1 /* $OpenBSD: moduli.c,v 1.12 2005/07/17 07:17:55 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
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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,
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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)
44 #include <openssl/bn.h>
50 /* need line long enough for largest moduli plus headers */
51 #define QLINESIZE (100+8192)
54 * Specifies the internal structure of the prime modulus.
56 #define QTYPE_UNKNOWN (0)
57 #define QTYPE_UNSTRUCTURED (1)
58 #define QTYPE_SAFE (2)
59 #define QTYPE_SCHNORR (3)
60 #define QTYPE_SOPHIE_GERMAIN (4)
61 #define QTYPE_STRONG (5)
63 /* Tests: decimal (bit field).
64 * Specifies the methods used in checking for primality.
65 * Usually, more than one test is used.
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)
76 * Specifies the number of the most significant bit (0 to M).
77 * WARNING: internally, usually 1 to N.
79 #define QSIZE_MINIMUM (511)
82 * Prime sieving defines
85 /* Constant: assuming 8 bit bytes and 32 bit words */
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)
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!
97 #define LARGE_MINIMUM (8UL) /* megabytes */
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).
103 #define LARGE_MAXIMUM (127UL) /* megabytes */
106 * Constant: when used with 32-bit integers, the largest sieve prime
107 * has to be less than 2**32.
109 #define SMALL_MAXIMUM (0xffffffffUL)
111 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
112 #define TINY_NUMBER (1UL<<16)
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 */
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)))
126 * Prime testing defines
129 /* Minimum number of primality tests to perform */
130 #define TRIAL_MINIMUM (4)
133 * Sieving data (XXX - move to struct)
137 static u_int32_t *TinySieve, tinybits;
139 /* sieve 2**30 in 2**16 parts */
140 static u_int32_t *SmallSieve, smallbits, smallbase;
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;
147 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
148 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
151 * print moduli out in consistent form,
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)
162 gtm = gmtime(&time_now);
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);
172 if (BN_print_fp(ofile, omodulus) < 1)
175 res = fprintf(ofile, "\n");
178 return (res > 0 ? 0 : -1);
183 ** Sieve p's and q's with small factors
186 sieve_large(u_int32_t s)
190 debug3("sieve_large %u", s);
192 /* r = largebase mod s */
193 r = BN_mod_word(largebase, s);
195 u = 0; /* s divides into largebase exactly */
197 u = s - r; /* largebase+u is first entry divisible by s */
199 if (u < largebits * 2) {
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
206 u += s; /* Make largebase+u odd, and u even */
208 /* Mark all multiples of 2*s */
209 for (u /= 2; u < largebits; u += s)
210 BIT_SET(LargeSieve, u);
216 u = 0; /* s divides p exactly */
218 u = s - r; /* p+u is first entry divisible by s */
220 if (u < largebits * 4) {
222 * The sieve omits p's divisible by 4, so ensure that
223 * largebase+u is not. Then, step through the sieve in
227 if (SMALL_MAXIMUM - u < s)
232 /* Mark all multiples of 4*s */
233 for (u /= 4; u < largebits; u += s)
234 BIT_SET(LargeSieve, u);
239 * list candidates for Sophie-Germain primes (where q = (p-1)/2)
240 * to standard output.
241 * The list is checked against small known primes (less than 2**30).
244 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
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;
254 largememory = memory;
257 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
258 error("Invalid memory amount (min %ld, max %ld)",
259 LARGE_MINIMUM, LARGE_MAXIMUM);
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.
267 if (power > TEST_MAXIMUM) {
268 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
270 } else if (power < TEST_MINIMUM) {
271 error("Too few bits: %u < %u", power, TEST_MINIMUM);
274 power--; /* decrement before squaring */
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.
282 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
285 * Need idea of how much memory is available. We don't have to use all
288 if (largememory > LARGE_MAXIMUM) {
289 logit("Limited memory: %u MB; limit %lu MB",
290 largememory, LARGE_MAXIMUM);
291 largememory = LARGE_MAXIMUM;
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);
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);
310 tinybits = tinywords << SHIFT_WORD;
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);
319 smallbits = smallwords << SHIFT_WORD;
322 * dynamically determine available memory
324 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
325 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
327 largebits = largewords << SHIFT_WORD;
328 largenumbers = largebits * 2; /* even numbers excluded */
330 /* validation check: count the number of primes tried */
335 * Generate random starting point for subprime search, or use
336 * specified parameter.
