Annotation of src/usr.bin/ssh/moduli.c, Revision 1.16
1.16 ! stevesk 1: /* $OpenBSD: moduli.c,v 1.15 2006/07/22 20:48:23 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:
1.16 ! stevesk 46: #include <stdlib.h>
1.15 stevesk 47: #include <string.h>
1.14 stevesk 48: #include <time.h>
49:
1.1 djm 50: #include "xmalloc.h"
51: #include "log.h"
52:
53: /*
54: * File output defines
55: */
56:
57: /* need line long enough for largest moduli plus headers */
1.9 deraadt 58: #define QLINESIZE (100+8192)
1.1 djm 59:
60: /* Type: decimal.
61: * Specifies the internal structure of the prime modulus.
62: */
1.9 deraadt 63: #define QTYPE_UNKNOWN (0)
64: #define QTYPE_UNSTRUCTURED (1)
65: #define QTYPE_SAFE (2)
1.10 dtucker 66: #define QTYPE_SCHNORR (3)
1.9 deraadt 67: #define QTYPE_SOPHIE_GERMAIN (4)
68: #define QTYPE_STRONG (5)
1.1 djm 69:
70: /* Tests: decimal (bit field).
71: * Specifies the methods used in checking for primality.
72: * Usually, more than one test is used.
73: */
1.9 deraadt 74: #define QTEST_UNTESTED (0x00)
75: #define QTEST_COMPOSITE (0x01)
76: #define QTEST_SIEVE (0x02)
77: #define QTEST_MILLER_RABIN (0x04)
78: #define QTEST_JACOBI (0x08)
79: #define QTEST_ELLIPTIC (0x10)
1.1 djm 80:
1.5 djm 81: /*
82: * Size: decimal.
1.1 djm 83: * Specifies the number of the most significant bit (0 to M).
1.5 djm 84: * WARNING: internally, usually 1 to N.
1.1 djm 85: */
1.9 deraadt 86: #define QSIZE_MINIMUM (511)
1.1 djm 87:
88: /*
89: * Prime sieving defines
90: */
91:
92: /* Constant: assuming 8 bit bytes and 32 bit words */
1.9 deraadt 93: #define SHIFT_BIT (3)
94: #define SHIFT_BYTE (2)
95: #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
96: #define SHIFT_MEGABYTE (20)
97: #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
1.1 djm 98:
99: /*
1.7 djm 100: * Using virtual memory can cause thrashing. This should be the largest
101: * number that is supported without a large amount of disk activity --
102: * that would increase the run time from hours to days or weeks!
103: */
1.9 deraadt 104: #define LARGE_MINIMUM (8UL) /* megabytes */
1.7 djm 105:
106: /*
107: * Do not increase this number beyond the unsigned integer bit size.
108: * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
109: */
1.9 deraadt 110: #define LARGE_MAXIMUM (127UL) /* megabytes */
1.7 djm 111:
112: /*
1.1 djm 113: * Constant: when used with 32-bit integers, the largest sieve prime
114: * has to be less than 2**32.
