moduli.c revision 124208 
1/* $OpenBSD: moduli.c,v 1.1 2003/07/28 09:49:56 djm Exp $ */
2/*
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>
6 * All rights reserved.
7 *
8 * Redistribution and use in source and binary forms, with or without
9 * modification, are permitted provided that the following conditions
10 * are met:
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.
16 *
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.
27 */
28
29/*
30 * Two-step process to generate safe primes for DHGEX
31 *
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.
35 *
36 * First step: generate candidate primes (memory intensive)
37 * Second step: test primes' safety (processor intensive)
38 */
39
40#include "includes.h"
41#include "moduli.h"
42#include "xmalloc.h"
43#include "log.h"
44
45#include <openssl/bn.h>
46
47
48/*
49 * Debugging defines
50 */
51
52/* define DEBUG_LARGE 1 */
53/* define DEBUG_SMALL 1 */
54/* define DEBUG_TEST  1 */
55
56/*
57 * File output defines
58 */
59
60/* need line long enough for largest moduli plus headers */
61#define QLINESIZE               (100+8192)
62
63/* Type: decimal.
64 * Specifies the internal structure of the prime modulus.
65 */
66#define QTYPE_UNKNOWN           (0)
67#define QTYPE_UNSTRUCTURED      (1)
68#define QTYPE_SAFE              (2)
69#define QTYPE_SCHNOOR           (3)
70#define QTYPE_SOPHIE_GERMAINE   (4)
71#define QTYPE_STRONG            (5)
72
73/* Tests: decimal (bit field).
74 * Specifies the methods used in checking for primality.
75 * Usually, more than one test is used.
76 */
77#define QTEST_UNTESTED          (0x00)
78#define QTEST_COMPOSITE         (0x01)
79#define QTEST_SIEVE             (0x02)
80#define QTEST_MILLER_RABIN      (0x04)
81#define QTEST_JACOBI            (0x08)
82#define QTEST_ELLIPTIC          (0x10)
83
84/* Size: decimal.
85 * Specifies the number of the most significant bit (0 to M).
86 ** WARNING: internally, usually 1 to N.
87 */
88#define QSIZE_MINIMUM           (511)
89
90/*
91 * Prime sieving defines
92 */
93
94/* Constant: assuming 8 bit bytes and 32 bit words */
95#define SHIFT_BIT       (3)
96#define SHIFT_BYTE      (2)
97#define SHIFT_WORD      (SHIFT_BIT+SHIFT_BYTE)
98#define SHIFT_MEGABYTE  (20)
99#define SHIFT_MEGAWORD  (SHIFT_MEGABYTE-SHIFT_BYTE)
100
101/*
102 * Constant: when used with 32-bit integers, the largest sieve prime
103 * has to be less than 2**32.
104 */
105#define SMALL_MAXIMUM   (0xffffffffUL)
106
107/* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
108#define TINY_NUMBER     (1UL<<16)
109
110/* Ensure enough bit space for testing 2*q. */
111#define TEST_MAXIMUM    (1UL<<16)
112#define TEST_MINIMUM    (QSIZE_MINIMUM + 1)
113/* real TEST_MINIMUM    (1UL << (SHIFT_WORD - TEST_POWER)) */
114#define TEST_POWER      (3)	/* 2**n, n < SHIFT_WORD */
115
116/* bit operations on 32-bit words */
117#define BIT_CLEAR(a,n)  ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
118#define BIT_SET(a,n)    ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
119#define BIT_TEST(a,n)   ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))
120
121/*
122 * Prime testing defines
123 */
124
125/*
126 * Sieving data (XXX - move to struct)
127 */
128
129/* sieve 2**16 */
130static u_int32_t *TinySieve, tinybits;
131
132/* sieve 2**30 in 2**16 parts */
133static u_int32_t *SmallSieve, smallbits, smallbase;
134
135/* sieve relative to the initial value */
136static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
137static u_int32_t largebits, largememory;	/* megabytes */
138static BIGNUM *largebase;
139
140
141/*
142 * print moduli out in consistent form,
143 */
144static int
145qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
146    u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
147{
148	struct tm *gtm;
149	time_t time_now;
150	int res;
151
152	time(&time_now);
153	gtm = gmtime(&time_now);
154
155	res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
156	    gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
157	    gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
158	    otype, otests, otries, osize, ogenerator);
159
160	if (res < 0)
161		return (-1);
162
163	if (BN_print_fp(ofile, omodulus) < 1)
164		return (-1);
165
166	res = fprintf(ofile, "\n");
167	fflush(ofile);
168
169	return (res > 0 ? 0 : -1);
170}
171
172
173/*
174 ** Sieve p's and q's with small factors
