1 /*- 2 * Copyright (c) 2014 Colin Percival 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 24 * SUCH DAMAGE. 25 */ 26 #include <sys/cdefs.h> 27 __FBSDID("$FreeBSD$"); 28 29 #include <assert.h> 30 #include <stddef.h> 31 #include <stdint.h> 32 33 #include "primes.h" 34 35 /* Return a * b % n, where 0 <= a, b < 2^63, 0 < n < 2^63. */ 36 static uint64_t 37 mulmod(uint64_t a, uint64_t b, uint64_t n) 38 { 39 uint64_t x = 0; 40 41 while (b != 0) { 42 if (b & 1) 43 x = (x + a) % n; 44 a = (a + a) % n; 45 b >>= 1; 46 } 47 48 return (x); 49 } 50 51 /* Return a^r % n, where 0 <= a < 2^63, 0 < n < 2^63. */ 52 static uint64_t 53 powmod(uint64_t a, uint64_t r, uint64_t n) 54 { 55 uint64_t x = 1; 56 57 while (r != 0) { 58 if (r & 1) 59 x = mulmod(a, x, n); 60 a = mulmod(a, a, n); 61 r >>= 1; 62 } 63 64 return (x); 65 } 66 67 /* Return non-zero if n is a strong pseudoprime to base p. */ 68 static int 69 spsp(uint64_t n, uint64_t p) 70 { 71 uint64_t x; 72 uint64_t r = n - 1; 73 int k = 0; 74 75 /* Compute n - 1 = 2^k * r. */ 76 while ((r & 1) == 0) { 77 k++; 78 r >>= 1; 79 } 80 81 /* Compute x = p^r mod n. If x = 1, n is a p-spsp. */ 82 x = powmod(p, r, n); 83 if (x == 1) 84 return (1); 85 86 /* Compute x^(2^i) for 0 <= i < n. If any are -1, n is a p-spsp. */ 87 while (k > 0) { 88 if (x == n - 1) 89 return (1); 90 x = powmod(x, 2, n); 91 k--; 92 } 93 94 /* Not a p-spsp. */ 95 return (0); 96 } 97 98 /* Test for primality using strong pseudoprime tests. */ 99 int 100 isprime(ubig _n) 101 { 102 uint64_t n = _n; 103 104 /* 105 * Values from: 106 * C. Pomerance, J.L. Selfridge, and S.S. Wagstaff, Jr., 107 * The pseudoprimes to 25 * 10^9, Math. Comp. 35(151):1003-1026, 1980. 108 */ 109 110 /* No SPSPs to base 2 less than 2047. */ 111 if (!spsp(n, 2)) 112 return (0); 113 if (n < 2047ULL) 114 return (1); 115 116 /* No SPSPs to bases 2,3 less than 1373653. */ 117 if (!spsp(n, 3)) 118 return (0); 119 if (n < 1373653ULL) 120 return (1); 121 122 /* No SPSPs to bases 2,3,5 less than 25326001. */ 123 if (!spsp(n, 5)) 124 return (0); 125 if (n < 25326001ULL) 126 return (1); 127 128 /* No SPSPs to bases 2,3,5,7 less than 3215031751. */ 129 if (!spsp(n, 7)) 130 return (0); 131 if (n < 3215031751ULL) 132 return (1); 133 134 /* 135 * Values from: 136 * G. Jaeschke, On strong pseudoprimes to several bases, 137 * Math. Comp. 61(204):915-926, 1993. 138 */ 139 140 /* No SPSPs to bases 2,3,5,7,11 less than 2152302898747. */ 141 if (!spsp(n, 11)) 142 return (0); 143 if (n < 2152302898747ULL) 144 return (1); 145 146 /* No SPSPs to bases 2,3,5,7,11,13 less than 3474749660383. */ 147 if (!spsp(n, 13)) 148 return (0); 149 if (n < 3474749660383ULL) 150 return (1); 151 152 /* No SPSPs to bases 2,3,5,7,11,13,17 less than 341550071728321. */ 153 if (!spsp(n, 17)) 154 return (0); 155 if (n < 341550071728321ULL) 156 return (1); 157 158 /* No SPSPs to bases 2,3,5,7,11,13,17,19 less than 341550071728321. */ 159 if (!spsp(n, 19)) 160 return (0); 161 if (n < 341550071728321ULL) 162 return (1); 163 164 /* 165 * Value from: 166 * Y. Jiang and Y. Deng, Strong pseudoprimes to the first eight prime 167 * bases, Math. Comp. 83(290):2915-2924, 2014. 168 */ 169 170 /* No SPSPs to bases 2..23 less than 3825123056546413051. */ 171 if (!spsp(n, 23)) 172 return (0); 173 if (n < 3825123056546413051) 174 return (1); 175 176 /* We can't handle values larger than this. */ 177 assert(n <= SPSPMAX); 178 179 /* UNREACHABLE */ 180 return (0); 181 } 182