi_tdiv_q_2exp(q, q, k); for(i=0 ; i < steps; i++ ) { ++*count; if( !i ) { mpi_set_ui( x, 2 ); } else { gcry_mpi_randomize( x, nbits, GCRY_WEAK_RANDOM ); /* make sure that the number is smaller than the prime * and keep the randomness of the high bit */ if( mpi_test_bit( x, nbits-2 ) ) { mpi_set_highbit( x, nbits-2 ); /* clear all higher bits */ } else { mpi_set_highbit( x, nbits-2 ); mpi_clear_bit( x, nbits-2 ); } assert( mpi_cmp( x, nminus1 ) < 0 && mpi_cmp_ui( x, 1 ) > 0 ); } gcry_mpi_powm( y, x, q, n); if( mpi_cmp_ui(y, 1) && mpi_cmp( y, nminus1 ) ) { for( j=1; j < k && mpi_cmp( y, nminus1 ); j++ ) { gcry_mpi_powm(y, y, a2, n); if( !mpi_cmp_ui( y, 1 ) ) goto leave; /* not a prime */ } if( mpi_cmp( y, nminus1 ) ) goto leave; /* not a prime */ } progress('+'); } rc = 1; /* may be a prime */ leave: mpi_free( x ); mpi_free( y ); mpi_free( z ); mpi_free( nminus1 ); mpi_free( q ); return rc; } static void m_out_of_n( char *array, int m, int n ) { int i=0, i1=0, j=0, jp=0, j1=0, k1=0, k2=0; if( !m || m >= n ) return; if( m == 1 ) { /* special case */ for(i=0; i < n; i++ ) if( array[i] ) { array[i++] = 0; if( i >= n ) i = 0; array[i] = 1; return; } BUG(); } for(j=1; j < n; j++ ) { if( array[n-1] == array[n-j-1] ) continue; j1 = j; break; } if( m & 1 ) { /* m is odd */ if( array[n-1] ) { if( j1 & 1 ) { k1 = n - j1; k2 = k1+2; if( k2 > n ) k2 = n; goto leave; } goto scan; } k2 = n - j1 - 1; if( k2 == 0 ) { k1 = i; k2 = n - j1; } else if( array[k2] && array[k2-1] ) k1 = n; else k1 = k2 + 1; } else { /* m is even */ if( !array[n-1] ) { k1 = n - j1; k2 = k1 + 1; goto leave; } if( !(j1 & 1) ) { k1 = n - j1; k2 = k1+2; if( k2 > n ) k2 = n; goto leave; } scan: jp = n - j1 - 1; for(i=1; i <= jp; i++ ) { i1 = jp + 2 - i; if( array[i1-1] ) { if( array[i1-2] ) { k1 = i1 - 1; k2 = n - j1; } else { k1 = i1 - 1; k2 = n + 1 - j1; } goto leave; } } k1 = 1; k2 = n + 1 - m; } leave: array[k1-1] = !array[k1-1]; array[k2-1] = !array[k2-1]; }