1 /* $OpenBSD: moduli.c,v 1.26 2012/07/06 00:41:59 dtucker Exp $ */
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>
8 * Redistribution and use in source and binary forms, with or without
9 * modification, are permitted provided that the following conditions
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.
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.
30 * Two-step process to generate safe primes for DHGEX
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.
36 * First step: generate candidate primes (memory intensive)
37 * Second step: test primes' safety (processor intensive)
42 #include <sys/param.h>
43 #include <sys/types.h>
45 #include <openssl/bn.h>
46 #include <openssl/dh.h>
60 #include "openbsd-compat/openssl-compat.h"
66 /* need line long enough for largest moduli plus headers */
67 #define QLINESIZE (100+8192)
71 * Specifies the number of the most significant bit (0 to M).
72 * WARNING: internally, usually 1 to N.
74 #define QSIZE_MINIMUM (511)
77 * Prime sieving defines
80 /* Constant: assuming 8 bit bytes and 32 bit words */
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)
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!
92 #define LARGE_MINIMUM (8UL) /* megabytes */
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).
98 #define LARGE_MAXIMUM (127UL) /* megabytes */
101 * Constant: when used with 32-bit integers, the largest sieve prime
102 * has to be less than 2**32.
104 #define SMALL_MAXIMUM (0xffffffffUL)
106 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
107 #define TINY_NUMBER (1UL<<16)
109 /* Ensure enough bit space for testing 2*q. */
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 */
115 /* bit operations on 32-bit words */
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)))
121 * Prime testing defines
124 /* Minimum number of primality tests to perform */
125 #define TRIAL_MINIMUM (4)
128 * Sieving data (XXX - move to struct)
132 static u_int32_t *TinySieve, tinybits;
134 /* sieve 2**30 in 2**16 parts */
135 static u_int32_t *SmallSieve, smallbits, smallbase;
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;
142 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
143 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t, char *, unsigned long,
147 * print moduli out in consistent form,
150 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
151 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
158 gtm = gmtime(&time_now);
160 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
161 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
162 gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
163 otype, otests, otries, osize, ogenerator);
168 if (BN_print_fp(ofile, omodulus) < 1)
171 res = fprintf(ofile, "\n");
174 return (res > 0 ? 0 : -1);
179 ** Sieve p's and q's with small factors
182 sieve_large(u_int32_t s)
186 debug3("sieve_large %u", s);
188 /* r = largebase mod s */
189 r = BN_mod_word(largebase, s);
191 u = 0; /* s divides into largebase exactly */
193 u = s - r; /* largebase+u is first entry divisible by s */
195 if (u < largebits * 2) {
197 * The sieve omits p's and q's divisible by 2, so ensure that
198 * largebase+u is odd. Then, step through the sieve in
202 u += s; /* Make largebase+u odd, and u even */
204 /* Mark all multiples of 2*s */
205 for (u /= 2; u < largebits; u += s)
206 BIT_SET(LargeSieve, u);
212 u = 0; /* s divides p exactly */
214 u = s - r; /* p+u is first entry divisible by s */
216 if (u < largebits * 4) {
218 * The sieve omits p's divisible by 4, so ensure that
219 * largebase+u is not. Then, step through the sieve in
223 if (SMALL_MAXIMUM - u < s)
228 /* Mark all multiples of 4*s */
229 for (u /= 4; u < largebits; u += s)
230 BIT_SET(LargeSieve, u);
235 * list candidates for Sophie-Germain primes (where q = (p-1)/2)
236 * to standard output.
237 * The list is checked against small known primes (less than 2**30).
240 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
243 u_int32_t j, r, s, t;
244 u_int32_t smallwords = TINY_NUMBER >> 6;
245 u_int32_t tinywords = TINY_NUMBER >> 6;
246 time_t time_start, time_stop;
250 largememory = memory;
253 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
254 error("Invalid memory amount (min %ld, max %ld)",
255 LARGE_MINIMUM, LARGE_MAXIMUM);
260 * Set power to the length in bits of the prime to be generated.
