moduli.c   [plain text]


/* $OpenBSD: moduli.c,v 1.5 2003/12/22 09:16:57 djm Exp $ */
/*
 * Copyright 1994 Phil Karn <karn@qualcomm.com>
 * Copyright 1996-1998, 2003 William Allen Simpson <wsimpson@greendragon.com>
 * Copyright 2000 Niels Provos <provos@citi.umich.edu>
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
 */

/*
 * Two-step process to generate safe primes for DHGEX
 *
 *  Sieve candidates for "safe" primes,
 *  suitable for use as Diffie-Hellman moduli;
 *  that is, where q = (p-1)/2 is also prime.
 *
 * First step: generate candidate primes (memory intensive)
 * Second step: test primes' safety (processor intensive)
 */

#include "includes.h"
#include "moduli.h"
#include "xmalloc.h"
#include "log.h"

#include <openssl/bn.h>

/*
 * File output defines
 */

/* need line long enough for largest moduli plus headers */
#define QLINESIZE               (100+8192)

/* Type: decimal.
 * Specifies the internal structure of the prime modulus.
 */
#define QTYPE_UNKNOWN           (0)
#define QTYPE_UNSTRUCTURED      (1)
#define QTYPE_SAFE              (2)
#define QTYPE_SCHNOOR           (3)
#define QTYPE_SOPHIE_GERMAINE   (4)
#define QTYPE_STRONG            (5)

/* Tests: decimal (bit field).
 * Specifies the methods used in checking for primality.
 * Usually, more than one test is used.
 */
#define QTEST_UNTESTED          (0x00)
#define QTEST_COMPOSITE         (0x01)
#define QTEST_SIEVE             (0x02)
#define QTEST_MILLER_RABIN      (0x04)
#define QTEST_JACOBI            (0x08)
#define QTEST_ELLIPTIC          (0x10)

/*
 * Size: decimal.
 * Specifies the number of the most significant bit (0 to M).
 * WARNING: internally, usually 1 to N.
 */
#define QSIZE_MINIMUM           (511)

/*
 * Prime sieving defines
 */

/* Constant: assuming 8 bit bytes and 32 bit words */
#define SHIFT_BIT       (3)
#define SHIFT_BYTE      (2)
#define SHIFT_WORD      (SHIFT_BIT+SHIFT_BYTE)
#define SHIFT_MEGABYTE  (20)
#define SHIFT_MEGAWORD  (SHIFT_MEGABYTE-SHIFT_BYTE)

/*
 * Constant: when used with 32-bit integers, the largest sieve prime
 * has to be less than 2**32.
 */
#define SMALL_MAXIMUM   (0xffffffffUL)

/* Constant: can sieve all primes less than 2**32, as 65537**2 > 2**32-1. */
#define TINY_NUMBER     (1UL<<16)

/* Ensure enough bit space for testing 2*q. */
#define TEST_MAXIMUM    (1UL<<16)
#define TEST_MINIMUM    (QSIZE_MINIMUM + 1)
/* real TEST_MINIMUM    (1UL << (SHIFT_WORD - TEST_POWER)) */
#define TEST_POWER      (3)	/* 2**n, n < SHIFT_WORD */

/* bit operations on 32-bit words */
#define BIT_CLEAR(a,n)  ((a)[(n)>>SHIFT_WORD] &= ~(1L << ((n) & 31)))
#define BIT_SET(a,n)    ((a)[(n)>>SHIFT_WORD] |= (1L << ((n) & 31)))
#define BIT_TEST(a,n)   ((a)[(n)>>SHIFT_WORD] & (1L << ((n) & 31)))

/*
 * Prime testing defines
 */

/*
 * Sieving data (XXX - move to struct)
 */

/* sieve 2**16 */
static u_int32_t *TinySieve, tinybits;

/* sieve 2**30 in 2**16 parts */
static u_int32_t *SmallSieve, smallbits, smallbase;

/* sieve relative to the initial value */
static u_int32_t *LargeSieve, largewords, largetries, largenumbers;
static u_int32_t largebits, largememory;	/* megabytes */
static BIGNUM *largebase;