115: */
1.9 deraadt 116: #define SMALL_MAXIMUM (0xffffffffUL)
1.1 djm 117:
118: /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
1.9 deraadt 119: #define TINY_NUMBER (1UL<<16)
1.1 djm 120:
121: /* Ensure enough bit space for testing 2*q. */
1.12 djm 122: #define TEST_MAXIMUM (1UL<<16)
123: #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
124: /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
125: #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
1.1 djm 126:
127: /* bit operations on 32-bit words */
1.12 djm 128: #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
129: #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
130: #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
1.1 djm 131:
132: /*
133: * Prime testing defines
134: */
135:
1.7 djm 136: /* Minimum number of primality tests to perform */
1.12 djm 137: #define TRIAL_MINIMUM (4)
1.7 djm 138:
1.1 djm 139: /*
140: * Sieving data (XXX - move to struct)
141: */
142:
143: /* sieve 2**16 */
144: static u_int32_t *TinySieve, tinybits;
145:
146: /* sieve 2**30 in 2**16 parts */
147: static u_int32_t *SmallSieve, smallbits, smallbase;
148:
149: /* sieve relative to the initial value */
150: static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
151: static u_int32_t largebits, largememory; /* megabytes */
152: static BIGNUM *largebase;
153:
1.11 avsm 154: int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
1.8 markus 155: int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
1.1 djm 156:
157: /*
158: * print moduli out in consistent form,
159: */
160: static int
161: qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
162: u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
163: {
164: struct tm *gtm;
165: time_t time_now;
166: int res;
167:
168: time(&time_now);
169: gtm = gmtime(&time_now);
1.2 djm 170:
1.1 djm 171: res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
172: gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
173: gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
174: otype, otests, otries, osize, ogenerator);
175:
176: if (res < 0)
177: return (-1);
178:
179: if (BN_print_fp(ofile, omodulus) < 1)
180: return (-1);
181:
182: res = fprintf(ofile, "\n");
183: fflush(ofile);
184:
185: return (res > 0 ? 0 : -1);
186: }
187:
188:
189: /*
190: ** Sieve p's and q's with small factors
191: */
192: static void
193: sieve_large(u_int32_t s)
194: {
195: u_int32_t r, u;
196:
1.5 djm 197: debug3("sieve_large %u", s);
1.1 djm 198: largetries++;
199: /* r = largebase mod s */
200: r = BN_mod_word(largebase, s);
201: if (r == 0)
202: u = 0; /* s divides into largebase exactly */
203: else
204: u = s - r; /* largebase+u is first entry divisible by s */
205:
206: if (u < largebits * 2) {
207: /*
208: * The sieve omits p's and q's divisible by 2, so ensure that
209: * largebase+u is odd. Then, step through the sieve in
210: * increments of 2*s
211: */
212: if (u & 0x1)
213: u += s; /* Make largebase+u odd, and u even */
214:
215: /* Mark all multiples of 2*s */
216: for (u /= 2; u < largebits; u += s)
217: BIT_SET(LargeSieve, u);
218: }
219:
220: /* r = p mod s */
221: r = (2 * r + 1) % s;
222: if (r == 0)
223: u = 0; /* s divides p exactly */
224: else
225: u = s - r; /* p+u is first entry divisible by s */
226:
227: if (u < largebits * 4) {
228: /*
229: * The sieve omits p's divisible by 4, so ensure that
230: * largebase+u is not. Then, step through the sieve in
231: * increments of 4*s
232: */
233: while (u & 0x3) {
234: if (SMALL_MAXIMUM - u < s)
235: return;
236: u += s;
237: }
238:
239: /* Mark all multiples of 4*s */
240: for (u /= 4; u < largebits; u += s)
241: BIT_SET(LargeSieve, u);
242: }
243: }
244:
245: /*
1.6 djm 246: * list candidates for Sophie-Germain primes (where q = (p-1)/2)
1.1 djm 247: * to standard output.
248: * The list is checked against small known primes (less than 2**30).
249: */
250: int
1.11 avsm 251: gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
1.1 djm 252: {
253: BIGNUM *q;
254: u_int32_t j, r, s, t;
255: u_int32_t smallwords = TINY_NUMBER >> 6;
256: u_int32_t tinywords = TINY_NUMBER >> 6;
257: time_t time_start, time_stop;
1.11 avsm 258: u_int32_t i;
259: int ret = 0;
1.1 djm 260:
261: largememory = memory;
262:
1.7 djm 263: if (memory != 0 &&
1.12 djm 264: (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
1.7 djm 265: error("Invalid memory amount (min %ld, max %ld)",
266: LARGE_MINIMUM, LARGE_MAXIMUM);
267: return (-1);
268: }
269:
1.1 djm 270: /*
1.2 djm 271: * Set power to the length in bits of the prime to be generated.
272: * This is changed to 1 less than the desired safe prime moduli p.
