Darren Tucker | 9ee09cf | 2011-11-04 10:52:43 +1100 | [diff] [blame^] | 1 | /* $OpenBSD: moduli.c,v 1.25 2011/10/19 00:06:10 djm Exp $ */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 2 | /* |
| 3 | * Copyright 1994 Phil Karn <karn@qualcomm.com> |
| 4 | * Copyright 1996-1998, 2003 William Allen Simpson <wsimpson@greendragon.com> |
| 5 | * Copyright 2000 Niels Provos <provos@citi.umich.edu> |
| 6 | * All rights reserved. |
| 7 | * |
| 8 | * Redistribution and use in source and binary forms, with or without |
| 9 | * modification, are permitted provided that the following conditions |
| 10 | * are met: |
| 11 | * 1. Redistributions of source code must retain the above copyright |
| 12 | * notice, this list of conditions and the following disclaimer. |
| 13 | * 2. Redistributions in binary form must reproduce the above copyright |
| 14 | * notice, this list of conditions and the following disclaimer in the |
| 15 | * documentation and/or other materials provided with the distribution. |
| 16 | * |
| 17 | * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR |
| 18 | * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES |
| 19 | * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. |
| 20 | * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT, |
| 21 | * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT |
| 22 | * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, |
| 23 | * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY |
| 24 | * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT |
| 25 | * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF |
| 26 | * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
| 27 | */ |
| 28 | |
| 29 | /* |
| 30 | * Two-step process to generate safe primes for DHGEX |
| 31 | * |
| 32 | * Sieve candidates for "safe" primes, |
| 33 | * suitable for use as Diffie-Hellman moduli; |
| 34 | * that is, where q = (p-1)/2 is also prime. |
| 35 | * |
| 36 | * First step: generate candidate primes (memory intensive) |
| 37 | * Second step: test primes' safety (processor intensive) |
| 38 | */ |
| 39 | |
| 40 | #include "includes.h" |
Damien Miller | 5598b4f | 2006-07-24 14:09:40 +1000 | [diff] [blame] | 41 | |
Damien Miller | 91f3eae | 2011-10-18 16:05:55 +1100 | [diff] [blame] | 42 | #include <sys/param.h> |
Damien Miller | 5598b4f | 2006-07-24 14:09:40 +1000 | [diff] [blame] | 43 | #include <sys/types.h> |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 44 | |
| 45 | #include <openssl/bn.h> |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 46 | #include <openssl/dh.h> |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 47 | |
Damien Miller | 91f3eae | 2011-10-18 16:05:55 +1100 | [diff] [blame] | 48 | #include <errno.h> |
Damien Miller | a7a73ee | 2006-08-05 11:37:59 +1000 | [diff] [blame] | 49 | #include <stdio.h> |
Damien Miller | e7a1e5c | 2006-08-05 11:34:19 +1000 | [diff] [blame] | 50 | #include <stdlib.h> |
Damien Miller | e3476ed | 2006-07-24 14:13:33 +1000 | [diff] [blame] | 51 | #include <string.h> |
Damien Miller | d783435 | 2006-08-05 12:39:39 +1000 | [diff] [blame] | 52 | #include <stdarg.h> |
Damien Miller | 5598b4f | 2006-07-24 14:09:40 +1000 | [diff] [blame] | 53 | #include <time.h> |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 54 | #include <unistd.h> |
Damien Miller | 5598b4f | 2006-07-24 14:09:40 +1000 | [diff] [blame] | 55 | |
| 56 | #include "xmalloc.h" |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 57 | #include "dh.h" |
Damien Miller | 5598b4f | 2006-07-24 14:09:40 +1000 | [diff] [blame] | 58 | #include "log.h" |
| 59 | |
Darren Tucker | ebdef76 | 2010-12-04 23:20:50 +1100 | [diff] [blame] | 60 | #include "openbsd-compat/openssl-compat.h" |
| 61 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 62 | /* |
| 63 | * File output defines |
| 64 | */ |
| 65 | |
| 66 | /* need line long enough for largest moduli plus headers */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 67 | #define QLINESIZE (100+8192) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 68 | |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 69 | /* |