175 */
176static void
177sieve_large(u_int32_t s)
178{
179	u_int32_t r, u;
180
181	debug2("sieve_large %u", s);
182	largetries++;
183	/* r = largebase mod s */
184	r = BN_mod_word(largebase, s);
185	if (r == 0)
186		u = 0; /* s divides into largebase exactly */
187	else
188		u = s - r; /* largebase+u is first entry divisible by s */
189
190	if (u < largebits * 2) {
191		/*
192		 * The sieve omits p's and q's divisible by 2, so ensure that
193		 * largebase+u is odd. Then, step through the sieve in
194		 * increments of 2*s
195		 */
196		if (u & 0x1)
197			u += s; /* Make largebase+u odd, and u even */
198
199		/* Mark all multiples of 2*s */
200		for (u /= 2; u < largebits; u += s)
201			BIT_SET(LargeSieve, u);
202	}
203
204	/* r = p mod s */
205	r = (2 * r + 1) % s;
206	if (r == 0)
207		u = 0; /* s divides p exactly */
208	else
209		u = s - r; /* p+u is first entry divisible by s */
210
211	if (u < largebits * 4) {
212		/*
213		 * The sieve omits p's divisible by 4, so ensure that
214		 * largebase+u is not. Then, step through the sieve in
215		 * increments of 4*s
216		 */
217		while (u & 0x3) {
218			if (SMALL_MAXIMUM - u < s)
219				return;
220			u += s;
221		}
222
223		/* Mark all multiples of 4*s */
224		for (u /= 4; u < largebits; u += s)
225			BIT_SET(LargeSieve, u);
226	}
227}
228
229/*
230 * list candidates for Sophie-Germaine primes (where q = (p-1)/2)
231 * to standard output.
232 * The list is checked against small known primes (less than 2**30).
233 */
234int
235gen_candidates(FILE *out, int memory, int power, BIGNUM *start)
236{
237	BIGNUM *q;
238	u_int32_t j, r, s, t;
239	u_int32_t smallwords = TINY_NUMBER >> 6;
240	u_int32_t tinywords = TINY_NUMBER >> 6;
241	time_t time_start, time_stop;
242	int i, ret = 0;
243
244	largememory = memory;
245
246	/*
247         * Set power to the length in bits of the prime to be generated.
248         * This is changed to 1 less than the desired safe prime moduli p.
249         */
250	if (power > TEST_MAXIMUM) {
251		error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
252		return (-1);
253	} else if (power < TEST_MINIMUM) {
254		error("Too few bits: %u < %u", power, TEST_MINIMUM);
255		return (-1);
256	}
257	power--; /* decrement before squaring */
258
259	/*
260         * The density of ordinary primes is on the order of 1/bits, so the
261         * density of safe primes should be about (1/bits)**2. Set test range
262         * to something well above bits**2 to be reasonably sure (but not
263         * guaranteed) of catching at least one safe prime.
264	 */
265	largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));
266
267	/*
268         * Need idea of how much memory is available. We don't have to use all
269         * of it.
270	 */
271	if (largememory > LARGE_MAXIMUM) {
272		logit("Limited memory: %u MB; limit %lu MB",
273		    largememory, LARGE_MAXIMUM);
274		largememory = LARGE_MAXIMUM;
275	}
276
277	if (largewords <= (largememory << SHIFT_MEGAWORD)) {
278		logit("Increased memory: %u MB; need %u bytes",
279		    largememory, (largewords << SHIFT_BYTE));
280		largewords = (largememory << SHIFT_MEGAWORD);
281	} else if (largememory > 0) {
282		logit("Decreased memory: %u MB; want %u bytes",
283		    largememory, (largewords << SHIFT_BYTE));
284		largewords = (largememory << SHIFT_MEGAWORD);
285	}
286
287	TinySieve = calloc(tinywords, sizeof(u_int32_t));
288	if (TinySieve == NULL) {
289		error("Insufficient memory for tiny sieve: need %u bytes",
290		    tinywords << SHIFT_BYTE);
291		exit(1);
292	}
293	tinybits = tinywords << SHIFT_WORD;
294
295	SmallSieve = calloc(smallwords, sizeof(u_int32_t));
296	if (SmallSieve == NULL) {
297		error("Insufficient memory for small sieve: need %u bytes",
298		    smallwords << SHIFT_BYTE);
299		xfree(TinySieve);
300		exit(1);
301	}
302	smallbits = smallwords << SHIFT_WORD;
303
304	/*
305	 * dynamically determine available memory
306	 */
307	while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
308		largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */
309
310	largebits = largewords << SHIFT_WORD;
311	largenumbers = largebits * 2;	/* even numbers excluded */
312
313	/* validation check: count the number of primes tried */
314	largetries = 0;
315	q = BN_new();
316
317	/*
318         * Generate random starting point for subprime search, or use
319         * specified parameter.