273: */
1.1 djm 274: if (power > TEST_MAXIMUM) {
275: error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
276: return (-1);
277: } else if (power < TEST_MINIMUM) {
278: error("Too few bits: %u < %u", power, TEST_MINIMUM);
279: return (-1);
280: }
281: power--; /* decrement before squaring */
282:
283: /*
1.2 djm 284: * The density of ordinary primes is on the order of 1/bits, so the
285: * density of safe primes should be about (1/bits)**2. Set test range
286: * to something well above bits**2 to be reasonably sure (but not
287: * guaranteed) of catching at least one safe prime.
1.1 djm 288: */
289: largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
290:
291: /*
1.2 djm 292: * Need idea of how much memory is available. We don't have to use all
293: * of it.
1.1 djm 294: */
295: if (largememory > LARGE_MAXIMUM) {
296: logit("Limited memory: %u MB; limit %lu MB",
297: largememory, LARGE_MAXIMUM);
298: largememory = LARGE_MAXIMUM;
299: }
300:
301: if (largewords <= (largememory << SHIFT_MEGAWORD)) {
302: logit("Increased memory: %u MB; need %u bytes",
303: largememory, (largewords << SHIFT_BYTE));
304: largewords = (largememory << SHIFT_MEGAWORD);
305: } else if (largememory > 0) {
306: logit("Decreased memory: %u MB; want %u bytes",
307: largememory, (largewords << SHIFT_BYTE));
308: largewords = (largememory << SHIFT_MEGAWORD);
309: }
310:
1.13 djm 311: TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
1.1 djm 312: tinybits = tinywords << SHIFT_WORD;
313:
1.13 djm 314: SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
1.1 djm 315: smallbits = smallwords << SHIFT_WORD;
316:
317: /*
318: * dynamically determine available memory
319: */
320: while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
321: largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
322:
323: largebits = largewords << SHIFT_WORD;
324: largenumbers = largebits * 2; /* even numbers excluded */
325:
326: /* validation check: count the number of primes tried */
327: largetries = 0;
328: q = BN_new();
329:
330: /*
1.2 djm 331: * Generate random starting point for subprime search, or use
332: * specified parameter.
1.1 djm 333: */
334: largebase = BN_new();
335: if (start == NULL)
336: BN_rand(largebase, power, 1, 1);
337: else
338: BN_copy(largebase, start);
339:
340: /* ensure odd */
341: BN_set_bit(largebase, 0);
342:
343: time(&time_start);
344:
1.2 djm 345: logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
1.1 djm 346: largenumbers, power);
347: debug2("start point: 0x%s", BN_bn2hex(largebase));
348:
349: /*
1.2 djm 350: * TinySieve
351: */
1.1 djm 352: for (i = 0; i < tinybits; i++) {
353: if (BIT_TEST(TinySieve, i))
354: continue; /* 2*i+3 is composite */
355:
356: /* The next tiny prime */
357: t = 2 * i + 3;
358:
359: /* Mark all multiples of t */
360: for (j = i + t; j < tinybits; j += t)
361: BIT_SET(TinySieve, j);
362:
363: sieve_large(t);
364: }
365:
366: /*
1.2 djm 367: * Start the small block search at the next possible prime. To avoid
368: * fencepost errors, the last pass is skipped.