338 largebase = BN_new();
340 BN_rand(largebase, power, 1, 1);
342 BN_copy(largebase, start);
345 BN_set_bit(largebase, 0);
349 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
350 largenumbers, power);
351 debug2("start point: 0x%s", BN_bn2hex(largebase));
356 for (i = 0; i < tinybits; i++) {
357 if (BIT_TEST(TinySieve, i))
358 continue; /* 2*i+3 is composite */
360 /* The next tiny prime */
363 /* Mark all multiples of t */
364 for (j = i + t; j < tinybits; j += t)
365 BIT_SET(TinySieve, j);
371 * Start the small block search at the next possible prime. To avoid
372 * fencepost errors, the last pass is skipped.
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 */
381 /* The next tiny prime */
386 s = 0; /* t divides into smallbase exactly */
388 /* smallbase+s is first entry divisible by t */
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
398 s += t; /* Make smallbase+s odd, and s even */
400 /* Mark all multiples of 2*t */
401 for (s /= 2; s < smallbits; s += t)
402 BIT_SET(SmallSieve, s);
408 for (i = 0; i < smallbits; i++) {
409 if (BIT_TEST(SmallSieve, i))
410 continue; /* 2*i+smallbase is composite */
412 /* The next small prime */
413 sieve_large((2 * i) + smallbase);
416 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
421 logit("%.24s Sieved with %u small primes in %ld seconds",
422 ctime(&time_stop), largetries, (long) (time_stop - time_start));
424 for (j = r = 0; j < largebits; j++) {
425 if (BIT_TEST(LargeSieve, j))
426 continue; /* Definitely composite, skip */
428 debug2("test q = largebase+%u", 2 * j);
429 BN_set_word(q, 2 * j);
430 BN_add(q, q, largebase);
431 if (qfileout(out, QTYPE_SOPHIE_GERMAIN, QTEST_SIEVE,
432 largetries, (power - 1) /* MSB */, (0), q) == -1) {
446 logit("%.24s Found %u candidates", ctime(&time_stop), r);
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
458 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted)
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;
468 if (trials < TRIAL_MINIMUM) {
469 error("Minimum primality trials is %d", TRIAL_MINIMUM);
479 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
480 ctime(&time_start), trials, generator_wanted);
483 lp = xmalloc(QLINESIZE + 1);
484 while (fgets(lp, QLINESIZE, in) != NULL) {
488 if (ll < 14 || *lp == '!' || *lp == '#') {
489 debug2("%10u: comment or short line", count_in);
493 /* XXX - fragile parser */
495 cp = &lp[14]; /* (skip) */
498 in_type = strtoul(cp, &cp, 10);
501 in_tests = strtoul(cp, &cp, 10);
503 if (in_tests & QTEST_COMPOSITE) {
504 debug2("%10u: known composite", count_in);
509 in_tries = strtoul(cp, &cp, 10);
511 /* size (most significant bit) */
512 in_size = strtoul(cp, &cp, 10);
514 /* generator (hex) */
515 generator_known = strtoul(cp, &cp, 16);
517 /* Skip white space */
518 cp += strspn(cp, " ");
522 case QTYPE_SOPHIE_GERMAIN:
523 debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
532 case QTYPE_UNSTRUCTURED:
537 debug2("%10u: (%u)", count_in, in_type);
544 debug2("Unknown prime type");
549 * due to earlier inconsistencies in interpretation, check
550 * the proposed bit size.
552 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
553 debug2("%10u: bit size %u mismatch", count_in, in_size);
556 if (in_size < QSIZE_MINIMUM) {
557 debug2("%10u: bit size %u too short", count_in, in_size);
561 if (in_tests & QTEST_MILLER_RABIN)
567 * guess unknown generator
569 if (generator_known == 0) {
570 if (BN_mod_word(p, 24) == 11)
572 else if (BN_mod_word(p, 12) == 5)
575 u_int32_t r = BN_mod_word(p, 10);
577 if (r == 3 || r == 7)
582 * skip tests when desired generator doesn't match
584 if (generator_wanted > 0 &&
585 generator_wanted != generator_known) {
586 debug2("%10u: generator %d != %d",
587 count_in, generator_known, generator_wanted);
592 * Primes with no known generator are useless for DH, so
595 if (generator_known == 0) {
596 debug2("%10u: no known generator", count_in);
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
607 * vast majority of composite q's.
609 if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
610 debug("%10u: q failed first possible prime test",
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
619 * will show up on the first Rabin-Miller iteration so it
620 * doesn't hurt to specify a high iteration count.
622 if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
623 debug("%10u: p is not prime", count_in);
626 debug("%10u: p is almost certainly prime", count_in);
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);
633 debug("%10u: q is almost certainly prime", count_in);
635 if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
636 in_tries, in_size, generator_known, p)) {
650 logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
651 ctime(&time_stop), count_out, count_possible,
652 (long) (time_stop - time_start));