261 * This is changed to 1 less than the desired safe prime moduli p.
263 if (power > TEST_MAXIMUM) {
264 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
266 } else if (power < TEST_MINIMUM) {
267 error("Too few bits: %u < %u", power, TEST_MINIMUM);
270 power--; /* decrement before squaring */
273 * The density of ordinary primes is on the order of 1/bits, so the
274 * density of safe primes should be about (1/bits)**2. Set test range
275 * to something well above bits**2 to be reasonably sure (but not
276 * guaranteed) of catching at least one safe prime.
278 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
281 * Need idea of how much memory is available. We don't have to use all
284 if (largememory > LARGE_MAXIMUM) {
285 logit("Limited memory: %u MB; limit %lu MB",
286 largememory, LARGE_MAXIMUM);
287 largememory = LARGE_MAXIMUM;
290 if (largewords <= (largememory << SHIFT_MEGAWORD)) {
291 logit("Increased memory: %u MB; need %u bytes",
292 largememory, (largewords << SHIFT_BYTE));
293 largewords = (largememory << SHIFT_MEGAWORD);
294 } else if (largememory > 0) {
295 logit("Decreased memory: %u MB; want %u bytes",
296 largememory, (largewords << SHIFT_BYTE));
297 largewords = (largememory << SHIFT_MEGAWORD);
300 TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
301 tinybits = tinywords << SHIFT_WORD;
303 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
304 smallbits = smallwords << SHIFT_WORD;
307 * dynamically determine available memory
309 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
310 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
312 largebits = largewords << SHIFT_WORD;
313 largenumbers = largebits * 2; /* even numbers excluded */
315 /* validation check: count the number of primes tried */
317 if ((q = BN_new()) == NULL)
318 fatal("BN_new failed");
321 * Generate random starting point for subprime search, or use
322 * specified parameter.
324 if ((largebase = BN_new()) == NULL)
325 fatal("BN_new failed");
327 if (BN_rand(largebase, power, 1, 1) == 0)
328 fatal("BN_rand failed");
330 if (BN_copy(largebase, start) == NULL)
331 fatal("BN_copy: failed");
335 if (BN_set_bit(largebase, 0) == 0)
336 fatal("BN_set_bit: failed");
340 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
341 largenumbers, power);
342 debug2("start point: 0x%s", BN_bn2hex(largebase));
347 for (i = 0; i < tinybits; i++) {
348 if (BIT_TEST(TinySieve, i))
349 continue; /* 2*i+3 is composite */
351 /* The next tiny prime */
354 /* Mark all multiples of t */
355 for (j = i + t; j < tinybits; j += t)
356 BIT_SET(TinySieve, j);
362 * Start the small block search at the next possible prime. To avoid
363 * fencepost errors, the last pass is skipped.
