1 /* $OpenBSD: moduli.c,v 1.22 2010/11/10 01:33:07 djm 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/types.h>
44 #include <openssl/bn.h>
45 #include <openssl/dh.h>
57 #include "openbsd-compat/openssl-compat.h"
63 /* need line long enough for largest moduli plus headers */
64 #define QLINESIZE (100+8192)
68 * Specifies the number of the most significant bit (0 to M).
69 * WARNING: internally, usually 1 to N.
71 #define QSIZE_MINIMUM (511)
74 * Prime sieving defines
77 /* Constant: assuming 8 bit bytes and 32 bit words */
79 #define SHIFT_BYTE (2)
80 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
81 #define SHIFT_MEGABYTE (20)
82 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
85 * Using virtual memory can cause thrashing. This should be the largest
86 * number that is supported without a large amount of disk activity --
87 * that would increase the run time from hours to days or weeks!
89 #define LARGE_MINIMUM (8UL) /* megabytes */
92 * Do not increase this number beyond the unsigned integer bit size.
93 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
95 #define LARGE_MAXIMUM (127UL) /* megabytes */
98 * Constant: when used with 32-bit integers, the largest sieve prime
99 * has to be less than 2**32.
101 #define SMALL_MAXIMUM (0xffffffffUL)
103 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
104 #define TINY_NUMBER (1UL<<16)
106 /* Ensure enough bit space for testing 2*q. */
107 #define TEST_MAXIMUM (1UL<<16)
108 #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
109 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
110 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
112 /* bit operations on 32-bit words */
113 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
114 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
115 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
118 * Prime testing defines
121 /* Minimum number of primality tests to perform */
122 #define TRIAL_MINIMUM (4)
125 * Sieving data (XXX - move to struct)
129 static u_int32_t *TinySieve, tinybits;
131 /* sieve 2**30 in 2**16 parts */
132 static u_int32_t *SmallSieve, smallbits, smallbase;
134 /* sieve relative to the initial value */
135 static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
136 static u_int32_t largebits, largememory; /* megabytes */
137 static BIGNUM *largebase;
139 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
140 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
143 * print moduli out in consistent form,
146 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
147 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
154 gtm = gmtime(&time_now);
156 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
157 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
158 gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
159 otype, otests, otries, osize, ogenerator);
164 if (BN_print_fp(ofile, omodulus) < 1)
167 res = fprintf(ofile, "\n");
170 return (res > 0 ? 0 : -1);
175 ** Sieve p's and q's with small factors
178 sieve_large(u_int32_t s)
182 debug3("sieve_large %u", s);
184 /* r = largebase mod s */
185 r = BN_mod_word(largebase, s);
187 u = 0; /* s divides into largebase exactly */
189 u = s - r; /* largebase+u is first entry divisible by s */
191 if (u < largebits * 2) {
193 * The sieve omits p's and q's divisible by 2, so ensure that
194 * largebase+u is odd. Then, step through the sieve in
198 u += s; /* Make largebase+u odd, and u even */
200 /* Mark all multiples of 2*s */
201 for (u /= 2; u < largebits; u += s)
202 BIT_SET(LargeSieve, u);
208 u = 0; /* s divides p exactly */
210 u = s - r; /* p+u is first entry divisible by s */
212 if (u < largebits * 4) {
214 * The sieve omits p's divisible by 4, so ensure that
215 * largebase+u is not. Then, step through the sieve in
219 if (SMALL_MAXIMUM - u < s)
224 /* Mark all multiples of 4*s */
225 for (u /= 4; u < largebits; u += s)
226 BIT_SET(LargeSieve, u);
231 * list candidates for Sophie-Germain primes (where q = (p-1)/2)
232 * to standard output.
233 * The list is checked against small known primes (less than 2**30).
236 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
239 u_int32_t j, r, s, t;
240 u_int32_t smallwords = TINY_NUMBER >> 6;
241 u_int32_t tinywords = TINY_NUMBER >> 6;
242 time_t time_start, time_stop;
246 largememory = memory;
249 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
250 error("Invalid memory amount (min %ld, max %ld)",
251 LARGE_MINIMUM, LARGE_MAXIMUM);
256 * Set power to the length in bits of the prime to be generated.
257 * This is changed to 1 less than the desired safe prime moduli p.