| 70 | * Size: decimal. |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 71 | * Specifies the number of the most significant bit (0 to M). |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 72 | * WARNING: internally, usually 1 to N. |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 73 | */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 74 | #define QSIZE_MINIMUM (511) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 75 | |
| 76 | /* |
| 77 | * Prime sieving defines |
| 78 | */ |
| 79 | |
| 80 | /* Constant: assuming 8 bit bytes and 32 bit words */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 81 | #define SHIFT_BIT (3) |
| 82 | #define SHIFT_BYTE (2) |
| 83 | #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE) |
| 84 | #define SHIFT_MEGABYTE (20) |
| 85 | #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 86 | |
| 87 | /* |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 88 | * Using virtual memory can cause thrashing. This should be the largest |
| 89 | * number that is supported without a large amount of disk activity -- |
| 90 | * that would increase the run time from hours to days or weeks! |
| 91 | */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 92 | #define LARGE_MINIMUM (8UL) /* megabytes */ |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 93 | |
| 94 | /* |
| 95 | * Do not increase this number beyond the unsigned integer bit size. |
| 96 | * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits). |
| 97 | */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 98 | #define LARGE_MAXIMUM (127UL) /* megabytes */ |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 99 | |
| 100 | /* |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 101 | * Constant: when used with 32-bit integers, the largest sieve prime |
| 102 | * has to be less than 2**32. |
| 103 | */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 104 | #define SMALL_MAXIMUM (0xffffffffUL) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 105 | |
| 106 | /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */ |
Darren Tucker | fc95970 | 2004-07-17 16:12:08 +1000 | [diff] [blame] | 107 | #define TINY_NUMBER (1UL<<16) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 108 | |
| 109 | /* Ensure enough bit space for testing 2*q. */ |
Damien Miller | 0dc1bef | 2005-07-17 17:22:45 +1000 | [diff] [blame] | 110 | #define TEST_MAXIMUM (1UL<<16) |
| 111 | #define TEST_MINIMUM (QSIZE_MINIMUM + 1) |
| 112 | /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */ |
| 113 | #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 114 | |
| 115 | /* bit operations on 32-bit words */ |
Damien Miller | 0dc1bef | 2005-07-17 17:22:45 +1000 | [diff] [blame] | 116 | #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31))) |
| 117 | #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31))) |
| 118 | #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31))) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 119 | |
| 120 | /* |
| 121 | * Prime testing defines |
| 122 | */ |
| 123 | |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 124 | /* Minimum number of primality tests to perform */ |
Damien Miller | 0dc1bef | 2005-07-17 17:22:45 +1000 | [diff] [blame] | 125 | #define TRIAL_MINIMUM (4) |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 126 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 127 | /* |
| 128 | * Sieving data (XXX - move to struct) |
| 129 | */ |
| 130 | |
| 131 | /* sieve 2**16 */ |
| 132 | static u_int32_t *TinySieve, tinybits; |
| 133 | |
| 134 | /* sieve 2**30 in 2**16 parts */ |
| 135 | static u_int32_t *SmallSieve, smallbits, smallbase; |
| 136 | |
| 137 | /* sieve relative to the initial value */ |
| 138 | static u_int32_t *LargeSieve, largewords, largetries, largenumbers; |
| 139 | static u_int32_t largebits, largememory; /* megabytes */ |
| 140 | static BIGNUM *largebase; |
| 141 | |
Damien Miller | b089fb5 | 2005-05-26 12:16:18 +1000 | [diff] [blame] | 142 | int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *); |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 143 | int prime_test(FILE *, FILE *, u_int32_t, u_int32_t, char *); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 144 | |
