1 /* $OpenBSD: moduli.c,v 1.17 2006/08/01 23:22:47 stevesk 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
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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,
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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>
58 /* need line long enough for largest moduli plus headers */
59 #define QLINESIZE (100+8192)
62 * Specifies the internal structure of the prime modulus.
64 #define QTYPE_UNKNOWN (0)
65 #define QTYPE_UNSTRUCTURED (1)
66 #define QTYPE_SAFE (2)
67 #define QTYPE_SCHNORR (3)
68 #define QTYPE_SOPHIE_GERMAIN (4)
69 #define QTYPE_STRONG (5)
71 /* Tests: decimal (bit field).
72 * Specifies the methods used in checking for primality.
73 * Usually, more than one test is used.
75 #define QTEST_UNTESTED (0x00)
76 #define QTEST_COMPOSITE (0x01)
77 #define QTEST_SIEVE (0x02)
78 #define QTEST_MILLER_RABIN (0x04)
79 #define QTEST_JACOBI (0x08)
80 #define QTEST_ELLIPTIC (0x10)
84 * Specifies the number of the most significant bit (0 to M).
85 * WARNING: internally, usually 1 to N.
87 #define QSIZE_MINIMUM (511)
90 * Prime sieving defines
93 /* Constant: assuming 8 bit bytes and 32 bit words */
95 #define SHIFT_BYTE (2)
96 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
97 #define SHIFT_MEGABYTE (20)
98 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
101 * Using virtual memory can cause thrashing. This should be the largest
102 * number that is supported without a large amount of disk activity --
103 * that would increase the run time from hours to days or weeks!
105 #define LARGE_MINIMUM (8UL) /* megabytes */
108 * Do not increase this number beyond the unsigned integer bit size.
109 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
111 #define LARGE_MAXIMUM (127UL) /* megabytes */
114 * Constant: when used with 32-bit integers, the largest sieve prime
115 * has to be less than 2**32.
117 #define SMALL_MAXIMUM (0xffffffffUL)
119 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
120 #define TINY_NUMBER (1UL<<16)
122 /* Ensure enough bit space for testing 2*q. */
123 #define TEST_MAXIMUM (1UL<<16)
124 #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
125 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
126 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
128 /* bit operations on 32-bit words */
129 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
130 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
131 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
134 * Prime testing defines
137 /* Minimum number of primality tests to perform */
138 #define TRIAL_MINIMUM (4)
141 * Sieving data (XXX - move to struct)
145 static u_int32_t *TinySieve, tinybits;
147 /* sieve 2**30 in 2**16 parts */
148 static u_int32_t *SmallSieve, smallbits, smallbase;
150 /* sieve relative to the initial value */
151 static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
152 static u_int32_t largebits, largememory; /* megabytes */
153 static BIGNUM *largebase;
155 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
156 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
159 * print moduli out in consistent form,
162 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
163 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
170 gtm = gmtime(&time_now);
172 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
173 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
174 gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
175 otype, otests, otries, osize, ogenerator);
180 if (BN_print_fp(ofile, omodulus) < 1)
183 res = fprintf(ofile, "\n");
186 return (res > 0 ? 0 : -1);
191 ** Sieve p's and q's with small factors
194 sieve_large(u_int32_t s)
198 debug3("sieve_large %u", s);
200 /* r = largebase mod s */
201 r = BN_mod_word(largebase, s);
203 u = 0; /* s divides into largebase exactly */
205 u = s - r; /* largebase+u is first entry divisible by s */
207 if (u < largebits * 2) {
209 * The sieve omits p's and q's divisible by 2, so ensure that
210 * largebase+u is odd. Then, step through the sieve in
214 u += s; /* Make largebase+u odd, and u even */
216 /* Mark all multiples of 2*s */
217 for (u /= 2; u < largebits; u += s)
218 BIT_SET(LargeSieve, u);
224 u = 0; /* s divides p exactly */
226 u = s - r; /* p+u is first entry divisible by s */
228 if (u < largebits * 4) {
230 * The sieve omits p's divisible by 4, so ensure that
231 * largebase+u is not. Then, step through the sieve in
235 if (SMALL_MAXIMUM - u < s)
240 /* Mark all multiples of 4*s */
241 for (u /= 4; u < largebits; u += s)
242 BIT_SET(LargeSieve, u);
247 * list candidates for Sophie-Germain primes (where q = (p-1)/2)
248 * to standard output.
