Annotation of src/usr.bin/ssh/moduli.c, Revision 1.15
1.15 ! stevesk 1: /* $OpenBSD: moduli.c,v 1.14 2006/07/22 19:08:54 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:
1.15 ! stevesk 46: #include <string.h>
1.14 stevesk 47: #include <time.h>
48:
1.1 djm 49: #include "xmalloc.h"
50: #include "log.h"
51:
52: /*
53: * File output defines
54: */
55:
56: /* need line long enough for largest moduli plus headers */
1.9 deraadt 57: #define QLINESIZE (100+8192)
1.1 djm 58:
59: /* Type: decimal.
60: * Specifies the internal structure of the prime modulus.
61: */
1.9 deraadt 62: #define QTYPE_UNKNOWN (0)
63: #define QTYPE_UNSTRUCTURED (1)
64: #define QTYPE_SAFE (2)
1.10 dtucker 65: #define QTYPE_SCHNORR (3)
1.9 deraadt 66: #define QTYPE_SOPHIE_GERMAIN (4)
67: #define QTYPE_STRONG (5)
1.1 djm 68:
69: /* Tests: decimal (bit field).
70: * Specifies the methods used in checking for primality.
71: * Usually, more than one test is used.
72: */
1.9 deraadt 73: #define QTEST_UNTESTED (0x00)
74: #define QTEST_COMPOSITE (0x01)
75: #define QTEST_SIEVE (0x02)
76: #define QTEST_MILLER_RABIN (0x04)
77: #define QTEST_JACOBI (0x08)
78: #define QTEST_ELLIPTIC (0x10)
1.1 djm 79:
1.5 djm 80: /*
81: * Size: decimal.
1.1 djm 82: * Specifies the number of the most significant bit (0 to M).
1.5 djm 83: * WARNING: internally, usually 1 to N.
1.1 djm 84: */
1.9 deraadt 85: #define QSIZE_MINIMUM (511)
1.1 djm 86:
87: /*
88: * Prime sieving defines
89: */
90:
91: /* Constant: assuming 8 bit bytes and 32 bit words */
1.9 deraadt 92: #define SHIFT_BIT (3)
93: #define SHIFT_BYTE (2)
94: #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
95: #define SHIFT_MEGABYTE (20)
96: #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
1.1 djm 97:
98: /*
1.7 djm 99: * Using virtual memory can cause thrashing. This should be the largest
100: * number that is supported without a large amount of disk activity --
101: * that would increase the run time from hours to days or weeks!
102: */
1.9 deraadt 103: #define LARGE_MINIMUM (8UL) /* megabytes */
1.7 djm 104:
105: /*
106: * Do not increase this number beyond the unsigned integer bit size.
107: * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
108: */
1.9 deraadt 109: #define LARGE_MAXIMUM (127UL) /* megabytes */
1.7 djm 110:
111: /*
1.1 djm 112: * Constant: when used with 32-bit integers, the largest sieve prime
113: * has to be less than 2**32.