| 145 | /* |
| 146 | * print moduli out in consistent form, |
| 147 | */ |
| 148 | static int |
| 149 | qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries, |
| 150 | u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus) |
| 151 | { |
| 152 | struct tm *gtm; |
| 153 | time_t time_now; |
| 154 | int res; |
| 155 | |
| 156 | time(&time_now); |
| 157 | gtm = gmtime(&time_now); |
Damien Miller | 787b2ec | 2003-11-21 23:56:47 +1100 | [diff] [blame] | 158 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 159 | res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ", |
| 160 | gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday, |
| 161 | gtm->tm_hour, gtm->tm_min, gtm->tm_sec, |
| 162 | otype, otests, otries, osize, ogenerator); |
| 163 | |
| 164 | if (res < 0) |
| 165 | return (-1); |
| 166 | |
| 167 | if (BN_print_fp(ofile, omodulus) < 1) |
| 168 | return (-1); |
| 169 | |
| 170 | res = fprintf(ofile, "\n"); |
| 171 | fflush(ofile); |
| 172 | |
| 173 | return (res > 0 ? 0 : -1); |
| 174 | } |
| 175 | |
| 176 | |
| 177 | /* |
| 178 | ** Sieve p's and q's with small factors |
| 179 | */ |
| 180 | static void |
| 181 | sieve_large(u_int32_t s) |
| 182 | { |
| 183 | u_int32_t r, u; |
| 184 | |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 185 | debug3("sieve_large %u", s); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 186 | largetries++; |
| 187 | /* r = largebase mod s */ |
| 188 | r = BN_mod_word(largebase, s); |
| 189 | if (r == 0) |
| 190 | u = 0; /* s divides into largebase exactly */ |
| 191 | else |
| 192 | u = s - r; /* largebase+u is first entry divisible by s */ |
| 193 | |
| 194 | if (u < largebits * 2) { |
| 195 | /* |
| 196 | * The sieve omits p's and q's divisible by 2, so ensure that |
| 197 | * largebase+u is odd. Then, step through the sieve in |
| 198 | * increments of 2*s |
| 199 | */ |
| 200 | if (u & 0x1) |
| 201 | u += s; /* Make largebase+u odd, and u even */ |
| 202 | |
| 203 | /* Mark all multiples of 2*s */ |
| 204 | for (u /= 2; u < largebits; u += s) |
| 205 | BIT_SET(LargeSieve, u); |
| 206 | } |
| 207 | |
| 208 | /* r = p mod s */ |
| 209 | r = (2 * r + 1) % s; |
| 210 | if (r == 0) |
| 211 | u = 0; /* s divides p exactly */ |
| 212 | else |
| 213 | u = s - r; /* p+u is first entry divisible by s */ |
| 214 | |
| 215 | if (u < largebits * 4) { |
| 216 | /* |
| 217 | * The sieve omits p's divisible by 4, so ensure that |
| 218 | * largebase+u is not. Then, step through the sieve in |
| 219 | * increments of 4*s |
| 220 | */ |
| 221 | while (u & 0x3) { |
| 222 | if (SMALL_MAXIMUM - u < s) |
| 223 | return; |
| 224 | u += s; |
| 225 | } |
| 226 | |
| 227 | /* Mark all multiples of 4*s */ |
| 228 | for (u /= 4; u < largebits; u += s) |
| 229 | BIT_SET(LargeSieve, u); |
| 230 | } |
| 231 | } |
| 232 | |
| 233 | /* |
Darren Tucker | 47abce4 | 2004-05-02 22:09:00 +1000 | [diff] [blame] | 234 | * list candidates for Sophie-Germain primes (where q = (p-1)/2) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 235 | * to standard output. |
| 236 | * The list is checked against small known primes (less than 2**30). |
| 237 | */ |
| 238 | int |
Damien Miller | b089fb5 | 2005-05-26 12:16:18 +1000 | [diff] [blame] | 239 | gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 240 | { |
| 241 | BIGNUM *q; |
| 242 | u_int32_t j, r, s, t; |
| 243 | u_int32_t smallwords = TINY_NUMBER >> 6; |
| 244 | u_int32_t tinywords = TINY_NUMBER >> 6; |
| 245 | time_t time_start, time_stop; |
Damien Miller | b089fb5 | 2005-05-26 12:16:18 +1000 | [diff] [blame] | 246 | u_int32_t i; |
| 247 | int ret = 0; |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 248 | |
| 249 | largememory = memory; |
| 250 | |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 251 | if (memory != 0 && |
Damien Miller | 0dc1bef | 2005-07-17 17:22:45 +1000 | [diff] [blame] | 252 | (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) { |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 253 | error("Invalid memory amount (min %ld, max %ld)", |
| 254 | LARGE_MINIMUM, LARGE_MAXIMUM); |
| 255 | return (-1); |
| 256 | } |
