Annotation of src/usr.bin/ssh/moduli.c, Revision 1.17
1.17 ! stevesk 1: /* $OpenBSD: moduli.c,v 1.16 2006/07/26 13:57:17 stevesk Exp $ */
1.1 djm 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"
1.14 stevesk 41:
42: #include <sys/types.h>
43:
44: #include <openssl/bn.h>
45:
1.17 ! stevesk 46: #include <stdio.h>
1.16 stevesk 47: #include <stdlib.h>
1.15 stevesk 48: #include <string.h>
1.14 stevesk 49: #include <time.h>
50:
1.1 djm 51: #include "xmalloc.h"
52: #include "log.h"
53:
54: /*
55: * File output defines
56: */
57:
58: /* need line long enough for largest moduli plus headers */
1.9 deraadt 59: #define QLINESIZE (100+8192)
1.1 djm 60:
61: /* Type: decimal.
62: * Specifies the internal structure of the prime modulus.
63: */
1.9 deraadt 64: #define QTYPE_UNKNOWN (0)
65: #define QTYPE_UNSTRUCTURED (1)
66: #define QTYPE_SAFE (2)
1.10 dtucker 67: #define QTYPE_SCHNORR (3)
1.9 deraadt 68: #define QTYPE_SOPHIE_GERMAIN (4)
69: #define QTYPE_STRONG (5)
1.1 djm 70:
71: /* Tests: decimal (bit field).
72: * Specifies the methods used in checking for primality.
73: * Usually, more than one test is used.
74: */
1.9 deraadt 75: #define QTEST_UNTESTED (0x00)
76: #define QTEST_COMPOSITE (0x01)
77: #define QTEST_SIEVE (0x02)
78: #define QTEST_MILLER_RABIN (0x04)
79: #define QTEST_JACOBI (0x08)
80: #define QTEST_ELLIPTIC (0x10)
1.1 djm 81:
1.5 djm 82: /*
83: * Size: decimal.
1.1 djm 84: * Specifies the number of the most significant bit (0 to M).
1.5 djm 85: * WARNING: internally, usually 1 to N.
1.1 djm 86: */
1.9 deraadt 87: #define QSIZE_MINIMUM (511)
1.1 djm 88:
89: /*
90: * Prime sieving defines
91: */
92:
93: /* Constant: assuming 8 bit bytes and 32 bit words */
1.9 deraadt 94: #define SHIFT_BIT (3)
95: #define SHIFT_BYTE (2)
96: #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
97: #define SHIFT_MEGABYTE (20)
98: #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
1.1 djm 99:
100: /*
1.7 djm 101: * Using virtual memory can cause thrashing. This should be the largest
102: * number that is supported without a large amount of disk activity --
103: * that would increase the run time from hours to days or weeks!
104: */
1.9 deraadt 105: #define LARGE_MINIMUM (8UL) /* megabytes */
1.7 djm 106:
107: /*
108: * Do not increase this number beyond the unsigned integer bit size.
109: * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
110: */
1.9 deraadt 111: #define LARGE_MAXIMUM (127UL) /* megabytes */
1.7 djm 112:
113: /*
1.1 djm 114: * Constant: when used with 32-bit integers, the largest sieve prime
115: * has to be less than 2**32.
