1 /* mpz_probab_prime_p -- 2 An implementation of the probabilistic primality test found in Knuth's 3 Seminumerical Algorithms book. If the function mpz_probab_prime_p() 4 returns 0 then n is not prime. If it returns 1, then n is 'probably' 5 prime. If it returns 2, n is surely prime. The probability of a false 6 positive is (1/4)**reps, where reps is the number of internal passes of the 7 probabilistic algorithm. Knuth indicates that 25 passes are reasonable. 8 9 Copyright 1991, 1993, 1994, 1996-2002, 2005, 2015, 2016 Free Software 10 Foundation, Inc. 11 12 This file is part of the GNU MP Library. 13 14 The GNU MP Library is free software; you can redistribute it and/or modify 15 it under the terms of either: 16 17 * the GNU Lesser General Public License as published by the Free 18 Software Foundation; either version 3 of the License, or (at your 19 option) any later version. 20 21 or 22 23 * the GNU General Public License as published by the Free Software 24 Foundation; either version 2 of the License, or (at your option) any 25 later version. 26 27 or both in parallel, as here. 28 29 The GNU MP Library is distributed in the hope that it will be useful, but 30 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 31 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 32 for more details. 33 34 You should have received copies of the GNU General Public License and the 35 GNU Lesser General Public License along with the GNU MP Library. If not, 36 see https://www.gnu.org/licenses/. */ 37 38 #include "gmp-impl.h" 39 #include "longlong.h" 40 41 static int isprime (unsigned long int); 42 43 44 /* MPN_MOD_OR_MODEXACT_1_ODD can be used instead of mpn_mod_1 for the trial 45 division. It gives a result which is not the actual remainder r but a 46 value congruent to r*2^n mod d. Since all the primes being tested are 47 odd, r*2^n mod p will be 0 if and only if r mod p is 0. */ 48 49 int 50 mpz_probab_prime_p (mpz_srcptr n, int reps) 51 { 52 mp_limb_t r; 53 mpz_t n2; 54 55 /* Handle small and negative n. */ 56 if (mpz_cmp_ui (n, 1000000L) <= 0) 57 { 58 if (mpz_cmpabs_ui (n, 1000000L) <= 0) 59 { 60 int is_prime; 61 unsigned long n0; 62 n0 = mpz_get_ui (n); 63 is_prime = n0 & (n0 > 1) ? isprime (n0) : n0 == 2; 64 return is_prime ? 2 : 0; 65 } 66 /* Negative number. Negate and fall out. */ 67 PTR(n2) = PTR(n); 68 SIZ(n2) = -SIZ(n); 69 n = n2; 70 } 71 72 /* If n is now even, it is not a prime. */ 73 if (mpz_even_p (n)) 74 return 0; 75 76 #if defined (PP) 77 /* Check if n has small factors. */ 78 #if defined (PP_INVERTED) 79 r = MPN_MOD_OR_PREINV_MOD_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP, 80 (mp_limb_t) PP_INVERTED); 81 #else 82 r = mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) PP); 83 #endif 84 if (r % 3 == 0 85 #if GMP_LIMB_BITS >= 4 86 || r % 5 == 0 87 #endif 88 #if GMP_LIMB_BITS >= 8 89 || r % 7 == 0 90 #endif 91 #if GMP_LIMB_BITS >= 16 92 || r % 11 == 0 || r % 13 == 0 93 #endif 94 #if GMP_LIMB_BITS >= 32 95 || r % 17 == 0 || r % 19 == 0 || r % 23 == 0 || r % 29 == 0 96 #endif 97 #if GMP_LIMB_BITS >= 64 98 || r % 31 == 0 || r % 37 == 0 || r % 41 == 0 || r % 43 == 0 99 || r % 47 == 0 || r % 53 == 0 100 #endif 101 ) 102 { 103 return 0; 104 } 105 #endif /* PP */ 106 107 /* Do more dividing. We collect small primes, using umul_ppmm, until we 108 overflow a single limb. We divide our number by the small primes product, 109 and look for factors in the remainder. */ 110 { 111 unsigned long int ln2; 112 unsigned long int q; 113 mp_limb_t p1, p0, p; 114 unsigned int primes[15]; 115 int nprimes; 116 117 nprimes = 0; 118 p = 1; 119 ln2 = mpz_sizeinbase (n, 2); /* FIXME: tune this limit */ 120 for (q = PP_FIRST_OMITTED; q < ln2; q += 2) 121 { 122 if (isprime (q)) 123 { 124 umul_ppmm (p1, p0, p, q); 125 if (p1 != 0) 126 { 127 r = MPN_MOD_OR_MODEXACT_1_ODD (PTR(n), (mp_size_t) SIZ(n), p); 128 while (--nprimes >= 0) 129 if (r % primes[nprimes] == 0) 130 { 131 ASSERT_ALWAYS (mpn_mod_1 (PTR(n), (mp_size_t) SIZ(n), (mp_limb_t) primes[nprimes]) == 0); 132 return 0; 133 } 134 p = q; 135 nprimes = 0; 136 } 137 else 138 { 139 p = p0; 140 } 141 primes[nprimes++] = q; 142 } 143 } 144 } 145 146 /* Perform a number of Miller-Rabin tests. */ 147 return mpz_millerrabin (n, reps); 148 } 149 150 static int 151 isprime (unsigned long int t) 152 { 153 unsigned long int q, r, d; 154 155 ASSERT (t >= 3 && (t & 1) != 0); 156 157 d = 3; 158 do { 159 q = t / d; 160 r = t - q * d; 161 if (q < d) 162 return 1; 163 d += 2; 164 } while (r != 0); 165 return 0; 166 } 167