369: */
1.1 djm 370: for (smallbase = TINY_NUMBER + 3;
1.12 djm 371: smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
372: smallbase += TINY_NUMBER) {
1.1 djm 373: for (i = 0; i < tinybits; i++) {
374: if (BIT_TEST(TinySieve, i))
375: continue; /* 2*i+3 is composite */
376:
377: /* The next tiny prime */
378: t = 2 * i + 3;
379: r = smallbase % t;
380:
381: if (r == 0) {
382: s = 0; /* t divides into smallbase exactly */
383: } else {
384: /* smallbase+s is first entry divisible by t */
385: s = t - r;
386: }
387:
388: /*
389: * The sieve omits even numbers, so ensure that
390: * smallbase+s is odd. Then, step through the sieve
391: * in increments of 2*t
392: */
393: if (s & 1)
394: s += t; /* Make smallbase+s odd, and s even */
395:
396: /* Mark all multiples of 2*t */
397: for (s /= 2; s < smallbits; s += t)
398: BIT_SET(SmallSieve, s);
399: }
400:
401: /*
1.2 djm 402: * SmallSieve
403: */
1.1 djm 404: for (i = 0; i < smallbits; i++) {
405: if (BIT_TEST(SmallSieve, i))
406: continue; /* 2*i+smallbase is composite */
407:
408: /* The next small prime */
409: sieve_large((2 * i) + smallbase);
410: }
411:
412: memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
413: }
414:
415: time(&time_stop);
416:
417: logit("%.24s Sieved with %u small primes in %ld seconds",
418: ctime(&time_stop), largetries, (long) (time_stop - time_start));
419:
420: for (j = r = 0; j < largebits; j++) {
421: if (BIT_TEST(LargeSieve, j))
422: continue; /* Definitely composite, skip */
423:
424: debug2("test q = largebase+%u", 2 * j);
425: BN_set_word(q, 2 * j);
426: BN_add(q, q, largebase);
1.6 djm 427: if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
1.1 djm 428: largetries, (power - 1) /* MSB */, (0), q) == -1) {
429: ret = -1;
430: break;
431: }
432:
433: r++; /* count q */
434: }
435:
436: time(&time_stop);
437:
438: xfree(LargeSieve);
439: xfree(SmallSieve);
440: xfree(TinySieve);
441:
442: logit("%.24s Found %u candidates", ctime(&time_stop), r);
443:
444: return (ret);
445: }
446:
447: /*
448: * perform a Miller-Rabin primality test
449: * on the list of candidates
450: * (checking both q and p)
451: * The result is a list of so-call "safe" primes
452: */
453: int
1.7 djm 454: prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
1.1 djm 455: {
456: BIGNUM *q, *p, *a;
457: BN_CTX *ctx;
458: char *cp, *lp;
459: u_int32_t count_in = 0, count_out = 0, count_possible = 0;
460: u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
461: time_t time_start, time_stop;
462: int res;
1.7 djm 463:
464: if (trials < TRIAL_MINIMUM) {
465: error("Minimum primality trials is %d", TRIAL_MINIMUM);
466: return (-1);
467: }
1.1 djm 468:
469: time(&time_start);
470:
471: p = BN_new();
472: q = BN_new();
473: ctx = BN_CTX_new();
474:
475: debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
476: ctime(&time_start), trials, generator_wanted);
477:
478: res = 0;
479: lp = xmalloc(QLINESIZE + 1);
480: while (fgets(lp, QLINESIZE, in) != NULL) {
481: int ll = strlen(lp);
482:
483: count_in++;
484: if (ll < 14 || *lp == '!' || *lp == '#') {
485: debug2("%10u: comment or short line", count_in);
486: continue;
487: }
488:
489: /* XXX - fragile parser */
490: /* time */
491: cp = &lp[14]; /* (skip) */
492:
493: /* type */
494: in_type = strtoul(cp, &cp, 10);
495:
496: /* tests */
497: in_tests = strtoul(cp, &cp, 10);
498:
499: if (in_tests & QTEST_COMPOSITE) {
500: debug2("%10u: known composite", count_in);
501: continue;
502: }
1.5 djm 503:
1.1 djm 504: /* tries */
505: in_tries = strtoul(cp, &cp, 10);
506:
507: /* size (most significant bit) */
508: in_size = strtoul(cp, &cp, 10);
509:
510: /* generator (hex) */
511: generator_known = strtoul(cp, &cp, 16);
512:
513: /* Skip white space */
514: cp += strspn(cp, " ");
515:
516: /* modulus (hex) */
517: switch (in_type) {
1.6 djm 518: case QTYPE_SOPHIE_GERMAIN:
519: debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
1.1 djm 520: a = q;
521: BN_hex2bn(&a, cp);
522: /* p = 2*q + 1 */
523: BN_lshift(p, q, 1);
524: BN_add_word(p, 1);
525: in_size += 1;
526: generator_known = 0;
527: break;
1.5 djm 528: case QTYPE_UNSTRUCTURED:
529: case QTYPE_SAFE:
1.10 dtucker 530: case QTYPE_SCHNORR:
1.5 djm 531: case QTYPE_STRONG:
532: case QTYPE_UNKNOWN:
1.1 djm 533: debug2("%10u: (%u)", count_in, in_type);
534: a = p;
535: BN_hex2bn(&a, cp);
536: /* q = (p-1) / 2 */
537: BN_rshift(q, p, 1);
538: break;
1.5 djm 539: default:
540: debug2("Unknown prime type");
541: break;
1.1 djm 542: }
543:
544: /*
545: * due to earlier inconsistencies in interpretation, check
546: * the proposed bit size.