116: */
1.9 deraadt 117: #define SMALL_MAXIMUM (0xffffffffUL)
1.1 djm 118:
119: /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
1.9 deraadt 120: #define TINY_NUMBER (1UL<<16)
1.1 djm 121:
122: /* Ensure enough bit space for testing 2*q. */
1.12 djm 123: #define TEST_MAXIMUM (1UL<<16)
124: #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
125: /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
126: #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
1.1 djm 127:
128: /* bit operations on 32-bit words */
1.12 djm 129: #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
130: #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
131: #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
1.1 djm 132:
133: /*
134: * Prime testing defines
135: */
136:
1.7 djm 137: /* Minimum number of primality tests to perform */
1.12 djm 138: #define TRIAL_MINIMUM (4)
1.7 djm 139:
1.1 djm 140: /*
141: * Sieving data (XXX - move to struct)
142: */
143:
144: /* sieve 2**16 */
145: static u_int32_t *TinySieve, tinybits;
146:
147: /* sieve 2**30 in 2**16 parts */
148: static u_int32_t *SmallSieve, smallbits, smallbase;
149:
150: /* sieve relative to the initial value */
151: static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
152: static u_int32_t largebits, largememory; /* megabytes */
153: static BIGNUM *largebase;
154:
1.11 avsm 155: int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
1.8 markus 156: int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
1.1 djm 157:
158: /*
159: * print moduli out in consistent form,
160: */
161: static int
162: qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
163: u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
164: {
165: struct tm *gtm;
166: time_t time_now;
167: int res;
168:
169: time(&time_now);
170: gtm = gmtime(&time_now);
1.2 djm 171:
1.1 djm 172: res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
173: gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
174: gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
175: otype, otests, otries, osize, ogenerator);
176:
177: if (res < 0)
178: return (-1);
179:
180: if (BN_print_fp(ofile, omodulus) < 1)
181: return (-1);
182:
183: res = fprintf(ofile, "\n");
184: fflush(ofile);
185:
186: return (res > 0 ? 0 : -1);
187: }
188:
189:
190: /*
191: ** Sieve p's and q's with small factors
192: */
193: static void
194: sieve_large(u_int32_t s)
195: {
196: u_int32_t r, u;
197:
1.5 djm 198: debug3("sieve_large %u", s);
1.1 djm 199: largetries++;
200: /* r = largebase mod s */
201: r = BN_mod_word(largebase, s);
202: if (r == 0)
203: u = 0; /* s divides into largebase exactly */
204: else
205: u = s - r; /* largebase+u is first entry divisible by s */
206:
207: if (u < largebits * 2) {
208: /*
209: * The sieve omits p's and q's divisible by 2, so ensure that
210: * largebase+u is odd. Then, step through the sieve in
211: * increments of 2*s
212: */
213: if (u & 0x1)
214: u += s; /* Make largebase+u odd, and u even */
215:
216: /* Mark all multiples of 2*s */
217: for (u /= 2; u < largebits; u += s)
218: BIT_SET(LargeSieve, u);
219: }
220:
221: /* r = p mod s */
222: r = (2 * r + 1) % s;
223: if (r == 0)
224: u = 0; /* s divides p exactly */
225: else
226: u = s - r; /* p+u is first entry divisible by s */
227:
228: if (u < largebits * 4) {
229: /*
230: * The sieve omits p's divisible by 4, so ensure that
231: * largebase+u is not. Then, step through the sieve in
232: * increments of 4*s
233: */
234: while (u & 0x3) {
235: if (SMALL_MAXIMUM - u < s)
236: return;
237: u += s;
238: }
239:
240: /* Mark all multiples of 4*s */
241: for (u /= 4; u < largebits; u += s)
242: BIT_SET(LargeSieve, u);
243: }
244: }
245:
246: /*
1.6 djm 247: * list candidates for Sophie-Germain primes (where q = (p-1)/2)
1.1 djm 248: * to standard output.
249: * The list is checked against small known primes (less than 2**30).
250: */
251: int
1.11 avsm 252: gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
1.1 djm 253: {
254: BIGNUM *q;
255: u_int32_t j, r, s, t;
256: u_int32_t smallwords = TINY_NUMBER >> 6;
257: u_int32_t tinywords = TINY_NUMBER >> 6;
258: time_t time_start, time_stop;
1.11 avsm 259: u_int32_t i;
260: int ret = 0;
1.1 djm 261:
262: largememory = memory;
263:
1.7 djm 264: if (memory != 0 &&
1.12 djm 265: (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
1.7 djm 266: error("Invalid memory amount (min %ld, max %ld)",
267: LARGE_MINIMUM, LARGE_MAXIMUM);
268: return (-1);
269: }
270:
1.1 djm 271: /*
1.2 djm 272: * Set power to the length in bits of the prime to be generated.
273: * This is changed to 1 less than the desired safe prime moduli p.