| 257 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 258 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 259 | * Set power to the length in bits of the prime to be generated. |
| 260 | * This is changed to 1 less than the desired safe prime moduli p. |
| 261 | */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 262 | if (power > TEST_MAXIMUM) { |
| 263 | error("Too many bits: %u > %lu", power, TEST_MAXIMUM); |
| 264 | return (-1); |
| 265 | } else if (power < TEST_MINIMUM) { |
| 266 | error("Too few bits: %u < %u", power, TEST_MINIMUM); |
| 267 | return (-1); |
| 268 | } |
| 269 | power--; /* decrement before squaring */ |
| 270 | |
| 271 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 272 | * The density of ordinary primes is on the order of 1/bits, so the |
| 273 | * density of safe primes should be about (1/bits)**2. Set test range |
| 274 | * to something well above bits**2 to be reasonably sure (but not |
| 275 | * guaranteed) of catching at least one safe prime. |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 276 | */ |
| 277 | largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER)); |
| 278 | |
| 279 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 280 | * Need idea of how much memory is available. We don't have to use all |
| 281 | * of it. |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 282 | */ |
| 283 | if (largememory > LARGE_MAXIMUM) { |
| 284 | logit("Limited memory: %u MB; limit %lu MB", |
| 285 | largememory, LARGE_MAXIMUM); |
| 286 | largememory = LARGE_MAXIMUM; |
| 287 | } |
| 288 | |
| 289 | if (largewords <= (largememory << SHIFT_MEGAWORD)) { |
| 290 | logit("Increased memory: %u MB; need %u bytes", |
| 291 | largememory, (largewords << SHIFT_BYTE)); |
| 292 | largewords = (largememory << SHIFT_MEGAWORD); |
| 293 | } else if (largememory > 0) { |
| 294 | logit("Decreased memory: %u MB; want %u bytes", |
| 295 | largememory, (largewords << SHIFT_BYTE)); |
| 296 | largewords = (largememory << SHIFT_MEGAWORD); |
| 297 | } |
| 298 | |
Damien Miller | 07d86be | 2006-03-26 14:19:21 +1100 | [diff] [blame] | 299 | TinySieve = xcalloc(tinywords, sizeof(u_int32_t)); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 300 | tinybits = tinywords << SHIFT_WORD; |
| 301 | |
Damien Miller | 07d86be | 2006-03-26 14:19:21 +1100 | [diff] [blame] | 302 | SmallSieve = xcalloc(smallwords, sizeof(u_int32_t)); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 303 | smallbits = smallwords << SHIFT_WORD; |
| 304 | |
| 305 | /* |
| 306 | * dynamically determine available memory |
| 307 | */ |
| 308 | while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL) |
| 309 | largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */ |
| 310 | |
| 311 | largebits = largewords << SHIFT_WORD; |
| 312 | largenumbers = largebits * 2; /* even numbers excluded */ |
| 313 | |
| 314 | /* validation check: count the number of primes tried */ |
| 315 | largetries = 0; |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 316 | if ((q = BN_new()) == NULL) |
| 317 | fatal("BN_new failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 318 | |
| 319 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 320 | * Generate random starting point for subprime search, or use |
| 321 | * specified parameter. |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 322 | */ |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 323 | if ((largebase = BN_new()) == NULL) |
| 324 | fatal("BN_new failed"); |
| 325 | if (start == NULL) { |
| 326 | if (BN_rand(largebase, power, 1, 1) == 0) |
| 327 | fatal("BN_rand failed"); |
| 328 | } else { |
| 329 | if (BN_copy(largebase, start) == NULL) |
| 330 | fatal("BN_copy: failed"); |
| 331 | } |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 332 | |
| 333 | /* ensure odd */ |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 334 | if (BN_set_bit(largebase, 0) == 0) |
| 335 | fatal("BN_set_bit: failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 336 | |
| 337 | time(&time_start); |
| 338 | |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 339 | logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start), |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 340 | largenumbers, power); |
