1 /* $OpenBSD: moduli.c,v 1.38 2022/05/01 23:20:30 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)
44 #include <sys/types.h>
46 #include <openssl/bn.h>
47 #include <openssl/dh.h>
63 #include "openbsd-compat/openssl-compat.h"
69 /* need line long enough for largest moduli plus headers */
70 #define QLINESIZE (100+8192)
74 * Specifies the number of the most significant bit (0 to M).
75 * WARNING: internally, usually 1 to N.
77 #define QSIZE_MINIMUM (511)
80 * Prime sieving defines
83 /* Constant: assuming 8 bit bytes and 32 bit words */
85 #define SHIFT_BYTE (2)
86 #define SHIFT_WORD (SHIFT_BIT+SHIFT_BYTE)
87 #define SHIFT_MEGABYTE (20)
88 #define SHIFT_MEGAWORD (SHIFT_MEGABYTE-SHIFT_BYTE)
91 * Using virtual memory can cause thrashing. This should be the largest
92 * number that is supported without a large amount of disk activity --
93 * that would increase the run time from hours to days or weeks!
95 #define LARGE_MINIMUM (8UL) /* megabytes */
98 * Do not increase this number beyond the unsigned integer bit size.
99 * Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
101 #define LARGE_MAXIMUM (127UL) /* megabytes */
104 * Constant: when used with 32-bit integers, the largest sieve prime
105 * has to be less than 2**32.
107 #define SMALL_MAXIMUM (0xffffffffUL)
109 /* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
110 #define TINY_NUMBER (1UL<<16)
112 /* Ensure enough bit space for testing 2*q. */
113 #define TEST_MAXIMUM (1UL<<16)
114 #define TEST_MINIMUM (QSIZE_MINIMUM + 1)
115 /* real TEST_MINIMUM (1UL << (SHIFT_WORD - TEST_POWER)) */
116 #define TEST_POWER (3) /* 2**n, n < SHIFT_WORD */
118 /* bit operations on 32-bit words */
119 #define BIT_CLEAR(a,n) ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
120 #define BIT_SET(a,n) ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
121 #define BIT_TEST(a,n) ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
124 * Prime testing defines
127 /* Minimum number of primality tests to perform */
128 #define TRIAL_MINIMUM (4)
131 * Sieving data (XXX - move to struct)
135 static u_int32_t *TinySieve, tinybits;
137 /* sieve 2**30 in 2**16 parts */
138 static u_int32_t *SmallSieve, smallbits, smallbase;
140 /* sieve relative to the initial value */
141 static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
142 static u_int32_t largebits, largememory; /* megabytes */
143 static BIGNUM *largebase;
145 int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
146 int prime_test(FILE *, FILE *, u_int32_t, u_int32_t, char *, unsigned long,
150 * print moduli out in consistent form,
153 qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
154 u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
161 gtm = gmtime(&time_now);
165 res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
166 gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
167 gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
168 otype, otests, otries, osize, ogenerator);
173 if (BN_print_fp(ofile, omodulus) < 1)
176 res = fprintf(ofile, "\n");
179 return (res > 0 ? 0 : -1);
184 ** Sieve p's and q's with small factors
187 sieve_large(u_int32_t s32)
189 u_int64_t r, u, s = s32;
191 debug3("sieve_large %u", s32);
193 /* r = largebase mod s */
194 r = BN_mod_word(largebase, s32);
196 u = 0; /* s divides into largebase exactly */
198 u = s - r; /* largebase+u is first entry divisible by s */
200 if (u < largebits * 2ULL) {
202 * The sieve omits p's and q's divisible by 2, so ensure that
203 * largebase+u is odd. Then, step through the sieve in
207 u += s; /* Make largebase+u odd, and u even */
209 /* Mark all multiples of 2*s */
210 for (u /= 2; u < largebits; u += s)
211 BIT_SET(LargeSieve, u);
217 u = 0; /* s divides p exactly */
219 u = s - r; /* p+u is first entry divisible by s */
221 if (u < largebits * 4ULL) {
223 * The sieve omits p's divisible by 4, so ensure that
224 * largebase+u is not. Then, step through the sieve in
228 if (SMALL_MAXIMUM - u < s)
233 /* Mark all multiples of 4*s */
234 for (u /= 4; u < largebits; u += s)
235 BIT_SET(LargeSieve, u);
240 * list candidates for Sophie-Germain primes (where q = (p-1)/2)
241 * to standard output.
242 * The list is checked against small known primes (less than 2**30).
245 gen_candidates(FILE *out, u_int32_t memory, u_int32_t power, BIGNUM *start)
248 u_int32_t j, r, s, t;
249 u_int32_t smallwords = TINY_NUMBER >> 6;
250 u_int32_t tinywords = TINY_NUMBER >> 6;
251 time_t time_start, time_stop;
255 largememory = memory;
258 (memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
259 error("Invalid memory amount (min %ld, max %ld)",
260 LARGE_MINIMUM, LARGE_MAXIMUM);
265 * Set power to the length in bits of the prime to be generated.
