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 #include <stddef.h> 28 #include <stdint.h> 29 30 #include "primes.h" 31 32 /* Return a * b % n, where 0 < n. */ 33 static uint64_t 34 mulmod(uint64_t a, uint64_t b, uint64_t n) 35 { 36 uint64_t x = 0; 37 uint64_t an = a % n; 38 39 while (b != 0) { 40 if (b & 1) { 41 x += an; 42 if ((x < an) || (x >= n)) 43 x -= n; 44 } 45 if (an + an < an) 46 an = an + an - n; 47 else if (an + an >= n) 48 an = an + an - n; 49 else 50 an = an + an; 51 b >>= 1; 52 } 53 54 return (x); 55 } 56 57 /* Return a^r % n, where 0 < n. */ 58 static uint64_t 59 powmod(uint64_t a, uint64_t r, uint64_t n) 60 { 61 uint64_t x = 1; 62 63 while (r != 0) { 64 if (r & 1) 65 x = mulmod(a, x, n); 66 a = mulmod(a, a, n); 67 r >>= 1; 68 } 69 70 return (x); 71 } 72 73 /* Return non-zero if n is a strong pseudoprime to base p. */ 74 static int 75 spsp(uint64_t n, uint64_t p) 76 { 77 uint64_t x; 78 uint64_t r = n - 1; 79 int k = 0; 80 81 /* Compute n - 1 = 2^k * r. */ 82 while ((r & 1) == 0) { 83 k++; 84 r >>= 1; 85 } 86 87 /* Compute x = p^r mod n. If x = 1, n is a p-spsp. */ 88 x = powmod(p, r, n); 89 if (x == 1) 90 return (1); 91 92 /* Compute x^(2^i) for 0 <= i < n. If any are -1, n is a p-spsp. */ 93 while (k > 0) { 94 if (x == n - 1) 95 return (1); 96 x = powmod(x, 2, n); 97 k--; 98 } 99 100 /* Not a p-spsp. */ 101 return (0); 102 } 103 104 /* Test for primality using strong pseudoprime tests. */ 105 int 106 isprime(ubig _n) 107 { 108 uint64_t n = _n; 109 110 /* 111 * Values from: 112 * C. Pomerance, J.L. Selfridge, and S.S. Wagstaff, Jr., 113 * The pseudoprimes to 25 * 10^9, Math. Comp. 35(151):1003-1026, 1980. 114 */ 115 116 /* No SPSPs to base 2 less than 2047. */ 117 if (!spsp(n, 2)) 118 return (0); 119 if (n < 2047ULL) 120 return (1); 121 122 /* No SPSPs to bases 2,3 less than 1373653. */ 123 if (!spsp(n, 3)) 124 return (0); 125 if (n < 1373653ULL) 126 return (1); 127 128 /* No SPSPs to bases 2,3,5 less than 25326001. */ 129 if (!spsp(n, 5)) 130 return (0); 131 if (n < 25326001ULL) 132 return (1); 133 134 /* No SPSPs to bases 2,3,5,7 less than 3215031751. */ 135 if (!spsp(n, 7)) 136 return (0); 137 if (n < 3215031751ULL) 138 return (1); 139 140 /* 141 * Values from: 142 * G. Jaeschke, On strong pseudoprimes to several bases, 143 * Math. Comp. 61(204):915-926, 1993. 144 */ 145 146 /* No SPSPs to bases 2,3,5,7,11 less than 2152302898747. */ 147 if (!spsp(n, 11)) 148 return (0); 149 if (n < 2152302898747ULL) 150 return (1); 151 152 /* No SPSPs to bases 2,3,5,7,11,13 less than 3474749660383. */ 153 if (!spsp(n, 13)) 154 return (0); 155 if (n < 3474749660383ULL) 156 return (1); 157 158 /* No SPSPs to bases 2,3,5,7,11,13,17 less than 341550071728321. */ 159 if (!spsp(n, 17)) 160 return (0); 161 if (n < 341550071728321ULL) 162 return (1); 163 164 /* No SPSPs to bases 2,3,5,7,11,13,17,19 less than 341550071728321. */ 165 if (!spsp(n, 19)) 166 return (0); 167 if (n < 341550071728321ULL) 168 return (1); 169 170 /* 171 * Value from: 172 * Y. Jiang and Y. Deng, Strong pseudoprimes to the first eight prime 173 * bases, Math. Comp. 83(290):2915-2924, 2014. 174 */ 175 176 /* No SPSPs to bases 2..23 less than 3825123056546413051. */ 177 if (!spsp(n, 23)) 178 return (0); 179 if (n < 3825123056546413051) 180 return (1); 181 182 /* 183 * Value from: 184 * J. Sorenson and J. Webster, Strong pseudoprimes to twelve prime 185 * bases, Math. Comp. 86(304):985-1003, 2017. 186 */ 187 188 /* No SPSPs to bases 2..37 less than 318665857834031151167461. */ 189 if (!spsp(n, 29)) 190 return (0); 191 if (!spsp(n, 31)) 192 return (0); 193 if (!spsp(n, 37)) 194 return (0); 195 196 /* All 64-bit values are less than 318665857834031151167461. */ 197 return (1); 198 } 199