/*
 * print moduli out in consistent form,
 */
static int
qfileout(FILE * ofile, u_int32_t otype, u_int32_t otests, u_int32_t otries,
    u_int32_t osize, u_int32_t ogenerator, BIGNUM * omodulus)
{
	struct tm *gtm;
	time_t time_now;
	int res;

	time(&time_now);
	gtm = gmtime(&time_now);

	res = fprintf(ofile, "%04d%02d%02d%02d%02d%02d %u %u %u %u %x ",
	    gtm->tm_year + 1900, gtm->tm_mon + 1, gtm->tm_mday,
	    gtm->tm_hour, gtm->tm_min, gtm->tm_sec,
	    otype, otests, otries, osize, ogenerator);

	if (res < 0)
		return (-1);

	if (BN_print_fp(ofile, omodulus) < 1)
		return (-1);

	res = fprintf(ofile, "\n");
	fflush(ofile);

	return (res > 0 ? 0 : -1);
}


/*
 ** Sieve p's and q's with small factors
 */
static void
sieve_large(u_int32_t s)
{
	u_int32_t r, u;

	debug3("sieve_large %u", s);
	largetries++;
	/* r = largebase mod s */
	r = BN_mod_word(largebase, s);
	if (r == 0)
		u = 0; /* s divides into largebase exactly */
	else
		u = s - r; /* largebase+u is first entry divisible by s */

	if (u < largebits * 2) {
		/*
		 * The sieve omits p's and q's divisible by 2, so ensure that
		 * largebase+u is odd. Then, step through the sieve in
		 * increments of 2*s
		 */
		if (u & 0x1)
			u += s; /* Make largebase+u odd, and u even */

		/* Mark all multiples of 2*s */
		for (u /= 2; u < largebits; u += s)
			BIT_SET(LargeSieve, u);
	}

	/* r = p mod s */
	r = (2 * r + 1) % s;
	if (r == 0)
		u = 0; /* s divides p exactly */
	else
		u = s - r; /* p+u is first entry divisible by s */

	if (u < largebits * 4) {
		/*
		 * The sieve omits p's divisible by 4, so ensure that
		 * largebase+u is not. Then, step through the sieve in
		 * increments of 4*s
		 */
		while (u & 0x3) {
			if (SMALL_MAXIMUM - u < s)
				return;
			u += s;
		}

		/* Mark all multiples of 4*s */
		for (u /= 4; u < largebits; u += s)
			BIT_SET(LargeSieve, u);
	}
}

/*
 * list candidates for Sophie-Germaine primes (where q = (p-1)/2)
 * to standard output.
 * The list is checked against small known primes (less than 2**30).
 */
int
gen_candidates(FILE *out, int memory, int power, BIGNUM *start)
{
	BIGNUM *q;
	u_int32_t j, r, s, t;
	u_int32_t smallwords = TINY_NUMBER >> 6;
	u_int32_t tinywords = TINY_NUMBER >> 6;
	time_t time_start, time_stop;
	int i, ret = 0;

	largememory = memory;

	/*
	 * Set power to the length in bits of the prime to be generated.
	 * This is changed to 1 less than the desired safe prime moduli p.
	 */
	if (power > TEST_MAXIMUM) {
		error("Too many bits: %u > %lu", power, TEST_MAXIMUM);
		return (-1);
	} else if (power < TEST_MINIMUM) {
		error("Too few bits: %u < %u", power, TEST_MINIMUM);
		return (-1);
	}
	power--; /* decrement before squaring */

	/*
	 * The density of ordinary primes is on the order of 1/bits, so the
	 * density of safe primes should be about (1/bits)**2. Set test range
	 * to something well above bits**2 to be reasonably sure (but not
	 * guaranteed) of catching at least one safe prime.
	 */
	largewords = ((power * power) >> (SHIFT_WORD - TEST_POWER));

	/*
	 * Need idea of how much memory is available. We don't have to use all
	 * of it.
	 */
	if (largememory > LARGE_MAXIMUM) {
		logit("Limited memory: %u MB; limit %lu MB",
		    largememory, LARGE_MAXIMUM);
		largememory = LARGE_MAXIMUM;
	}

	if (largewords <= (largememory << SHIFT_MEGAWORD)) {
		logit("Increased memory: %u MB; need %u bytes",
		    largememory, (largewords << SHIFT_BYTE));
		largewords = (largememory << SHIFT_MEGAWORD);
	} else if (largememory > 0) {
		logit("Decreased memory: %u MB; want %u bytes",
		    largememory, (largewords << SHIFT_BYTE));
		largewords = (largememory << SHIFT_MEGAWORD);
	}

	TinySieve = calloc(tinywords, sizeof(u_int32_t));
	if (TinySieve == NULL) {
		error("Insufficient memory for tiny sieve: need %u bytes",
		    tinywords << SHIFT_BYTE);
		exit(1);
	}
	tinybits = tinywords << SHIFT_WORD;