274: */
1.1 djm 275: if (power > TEST_MAXIMUM) {
276: error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
277: return (-1);
278: } else if (power < TEST_MINIMUM) {
279: error("Too few bits: %u < %u", power, TEST_MINIMUM);
280: return (-1);
281: }
282: power--; /* decrement before squaring */
283:
284: /*
1.2 djm 285: * The density of ordinary primes is on the order of 1/bits, so the
286: * density of safe primes should be about (1/bits)**2. Set test range
287: * to something well above bits**2 to be reasonably sure (but not
288: * guaranteed) of catching at least one safe prime.
1.1 djm 289: */
290: largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
291:
292: /*
1.2 djm 293: * Need idea of how much memory is available. We don't have to use all
294: * of it.
1.1 djm 295: */
296: if (largememory > LARGE_MAXIMUM) {
297: logit("Limited memory: %u MB; limit %lu MB",
298: largememory, LARGE_MAXIMUM);
299: largememory = LARGE_MAXIMUM;
300: }
301:
302: if (largewords <= (largememory << SHIFT_MEGAWORD)) {
303: logit("Increased memory: %u MB; need %u bytes",
304: largememory, (largewords << SHIFT_BYTE));
305: largewords = (largememory << SHIFT_MEGAWORD);
306: } else if (largememory > 0) {
307: logit("Decreased memory: %u MB; want %u bytes",
308: largememory, (largewords << SHIFT_BYTE));
309: largewords = (largememory << SHIFT_MEGAWORD);
310: }
311:
1.13 djm 312: TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
1.1 djm 313: tinybits = tinywords << SHIFT_WORD;
314:
1.13 djm 315: SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
1.1 djm 316: smallbits = smallwords << SHIFT_WORD;
317:
318: /*
319: * dynamically determine available memory
320: */
321: while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
322: largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
323:
324: largebits = largewords << SHIFT_WORD;
325: largenumbers = largebits * 2; /* even numbers excluded */
326:
327: /* validation check: count the number of primes tried */
328: largetries = 0;
329: q = BN_new();
330:
331: /*
1.2 djm 332: * Generate random starting point for subprime search, or use
333: * specified parameter.
1.1 djm 334: */
335: largebase = BN_new();
336: if (start == NULL)
337: BN_rand(largebase, power, 1, 1);
338: else
339: BN_copy(largebase, start);
340:
341: /* ensure odd */
342: BN_set_bit(largebase, 0);
343:
344: time(&time_start);
345:
1.2 djm 346: logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
1.1 djm 347: largenumbers, power);
348: debug2("start point: 0x%s", BN_bn2hex(largebase));
349:
350: /*
1.2 djm 351: * TinySieve
352: */
1.1 djm 353: for (i = 0; i < tinybits; i++) {
354: if (BIT_TEST(TinySieve, i))
355: continue; /* 2*i+3 is composite */
356:
357: /* The next tiny prime */
358: t = 2 * i + 3;
359:
360: /* Mark all multiples of t */
361: for (j = i + t; j < tinybits; j += t)
362: BIT_SET(TinySieve, j);
363:
364: sieve_large(t);
365: }
366:
367: /*
1.2 djm 368: * Start the small block search at the next possible prime. To avoid
369: * fencepost errors, the last pass is skipped.