547: */
1.11 avsm 548: if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
1.1 djm 549: debug2("%10u: bit size %u mismatch", count_in, in_size);
550: continue;
551: }
552: if (in_size < QSIZE_MINIMUM) {
553: debug2("%10u: bit size %u too short", count_in, in_size);
554: continue;
555: }
556:
557: if (in_tests & QTEST_MILLER_RABIN)
558: in_tries += trials;
559: else
560: in_tries = trials;
1.5 djm 561:
1.1 djm 562: /*
563: * guess unknown generator
564: */
565: if (generator_known == 0) {
566: if (BN_mod_word(p, 24) == 11)
567: generator_known = 2;
568: else if (BN_mod_word(p, 12) == 5)
569: generator_known = 3;
570: else {
571: u_int32_t r = BN_mod_word(p, 10);
572:
1.5 djm 573: if (r == 3 || r == 7)
1.1 djm 574: generator_known = 5;
575: }
576: }
577: /*
578: * skip tests when desired generator doesn't match
579: */
580: if (generator_wanted > 0 &&
581: generator_wanted != generator_known) {
582: debug2("%10u: generator %d != %d",
583: count_in, generator_known, generator_wanted);
1.4 dtucker 584: continue;
585: }
586:
587: /*
588: * Primes with no known generator are useless for DH, so
589: * skip those.
590: */
591: if (generator_known == 0) {
592: debug2("%10u: no known generator", count_in);
1.1 djm 593: continue;
594: }
595:
596: count_possible++;
597:
598: /*
1.2 djm 599: * The (1/4)^N performance bound on Miller-Rabin is
600: * extremely pessimistic, so don't spend a lot of time
601: * really verifying that q is prime until after we know
602: * that p is also prime. A single pass will weed out the
1.1 djm 603: * vast majority of composite q's.
604: */
605: if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
1.5 djm 606: debug("%10u: q failed first possible prime test",
1.1 djm 607: count_in);
608: continue;
609: }
1.2 djm 610:
1.1 djm 611: /*
1.2 djm 612: * q is possibly prime, so go ahead and really make sure
613: * that p is prime. If it is, then we can go back and do
614: * the same for q. If p is composite, chances are that
1.1 djm 615: * will show up on the first Rabin-Miller iteration so it
616: * doesn't hurt to specify a high iteration count.
617: */
618: if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
1.5 djm 619: debug("%10u: p is not prime", count_in);
1.1 djm 620: continue;
621: }
622: debug("%10u: p is almost certainly prime", count_in);
623:
624: /* recheck q more rigorously */
625: if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
626: debug("%10u: q is not prime", count_in);
627: continue;
628: }
629: debug("%10u: q is almost certainly prime", count_in);
630:
1.2 djm 631: if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
1.1 djm 632: in_tries, in_size, generator_known, p)) {
633: res = -1;
634: break;
635: }
636:
637: count_out++;
638: }
639:
640: time(&time_stop);
641: xfree(lp);
642: BN_free(p);
643: BN_free(q);
644: BN_CTX_free(ctx);
645:
646: logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
1.2 djm 647: ctime(&time_stop), count_out, count_possible,
1.1 djm 648: (long) (time_stop - time_start));
649:
650: return (res);
651: }
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