| 341 | debug2("start point: 0x%s", BN_bn2hex(largebase)); |
| 342 | |
| 343 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 344 | * TinySieve |
| 345 | */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 346 | for (i = 0; i < tinybits; i++) { |
| 347 | if (BIT_TEST(TinySieve, i)) |
| 348 | continue; /* 2*i+3 is composite */ |
| 349 | |
| 350 | /* The next tiny prime */ |
| 351 | t = 2 * i + 3; |
| 352 | |
| 353 | /* Mark all multiples of t */ |
| 354 | for (j = i + t; j < tinybits; j += t) |
| 355 | BIT_SET(TinySieve, j); |
| 356 | |
| 357 | sieve_large(t); |
| 358 | } |
| 359 | |
| 360 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 361 | * Start the small block search at the next possible prime. To avoid |
| 362 | * fencepost errors, the last pass is skipped. |
| 363 | */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 364 | for (smallbase = TINY_NUMBER + 3; |
Damien Miller | 0dc1bef | 2005-07-17 17:22:45 +1000 | [diff] [blame] | 365 | smallbase < (SMALL_MAXIMUM - TINY_NUMBER); |
| 366 | smallbase += TINY_NUMBER) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 367 | for (i = 0; i < tinybits; i++) { |
| 368 | if (BIT_TEST(TinySieve, i)) |
| 369 | continue; /* 2*i+3 is composite */ |
| 370 | |
| 371 | /* The next tiny prime */ |
| 372 | t = 2 * i + 3; |
| 373 | r = smallbase % t; |
| 374 | |
| 375 | if (r == 0) { |
| 376 | s = 0; /* t divides into smallbase exactly */ |
| 377 | } else { |
| 378 | /* smallbase+s is first entry divisible by t */ |
| 379 | s = t - r; |
| 380 | } |
| 381 | |
| 382 | /* |
| 383 | * The sieve omits even numbers, so ensure that |
| 384 | * smallbase+s is odd. Then, step through the sieve |
| 385 | * in increments of 2*t |
| 386 | */ |
| 387 | if (s & 1) |
| 388 | s += t; /* Make smallbase+s odd, and s even */ |
| 389 | |
| 390 | /* Mark all multiples of 2*t */ |
| 391 | for (s /= 2; s < smallbits; s += t) |
| 392 | BIT_SET(SmallSieve, s); |
| 393 | } |
| 394 | |
| 395 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 396 | * SmallSieve |
| 397 | */ |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 398 | for (i = 0; i < smallbits; i++) { |
| 399 | if (BIT_TEST(SmallSieve, i)) |
| 400 | continue; /* 2*i+smallbase is composite */ |
| 401 | |
| 402 | /* The next small prime */ |
| 403 | sieve_large((2 * i) + smallbase); |
| 404 | } |
| 405 | |
| 406 | memset(SmallSieve, 0, smallwords << SHIFT_BYTE); |
| 407 | } |
| 408 | |
| 409 | time(&time_stop); |
| 410 | |
| 411 | logit("%.24s Sieved with %u small primes in %ld seconds", |
| 412 | ctime(&time_stop), largetries, (long) (time_stop - time_start)); |
| 413 | |
| 414 | for (j = r = 0; j < largebits; j++) { |
| 415 | if (BIT_TEST(LargeSieve, j)) |
| 416 | continue; /* Definitely composite, skip */ |
| 417 | |
| 418 | debug2("test q = largebase+%u", 2 * j); |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 419 | if (BN_set_word(q, 2 * j) == 0) |
| 420 | fatal("BN_set_word failed"); |
| 421 | if (BN_add(q, q, largebase) == 0) |
| 422 | fatal("BN_add failed"); |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 423 | if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN, |
| 424 | MODULI_TESTS_SIEVE, largetries, |
| 425 | (power - 1) /* MSB */, (0), q) == -1) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 426 | ret = -1; |
| 427 | break; |
| 428 | } |
| 429 | |
| 430 | r++; /* count q */ |
| 431 | } |
| 432 | |
| 433 | time(&time_stop); |
| 434 | |
| 435 | xfree(LargeSieve); |
| 436 | xfree(SmallSieve); |
| 437 | xfree(TinySieve); |
| 438 | |
| 439 | logit("%.24s Found %u candidates", ctime(&time_stop), r); |
| 440 | |
| 441 | return (ret); |
| 442 | } |
| 443 | |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 444 | static void |
| 445 | write_checkpoint(char *cpfile, u_int32_t lineno) |
| 446 | { |
| 447 | FILE *fp; |
Darren Tucker | 9ee09cf | 2011-11-04 10:52:43 +1100 | [diff] [blame^] | 448 | char tmp[MAXPATHLEN]; |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 449 | int r; |
| 450 | |