370: */
1.1 djm 371: for (smallbase = TINY_NUMBER + 3;
1.12 djm 372: smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
373: smallbase += TINY_NUMBER) {
1.1 djm 374: for (i = 0; i < tinybits; i++) {
375: if (BIT_TEST(TinySieve, i))
376: continue; /* 2*i+3 is composite */
377:
378: /* The next tiny prime */
379: t = 2 * i + 3;
380: r = smallbase % t;
381:
382: if (r == 0) {
383: s = 0; /* t divides into smallbase exactly */
384: } else {
385: /* smallbase+s is first entry divisible by t */
386: s = t - r;
387: }
388:
389: /*
390: * The sieve omits even numbers, so ensure that
391: * smallbase+s is odd. Then, step through the sieve
392: * in increments of 2*t
393: */
394: if (s & 1)
395: s += t; /* Make smallbase+s odd, and s even */
396:
397: /* Mark all multiples of 2*t */
398: for (s /= 2; s < smallbits; s += t)
399: BIT_SET(SmallSieve, s);
400: }
401:
402: /*
1.2 djm 403: * SmallSieve
404: */
1.1 djm 405: for (i = 0; i < smallbits; i++) {
406: if (BIT_TEST(SmallSieve, i))
407: continue; /* 2*i+smallbase is composite */
408:
409: /* The next small prime */
410: sieve_large((2 * i) + smallbase);
411: }
412:
413: memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
414: }
415:
416: time(&time_stop);
417:
418: logit("%.24s Sieved with %u small primes in %ld seconds",
419: ctime(&time_stop), largetries, (long) (time_stop - time_start));
420:
421: for (j = r = 0; j < largebits; j++) {
422: if (BIT_TEST(LargeSieve, j))
423: continue; /* Definitely composite, skip */
424:
425: debug2("test q = largebase+%u", 2 * j);
426: BN_set_word(q, 2 * j);
427: BN_add(q, q, largebase);
1.6 djm 428: if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
1.1 djm 429: largetries, (power - 1) /* MSB */, (0), q) == -1) {
430: ret = -1;
431: break;
432: }
433:
434: r++; /* count q */
435: }
436:
437: time(&time_stop);
438:
439: xfree(LargeSieve);
440: xfree(SmallSieve);
441: xfree(TinySieve);
442:
443: logit("%.24s Found %u candidates", ctime(&time_stop), r);
444:
445: return (ret);
446: }
447:
448: /*
449: * perform a Miller-Rabin primality test
450: * on the list of candidates
451: * (checking both q and p)
452: * The result is a list of so-call "safe" primes
453: */
454: int
1.7 djm 455: prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
1.1 djm 456: {
457: BIGNUM *q, *p, *a;
458: BN_CTX *ctx;
459: char *cp, *lp;
460: u_int32_t count_in = 0, count_out = 0, count_possible = 0;
461: u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
462: time_t time_start, time_stop;
463: int res;
1.7 djm 464:
465: if (trials < TRIAL_MINIMUM) {
466: error("Minimum primality trials is %d", TRIAL_MINIMUM);
467: return (-1);
468: }
1.1 djm 469:
470: time(&time_start);
471:
472: p = BN_new();
473: q = BN_new();
474: ctx = BN_CTX_new();
475:
476: debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
477: ctime(&time_start), trials, generator_wanted);
478:
479: res = 0;
480: lp = xmalloc(QLINESIZE + 1);
481: while (fgets(lp, QLINESIZE, in) != NULL) {
482: int ll = strlen(lp);
483:
484: count_in++;
485: if (ll < 14 || *lp == '!' || *lp == '#') {
486: debug2("%10u: comment or short line", count_in);
487: continue;
488: }
489:
490: /* XXX - fragile parser */
491: /* time */
492: cp = &lp[14]; /* (skip) */
493:
494: /* type */
495: in_type = strtoul(cp, &cp, 10);
496:
497: /* tests */
498: in_tests = strtoul(cp, &cp, 10);
499:
500: if (in_tests & QTEST_COMPOSITE) {
501: debug2("%10u: known composite", count_in);
502: continue;
503: }
1.5 djm 504:
1.1 djm 505: /* tries */
506: in_tries = strtoul(cp, &cp, 10);
507:
508: /* size (most significant bit) */
509: in_size = strtoul(cp, &cp, 10);
510:
511: /* generator (hex) */
512: generator_known = strtoul(cp, &cp, 16);
513:
514: /* Skip white space */
515: cp += strspn(cp, " ");
516:
517: /* modulus (hex) */
518: switch (in_type) {
1.6 djm 519: case QTYPE_SOPHIE_GERMAIN:
520: debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
1.1 djm 521: a = q;
522: BN_hex2bn(&a, cp);
523: /* p = 2*q + 1 */
524: BN_lshift(p, q, 1);
525: BN_add_word(p, 1);
526: in_size += 1;
527: generator_known = 0;
528: break;
1.5 djm 529: case QTYPE_UNSTRUCTURED:
530: case QTYPE_SAFE:
1.10 dtucker 531: case QTYPE_SCHNORR:
1.5 djm 532: case QTYPE_STRONG:
533: case QTYPE_UNKNOWN:
1.1 djm 534: debug2("%10u: (%u)", count_in, in_type);
535: a = p;
536: BN_hex2bn(&a, cp);
537: /* q = (p-1) / 2 */
538: BN_rshift(q, p, 1);
539: break;
1.5 djm 540: default:
541: debug2("Unknown prime type");
542: break;
1.1 djm 543: }
544:
545: /*
546: * due to earlier inconsistencies in interpretation, check
547: * the proposed bit size.