320	 */
321	largebase = BN_new();
322	if (start == NULL)
323		BN_rand(largebase, power, 1, 1);
324	else
325		BN_copy(largebase, start);
326
327	/* ensure odd */
328	BN_set_bit(largebase, 0);
329
330	time(&time_start);
331
332	logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
333	    largenumbers, power);
334	debug2("start point: 0x%s", BN_bn2hex(largebase));
335
336	/*
337         * TinySieve
338         */
339	for (i = 0; i < tinybits; i++) {
340		if (BIT_TEST(TinySieve, i))
341			continue; /* 2*i+3 is composite */
342
343		/* The next tiny prime */
344		t = 2 * i + 3;
345
346		/* Mark all multiples of t */
347		for (j = i + t; j < tinybits; j += t)
348			BIT_SET(TinySieve, j);
349
350		sieve_large(t);
351	}
352
353	/*
354         * Start the small block search at the next possible prime. To avoid
355         * fencepost errors, the last pass is skipped.
356         */
357	for (smallbase = TINY_NUMBER + 3;
358	     smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
359	     smallbase += TINY_NUMBER) {
360		for (i = 0; i < tinybits; i++) {
361			if (BIT_TEST(TinySieve, i))
362				continue; /* 2*i+3 is composite */
363
364			/* The next tiny prime */
365			t = 2 * i + 3;
366			r = smallbase % t;
367
368			if (r == 0) {
369				s = 0; /* t divides into smallbase exactly */
370			} else {
371				/* smallbase+s is first entry divisible by t */
372				s = t - r;
373			}
374
375			/*
376			 * The sieve omits even numbers, so ensure that
377			 * smallbase+s is odd. Then, step through the sieve
378			 * in increments of 2*t
379			 */
380			if (s & 1)
381				s += t; /* Make smallbase+s odd, and s even */
382
383			/* Mark all multiples of 2*t */
384			for (s /= 2; s < smallbits; s += t)
385				BIT_SET(SmallSieve, s);
386		}
387
388		/*
389                 * SmallSieve
390                 */
391		for (i = 0; i < smallbits; i++) {
392			if (BIT_TEST(SmallSieve, i))
393				continue; /* 2*i+smallbase is composite */
394
395			/* The next small prime */
396			sieve_large((2 * i) + smallbase);
397		}
398
399		memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
400	}
401
402	time(&time_stop);
403
404	logit("%.24s Sieved with %u small primes in %ld seconds",
405	    ctime(&time_stop), largetries, (long) (time_stop - time_start));
406
407	for (j = r = 0; j < largebits; j++) {
408		if (BIT_TEST(LargeSieve, j))
409			continue; /* Definitely composite, skip */
410
411		debug2("test q = largebase+%u", 2 * j);
412		BN_set_word(q, 2 * j);
413		BN_add(q, q, largebase);
414		if (qfileout(out, QTYPE_SOPHIE_GERMAINE, QTEST_SIEVE,
415		    largetries, (power - 1) /* MSB */, (0), q) == -1) {
416			ret = -1;
417			break;
418		}
419
420		r++; /* count q */
421	}
422
423	time(&time_stop);
424
425	xfree(LargeSieve);
426	xfree(SmallSieve);
427	xfree(TinySieve);
428
429	logit("%.24s Found %u candidates", ctime(&time_stop), r);
430
431	return (ret);
432}
433
434/*
435 * perform a Miller-Rabin primality test
436 * on the list of candidates
437 * (checking both q and p)
438 * The result is a list of so-call "safe" primes
439 */
440int
441prime_test(FILE *in, FILE *out, u_int32_t trials,
442    u_int32_t generator_wanted)
443{
444	BIGNUM *q, *p, *a;
445	BN_CTX *ctx;
446	char *cp, *lp;
447	u_int32_t count_in = 0, count_out = 0, count_possible = 0;
448	u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
449	time_t time_start, time_stop;
450	int res;
451
452	time(&time_start);
453
454	p = BN_new();
455	q = BN_new();
456	ctx = BN_CTX_new();
457
458	debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
459	    ctime(&time_start), trials, generator_wanted);
460
461	res = 0;
462	lp = xmalloc(QLINESIZE + 1);
463	while (fgets(lp, QLINESIZE, in) != NULL) {
464		int ll = strlen(lp);
465
466		count_in++;
467		if (ll < 14 || *lp == '!' || *lp == '#') {
468			debug2("%10u: comment or short line", count_in);
469			continue;
470		}
471