259 if (power > TEST_MAXIMUM) {
260 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
262 } else if (power < TEST_MINIMUM) {
263 error("Too few bits: %u < %u", power, TEST_MINIMUM);
266 power--; /* decrement before squaring */
269 * The density of ordinary primes is on the order of 1/bits, so the
270 * density of safe primes should be about (1/bits)**2. Set test range
271 * to something well above bits**2 to be reasonably sure (but not
272 * guaranteed) of catching at least one safe prime.
274 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
277 * Need idea of how much memory is available. We don't have to use all
280 if (largememory > LARGE_MAXIMUM) {
281 logit("Limited memory: %u MB; limit %lu MB",
282 largememory, LARGE_MAXIMUM);
283 largememory = LARGE_MAXIMUM;
286 if (largewords <= (largememory << SHIFT_MEGAWORD)) {
287 logit("Increased memory: %u MB; need %u bytes",
288 largememory, (largewords << SHIFT_BYTE));
289 largewords = (largememory << SHIFT_MEGAWORD);
290 } else if (largememory > 0) {
291 logit("Decreased memory: %u MB; want %u bytes",
292 largememory, (largewords << SHIFT_BYTE));
293 largewords = (largememory << SHIFT_MEGAWORD);
296 TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
297 tinybits = tinywords << SHIFT_WORD;
299 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
300 smallbits = smallwords << SHIFT_WORD;
303 * dynamically determine available memory
305 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
306 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
308 largebits = largewords << SHIFT_WORD;
309 largenumbers = largebits * 2; /* even numbers excluded */
311 /* validation check: count the number of primes tried */
313 if ((q = BN_new()) == NULL)
314 fatal("BN_new failed");
317 * Generate random starting point for subprime search, or use
318 * specified parameter.
320 if ((largebase = BN_new()) == NULL)
321 fatal("BN_new failed");
323 if (BN_rand(largebase, power, 1, 1) == 0)
324 fatal("BN_rand failed");
326 if (BN_copy(largebase, start) == NULL)
327 fatal("BN_copy: failed");
331 if (BN_set_bit(largebase, 0) == 0)
332 fatal("BN_set_bit: failed");
336 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
337 largenumbers, power);
338 debug2("start point: 0x%s", BN_bn2hex(largebase));
343 for (i = 0; i < tinybits; i++) {
344 if (BIT_TEST(TinySieve, i))
345 continue; /* 2*i+3 is composite */
347 /* The next tiny prime */
350 /* Mark all multiples of t */
351 for (j = i + t; j < tinybits; j += t)
352 BIT_SET(TinySieve, j);
358 * Start the small block search at the next possible prime. To avoid
359 * fencepost errors, the last pass is skipped.
361 for (smallbase = TINY_NUMBER + 3;
362 smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
363 smallbase += TINY_NUMBER) {
364 for (i = 0; i < tinybits; i++) {
365 if (BIT_TEST(TinySieve, i))
366 continue; /* 2*i+3 is composite */
368 /* The next tiny prime */
373 s = 0; /* t divides into smallbase exactly */
375 /* smallbase+s is first entry divisible by t */
380 * The sieve omits even numbers, so ensure that
381 * smallbase+s is odd. Then, step through the sieve
382 * in increments of 2*t
385 s += t; /* Make smallbase+s odd, and s even */
387 /* Mark all multiples of 2*t */
388 for (s /= 2; s < smallbits; s += t)
389 BIT_SET(SmallSieve, s);
395 for (i = 0; i < smallbits; i++) {
396 if (BIT_TEST(SmallSieve, i))
397 continue; /* 2*i+smallbase is composite */
399 /* The next small prime */
400 sieve_large((2 * i) + smallbase);
403 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
408 logit("%.24s Sieved with %u small primes in %ld seconds",
409 ctime(&time_stop), largetries, (long) (time_stop - time_start));
411 for (j = r = 0; j < largebits; j++) {
412 if (BIT_TEST(LargeSieve, j))
413 continue; /* Definitely composite, skip */
415 debug2("test q = largebase+%u", 2 * j);
416 if (BN_set_word(q, 2 * j) == 0)
417 fatal("BN_set_word failed");
418 if (BN_add(q, q, largebase) == 0)
419 fatal("BN_add failed");
420 if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN,
421 MODULI_TESTS_SIEVE, largetries,