Darren Tucker | 9ee09cf | 2011-11-04 10:52:43 +1100 | [diff] [blame^] | 451 | r = snprintf(tmp, sizeof(tmp), "%s.XXXXXXXXXX", cpfile); |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 452 | if (r == -1 || r >= MAXPATHLEN) { |
| 453 | logit("write_checkpoint: temp pathname too long"); |
| 454 | return; |
| 455 | } |
Darren Tucker | 9ee09cf | 2011-11-04 10:52:43 +1100 | [diff] [blame^] | 456 | if ((r = mkstemp(tmp)) == -1) { |
| 457 | logit("mkstemp(%s): %s", tmp, strerror(errno)); |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 458 | return; |
| 459 | } |
| 460 | if ((fp = fdopen(r, "w")) == NULL) { |
| 461 | logit("write_checkpoint: fdopen: %s", strerror(errno)); |
| 462 | close(r); |
| 463 | return; |
| 464 | } |
| 465 | if (fprintf(fp, "%lu\n", (unsigned long)lineno) > 0 && fclose(fp) == 0 |
Darren Tucker | 9ee09cf | 2011-11-04 10:52:43 +1100 | [diff] [blame^] | 466 | && rename(tmp, cpfile) == 0) |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 467 | debug3("wrote checkpoint line %lu to '%s'", |
| 468 | (unsigned long)lineno, cpfile); |
| 469 | else |
| 470 | logit("failed to write to checkpoint file '%s': %s", cpfile, |
| 471 | strerror(errno)); |
| 472 | } |
| 473 | |
| 474 | static unsigned long |
| 475 | read_checkpoint(char *cpfile) |
| 476 | { |
| 477 | FILE *fp; |
| 478 | unsigned long lineno = 0; |
| 479 | |
| 480 | if ((fp = fopen(cpfile, "r")) == NULL) |
| 481 | return 0; |
| 482 | if (fscanf(fp, "%lu\n", &lineno) < 1) |
| 483 | logit("Failed to load checkpoint from '%s'", cpfile); |
| 484 | else |
| 485 | logit("Loaded checkpoint from '%s' line %lu", cpfile, lineno); |
| 486 | fclose(fp); |
| 487 | return lineno; |
| 488 | } |
| 489 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 490 | /* |
| 491 | * perform a Miller-Rabin primality test |
| 492 | * on the list of candidates |
| 493 | * (checking both q and p) |
| 494 | * The result is a list of so-call "safe" primes |
| 495 | */ |
| 496 | int |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 497 | prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted, |
| 498 | char *checkpoint_file) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 499 | { |
| 500 | BIGNUM *q, *p, *a; |
| 501 | BN_CTX *ctx; |
| 502 | char *cp, *lp; |
| 503 | u_int32_t count_in = 0, count_out = 0, count_possible = 0; |
| 504 | u_int32_t generator_known, in_tests, in_tries, in_type, in_size; |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 505 | unsigned long last_processed = 0; |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 506 | time_t time_start, time_stop; |
| 507 | int res; |
| 508 | |
Darren Tucker | 770fc01 | 2004-05-13 16:24:32 +1000 | [diff] [blame] | 509 | if (trials < TRIAL_MINIMUM) { |
| 510 | error("Minimum primality trials is %d", TRIAL_MINIMUM); |
| 511 | return (-1); |
| 512 | } |
| 513 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 514 | time(&time_start); |
| 515 | |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 516 | if ((p = BN_new()) == NULL) |
| 517 | fatal("BN_new failed"); |
| 518 | if ((q = BN_new()) == NULL) |
| 519 | fatal("BN_new failed"); |
| 520 | if ((ctx = BN_CTX_new()) == NULL) |
| 521 | fatal("BN_CTX_new failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 522 | |
| 523 | debug2("%.24s Final %u Miller-Rabin trials (%x generator)", |
| 524 | ctime(&time_start), trials, generator_wanted); |
| 525 | |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 526 | if (checkpoint_file != NULL) |
| 527 | last_processed = read_checkpoint(checkpoint_file); |
| 528 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 529 | res = 0; |
| 530 | lp = xmalloc(QLINESIZE + 1); |
Darren Tucker | 90aaed4 | 2007-02-25 20:38:55 +1100 | [diff] [blame] | 531 | while (fgets(lp, QLINESIZE + 1, in) != NULL) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 532 | count_in++; |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 533 | if (checkpoint_file != NULL) { |
| 534 | if (count_in <= last_processed) { |
| 535 | debug3("skipping line %u, before checkpoint", |
| 536 | count_in); |
| 537 | continue; |
| 538 | } |
| 539 | write_checkpoint(checkpoint_file, count_in); |
| 540 | } |