114: */
1.9 deraadt 115: #define SMALL_MAXIMUM (0xffffffffUL)
1.1 djm 116:
117: /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
1.9 deraadt 118: #define TINY_NUMBER (1UL<<16)
1.1 djm 119:
120: /* Ensure enough bit space for testing 2*q. */
1.12 djm 121: #define TEST_MAXIMUM (1UL<<16)
122: #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
123: /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
124: #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
1.1 djm 125:
126: /* bit operations on 32-bit words */
1.12 djm 127: #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
128: #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
129: #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
1.1 djm 130:
131: /*
132: * Prime testing defines
133: */
134:
1.7 djm 135: /* Minimum number of primality tests to perform */
1.12 djm 136: #define TRIAL_MINIMUM (4)
1.7 djm 137:
1.1 djm 138: /*
139: * Sieving data (XXX - move to struct)
140: */
141:
142: /* sieve 2**16 */
143: static u_int32_t *TinySieve, tinybits;
144:
145: /* sieve 2**30 in 2**16 parts */
146: static u_int32_t *SmallSieve, smallbits, smallbase;
147:
148: /* sieve relative to the initial value */
149: static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
150: static u_int32_t largebits, largememory; /* megabytes */
151: static BIGNUM *largebase;
152:
1.11 avsm 153: int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
1.8 markus 154: int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
1.1 djm 155:
156: /*
157: * print moduli out in consistent form,
158: */
159: static int
160: qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
161: u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
162: {
163: struct tm *gtm;
164: time_t time_now;
165: int res;
166:
167: time(&time_now);
168: gtm = gmtime(&time_now);
1.2 djm 169:
1.1 djm 170: res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
171: gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
172: gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
173: otype, otests, otries, osize, ogenerator);
174:
175: if (res < 0)
176: return (-1);
177:
178: if (BN_print_fp(ofile, omodulus) < 1)
179: return (-1);
180:
181: res = fprintf(ofile, "\n");
182: fflush(ofile);
183:
184: return (res > 0 ? 0 : -1);
185: }
186:
187:
188: /*
189: ** Sieve p's and q's with small factors
190: */
191: static void
192: sieve_large(u_int32_t s)
193: {
194: u_int32_t r, u;
195:
1.5 djm 196: debug3("sieve_large %u", s);
1.1 djm 197: largetries++;
198: /* r = largebase mod s */
199: r = BN_mod_word(largebase, s);
200: if (r == 0)
201: u = 0; /* s divides into largebase exactly */
202: else
203: u = s - r; /* largebase+u is first entry divisible by s */
204:
205: if (u < largebits * 2) {
206: /*
207: * The sieve omits p's and q's divisible by 2, so ensure that
208: * largebase+u is odd. Then, step through the sieve in
209: * increments of 2*s
210: */
211: if (u & 0x1)
212: u += s; /* Make largebase+u odd, and u even */
213:
214: /* Mark all multiples of 2*s */
215: for (u /= 2; u < largebits; u += s)
216: BIT_SET(LargeSieve, u);
217: }
218:
219: /* r = p mod s */
220: r = (2 * r + 1) % s;
221: if (r == 0)
222: u = 0; /* s divides p exactly */
223: else
224: u = s - r; /* p+u is first entry divisible by s */
225:
226: if (u < largebits * 4) {
227: /*
228: * The sieve omits p's divisible by 4, so ensure that
229: * largebase+u is not. Then, step through the sieve in
230: * increments of 4*s
231: */
232: while (u & 0x3) {
233: if (SMALL_MAXIMUM - u < s)
234: return;
235: u += s;
236: }
237:
238: /* Mark all multiples of 4*s */
239: for (u /= 4; u < largebits; u += s)
240: BIT_SET(LargeSieve, u);
241: }
242: }
243:
244: /*
1.6 djm 245: * list candidates for Sophie-Germain primes (where q = (p-1)/2)
1.1 djm 246: * to standard output.
247: * The list is checked against small known primes (less than 2**30).
248: */
249: int
1.11 avsm 250: gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
1.1 djm 251: {
252: BIGNUM *q;
253: u_int32_t j, r, s, t;
254: u_int32_t smallwords = TINY_NUMBER >> 6;
255: u_int32_t tinywords = TINY_NUMBER >> 6;
256: time_t time_start, time_stop;
1.11 avsm 257: u_int32_t i;
258: int ret = 0;
1.1 djm 259:
260: largememory = memory;
261:
1.7 djm 262: if (memory != 0 &&
1.12 djm 263: (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
1.7 djm 264: error("Invalid memory amount (min %ld, max %ld)",
265: LARGE_MINIMUM, LARGE_MAXIMUM);
266: return (-1);
267: }
268:
1.1 djm 269: /*
1.2 djm 270: * Set power to the length in bits of the prime to be generated.
271: * This is changed to 1 less than the desired safe prime moduli p.