422 (power - 1) /* MSB */, (0), q) == -1) {
436 logit("%.24s Found %u candidates", ctime(&time_stop), r);
442 * perform a Miller-Rabin primality test
443 * on the list of candidates
444 * (checking both q and p)
445 * The result is a list of so-call "safe" primes
448 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
453 u_int32_t count_in = 0, count_out = 0, count_possible = 0;
454 u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
455 time_t time_start, time_stop;
458 if (trials < TRIAL_MINIMUM) {
459 error("Minimum primality trials is %d", TRIAL_MINIMUM);
465 if ((p = BN_new()) == NULL)
466 fatal("BN_new failed");
467 if ((q = BN_new()) == NULL)
468 fatal("BN_new failed");
469 if ((ctx = BN_CTX_new()) == NULL)
470 fatal("BN_CTX_new failed");
472 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
473 ctime(&time_start), trials, generator_wanted);
476 lp = xmalloc(QLINESIZE + 1);
477 while (fgets(lp, QLINESIZE + 1, in) != NULL) {
479 if (strlen(lp) < 14 || *lp == '!' || *lp == '#') {
480 debug2("%10u: comment or short line", count_in);
484 /* XXX - fragile parser */
486 cp = &lp[14]; /* (skip) */
489 in_type = strtoul(cp, &cp, 10);
492 in_tests = strtoul(cp, &cp, 10);
494 if (in_tests & MODULI_TESTS_COMPOSITE) {
495 debug2("%10u: known composite", count_in);
500 in_tries = strtoul(cp, &cp, 10);
502 /* size (most significant bit) */
503 in_size = strtoul(cp, &cp, 10);
505 /* generator (hex) */
506 generator_known = strtoul(cp, &cp, 16);
508 /* Skip white space */
509 cp += strspn(cp, " ");
513 case MODULI_TYPE_SOPHIE_GERMAIN:
514 debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
516 if (BN_hex2bn(&a, cp) == 0)
517 fatal("BN_hex2bn failed");
519 if (BN_lshift(p, q, 1) == 0)
520 fatal("BN_lshift failed");
521 if (BN_add_word(p, 1) == 0)
522 fatal("BN_add_word failed");
526 case MODULI_TYPE_UNSTRUCTURED:
527 case MODULI_TYPE_SAFE:
528 case MODULI_TYPE_SCHNORR:
529 case MODULI_TYPE_STRONG:
530 case MODULI_TYPE_UNKNOWN:
531 debug2("%10u: (%u)", count_in, in_type);
533 if (BN_hex2bn(&a, cp) == 0)
534 fatal("BN_hex2bn failed");
536 if (BN_rshift(q, p, 1) == 0)
537 fatal("BN_rshift failed");
540 debug2("Unknown prime type");
545 * due to earlier inconsistencies in interpretation, check
546 * the proposed bit size.
548 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
549 debug2("%10u: bit size %u mismatch", count_in, in_size);
552 if (in_size < QSIZE_MINIMUM) {
553 debug2("%10u: bit size %u too short", count_in, in_size);
557 if (in_tests & MODULI_TESTS_MILLER_RABIN)
563 * guess unknown generator
565 if (generator_known == 0) {
566 if (BN_mod_word(p, 24) == 11)
568 else if (BN_mod_word(p, 12) == 5)
571 u_int32_t r = BN_mod_word(p, 10);
573 if (r == 3 || r == 7)
578 * skip tests when desired generator doesn't match
580 if (generator_wanted > 0 &&
581 generator_wanted != generator_known) {
582 debug2("%10u: generator %d != %d",
583 count_in, generator_known, generator_wanted);
588 * Primes with no known generator are useless for DH, so
591 if (generator_known == 0) {
592 debug2("%10u: no known generator", count_in);
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
603 * vast majority of composite q's.
605 if (BN_is_prime_ex(q, 1, ctx, NULL) <= 0) {
606 debug("%10u: q failed first possible prime test",
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
615 * will show up on the first Rabin-Miller iteration so it
616 * doesn't hurt to specify a high iteration count.
618 if (!BN_is_prime_ex(p, trials, ctx, NULL)) {
619 debug("%10u: p is not prime", count_in);
622 debug("%10u: p is almost certainly prime", count_in);
624 /* recheck q more rigorously */
625 if (!BN_is_prime_ex(q, trials - 1, ctx, NULL)) {
626 debug("%10u: q is not prime", count_in);
629 debug("%10u: q is almost certainly prime", count_in);
631 if (qfileout(out, MODULI_TYPE_SAFE,
632 in_tests | MODULI_TESTS_MILLER_RABIN,
633 in_tries, in_size, generator_known, p)) {
647 logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
648 ctime(&time_stop), count_out, count_possible,
649 (long) (time_stop - time_start));