Darren Tucker | 90aaed4 | 2007-02-25 20:38:55 +1100 | [diff] [blame] | 541 | if (strlen(lp) < 14 || *lp == '!' || *lp == '#') { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 542 | debug2("%10u: comment or short line", count_in); |
| 543 | continue; |
| 544 | } |
| 545 | |
| 546 | /* XXX - fragile parser */ |
| 547 | /* time */ |
| 548 | cp = &lp[14]; /* (skip) */ |
| 549 | |
| 550 | /* type */ |
| 551 | in_type = strtoul(cp, &cp, 10); |
| 552 | |
| 553 | /* tests */ |
| 554 | in_tests = strtoul(cp, &cp, 10); |
| 555 | |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 556 | if (in_tests & MODULI_TESTS_COMPOSITE) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 557 | debug2("%10u: known composite", count_in); |
| 558 | continue; |
| 559 | } |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 560 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 561 | /* tries */ |
| 562 | in_tries = strtoul(cp, &cp, 10); |
| 563 | |
| 564 | /* size (most significant bit) */ |
| 565 | in_size = strtoul(cp, &cp, 10); |
| 566 | |
| 567 | /* generator (hex) */ |
| 568 | generator_known = strtoul(cp, &cp, 16); |
| 569 | |
| 570 | /* Skip white space */ |
| 571 | cp += strspn(cp, " "); |
| 572 | |
| 573 | /* modulus (hex) */ |
| 574 | switch (in_type) { |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 575 | case MODULI_TYPE_SOPHIE_GERMAIN: |
Darren Tucker | 47abce4 | 2004-05-02 22:09:00 +1000 | [diff] [blame] | 576 | debug2("%10u: (%u) Sophie-Germain", count_in, in_type); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 577 | a = q; |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 578 | if (BN_hex2bn(&a, cp) == 0) |
| 579 | fatal("BN_hex2bn failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 580 | /* p = 2*q + 1 */ |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 581 | if (BN_lshift(p, q, 1) == 0) |
| 582 | fatal("BN_lshift failed"); |
| 583 | if (BN_add_word(p, 1) == 0) |
| 584 | fatal("BN_add_word failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 585 | in_size += 1; |
| 586 | generator_known = 0; |
| 587 | break; |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 588 | case MODULI_TYPE_UNSTRUCTURED: |
| 589 | case MODULI_TYPE_SAFE: |
| 590 | case MODULI_TYPE_SCHNORR: |
| 591 | case MODULI_TYPE_STRONG: |
| 592 | case MODULI_TYPE_UNKNOWN: |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 593 | debug2("%10u: (%u)", count_in, in_type); |
| 594 | a = p; |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 595 | if (BN_hex2bn(&a, cp) == 0) |
| 596 | fatal("BN_hex2bn failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 597 | /* q = (p-1) / 2 */ |
Darren Tucker | 0bc8557 | 2006-11-07 23:14:41 +1100 | [diff] [blame] | 598 | if (BN_rshift(q, p, 1) == 0) |
| 599 | fatal("BN_rshift failed"); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 600 | break; |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 601 | default: |
| 602 | debug2("Unknown prime type"); |
| 603 | break; |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 604 | } |
| 605 | |
| 606 | /* |
| 607 | * due to earlier inconsistencies in interpretation, check |
| 608 | * the proposed bit size. |
| 609 | */ |
Damien Miller | b089fb5 | 2005-05-26 12:16:18 +1000 | [diff] [blame] | 610 | if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 611 | debug2("%10u: bit size %u mismatch", count_in, in_size); |
| 612 | continue; |
| 613 | } |
| 614 | if (in_size < QSIZE_MINIMUM) { |
| 615 | debug2("%10u: bit size %u too short", count_in, in_size); |
| 616 | continue; |
| 617 | } |
| 618 | |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 619 | if (in_tests & MODULI_TESTS_MILLER_RABIN) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 620 | in_tries += trials; |
| 621 | else |
| 622 | in_tries = trials; |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 623 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 624 | /* |
| 625 | * guess unknown generator |
| 626 | */ |
| 627 | if (generator_known == 0) { |
| 628 | if (BN_mod_word(p, 24) == 11) |
| 629 | generator_known = 2; |
| 630 | else if (BN_mod_word(p, 12) == 5) |
| 631 | generator_known = 3; |
| 632 | else { |
| 633 | u_int32_t r = BN_mod_word(p, 10); |
