1 /* mpz_millerrabin(n,reps) -- An implementation of the probabilistic primality 2 test found in Knuth's Seminumerical Algorithms book. If the function 3 mpz_millerrabin() returns 0 then n is not prime. If it returns 1, then n is 4 'probably' prime. The probability of a false positive is (1/4)**reps, where 5 reps is the number of internal passes of the probabilistic algorithm. Knuth 6 indicates that 25 passes are reasonable. 7 8 THE FUNCTIONS IN THIS FILE ARE FOR INTERNAL USE ONLY. THEY'RE ALMOST 9 CERTAIN TO BE SUBJECT TO INCOMPATIBLE CHANGES OR DISAPPEAR COMPLETELY IN 10 FUTURE GNU MP RELEASES. 11 12 Copyright 1991, 1993, 1994, 1996-2002, 2005, 2014 Free Software 13 Foundation, Inc. 14 15 Contributed by John Amanatides. 16 17 This file is part of the GNU MP Library. 18 19 The GNU MP Library is free software; you can redistribute it and/or modify 20 it under the terms of either: 21 22 * the GNU Lesser General Public License as published by the Free 23 Software Foundation; either version 3 of the License, or (at your 24 option) any later version. 25 26 or 27 28 * the GNU General Public License as published by the Free Software 29 Foundation; either version 2 of the License, or (at your option) any 30 later version. 31 32 or both in parallel, as here. 33 34 The GNU MP Library is distributed in the hope that it will be useful, but 35 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY 36 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 37 for more details. 38 39 You should have received copies of the GNU General Public License and the 40 GNU Lesser General Public License along with the GNU MP Library. If not, 41 see https://www.gnu.org/licenses/. */ 42 43 #include "gmp.h" 44 #include "gmp-impl.h" 45 46 static int millerrabin (mpz_srcptr, mpz_srcptr, 47 mpz_ptr, mpz_ptr, 48 mpz_srcptr, unsigned long int); 49 50 int 51 mpz_millerrabin (mpz_srcptr n, int reps) 52 { 53 int r; 54 mpz_t nm1, nm3, x, y, q; 55 unsigned long int k; 56 gmp_randstate_t rstate; 57 int is_prime; 58 TMP_DECL; 59 TMP_MARK; 60 61 MPZ_TMP_INIT (nm1, SIZ (n) + 1); 62 mpz_sub_ui (nm1, n, 1L); 63 64 MPZ_TMP_INIT (x, SIZ (n) + 1); 65 MPZ_TMP_INIT (y, 2 * SIZ (n)); /* mpz_powm_ui needs excessive memory!!! */ 66 67 /* Perform a Fermat test. */ 68 mpz_set_ui (x, 210L); 69 mpz_powm (y, x, nm1, n); 70 if (mpz_cmp_ui (y, 1L) != 0) 71 { 72 TMP_FREE; 73 return 0; 74 } 75 76 MPZ_TMP_INIT (q, SIZ (n)); 77 78 /* Find q and k, where q is odd and n = 1 + 2**k * q. */ 79 k = mpz_scan1 (nm1, 0L); 80 mpz_tdiv_q_2exp (q, nm1, k); 81 82 /* n-3 */ 83 MPZ_TMP_INIT (nm3, SIZ (n) + 1); 84 mpz_sub_ui (nm3, n, 3L); 85 ASSERT (mpz_cmp_ui (nm3, 1L) >= 0); 86 87 gmp_randinit_default (rstate); 88 89 is_prime = 1; 90 for (r = 0; r < reps && is_prime; r++) 91 { 92 /* 2 to n-2 inclusive, don't want 1, 0 or -1 */ 93 mpz_urandomm (x, rstate, nm3); 94 mpz_add_ui (x, x, 2L); 95 96 is_prime = millerrabin (n, nm1, x, y, q, k); 97 } 98 99 gmp_randclear (rstate); 100 101 TMP_FREE; 102 return is_prime; 103 } 104 105 static int 106 millerrabin (mpz_srcptr n, mpz_srcptr nm1, mpz_ptr x, mpz_ptr y, 107 mpz_srcptr q, unsigned long int k) 108 { 109 unsigned long int i; 110 111 mpz_powm (y, x, q, n); 112 113 if (mpz_cmp_ui (y, 1L) == 0 || mpz_cmp (y, nm1) == 0) 114 return 1; 115 116 for (i = 1; i < k; i++) 117 { 118 mpz_powm_ui (y, y, 2L, n); 119 if (mpz_cmp (y, nm1) == 0) 120 return 1; 121 /* y == 1 means that the previous y was a non-trivial square root 122 of 1 (mod n). y == 0 means that n is a power of the base. 123 In either case, n is not prime. */ 124 if (mpz_cmp_ui (y, 1L) <= 0) 125 return 0; 126 } 127 return 0; 128 } 129