2 * Copyright (c) 2014 Colin Percival
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
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.
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
26 #include <sys/cdefs.h>
27 __FBSDID("$FreeBSD$");
34 /* Return a * b % n, where 0 < n. */
36 mulmod(uint64_t a, uint64_t b, uint64_t n)
44 if ((x < an) || (x >= n))
49 else if (an + an >= n)
59 /* Return a^r % n, where 0 < n. */
61 powmod(uint64_t a, uint64_t r, uint64_t n)
75 /* Return non-zero if n is a strong pseudoprime to base p. */
77 spsp(uint64_t n, uint64_t p)
83 /* Compute n - 1 = 2^k * r. */
84 while ((r & 1) == 0) {
89 /* Compute x = p^r mod n. If x = 1, n is a p-spsp. */
94 /* Compute x^(2^i) for 0 <= i < n. If any are -1, n is a p-spsp. */
106 /* Test for primality using strong pseudoprime tests. */
114 * C. Pomerance, J.L. Selfridge, and S.S. Wagstaff, Jr.,
115 * The pseudoprimes to 25 * 10^9, Math. Comp. 35(151):1003-1026, 1980.
118 /* No SPSPs to base 2 less than 2047. */
124 /* No SPSPs to bases 2,3 less than 1373653. */
130 /* No SPSPs to bases 2,3,5 less than 25326001. */
136 /* No SPSPs to bases 2,3,5,7 less than 3215031751. */
139 if (n < 3215031751ULL)
144 * G. Jaeschke, On strong pseudoprimes to several bases,
145 * Math. Comp. 61(204):915-926, 1993.
148 /* No SPSPs to bases 2,3,5,7,11 less than 2152302898747. */
151 if (n < 2152302898747ULL)
154 /* No SPSPs to bases 2,3,5,7,11,13 less than 3474749660383. */
157 if (n < 3474749660383ULL)
160 /* No SPSPs to bases 2,3,5,7,11,13,17 less than 341550071728321. */
163 if (n < 341550071728321ULL)
166 /* No SPSPs to bases 2,3,5,7,11,13,17,19 less than 341550071728321. */
169 if (n < 341550071728321ULL)
174 * Y. Jiang and Y. Deng, Strong pseudoprimes to the first eight prime
175 * bases, Math. Comp. 83(290):2915-2924, 2014.
178 /* No SPSPs to bases 2..23 less than 3825123056546413051. */
181 if (n < 3825123056546413051)
186 * J. Sorenson and J. Webster, Strong pseudoprimes to twelve prime
187 * bases, Math. Comp. 86(304):985-1003, 2017.
190 /* No SPSPs to bases 2..37 less than 318665857834031151167461. */
198 /* All 64-bit values are less than 318665857834031151167461. */