| 634 | |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 635 | if (r == 3 || r == 7) |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 636 | generator_known = 5; |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 637 | } |
| 638 | } |
| 639 | /* |
| 640 | * skip tests when desired generator doesn't match |
| 641 | */ |
| 642 | if (generator_wanted > 0 && |
| 643 | generator_wanted != generator_known) { |
| 644 | debug2("%10u: generator %d != %d", |
| 645 | count_in, generator_known, generator_wanted); |
| 646 | continue; |
| 647 | } |
| 648 | |
Darren Tucker | 5cd9d44 | 2003-12-10 00:54:38 +1100 | [diff] [blame] | 649 | /* |
| 650 | * Primes with no known generator are useless for DH, so |
| 651 | * skip those. |
| 652 | */ |
| 653 | if (generator_known == 0) { |
| 654 | debug2("%10u: no known generator", count_in); |
| 655 | continue; |
| 656 | } |
| 657 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 658 | count_possible++; |
| 659 | |
| 660 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 661 | * The (1/4)^N performance bound on Miller-Rabin is |
| 662 | * extremely pessimistic, so don't spend a lot of time |
| 663 | * really verifying that q is prime until after we know |
| 664 | * that p is also prime. A single pass will weed out the |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 665 | * vast majority of composite q's. |
| 666 | */ |
Damien Miller | 4499f4c | 2010-11-20 15:15:49 +1100 | [diff] [blame] | 667 | if (BN_is_prime_ex(q, 1, ctx, NULL) <= 0) { |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 668 | debug("%10u: q failed first possible prime test", |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 669 | count_in); |
| 670 | continue; |
| 671 | } |
Damien Miller | 787b2ec | 2003-11-21 23:56:47 +1100 | [diff] [blame] | 672 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 673 | /* |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 674 | * q is possibly prime, so go ahead and really make sure |
| 675 | * that p is prime. If it is, then we can go back and do |
| 676 | * the same for q. If p is composite, chances are that |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 677 | * will show up on the first Rabin-Miller iteration so it |
| 678 | * doesn't hurt to specify a high iteration count. |
| 679 | */ |
Damien Miller | 4499f4c | 2010-11-20 15:15:49 +1100 | [diff] [blame] | 680 | if (!BN_is_prime_ex(p, trials, ctx, NULL)) { |
Darren Tucker | 06930c7 | 2003-12-31 11:34:51 +1100 | [diff] [blame] | 681 | debug("%10u: p is not prime", count_in); |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 682 | continue; |
| 683 | } |
| 684 | debug("%10u: p is almost certainly prime", count_in); |
| 685 | |
| 686 | /* recheck q more rigorously */ |
Damien Miller | 4499f4c | 2010-11-20 15:15:49 +1100 | [diff] [blame] | 687 | if (!BN_is_prime_ex(q, trials - 1, ctx, NULL)) { |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 688 | debug("%10u: q is not prime", count_in); |
| 689 | continue; |
| 690 | } |
| 691 | debug("%10u: q is almost certainly prime", count_in); |
| 692 | |
Damien Miller | 2e9cf49 | 2008-06-29 22:47:04 +1000 | [diff] [blame] | 693 | if (qfileout(out, MODULI_TYPE_SAFE, |
| 694 | in_tests | MODULI_TESTS_MILLER_RABIN, |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 695 | in_tries, in_size, generator_known, p)) { |
| 696 | res = -1; |
| 697 | break; |
| 698 | } |
| 699 | |
| 700 | count_out++; |
| 701 | } |
| 702 | |
| 703 | time(&time_stop); |
| 704 | xfree(lp); |
| 705 | BN_free(p); |
| 706 | BN_free(q); |
| 707 | BN_CTX_free(ctx); |
| 708 | |
Damien Miller | 390d056 | 2011-10-18 16:05:19 +1100 | [diff] [blame] | 709 | if (checkpoint_file != NULL) |
| 710 | unlink(checkpoint_file); |
| 711 | |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 712 | logit("%.24s Found %u safe primes of %u candidates in %ld seconds", |
Damien Miller | a8e06ce | 2003-11-21 23:48:55 +1100 | [diff] [blame] | 713 | ctime(&time_stop), count_out, count_possible, |
Darren Tucker | b2f9d41 | 2003-08-02 23:51:38 +1000 | [diff] [blame] | 714 | (long) (time_stop - time_start)); |
| 715 | |
| 716 | return (res); |
| 717 | } |