	SmallSieve = calloc(smallwords, sizeof(u_int32_t));
	if (SmallSieve == NULL) {
		error("Insufficient memory for small sieve: need %u bytes",
		    smallwords << SHIFT_BYTE);
		xfree(TinySieve);
		exit(1);
	}
	smallbits = smallwords << SHIFT_WORD;

	/*
	 * dynamically determine available memory
	 */
	while ((LargeSieve = calloc(largewords, sizeof(u_int32_t))) == NULL)
		largewords -= (1L << (SHIFT_MEGAWORD - 2)); /* 1/4 MB chunks */

	largebits = largewords << SHIFT_WORD;
	largenumbers = largebits * 2;	/* even numbers excluded */

	/* validation check: count the number of primes tried */
	largetries = 0;
	q = BN_new();

	/*
	 * Generate random starting point for subprime search, or use
	 * specified parameter.
	 */
	largebase = BN_new();
	if (start == NULL)
		BN_rand(largebase, power, 1, 1);
	else
		BN_copy(largebase, start);

	/* ensure odd */
	BN_set_bit(largebase, 0);

	time(&time_start);

	logit("%.24s Sieve next %u plus %u-bit", ctime(&time_start),
	    largenumbers, power);
	debug2("start point: 0x%s", BN_bn2hex(largebase));

	/*
	 * TinySieve
	 */
	for (i = 0; i < tinybits; i++) {
		if (BIT_TEST(TinySieve, i))
			continue; /* 2*i+3 is composite */

		/* The next tiny prime */
		t = 2 * i + 3;

		/* Mark all multiples of t */
		for (j = i + t; j < tinybits; j += t)
			BIT_SET(TinySieve, j);

		sieve_large(t);
	}

	/*
	 * Start the small block search at the next possible prime. To avoid
	 * fencepost errors, the last pass is skipped.
	 */
	for (smallbase = TINY_NUMBER + 3;
	     smallbase < (SMALL_MAXIMUM - TINY_NUMBER);
	     smallbase += TINY_NUMBER) {
		for (i = 0; i < tinybits; i++) {
			if (BIT_TEST(TinySieve, i))
				continue; /* 2*i+3 is composite */

			/* The next tiny prime */
			t = 2 * i + 3;
			r = smallbase % t;

			if (r == 0) {
				s = 0; /* t divides into smallbase exactly */
			} else {
				/* smallbase+s is first entry divisible by t */
				s = t - r;
			}

			/*
			 * The sieve omits even numbers, so ensure that
			 * smallbase+s is odd. Then, step through the sieve
			 * in increments of 2*t
			 */
			if (s & 1)
				s += t; /* Make smallbase+s odd, and s even */

			/* Mark all multiples of 2*t */
			for (s /= 2; s < smallbits; s += t)
				BIT_SET(SmallSieve, s);
		}

		/*
		 * SmallSieve
		 */
		for (i = 0; i < smallbits; i++) {
			if (BIT_TEST(SmallSieve, i))
				continue; /* 2*i+smallbase is composite */

			/* The next small prime */
			sieve_large((2 * i) + smallbase);
		}

		memset(SmallSieve, 0, smallwords << SHIFT_BYTE);
	}

	time(&time_stop);

	logit("%.24s Sieved with %u small primes in %ld seconds",
	    ctime(&time_stop), largetries, (long) (time_stop - time_start));

	for (j = r = 0; j < largebits; j++) {
		if (BIT_TEST(LargeSieve, j))
			continue; /* Definitely composite, skip */

		debug2("test q = largebase+%u", 2 * j);
		BN_set_word(q, 2 * j);
		BN_add(q, q, largebase);
		if (qfileout(out, QTYPE_SOPHIE_GERMAINE, QTEST_SIEVE,
		    largetries, (power - 1) /* MSB */, (0), q) == -1) {
			ret = -1;
			break;
		}

		r++; /* count q */
	}

	time(&time_stop);

	xfree(LargeSieve);
	xfree(SmallSieve);
	xfree(TinySieve);

	logit("%.24s Found %u candidates", ctime(&time_stop), r);