272: */
1.1 djm 273: if (power > TEST_MAXIMUM) {
274: error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
275: return (-1);
276: } else if (power < TEST_MINIMUM) {
277: error("Too few bits: %u < %u", power, TEST_MINIMUM);
278: return (-1);
279: }
280: power--; /* decrement before squaring */
281:
282: /*
1.2 djm 283: * The density of ordinary primes is on the order of 1/bits, so the
284: * density of safe primes should be about (1/bits)**2. Set test range
285: * to something well above bits**2 to be reasonably sure (but not
286: * guaranteed) of catching at least one safe prime.
1.1 djm 287: */
288: largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
289:
290: /*
1.2 djm 291: * Need idea of how much memory is available. We don't have to use all
292: * of it.
1.1 djm 293: */
294: if (largememory > LARGE_MAXIMUM) {
295: logit("Limited memory: %u MB; limit %lu MB",
296: largememory, LARGE_MAXIMUM);
297: largememory = LARGE_MAXIMUM;
298: }
299:
300: if (largewords <= (largememory << SHIFT_MEGAWORD)) {
301: logit("Increased memory: %u MB; need %u bytes",
302: largememory, (largewords << SHIFT_BYTE));
303: largewords = (largememory << SHIFT_MEGAWORD);
304: } else if (largememory > 0) {
305: logit("Decreased memory: %u MB; want %u bytes",
306: largememory, (largewords << SHIFT_BYTE));
307: largewords = (largememory << SHIFT_MEGAWORD);
308: }
309:
1.13 djm 310: TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
1.1 djm 311: tinybits = tinywords << SHIFT_WORD;
312:
1.13 djm 313: SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
1.1 djm 314: smallbits = smallwords << SHIFT_WORD;
315:
316: /*
317: * dynamically determine available memory
318: */
319: while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
320: largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
321:
322: largebits = largewords << SHIFT_WORD;
323: largenumbers = largebits * 2; /* even numbers excluded */
324:
325: /* validation check: count the number of primes tried */
326: largetries = 0;
327: q = BN_new();
328:
329: /*
1.2 djm 330: * Generate random starting point for subprime search, or use
331: * specified parameter.
1.1 djm 332: */
333: largebase = BN_new();
334: if (start == NULL)
335: BN_rand(largebase, power, 1, 1);
336: else
337: BN_copy(largebase, start);
338:
339: /* ensure odd */
340: BN_set_bit(largebase, 0);
341:
342: time(&time_start);
343:
1.2 djm 344: logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
1.1 djm 345: largenumbers, power);
346: debug2("start point: 0x%s", BN_bn2hex(largebase));
347:
348: /*
1.2 djm 349: * TinySieve
350: */
1.1 djm 351: for (i = 0; i < tinybits; i++) {
352: if (BIT_TEST(TinySieve, i))
353: continue; /* 2*i+3 is composite */
354:
355: /* The next tiny prime */
356: t = 2 * i + 3;
357:
358: /* Mark all multiples of t */
359: for (j = i + t; j < tinybits; j += t)
360: BIT_SET(TinySieve, j);
361:
362: sieve_large(t);
363: }
364:
365: /*
1.2 djm 366: * Start the small block search at the next possible prime. To avoid
367: * fencepost errors, the last pass is skipped.
