Annotation of src/usr.bin/ssh/moduli.c, Revision 1.11
1.11 ! avsm 1: /* $OpenBSD: moduli.c,v 1.10 2005/01/17 03:25:46 dtucker 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"
41: #include "xmalloc.h"
42: #include "log.h"
43:
44: #include <openssl/bn.h>
45:
46: /*
47: * File output defines
48: */
49:
50: /* need line long enough for largest moduli plus headers */
1.9 deraadt 51: #define QLINESIZE (100+8192)
1.1 djm 52:
53: /* Type: decimal.
54: * Specifies the internal structure of the prime modulus.
55: */
1.9 deraadt 56: #define QTYPE_UNKNOWN (0)
57: #define QTYPE_UNSTRUCTURED (1)
58: #define QTYPE_SAFE (2)
1.10 dtucker 59: #define QTYPE_SCHNORR (3)
1.9 deraadt 60: #define QTYPE_SOPHIE_GERMAIN (4)
61: #define QTYPE_STRONG (5)
1.1 djm 62:
63: /* Tests: decimal (bit field).
64: * Specifies the methods used in checking for primality.
65: * Usually, more than one test is used.
66: */
1.9 deraadt 67: #define QTEST_UNTESTED (0x00)
68: #define QTEST_COMPOSITE (0x01)
69: #define QTEST_SIEVE (0x02)
70: #define QTEST_MILLER_RABIN (0x04)
71: #define QTEST_JACOBI (0x08)
72: #define QTEST_ELLIPTIC (0x10)
1.1 djm 73:
1.5 djm 74: /*
75: * Size: decimal.
1.1 djm 76: * Specifies the number of the most significant bit (0 to M).
1.5 djm 77: * WARNING: internally, usually 1 to N.
1.1 djm 78: */
1.9 deraadt 79: #define QSIZE_MINIMUM (511)
1.1 djm 80:
81: /*
82: * Prime sieving defines
83: */
84:
85: /* Constant: assuming 8 bit bytes and 32 bit words */
1.9 deraadt 86: #define SHIFT_BIT (3)
87: #define SHIFT_BYTE (2)
88: #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
89: #define SHIFT_MEGABYTE (20)
90: #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
1.1 djm 91:
92: /*
1.7 djm 93: * Using virtual memory can cause thrashing. This should be the largest
94: * number that is supported without a large amount of disk activity --
95: * that would increase the run time from hours to days or weeks!
96: */
1.9 deraadt 97: #define LARGE_MINIMUM (8UL) /* megabytes */
1.7 djm 98:
99: /*
100: * Do not increase this number beyond the unsigned integer bit size.
101: * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
102: */
1.9 deraadt 103: #define LARGE_MAXIMUM (127UL) /* megabytes */
1.7 djm 104:
105: /*
1.1 djm 106: * Constant: when used with 32-bit integers, the largest sieve prime
107: * has to be less than 2**32.