365 for (smallbase = TINY_NUMBER + 3;
366 smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
367 smallbase += TINY_NUMBER) {
368 for (i = 0; i < tinybits; i++) {
369 if (BIT_TEST(TinySieve, i))
370 continue; /* 2*i+3 is composite */
372 /* The next tiny prime */
377 s = 0; /* t divides into smallbase exactly */
379 /* smallbase+s is first entry divisible by t */
384 * The sieve omits even numbers, so ensure that
385 * smallbase+s is odd. Then, step through the sieve
386 * in increments of 2*t
389 s += t; /* Make smallbase+s odd, and s even */
391 /* Mark all multiples of 2*t */
392 for (s /= 2; s < smallbits; s += t)
393 BIT_SET(SmallSieve, s);
399 for (i = 0; i < smallbits; i++) {
400 if (BIT_TEST(SmallSieve, i))
401 continue; /* 2*i+smallbase is composite */
403 /* The next small prime */
404 sieve_large((2 * i) + smallbase);
407 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
412 logit("%.24s Sieved with %u small primes in %ld seconds",
413 ctime(&time_stop), largetries, (long) (time_stop - time_start));
415 for (j = r = 0; j < largebits; j++) {
416 if (BIT_TEST(LargeSieve, j))
417 continue; /* Definitely composite, skip */
419 debug2("test q = largebase+%u", 2 * j);
420 if (BN_set_word(q, 2 * j) == 0)
421 fatal("BN_set_word failed");
422 if (BN_add(q, q, largebase) == 0)
423 fatal("BN_add failed");
424 if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN,
425 MODULI_TESTS_SIEVE, largetries,
426 (power - 1) /* MSB */, (0), q) == -1) {
440 logit("%.24s Found %u candidates", ctime(&time_stop), r);
446 write_checkpoint(char *cpfile, u_int32_t lineno)
449 char tmp[MAXPATHLEN];
452 r = snprintf(tmp, sizeof(tmp), "%s.XXXXXXXXXX", cpfile);
453 if (r == -1 || r >= MAXPATHLEN) {
454 logit("write_checkpoint: temp pathname too long");
457 if ((r = mkstemp(tmp)) == -1) {
458 logit("mkstemp(%s): %s", tmp, strerror(errno));
461 if ((fp = fdopen(r, "w")) == NULL) {
462 logit("write_checkpoint: fdopen: %s", strerror(errno));
466 if (fprintf(fp, "%lu\n", (unsigned long)lineno) > 0 && fclose(fp) == 0
467 && rename(tmp, cpfile) == 0)
468 debug3("wrote checkpoint line %lu to '%s'",
469 (unsigned long)lineno, cpfile);
471 logit("failed to write to checkpoint file '%s': %s", cpfile,
476 read_checkpoint(char *cpfile)
479 unsigned long lineno = 0;
481 if ((fp = fopen(cpfile, "r")) == NULL)
483 if (fscanf(fp, "%lu\n", &lineno) < 1)
484 logit("Failed to load checkpoint from '%s'", cpfile);
486 logit("Loaded checkpoint from '%s' line %lu", cpfile, lineno);
492 * perform a Miller-Rabin primality test
493 * on the list of candidates
494 * (checking both q and p)
495 * The result is a list of so-call "safe" primes
498 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted,
499 char *checkpoint_file, unsigned long start_lineno, unsigned long num_lines)
504 u_int32_t count_in = 0, count_out = 0, count_possible = 0;
505 u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
506 unsigned long last_processed = 0, end_lineno;
507 time_t time_start, time_stop;
510 if (trials < TRIAL_MINIMUM) {
511 error("Minimum primality trials is %d", TRIAL_MINIMUM);
517 if ((p = BN_new()) == NULL)
518 fatal("BN_new failed");
519 if ((q = BN_new()) == NULL)
520 fatal("BN_new failed");
521 if ((ctx = BN_CTX_new()) == NULL)
522 fatal("BN_CTX_new failed");
524 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
525 ctime(&time_start), trials, generator_wanted);
527 if (checkpoint_file != NULL)
528 last_processed = read_checkpoint(checkpoint_file);
529 if (start_lineno > last_processed)
530 last_processed = start_lineno;
532 end_lineno = ULONG_MAX;
534 end_lineno = last_processed + num_lines;
535 debug2("process line %lu to line %lu", last_processed, end_lineno);
538 lp = xmalloc(QLINESIZE + 1);