266 * This is changed to 1 less than the desired safe prime moduli p.
268 if (power > TEST_MAXIMUM) {
269 error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
271 } else if (power < TEST_MINIMUM) {
272 error("Too few bits: %u < %u", power, TEST_MINIMUM);
275 power--; /* decrement before squaring */
278 * The density of ordinary primes is on the order of 1/bits, so the
279 * density of safe primes should be about (1/bits)**2. Set test range
280 * to something well above bits**2 to be reasonably sure (but not
281 * guaranteed) of catching at least one safe prime.
283 largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
286 * Need idea of how much memory is available. We don't have to use all
289 if (largememory > LARGE_MAXIMUM) {
290 logit("Limited memory: %u MB; limit %lu MB",
291 largememory, LARGE_MAXIMUM);
292 largememory = LARGE_MAXIMUM;
295 if (largewords <= (largememory << SHIFT_MEGAWORD)) {
296 logit("Increased memory: %u MB; need %u bytes",
297 largememory, (largewords << SHIFT_BYTE));
298 largewords = (largememory << SHIFT_MEGAWORD);
299 } else if (largememory > 0) {
300 logit("Decreased memory: %u MB; want %u bytes",
301 largememory, (largewords << SHIFT_BYTE));
302 largewords = (largememory << SHIFT_MEGAWORD);
305 TinySieve = xcalloc(tinywords, sizeof(u_int32_t));
306 tinybits = tinywords << SHIFT_WORD;
308 SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
309 smallbits = smallwords << SHIFT_WORD;
312 * dynamically determine available memory
314 while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
315 largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
317 largebits = largewords << SHIFT_WORD;
318 largenumbers = largebits * 2; /* even numbers excluded */
320 /* validation check: count the number of primes tried */
322 if ((q = BN_new()) == NULL)
323 fatal("BN_new failed");
326 * Generate random starting point for subprime search, or use
327 * specified parameter.
329 if ((largebase = BN_new()) == NULL)
330 fatal("BN_new failed");
332 if (BN_rand(largebase, power, 1, 1) == 0)
333 fatal("BN_rand failed");
335 if (BN_copy(largebase, start) == NULL)
336 fatal("BN_copy: failed");
340 if (BN_set_bit(largebase, 0) == 0)
341 fatal("BN_set_bit: failed");
345 logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
346 largenumbers, power);
347 debug2("start point: 0x%s", BN_bn2hex(largebase));
352 for (i = 0; i < tinybits; i++) {
353 if (BIT_TEST(TinySieve, i))
354 continue; /* 2*i+3 is composite */
356 /* The next tiny prime */
359 /* Mark all multiples of t */
360 for (j = i + t; j < tinybits; j += t)
361 BIT_SET(TinySieve, j);
367 * Start the small block search at the next possible prime. To avoid
368 * fencepost errors, the last pass is skipped.
370 for (smallbase = TINY_NUMBER + 3;
371 smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
372 smallbase += TINY_NUMBER) {
373 for (i = 0; i < tinybits; i++) {
374 if (BIT_TEST(TinySieve, i))
375 continue; /* 2*i+3 is composite */
377 /* The next tiny prime */
382 s = 0; /* t divides into smallbase exactly */
384 /* smallbase+s is first entry divisible by t */
389 * The sieve omits even numbers, so ensure that
390 * smallbase+s is odd. Then, step through the sieve
391 * in increments of 2*t
394 s += t; /* Make smallbase+s odd, and s even */
396 /* Mark all multiples of 2*t */
397 for (s /= 2; s < smallbits; s += t)
398 BIT_SET(SmallSieve, s);
404 for (i = 0; i < smallbits; i++) {
405 if (BIT_TEST(SmallSieve, i))
406 continue; /* 2*i+smallbase is composite */
408 /* The next small prime */
409 sieve_large((2 * i) + smallbase);
412 memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
417 logit("%.24s Sieved with %u small primes in %lld seconds",
418 ctime(&time_stop), largetries, (long long)(time_stop - time_start));
420 for (j = r = 0; j < largebits; j++) {
421 if (BIT_TEST(LargeSieve, j))
422 continue; /* Definitely composite, skip */
424 debug2("test q = largebase+%u", 2 * j);
425 if (BN_set_word(q, 2 * j) == 0)
426 fatal("BN_set_word failed");
427 if (BN_add(q, q, largebase) == 0)