108: */
1.9 deraadt 109: #define SMALL_MAXIMUM (0xffffffffUL)
1.1 djm 110:
111: /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
1.9 deraadt 112: #define TINY_NUMBER (1UL<<16)
1.1 djm 113:
114: /* Ensure enough bit space for testing 2*q. */
115: #define TEST_MAXIMUM (1UL<<16)
116: #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
117: /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
118: #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
119:
120: /* bit operations on 32-bit words */
121: #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
122: #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
123: #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
124:
125: /*
126: * Prime testing defines
127: */
128:
1.7 djm 129: /* Minimum number of primality tests to perform */
130: #define TRIAL_MINIMUM (4)
131:
1.1 djm 132: /*
133: * Sieving data (XXX - move to struct)
134: */
135:
136: /* sieve 2**16 */
137: static u_int32_t *TinySieve, tinybits;
138:
139: /* sieve 2**30 in 2**16 parts */
140: static u_int32_t *SmallSieve, smallbits, smallbase;
141:
142: /* sieve relative to the initial value */
143: static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
144: static u_int32_t largebits, largememory; /* megabytes */
145: static BIGNUM *largebase;
146:
1.11 ! avsm 147: int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
1.8 markus 148: int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
1.1 djm 149:
150: /*
151: * print moduli out in consistent form,
152: */
153: static int
154: qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
155: u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
156: {
157: struct tm *gtm;
158: time_t time_now;
159: int res;
160:
161: time(&time_now);
162: gtm = gmtime(&time_now);
1.2 djm 163:
1.1 djm 164: res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
165: gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
166: gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
167: otype, otests, otries, osize, ogenerator);
168:
169: if (res < 0)
170: return (-1);
171:
172: if (BN_print_fp(ofile, omodulus) < 1)
173: return (-1);
174:
175: res = fprintf(ofile, "\n");
176: fflush(ofile);
177:
178: return (res > 0 ? 0 : -1);
179: }
180:
181:
182: /*
183: ** Sieve p's and q's with small factors
184: */
185: static void
186: sieve_large(u_int32_t s)
187: {
188: u_int32_t r, u;
189:
1.5 djm 190: debug3("sieve_large %u", s);
1.1 djm 191: largetries++;
192: /* r = largebase mod s */
193: r = BN_mod_word(largebase, s);
194: if (r == 0)
195: u = 0; /* s divides into largebase exactly */
196: else
197: u = s - r; /* largebase+u is first entry divisible by s */
198:
199: if (u < largebits * 2) {
200: /*
201: * The sieve omits p's and q's divisible by 2, so ensure that
202: * largebase+u is odd. Then, step through the sieve in
203: * increments of 2*s
204: */
205: if (u & 0x1)
206: u += s; /* Make largebase+u odd, and u even */
207:
208: /* Mark all multiples of 2*s */
209: for (u /= 2; u < largebits; u += s)
210: BIT_SET(LargeSieve, u);
211: }
212:
213: /* r = p mod s */
214: r = (2 * r + 1) % s;
215: if (r == 0)
216: u = 0; /* s divides p exactly */
217: else
218: u = s - r; /* p+u is first entry divisible by s */
219:
220: if (u < largebits * 4) {
221: /*
222: * The sieve omits p's divisible by 4, so ensure that
223: * largebase+u is not. Then, step through the sieve in
224: * increments of 4*s
225: */
226: while (u & 0x3) {
227: if (SMALL_MAXIMUM - u < s)
228: return;
229: u += s;
230: }
231:
232: /* Mark all multiples of 4*s */
233: for (u /= 4; u < largebits; u += s)
234: BIT_SET(LargeSieve, u);
235: }
236: }
237:
238: /*
1.6 djm 239: * list candidates for Sophie-Germain primes (where q = (p-1)/2)
1.1 djm 240: * to standard output.
241: * The list is checked against small known primes (less than 2**30).
242: */
243: int
1.11 ! avsm 244: gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
1.1 djm 245: {
246: BIGNUM *q;
247: u_int32_t j, r, s, t;
248: u_int32_t smallwords = TINY_NUMBER >> 6;
249: u_int32_t tinywords = TINY_NUMBER >> 6;
250: time_t time_start, time_stop;
1.11 ! avsm 251: u_int32_t i;
! 252: int ret = 0;
1.1 djm 253:
254: largememory = memory;
255:
1.7 djm 256: if (memory != 0 &&
257: (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
258: error("Invalid memory amount (min %ld, max %ld)",
259: LARGE_MINIMUM, LARGE_MAXIMUM);
260: return (-1);
261: }
262:
1.1 djm 263: /*
1.2 djm 264: * Set power to the length in bits of the prime to be generated.
265: * This is changed to 1 less than the desired safe prime moduli p.
266: */
1.1 djm 267: if (power > TEST_MAXIMUM) {
268: error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
269: return (-1);
270: } else if (power < TEST_MINIMUM) {
271: error("Too few bits: %u < %u", power, TEST_MINIMUM);
272: return (-1);
273: }
274: power--; /* decrement before squaring */
275:
276: /*
1.2 djm 277: * The density of ordinary primes is on the order of 1/bits, so the
278: * density of safe primes should be about (1/bits)**2. Set test range
279: * to something well above bits**2 to be reasonably sure (but not
280: * guaranteed) of catching at least one safe prime.