472		/* XXX - fragile parser */
473		/* time */
474		cp = &lp[14];	/* (skip) */
475
476		/* type */
477		in_type = strtoul(cp, &cp, 10);
478
479		/* tests */
480		in_tests = strtoul(cp, &cp, 10);
481
482		if (in_tests & QTEST_COMPOSITE) {
483			debug2("%10u: known composite", count_in);
484			continue;
485		}
486		/* tries */
487		in_tries = strtoul(cp, &cp, 10);
488
489		/* size (most significant bit) */
490		in_size = strtoul(cp, &cp, 10);
491
492		/* generator (hex) */
493		generator_known = strtoul(cp, &cp, 16);
494
495		/* Skip white space */
496		cp += strspn(cp, " ");
497
498		/* modulus (hex) */
499		switch (in_type) {
500		case QTYPE_SOPHIE_GERMAINE:
501			debug2("%10u: (%u) Sophie-Germaine", count_in, in_type);
502			a = q;
503			BN_hex2bn(&a, cp);
504			/* p = 2*q + 1 */
505			BN_lshift(p, q, 1);
506			BN_add_word(p, 1);
507			in_size += 1;
508			generator_known = 0;
509			break;
510		default:
511			debug2("%10u: (%u)", count_in, in_type);
512			a = p;
513			BN_hex2bn(&a, cp);
514			/* q = (p-1) / 2 */
515			BN_rshift(q, p, 1);
516			break;
517		}
518
519		/*
520		 * due to earlier inconsistencies in interpretation, check
521		 * the proposed bit size.
522		 */
523		if (BN_num_bits(p) != (in_size + 1)) {
524			debug2("%10u: bit size %u mismatch", count_in, in_size);
525			continue;
526		}
527		if (in_size < QSIZE_MINIMUM) {
528			debug2("%10u: bit size %u too short", count_in, in_size);
529			continue;
530		}
531
532		if (in_tests & QTEST_MILLER_RABIN)
533			in_tries += trials;
534		else
535			in_tries = trials;
536		/*
537		 * guess unknown generator
538		 */
539		if (generator_known == 0) {
540			if (BN_mod_word(p, 24) == 11)
541				generator_known = 2;
542			else if (BN_mod_word(p, 12) == 5)
543				generator_known = 3;
544			else {
545				u_int32_t r = BN_mod_word(p, 10);
546
547				if (r == 3 || r == 7) {
548					generator_known = 5;
549				}
550			}
551		}
552		/*
553		 * skip tests when desired generator doesn't match
554		 */
555		if (generator_wanted > 0 &&
556		    generator_wanted != generator_known) {
557			debug2("%10u: generator %d != %d",
558			    count_in, generator_known, generator_wanted);
559			continue;
560		}
561
562		count_possible++;
563
564		/*
565		 * The (1/4)^N performance bound on Miller-Rabin is
566		 * extremely pessimistic, so don't spend a lot of time
567		 * really verifying that q is prime until after we know
568		 * that p is also prime. A single pass will weed out the
569		 * vast majority of composite q's.
570		 */
571		if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
572			debug2("%10u: q failed first possible prime test",
573			    count_in);
574			continue;
575		}
576
577		/*
578		 * q is possibly prime, so go ahead and really make sure
579		 * that p is prime. If it is, then we can go back and do
580		 * the same for q. If p is composite, chances are that
581		 * will show up on the first Rabin-Miller iteration so it
582		 * doesn't hurt to specify a high iteration count.
583		 */
584		if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
585			debug2("%10u: p is not prime", count_in);
586			continue;
587		}
588		debug("%10u: p is almost certainly prime", count_in);
589
590		/* recheck q more rigorously */
591		if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
592			debug("%10u: q is not prime", count_in);
593			continue;
594		}
595		debug("%10u: q is almost certainly prime", count_in);
596
597		if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
598		    in_tries, in_size, generator_known, p)) {
599			res = -1;
600			break;
601		}
602
603		count_out++;
604	}
605
606	time(&time_stop);
607	xfree(lp);
608	BN_free(p);
609	BN_free(q);
610	BN_CTX_free(ctx);
611
612	logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
613	    ctime(&time_stop), count_out, count_possible,
614	    (long) (time_stop - time_start));
615
616	return (res);
617}
618