428 fatal("BN_add failed");
429 if (qfileout(out, MODULI_TYPE_SOPHIE_GERMAIN,
430 MODULI_TESTS_SIEVE, largetries,
431 (power - 1) /* MSB */, (0), q) == -1) {
445 logit("%.24s Found %u candidates", ctime(&time_stop), r);
451 write_checkpoint(char *cpfile, u_int32_t lineno)
457 r = snprintf(tmp, sizeof(tmp), "%s.XXXXXXXXXX", cpfile);
458 if (r < 0 || r >= PATH_MAX) {
459 logit("write_checkpoint: temp pathname too long");
462 if ((r = mkstemp(tmp)) == -1) {
463 logit("mkstemp(%s): %s", tmp, strerror(errno));
466 if ((fp = fdopen(r, "w")) == NULL) {
467 logit("write_checkpoint: fdopen: %s", strerror(errno));
472 if (fprintf(fp, "%lu\n", (unsigned long)lineno) > 0 && fclose(fp) == 0
473 && rename(tmp, cpfile) == 0)
474 debug3("wrote checkpoint line %lu to '%s'",
475 (unsigned long)lineno, cpfile);
477 logit("failed to write to checkpoint file '%s': %s", cpfile,
482 read_checkpoint(char *cpfile)
485 unsigned long lineno = 0;
487 if ((fp = fopen(cpfile, "r")) == NULL)
489 if (fscanf(fp, "%lu\n", &lineno) < 1)
490 logit("Failed to load checkpoint from '%s'", cpfile);
492 logit("Loaded checkpoint from '%s' line %lu", cpfile, lineno);
500 unsigned long count = 0;
501 char lp[QLINESIZE + 1];
503 if (fseek(f, 0, SEEK_SET) != 0) {
504 debug("input file is not seekable");
507 while (fgets(lp, QLINESIZE + 1, f) != NULL)
510 debug("input file has %lu lines", count);
515 fmt_time(time_t seconds)
518 static char buf[128];
520 min = (seconds / 60) % 60;
521 hr = (seconds / 60 / 60) % 24;
522 day = seconds / 60 / 60 / 24;
524 snprintf(buf, sizeof buf, "%dd %d:%02d", day, hr, min);
526 snprintf(buf, sizeof buf, "%d:%02d", hr, min);
531 print_progress(unsigned long start_lineno, unsigned long current_lineno,
532 unsigned long end_lineno)
534 static time_t time_start, time_prev;
535 time_t time_now, elapsed;
536 unsigned long num_to_process, processed, remaining, percent, eta;
537 double time_per_line;
540 time_now = monotime();
541 if (time_start == 0) {
542 time_start = time_prev = time_now;
545 /* print progress after 1m then once per 5m */
546 if (time_now - time_prev < 5 * 60)
548 time_prev = time_now;
549 elapsed = time_now - time_start;
550 processed = current_lineno - start_lineno;
551 remaining = end_lineno - current_lineno;
552 num_to_process = end_lineno - start_lineno;
553 time_per_line = (double)elapsed / processed;
554 /* if we don't know how many we're processing just report count+time */
556 if (end_lineno == ULONG_MAX) {
557 logit("%.24s processed %lu in %s", ctime(&time_now),
558 processed, fmt_time(elapsed));
561 percent = 100 * processed / num_to_process;
562 eta = time_per_line * remaining;
563 eta_str = xstrdup(fmt_time(eta));
564 logit("%.24s processed %lu of %lu (%lu%%) in %s, ETA %s",
565 ctime(&time_now), processed, num_to_process, percent,
566 fmt_time(elapsed), eta_str);
571 * perform a Miller-Rabin primality test
572 * on the list of candidates
573 * (checking both q and p)
574 * The result is a list of so-call "safe" primes
577 prime_test(FILE *in, FILE *out, u_int32_t trials, u_int32_t generator_wanted,
578 char *checkpoint_file, unsigned long start_lineno, unsigned long num_lines)
582 u_int32_t count_in = 0, count_out = 0, count_possible = 0;
583 u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
584 unsigned long last_processed = 0, end_lineno;
585 time_t time_start, time_stop;
588 if (trials < TRIAL_MINIMUM) {
589 error("Minimum primality trials is %d", TRIAL_MINIMUM);
594 end_lineno = count_lines(in);
596 end_lineno = start_lineno + num_lines;
600 if ((p = BN_new()) == NULL)
601 fatal("BN_new failed");
602 if ((q = BN_new()) == NULL)
603 fatal("BN_new failed");
605 debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
606 ctime(&time_start), trials, generator_wanted);
608 if (checkpoint_file != NULL)
609 last_processed = read_checkpoint(checkpoint_file);
610 last_processed = start_lineno = MAXIMUM(last_processed, start_lineno);
611 if (end_lineno == ULONG_MAX)
612 debug("process from line %lu from pipe", last_processed);