1.1 djm 281: */
282: largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
283:
284: /*
1.2 djm 285: * Need idea of how much memory is available. We don't have to use all
286: * of it.
1.1 djm 287: */
288: if (largememory > LARGE_MAXIMUM) {
289: logit("Limited memory: %u MB; limit %lu MB",
290: largememory, LARGE_MAXIMUM);
291: largememory = LARGE_MAXIMUM;
292: }
293:
294: if (largewords <= (largememory << SHIFT_MEGAWORD)) {
295: logit("Increased memory: %u MB; need %u bytes",
296: largememory, (largewords << SHIFT_BYTE));
297: largewords = (largememory << SHIFT_MEGAWORD);
298: } else if (largememory > 0) {
299: logit("Decreased memory: %u MB; want %u bytes",
300: largememory, (largewords << SHIFT_BYTE));
301: largewords = (largememory << SHIFT_MEGAWORD);
302: }
303:
304: TinySieve = calloc(tinywords, sizeof(u_int32_t));
305: if (TinySieve == NULL) {
306: error("Insufficient memory for tiny sieve: need %u bytes",
307: tinywords << SHIFT_BYTE);
308: exit(1);
309: }
310: tinybits = tinywords << SHIFT_WORD;
311:
312: SmallSieve = calloc(smallwords, sizeof(u_int32_t));
313: if (SmallSieve == NULL) {
314: error("Insufficient memory for small sieve: need %u bytes",
315: smallwords << SHIFT_BYTE);
316: xfree(TinySieve);
317: exit(1);
318: }
319: smallbits = smallwords << SHIFT_WORD;
320:
321: /*
322: * dynamically determine available memory
323: */
324: while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
325: largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
326:
327: largebits = largewords << SHIFT_WORD;
328: largenumbers = largebits * 2; /* even numbers excluded */
329:
330: /* validation check: count the number of primes tried */
331: largetries = 0;
332: q = BN_new();
333:
334: /*
1.2 djm 335: * Generate random starting point for subprime search, or use
336: * specified parameter.
1.1 djm 337: */
338: largebase = BN_new();
339: if (start == NULL)
340: BN_rand(largebase, power, 1, 1);
341: else
342: BN_copy(largebase, start);
343:
344: /* ensure odd */
345: BN_set_bit(largebase, 0);
346:
347: time(&time_start);
348:
1.2 djm 349: logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
1.1 djm 350: largenumbers, power);
351: debug2("start point: 0x%s", BN_bn2hex(largebase));
352:
353: /*
1.2 djm 354: * TinySieve
355: */
1.1 djm 356: for (i = 0; i < tinybits; i++) {
357: if (BIT_TEST(TinySieve, i))
358: continue; /* 2*i+3 is composite */
359:
360: /* The next tiny prime */
361: t = 2 * i + 3;
362:
363: /* Mark all multiples of t */
364: for (j = i + t; j < tinybits; j += t)
365: BIT_SET(TinySieve, j);
366:
367: sieve_large(t);
368: }
369:
370: /*
1.2 djm 371: * Start the small block search at the next possible prime. To avoid
372: * fencepost errors, the last pass is skipped.