548: */
1.11 avsm 549: if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
1.1 djm 550: debug2("%10u: bit size %u mismatch", count_in, in_size);
551: continue;
552: }
553: if (in_size < QSIZE_MINIMUM) {
554: debug2("%10u: bit size %u too short", count_in, in_size);
555: continue;
556: }
557:
558: if (in_tests & QTEST_MILLER_RABIN)
559: in_tries += trials;
560: else
561: in_tries = trials;
1.5 djm 562:
1.1 djm 563: /*
564: * guess unknown generator
565: */
566: if (generator_known == 0) {
567: if (BN_mod_word(p, 24) == 11)
568: generator_known = 2;
569: else if (BN_mod_word(p, 12) == 5)
570: generator_known = 3;
571: else {
572: u_int32_t r = BN_mod_word(p, 10);
573:
1.5 djm 574: if (r == 3 || r == 7)
1.1 djm 575: generator_known = 5;
576: }
577: }
578: /*
579: * skip tests when desired generator doesn't match
580: */
581: if (generator_wanted > 0 &&
582: generator_wanted != generator_known) {
583: debug2("%10u: generator %d != %d",
584: count_in, generator_known, generator_wanted);
1.4 dtucker 585: continue;
586: }
587:
588: /*
589: * Primes with no known generator are useless for DH, so
590: * skip those.
591: */
592: if (generator_known == 0) {
593: debug2("%10u: no known generator", count_in);
1.1 djm 594: continue;
595: }
596:
597: count_possible++;
598:
599: /*
1.2 djm 600: * The (1/4)^N performance bound on Miller-Rabin is
601: * extremely pessimistic, so don't spend a lot of time
602: * really verifying that q is prime until after we know
603: * that p is also prime. A single pass will weed out the
1.1 djm 604: * vast majority of composite q's.
605: */
606: if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
1.5 djm 607: debug("%10u: q failed first possible prime test",
1.1 djm 608: count_in);
609: continue;
610: }
1.2 djm 611:
1.1 djm 612: /*
1.2 djm 613: * q is possibly prime, so go ahead and really make sure
614: * that p is prime. If it is, then we can go back and do
615: * the same for q. If p is composite, chances are that
1.1 djm 616: * will show up on the first Rabin-Miller iteration so it
617: * doesn't hurt to specify a high iteration count.
618: */
619: if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
1.5 djm 620: debug("%10u: p is not prime", count_in);
1.1 djm 621: continue;
622: }
623: debug("%10u: p is almost certainly prime", count_in);
624:
625: /* recheck q more rigorously */
626: if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
627: debug("%10u: q is not prime", count_in);
628: continue;
629: }
630: debug("%10u: q is almost certainly prime", count_in);
631:
1.2 djm 632: if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
1.1 djm 633: in_tries, in_size, generator_known, p)) {
634: res = -1;
635: break;
636: }
637:
638: count_out++;
639: }
640:
641: time(&time_stop);
642: xfree(lp);
643: BN_free(p);
644: BN_free(q);
645: BN_CTX_free(ctx);
646:
647: logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
1.2 djm 648: ctime(&time_stop), count_out, count_possible,
1.1 djm 649: (long) (time_stop - time_start));
650:
651: return (res);
652: }
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