614 debug("process from line %lu to line %lu", last_processed,
618 lp = xmalloc(QLINESIZE + 1);
619 while (fgets(lp, QLINESIZE + 1, in) != NULL && count_in < end_lineno) {
621 if (count_in <= last_processed) {
622 debug3("skipping line %u, before checkpoint or "
623 "specified start line", count_in);
626 if (checkpoint_file != NULL)
627 write_checkpoint(checkpoint_file, count_in);
628 print_progress(start_lineno, count_in, end_lineno);
629 if (strlen(lp) < 14 || *lp == '!' || *lp == '#') {
630 debug2("%10u: comment or short line", count_in);
634 /* XXX - fragile parser */
636 cp = &lp[14]; /* (skip) */
639 in_type = strtoul(cp, &cp, 10);
642 in_tests = strtoul(cp, &cp, 10);
644 if (in_tests & MODULI_TESTS_COMPOSITE) {
645 debug2("%10u: known composite", count_in);
650 in_tries = strtoul(cp, &cp, 10);
652 /* size (most significant bit) */
653 in_size = strtoul(cp, &cp, 10);
655 /* generator (hex) */
656 generator_known = strtoul(cp, &cp, 16);
658 /* Skip white space */
659 cp += strspn(cp, " ");
663 case MODULI_TYPE_SOPHIE_GERMAIN:
664 debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
666 if (BN_hex2bn(&a, cp) == 0)
667 fatal("BN_hex2bn failed");
669 if (BN_lshift(p, q, 1) == 0)
670 fatal("BN_lshift failed");
671 if (BN_add_word(p, 1) == 0)
672 fatal("BN_add_word failed");
676 case MODULI_TYPE_UNSTRUCTURED:
677 case MODULI_TYPE_SAFE:
678 case MODULI_TYPE_SCHNORR:
679 case MODULI_TYPE_STRONG:
680 case MODULI_TYPE_UNKNOWN:
681 debug2("%10u: (%u)", count_in, in_type);
683 if (BN_hex2bn(&a, cp) == 0)
684 fatal("BN_hex2bn failed");
686 if (BN_rshift(q, p, 1) == 0)
687 fatal("BN_rshift failed");
690 debug2("Unknown prime type");
695 * due to earlier inconsistencies in interpretation, check
696 * the proposed bit size.
698 if ((u_int32_t)BN_num_bits(p) != (in_size + 1)) {
699 debug2("%10u: bit size %u mismatch", count_in, in_size);
702 if (in_size < QSIZE_MINIMUM) {
703 debug2("%10u: bit size %u too short", count_in, in_size);
707 if (in_tests & MODULI_TESTS_MILLER_RABIN)
713 * guess unknown generator
715 if (generator_known == 0) {
716 if (BN_mod_word(p, 24) == 11)
719 u_int32_t r = BN_mod_word(p, 10);
721 if (r == 3 || r == 7)
726 * skip tests when desired generator doesn't match
728 if (generator_wanted > 0 &&
729 generator_wanted != generator_known) {
730 debug2("%10u: generator %d != %d",
731 count_in, generator_known, generator_wanted);
736 * Primes with no known generator are useless for DH, so
739 if (generator_known == 0) {
740 debug2("%10u: no known generator", count_in);
747 * The (1/4)^N performance bound on Miller-Rabin is
748 * extremely pessimistic, so don't spend a lot of time
749 * really verifying that q is prime until after we know
750 * that p is also prime. A single pass will weed out the
751 * vast majority of composite q's.
753 is_prime = BN_is_prime_ex(q, 1, NULL, NULL);
755 fatal("BN_is_prime_ex failed");
757 debug("%10u: q failed first possible prime test",
763 * q is possibly prime, so go ahead and really make sure
764 * that p is prime. If it is, then we can go back and do
765 * the same for q. If p is composite, chances are that
766 * will show up on the first Rabin-Miller iteration so it
767 * doesn't hurt to specify a high iteration count.
769 is_prime = BN_is_prime_ex(p, trials, NULL, NULL);
771 fatal("BN_is_prime_ex failed");
773 debug("%10u: p is not prime", count_in);
776 debug("%10u: p is almost certainly prime", count_in);
778 /* recheck q more rigorously */
779 is_prime = BN_is_prime_ex(q, trials - 1, NULL, NULL);
781 fatal("BN_is_prime_ex failed");
783 debug("%10u: q is not prime", count_in);
786 debug("%10u: q is almost certainly prime", count_in);
788 if (qfileout(out, MODULI_TYPE_SAFE,
789 in_tests | MODULI_TESTS_MILLER_RABIN,
790 in_tries, in_size, generator_known, p)) {
803 if (checkpoint_file != NULL)
804 unlink(checkpoint_file);
806 logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
807 ctime(&time_stop), count_out, count_possible,
808 (long) (time_stop - time_start));
813 #endif /* WITH_OPENSSL */