539 while (fgets(lp, QLINESIZE + 1, in) != NULL && count_in < end_lineno) {
541 if (checkpoint_file != NULL) {
542 if (count_in <= last_processed) {
543 debug3("skipping line %u, before checkpoint",
547 write_checkpoint(checkpoint_file, count_in);
549 if (strlen(lp) < 14 || *lp == '!' || *lp == '#') {
550 debug2("%10u: comment or short line", count_in);
554 /* XXX - fragile parser */
556 cp = &lp[14]; /* (skip) */
559 in_type = strtoul(cp, &cp, 10);
562 in_tests = strtoul(cp, &cp, 10);
564 if (in_tests & MODULI_TESTS_COMPOSITE) {
565 debug2("%10u: known composite", count_in);
570 in_tries = strtoul(cp, &cp, 10);
572 /* size (most significant bit) */
573 in_size = strtoul(cp, &cp, 10);
575 /* generator (hex) */
576 generator_known = strtoul(cp, &cp, 16);
578 /* Skip white space */
579 cp += strspn(cp, " ");
583 case MODULI_TYPE_SOPHIE_GERMAIN:
584 debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
586 if (BN_hex2bn(&a, cp) == 0)
587 fatal("BN_hex2bn failed");
589 if (BN_lshift(p, q, 1) == 0)
590 fatal("BN_lshift failed");
591 if (BN_add_word(p, 1) == 0)
592 fatal("BN_add_word failed");
596 case MODULI_TYPE_UNSTRUCTURED:
597 case MODULI_TYPE_SAFE:
598 case MODULI_TYPE_SCHNORR:
599 case MODULI_TYPE_STRONG:
600 case MODULI_TYPE_UNKNOWN:
601 debug2("%10u: (%u)", count_in, in_type);
603 if (BN_hex2bn(&a, cp) == 0)
604 fatal("BN_hex2bn failed");
606 if (BN_rshift(q, p, 1) == 0)
607 fatal("BN_rshift failed");
610 debug2("Unknown prime type");
615 * due to earlier inconsistencies in interpretation, check
616 * the proposed bit size.
618 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
619 debug2("%10u: bit size %u mismatch", count_in, in_size);
622 if (in_size < QSIZE_MINIMUM) {
623 debug2("%10u: bit size %u too short", count_in, in_size);
627 if (in_tests & MODULI_TESTS_MILLER_RABIN)
633 * guess unknown generator
635 if (generator_known == 0) {
636 if (BN_mod_word(p, 24) == 11)
638 else if (BN_mod_word(p, 12) == 5)
641 u_int32_t r = BN_mod_word(p, 10);
643 if (r == 3 || r == 7)
648 * skip tests when desired generator doesn't match
650 if (generator_wanted > 0 &&
651 generator_wanted != generator_known) {
652 debug2("%10u: generator %d != %d",
653 count_in, generator_known, generator_wanted);
658 * Primes with no known generator are useless for DH, so
661 if (generator_known == 0) {
662 debug2("%10u: no known generator", count_in);
669 * The (1/4)^N performance bound on Miller-Rabin is
670 * extremely pessimistic, so don't spend a lot of time
671 * really verifying that q is prime until after we know
672 * that p is also prime. A single pass will weed out the
673 * vast majority of composite q's.
675 if (BN_is_prime_ex(q, 1, ctx, NULL) <= 0) {
676 debug("%10u: q failed first possible prime test",
682 * q is possibly prime, so go ahead and really make sure
683 * that p is prime. If it is, then we can go back and do
684 * the same for q. If p is composite, chances are that
685 * will show up on the first Rabin-Miller iteration so it
686 * doesn't hurt to specify a high iteration count.
688 if (!BN_is_prime_ex(p, trials, ctx, NULL)) {
689 debug("%10u: p is not prime", count_in);
692 debug("%10u: p is almost certainly prime", count_in);
694 /* recheck q more rigorously */
695 if (!BN_is_prime_ex(q, trials - 1, ctx, NULL)) {
696 debug("%10u: q is not prime", count_in);
699 debug("%10u: q is almost certainly prime", count_in);
701 if (qfileout(out, MODULI_TYPE_SAFE,
702 in_tests | MODULI_TESTS_MILLER_RABIN,
703 in_tries, in_size, generator_known, p)) {
717 if (checkpoint_file != NULL)
718 unlink(checkpoint_file);
720 logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
721 ctime(&time_stop), count_out, count_possible,
722 (long) (time_stop - time_start));