373: */
1.1 djm 374: for (smallbase = TINY_NUMBER + 3;
375: smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
376: smallbase += TINY_NUMBER) {
377: for (i = 0; i < tinybits; i++) {
378: if (BIT_TEST(TinySieve, i))
379: continue; /* 2*i+3 is composite */
380:
381: /* The next tiny prime */
382: t = 2 * i + 3;
383: r = smallbase % t;
384:
385: if (r == 0) {
386: s = 0; /* t divides into smallbase exactly */
387: } else {
388: /* smallbase+s is first entry divisible by t */
389: s = t - r;
390: }
391:
392: /*
393: * The sieve omits even numbers, so ensure that
394: * smallbase+s is odd. Then, step through the sieve
395: * in increments of 2*t
396: */
397: if (s & 1)
398: s += t; /* Make smallbase+s odd, and s even */
399:
400: /* Mark all multiples of 2*t */
401: for (s /= 2; s < smallbits; s += t)
402: BIT_SET(SmallSieve, s);
403: }
404:
405: /*
1.2 djm 406: * SmallSieve
407: */
1.1 djm 408: for (i = 0; i < smallbits; i++) {
409: if (BIT_TEST(SmallSieve, i))
410: continue; /* 2*i+smallbase is composite */
411:
412: /* The next small prime */
413: sieve_large((2 * i) + smallbase);
414: }
415:
416: memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
417: }
418:
419: time(&time_stop);
420:
421: logit("%.24s Sieved with %u small primes in %ld seconds",
422: ctime(&time_stop), largetries, (long) (time_stop - time_start));
423:
424: for (j = r = 0; j < largebits; j++) {
425: if (BIT_TEST(LargeSieve, j))
426: continue; /* Definitely composite, skip */
427:
428: debug2("test q = largebase+%u", 2 * j);
429: BN_set_word(q, 2 * j);
430: BN_add(q, q, largebase);
1.6 djm 431: if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
1.1 djm 432: largetries, (power - 1) /* MSB */, (0), q) == -1) {
433: ret = -1;
434: break;
435: }
436:
437: r++; /* count q */
438: }
439:
440: time(&time_stop);
441:
442: xfree(LargeSieve);
443: xfree(SmallSieve);
444: xfree(TinySieve);
445:
446: logit("%.24s Found %u candidates", ctime(&time_stop), r);
447:
448: return (ret);
449: }
450:
451: /*
452: * perform a Miller-Rabin primality test
453: * on the list of candidates
454: * (checking both q and p)
455: * The result is a list of so-call "safe" primes
456: */
457: int
1.7 djm 458: prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
1.1 djm 459: {
460: BIGNUM *q, *p, *a;
461: BN_CTX *ctx;
462: char *cp, *lp;
463: u_int32_t count_in = 0, count_out = 0, count_possible = 0;
464: u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
465: time_t time_start, time_stop;
466: int res;
1.7 djm 467:
468: if (trials < TRIAL_MINIMUM) {
469: error("Minimum primality trials is %d", TRIAL_MINIMUM);
470: return (-1);
471: }
1.1 djm 472:
473: time(&time_start);
474:
475: p = BN_new();
476: q = BN_new();
477: ctx = BN_CTX_new();
478:
479: debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
480: ctime(&time_start), trials, generator_wanted);
481:
482: res = 0;
483: lp = xmalloc(QLINESIZE + 1);
484: while (fgets(lp, QLINESIZE, in) != NULL) {
485: int ll = strlen(lp);
486:
487: count_in++;
488: if (ll < 14 || *lp == '!' || *lp == '#') {
489: debug2("%10u: comment or short line", count_in);
490: continue;
491: }
492:
493: /* XXX - fragile parser */
494: /* time */
495: cp = &lp[14]; /* (skip) */
496:
497: /* type */
498: in_type = strtoul(cp, &cp, 10);
499:
500: /* tests */
501: in_tests = strtoul(cp, &cp, 10);
502:
503: if (in_tests & QTEST_COMPOSITE) {
504: debug2("%10u: known composite", count_in);
505: continue;
506: }
1.5 djm 507:
1.1 djm 508: /* tries */
509: in_tries = strtoul(cp, &cp, 10);
510:
511: /* size (most significant bit) */
512: in_size = strtoul(cp, &cp, 10);
513:
514: /* generator (hex) */
515: generator_known = strtoul(cp, &cp, 16);
516:
517: /* Skip white space */
518: cp += strspn(cp, " ");
519:
520: /* modulus (hex) */
521: switch (in_type) {
1.6 djm 522: case QTYPE_SOPHIE_GERMAIN:
523: debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
1.1 djm 524: a = q;
525: BN_hex2bn(&a, cp);
526: /* p = 2*q + 1 */
527: BN_lshift(p, q, 1);
528: BN_add_word(p, 1);
529: in_size += 1;
530: generator_known = 0;
531: break;
1.5 djm 532: case QTYPE_UNSTRUCTURED:
533: case QTYPE_SAFE:
1.10 dtucker 534: case QTYPE_SCHNORR:
1.5 djm 535: case QTYPE_STRONG:
536: case QTYPE_UNKNOWN:
1.1 djm 537: debug2("%10u: (%u)", count_in, in_type);
538: a = p;
539: BN_hex2bn(&a, cp);
540: /* q = (p-1) / 2 */
541: BN_rshift(q, p, 1);
542: break;
1.5 djm 543: default:
544: debug2("Unknown prime type");
545: break;
1.1 djm 546: }
547:
548: /*
549: * due to earlier inconsistencies in interpretation, check
550: * the proposed bit size.