368: */
1.1 djm 369: for (smallbase = TINY_NUMBER + 3;
1.12 djm 370: smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
371: smallbase += TINY_NUMBER) {
1.1 djm 372: for (i = 0; i < tinybits; i++) {
373: if (BIT_TEST(TinySieve, i))
374: continue; /* 2*i+3 is composite */
375:
376: /* The next tiny prime */
377: t = 2 * i + 3;
378: r = smallbase % t;
379:
380: if (r == 0) {
381: s = 0; /* t divides into smallbase exactly */
382: } else {
383: /* smallbase+s is first entry divisible by t */
384: s = t - r;
385: }
386:
387: /*
388: * The sieve omits even numbers, so ensure that
389: * smallbase+s is odd. Then, step through the sieve
390: * in increments of 2*t
391: */
392: if (s & 1)
393: s += t; /* Make smallbase+s odd, and s even */
394:
395: /* Mark all multiples of 2*t */
396: for (s /= 2; s < smallbits; s += t)
397: BIT_SET(SmallSieve, s);
398: }
399:
400: /*
1.2 djm 401: * SmallSieve
402: */
1.1 djm 403: for (i = 0; i < smallbits; i++) {
404: if (BIT_TEST(SmallSieve, i))
405: continue; /* 2*i+smallbase is composite */
406:
407: /* The next small prime */
408: sieve_large((2 * i) + smallbase);
409: }
410:
411: memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
412: }
413:
414: time(&time_stop);
415:
416: logit("%.24s Sieved with %u small primes in %ld seconds",
417: ctime(&time_stop), largetries, (long) (time_stop - time_start));
418:
419: for (j = r = 0; j < largebits; j++) {
420: if (BIT_TEST(LargeSieve, j))
421: continue; /* Definitely composite, skip */
422:
423: debug2("test q = largebase+%u", 2 * j);
424: BN_set_word(q, 2 * j);
425: BN_add(q, q, largebase);
1.6 djm 426: if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
1.1 djm 427: largetries, (power - 1) /* MSB */, (0), q) == -1) {
428: ret = -1;
429: break;
430: }
431:
432: r++; /* count q */
433: }
434:
435: time(&time_stop);
436:
437: xfree(LargeSieve);
438: xfree(SmallSieve);
439: xfree(TinySieve);
440:
441: logit("%.24s Found %u candidates", ctime(&time_stop), r);
442:
443: return (ret);
444: }
445:
446: /*
447: * perform a Miller-Rabin primality test
448: * on the list of candidates
449: * (checking both q and p)
450: * The result is a list of so-call "safe" primes
451: */
452: int
1.7 djm 453: prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
1.1 djm 454: {
455: BIGNUM *q, *p, *a;
456: BN_CTX *ctx;
457: char *cp, *lp;
458: u_int32_t count_in = 0, count_out = 0, count_possible = 0;
459: u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
460: time_t time_start, time_stop;
461: int res;
1.7 djm 462:
463: if (trials < TRIAL_MINIMUM) {
464: error("Minimum primality trials is %d", TRIAL_MINIMUM);
465: return (-1);
466: }
1.1 djm 467:
468: time(&time_start);
469:
470: p = BN_new();
471: q = BN_new();
472: ctx = BN_CTX_new();
473:
474: debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
475: ctime(&time_start), trials, generator_wanted);
476:
477: res = 0;
478: lp = xmalloc(QLINESIZE + 1);
479: while (fgets(lp, QLINESIZE, in) != NULL) {
480: int ll = strlen(lp);
481:
482: count_in++;
483: if (ll < 14 || *lp == '!' || *lp == '#') {
484: debug2("%10u: comment or short line", count_in);
485: continue;
486: }
487:
488: /* XXX - fragile parser */
489: /* time */
490: cp = &lp[14]; /* (skip) */
491:
492: /* type */
493: in_type = strtoul(cp, &cp, 10);
494:
495: /* tests */
496: in_tests = strtoul(cp, &cp, 10);
497:
498: if (in_tests & QTEST_COMPOSITE) {
499: debug2("%10u: known composite", count_in);
500: continue;
501: }
1.5 djm 502:
1.1 djm 503: /* tries */
504: in_tries = strtoul(cp, &cp, 10);
505:
506: /* size (most significant bit) */
507: in_size = strtoul(cp, &cp, 10);
508:
509: /* generator (hex) */
510: generator_known = strtoul(cp, &cp, 16);
511:
512: /* Skip white space */
513: cp += strspn(cp, " ");
514:
515: /* modulus (hex) */
516: switch (in_type) {
1.6 djm 517: case QTYPE_SOPHIE_GERMAIN:
518: debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
1.1 djm 519: a = q;
520: BN_hex2bn(&a, cp);
521: /* p = 2*q + 1 */
522: BN_lshift(p, q, 1);
523: BN_add_word(p, 1);
524: in_size += 1;
525: generator_known = 0;
526: break;
1.5 djm 527: case QTYPE_UNSTRUCTURED:
528: case QTYPE_SAFE:
1.10 dtucker 529: case QTYPE_SCHNORR:
1.5 djm 530: case QTYPE_STRONG:
531: case QTYPE_UNKNOWN:
1.1 djm 532: debug2("%10u: (%u)", count_in, in_type);
533: a = p;
534: BN_hex2bn(&a, cp);
535: /* q = (p-1) / 2 */
536: BN_rshift(q, p, 1);
537: break;
1.5 djm 538: default:
539: debug2("Unknown prime type");
540: break;
1.1 djm 541: }
542:
543: /*
544: * due to earlier inconsistencies in interpretation, check
545: * the proposed bit size.