249 * The list is checked against small known primes (less than 2**30).
252 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
255 u_int32_t j, r, s, t;
256 u_int32_t smallwords = TINY_NUMBER >> 6;
257 u_int32_t tinywords = TINY_NUMBER >> 6;
258 time_t time_start, time_stop;
262 largememory = memory;
265 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
266 error("Invalid memory amount (min %ld, max %ld)",
267 LARGE_MINIMUM, LARGE_MAXIMUM);
272 * Set power to the length in bits of the prime to be generated.
273 * This is changed to 1 less than the desired safe prime moduli p.
275 if (power > TEST_MAXIMUM) {
276 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
278 } else if (power < TEST_MINIMUM) {
279 error("Too few bits: %u < %u", power, TEST_MINIMUM);
282 power--; /* decrement before squaring */
285 * The density of ordinary primes is on the order of 1/bits, so the
286 * density of safe primes should be about (1/bits)**2. Set test range
287 * to something well above bits**2 to be reasonably sure (but not
288 * guaranteed) of catching at least one safe prime.
290 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
293 * Need idea of how much memory is available. We don't have to use all
296 if (largememory > LARGE_MAXIMUM) {
297 logit("Limited memory: %u MB; limit %lu MB",
298 largememory, LARGE_MAXIMUM);
299 largememory = LARGE_MAXIMUM;
302 if (largewords <= (largememory << SHIFT_MEGAWORD)) {
303 logit("Increased memory: %u MB; need %u bytes",
304 largememory, (largewords << SHIFT_BYTE));
305 largewords = (largememory << SHIFT_MEGAWORD);
306 } else if (largememory > 0) {
307 logit("Decreased memory: %u MB; want %u bytes",
308 largememory, (largewords << SHIFT_BYTE));
309 largewords = (largememory << SHIFT_MEGAWORD);
312 TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
313 tinybits = tinywords << SHIFT_WORD;
315 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
316 smallbits = smallwords << SHIFT_WORD;
319 * dynamically determine available memory
321 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
322 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
324 largebits = largewords << SHIFT_WORD;
325 largenumbers = largebits * 2; /* even numbers excluded */
327 /* validation check: count the number of primes tried */
332 * Generate random starting point for subprime search, or use
333 * specified parameter.
335 largebase = BN_new();
337 BN_rand(largebase, power, 1, 1);
339 BN_copy(largebase, start);
342 BN_set_bit(largebase, 0);
346 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
347 largenumbers, power);
348 debug2("start point: 0x%s", BN_bn2hex(largebase));
353 for (i = 0; i < tinybits; i++) {
354 if (BIT_TEST(TinySieve, i))
355 continue; /* 2*i+3 is composite */
357 /* The next tiny prime */
360 /* Mark all multiples of t */
361 for (j = i + t; j < tinybits; j += t)
362 BIT_SET(TinySieve, j);
368 * Start the small block search at the next possible prime. To avoid
369 * fencepost errors, the last pass is skipped.