551: */
1.11 ! avsm 552: if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
1.1 djm 553: debug2("%10u: bit size %u mismatch", count_in, in_size);
554: continue;
555: }
556: if (in_size < QSIZE_MINIMUM) {
557: debug2("%10u: bit size %u too short", count_in, in_size);
558: continue;
559: }
560:
561: if (in_tests & QTEST_MILLER_RABIN)
562: in_tries += trials;
563: else
564: in_tries = trials;
1.5 djm 565:
1.1 djm 566: /*
567: * guess unknown generator
568: */
569: if (generator_known == 0) {
570: if (BN_mod_word(p, 24) == 11)
571: generator_known = 2;
572: else if (BN_mod_word(p, 12) == 5)
573: generator_known = 3;
574: else {
575: u_int32_t r = BN_mod_word(p, 10);
576:
1.5 djm 577: if (r == 3 || r == 7)
1.1 djm 578: generator_known = 5;
579: }
580: }
581: /*
582: * skip tests when desired generator doesn't match
583: */
584: if (generator_wanted > 0 &&
585: generator_wanted != generator_known) {
586: debug2("%10u: generator %d != %d",
587: count_in, generator_known, generator_wanted);
1.4 dtucker 588: continue;
589: }
590:
591: /*
592: * Primes with no known generator are useless for DH, so
593: * skip those.
594: */
595: if (generator_known == 0) {
596: debug2("%10u: no known generator", count_in);
1.1 djm 597: continue;
598: }
599:
600: count_possible++;
601:
602: /*
1.2 djm 603: * The (1/4)^N performance bound on Miller-Rabin is
604: * extremely pessimistic, so don't spend a lot of time
605: * really verifying that q is prime until after we know
606: * that p is also prime. A single pass will weed out the
1.1 djm 607: * vast majority of composite q's.
608: */
609: if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
1.5 djm 610: debug("%10u: q failed first possible prime test",
1.1 djm 611: count_in);
612: continue;
613: }
1.2 djm 614:
1.1 djm 615: /*
1.2 djm 616: * q is possibly prime, so go ahead and really make sure
617: * that p is prime. If it is, then we can go back and do
618: * the same for q. If p is composite, chances are that
1.1 djm 619: * will show up on the first Rabin-Miller iteration so it
620: * doesn't hurt to specify a high iteration count.
621: */
622: if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
1.5 djm 623: debug("%10u: p is not prime", count_in);
1.1 djm 624: continue;
625: }
626: debug("%10u: p is almost certainly prime", count_in);
627:
628: /* recheck q more rigorously */
629: if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
630: debug("%10u: q is not prime", count_in);
631: continue;
632: }
633: debug("%10u: q is almost certainly prime", count_in);
634:
1.2 djm 635: if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
1.1 djm 636: in_tries, in_size, generator_known, p)) {
637: res = -1;
638: break;
639: }
640:
641: count_out++;
642: }
643:
644: time(&time_stop);
645: xfree(lp);
646: BN_free(p);
647: BN_free(q);
648: BN_CTX_free(ctx);
649:
650: logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
1.2 djm 651: ctime(&time_stop), count_out, count_possible,
1.1 djm 652: (long) (time_stop - time_start));
653:
654: return (res);
655: }
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