546: */
1.11 avsm 547: if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
1.1 djm 548: debug2("%10u: bit size %u mismatch", count_in, in_size);
549: continue;
550: }
551: if (in_size < QSIZE_MINIMUM) {
552: debug2("%10u: bit size %u too short", count_in, in_size);
553: continue;
554: }
555:
556: if (in_tests & QTEST_MILLER_RABIN)
557: in_tries += trials;
558: else
559: in_tries = trials;
1.5 djm 560:
1.1 djm 561: /*
562: * guess unknown generator
563: */
564: if (generator_known == 0) {
565: if (BN_mod_word(p, 24) == 11)
566: generator_known = 2;
567: else if (BN_mod_word(p, 12) == 5)
568: generator_known = 3;
569: else {
570: u_int32_t r = BN_mod_word(p, 10);
571:
1.5 djm 572: if (r == 3 || r == 7)
1.1 djm 573: generator_known = 5;
574: }
575: }
576: /*
577: * skip tests when desired generator doesn't match
578: */
579: if (generator_wanted > 0 &&
580: generator_wanted != generator_known) {
581: debug2("%10u: generator %d != %d",
582: count_in, generator_known, generator_wanted);
1.4 dtucker 583: continue;
584: }
585:
586: /*
587: * Primes with no known generator are useless for DH, so
588: * skip those.
589: */
590: if (generator_known == 0) {
591: debug2("%10u: no known generator", count_in);
1.1 djm 592: continue;
593: }
594:
595: count_possible++;
596:
597: /*
1.2 djm 598: * The (1/4)^N performance bound on Miller-Rabin is
599: * extremely pessimistic, so don't spend a lot of time
600: * really verifying that q is prime until after we know
601: * that p is also prime. A single pass will weed out the
1.1 djm 602: * vast majority of composite q's.
603: */
604: if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
1.5 djm 605: debug("%10u: q failed first possible prime test",
1.1 djm 606: count_in);
607: continue;
608: }
1.2 djm 609:
1.1 djm 610: /*
1.2 djm 611: * q is possibly prime, so go ahead and really make sure
612: * that p is prime. If it is, then we can go back and do
613: * the same for q. If p is composite, chances are that
1.1 djm 614: * will show up on the first Rabin-Miller iteration so it
615: * doesn't hurt to specify a high iteration count.
616: */
617: if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
1.5 djm 618: debug("%10u: p is not prime", count_in);
1.1 djm 619: continue;
620: }
621: debug("%10u: p is almost certainly prime", count_in);
622:
623: /* recheck q more rigorously */
624: if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
625: debug("%10u: q is not prime", count_in);
626: continue;
627: }
628: debug("%10u: q is almost certainly prime", count_in);
629:
1.2 djm 630: if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
1.1 djm 631: in_tries, in_size, generator_known, p)) {
632: res = -1;
633: break;
634: }
635:
636: count_out++;
637: }
638:
639: time(&time_stop);
640: xfree(lp);
641: BN_free(p);
642: BN_free(q);
643: BN_CTX_free(ctx);
644:
645: logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
1.2 djm 646: ctime(&time_stop), count_out, count_possible,
1.1 djm 647: (long) (time_stop - time_start));
648:
649: return (res);
650: }
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