371 for (smallbase = TINY_NUMBER + 3;
372 smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
373 smallbase += TINY_NUMBER) {
374 for (i = 0; i < tinybits; i++) {
375 if (BIT_TEST(TinySieve, i))
376 continue; /* 2*i+3 is composite */
378 /* The next tiny prime */
383 s = 0; /* t divides into smallbase exactly */
385 /* smallbase+s is first entry divisible by t */
390 * The sieve omits even numbers, so ensure that
391 * smallbase+s is odd. Then, step through the sieve
392 * in increments of 2*t
395 s += t; /* Make smallbase+s odd, and s even */
397 /* Mark all multiples of 2*t */
398 for (s /= 2; s < smallbits; s += t)
399 BIT_SET(SmallSieve, s);
405 for (i = 0; i < smallbits; i++) {
406 if (BIT_TEST(SmallSieve, i))
407 continue; /* 2*i+smallbase is composite */
409 /* The next small prime */
410 sieve_large((2 * i) + smallbase);
413 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
418 logit("%.24s Sieved with %u small primes in %ld seconds",
419 ctime(&time_stop), largetries, (long) (time_stop - time_start));
421 for (j = r = 0; j < largebits; j++) {
422 if (BIT_TEST(LargeSieve, j))
423 continue; /* Definitely composite, skip */
425 debug2("test q = largebase+%u", 2 * j);
426 BN_set_word(q, 2 * j);
427 BN_add(q, q, largebase);
428 if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
429 largetries, (power - 1) /* MSB */, (0), q) == -1) {
443 logit("%.24s Found %u candidates", ctime(&time_stop), r);
449 * perform a Miller-Rabin primality test
450 * on the list of candidates
451 * (checking both q and p)
452 * The result is a list of so-call "safe" primes
455 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
460 u_int32_t count_in = 0, count_out = 0, count_possible = 0;
461 u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
462 time_t time_start, time_stop;
465 if (trials < TRIAL_MINIMUM) {
466 error("Minimum primality trials is %d", TRIAL_MINIMUM);
476 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
477 ctime(&time_start), trials, generator_wanted);
480 lp = xmalloc(QLINESIZE + 1);
481 while (fgets(lp, QLINESIZE, in) != NULL) {
485 if (ll < 14 || *lp == '!' || *lp == '#') {
486 debug2("%10u: comment or short line", count_in);
490 /* XXX - fragile parser */
492 cp = &lp[14]; /* (skip) */
495 in_type = strtoul(cp, &cp, 10);
498 in_tests = strtoul(cp, &cp, 10);
500 if (in_tests & QTEST_COMPOSITE) {
501 debug2("%10u: known composite", count_in);
506 in_tries = strtoul(cp, &cp, 10);
508 /* size (most significant bit) */
509 in_size = strtoul(cp, &cp, 10);
511 /* generator (hex) */
512 generator_known = strtoul(cp, &cp, 16);
514 /* Skip white space */
515 cp += strspn(cp, " ");
519 case QTYPE_SOPHIE_GERMAIN:
520 debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
529 case QTYPE_UNSTRUCTURED:
534 debug2("%10u: (%u)", count_in, in_type);
541 debug2("Unknown prime type");
546 * due to earlier inconsistencies in interpretation, check
547 * the proposed bit size.
549 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
550 debug2("%10u: bit size %u mismatch", count_in, in_size);
553 if (in_size < QSIZE_MINIMUM) {
554 debug2("%10u: bit size %u too short", count_in, in_size);
558 if (in_tests & QTEST_MILLER_RABIN)
564 * guess unknown generator
566 if (generator_known == 0) {
567 if (BN_mod_word(p, 24) == 11)
569 else if (BN_mod_word(p, 12) == 5)
572 u_int32_t r = BN_mod_word(p, 10);
574 if (r == 3 || r == 7)
579 * skip tests when desired generator doesn't match
581 if (generator_wanted > 0 &&
582 generator_wanted != generator_known) {
583 debug2("%10u: generator %d != %d",
584 count_in, generator_known, generator_wanted);
589 * Primes with no known generator are useless for DH, so
592 if (generator_known == 0) {
593 debug2("%10u: no known generator", count_in);
600 * The (1/4)^N performance bound on Miller-Rabin is
601 * extremely pessimistic, so don't spend a lot of time
602 * really verifying that q is prime until after we know
603 * that p is also prime. A single pass will weed out the
604 * vast majority of composite q's.
606 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
607 debug("%10u: q failed first possible prime test",
613 * q is possibly prime, so go ahead and really make sure
614 * that p is prime. If it is, then we can go back and do
615 * the same for q. If p is composite, chances are that
616 * will show up on the first Rabin-Miller iteration so it
617 * doesn't hurt to specify a high iteration count.
619 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
620 debug("%10u: p is not prime", count_in);
623 debug("%10u: p is almost certainly prime", count_in);
625 /* recheck q more rigorously */
626 if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
627 debug("%10u: q is not prime", count_in);
630 debug("%10u: q is almost certainly prime", count_in);
632 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_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));