	return (ret);
}

/*
 * perform a Miller-Rabin primality test
 * on the list of candidates
 * (checking both q and p)
 * The result is a list of so-call "safe" primes
 */
int
prime_test(FILE *in, FILE *out, u_int32_t trials,
    u_int32_t generator_wanted)
{
	BIGNUM *q, *p, *a;
	BN_CTX *ctx;
	char *cp, *lp;
	u_int32_t count_in = 0, count_out = 0, count_possible = 0;
	u_int32_t generator_known, in_tests, in_tries, in_type, in_size;
	time_t time_start, time_stop;
	int res;

	time(&time_start);

	p = BN_new();
	q = BN_new();
	ctx = BN_CTX_new();

	debug2("%.24s Final %u Miller-Rabin trials (%x generator)",
	    ctime(&time_start), trials, generator_wanted);

	res = 0;
	lp = xmalloc(QLINESIZE + 1);
	while (fgets(lp, QLINESIZE, in) != NULL) {
		int ll = strlen(lp);

		count_in++;
		if (ll < 14 || *lp == '!' || *lp == '#') {
			debug2("%10u: comment or short line", count_in);
			continue;
		}

		/* XXX - fragile parser */
		/* time */
		cp = &lp[14];	/* (skip) */

		/* type */
		in_type = strtoul(cp, &cp, 10);

		/* tests */
		in_tests = strtoul(cp, &cp, 10);

		if (in_tests & QTEST_COMPOSITE) {
			debug2("%10u: known composite", count_in);
			continue;
		}

		/* tries */
		in_tries = strtoul(cp, &cp, 10);

		/* size (most significant bit) */
		in_size = strtoul(cp, &cp, 10);

		/* generator (hex) */
		generator_known = strtoul(cp, &cp, 16);

		/* Skip white space */
		cp += strspn(cp, " ");

		/* modulus (hex) */
		switch (in_type) {
		case QTYPE_SOPHIE_GERMAINE:
			debug2("%10u: (%u) Sophie-Germaine", count_in, in_type);
			a = q;
			BN_hex2bn(&a, cp);
			/* p = 2*q + 1 */
			BN_lshift(p, q, 1);
			BN_add_word(p, 1);
			in_size += 1;
			generator_known = 0;
			break;
		case QTYPE_UNSTRUCTURED:
		case QTYPE_SAFE:
		case QTYPE_SCHNOOR:
		case QTYPE_STRONG:
		case QTYPE_UNKNOWN:
			debug2("%10u: (%u)", count_in, in_type);
			a = p;
			BN_hex2bn(&a, cp);
			/* q = (p-1) / 2 */
			BN_rshift(q, p, 1);
			break;
		default:
			debug2("Unknown prime type");
			break;
		}

		/*
		 * due to earlier inconsistencies in interpretation, check
		 * the proposed bit size.
		 */
		if (BN_num_bits(p) != (in_size + 1)) {
			debug2("%10u: bit size %u mismatch", count_in, in_size);
			continue;
		}
		if (in_size < QSIZE_MINIMUM) {
			debug2("%10u: bit size %u too short", count_in, in_size);
			continue;
		}

		if (in_tests & QTEST_MILLER_RABIN)
			in_tries += trials;
		else
			in_tries = trials;

		/*
		 * guess unknown generator
		 */
		if (generator_known == 0) {
			if (BN_mod_word(p, 24) == 11)
				generator_known = 2;
			else if (BN_mod_word(p, 12) == 5)
				generator_known = 3;
			else {
				u_int32_t r = BN_mod_word(p, 10);

				if (r == 3 || r == 7)
					generator_known = 5;
			}
		}
		/*
		 * skip tests when desired generator doesn't match
		 */
		if (generator_wanted > 0 &&
		    generator_wanted != generator_known) {
			debug2("%10u: generator %d != %d",
			    count_in, generator_known, generator_wanted);
			continue;
		}

		/*
		 * Primes with no known generator are useless for DH, so
		 * skip those.
		 */
		if (generator_known == 0) {
			debug2("%10u: no known generator", count_in);
			continue;
		}

		count_possible++;

		/*
		 * The (1/4)^N performance bound on Miller-Rabin is
		 * extremely pessimistic, so don't spend a lot of time
		 * really verifying that q is prime until after we know
		 * that p is also prime. A single pass will weed out the
		 * vast majority of composite q's.
		 */
		if (BN_is_prime(q, 1, NULL, ctx, NULL) <= 0) {
			debug("%10u: q failed first possible prime test",
			    count_in);
			continue;
		}

		/*
		 * q is possibly prime, so go ahead and really make sure
		 * that p is prime. If it is, then we can go back and do
		 * the same for q. If p is composite, chances are that
		 * will show up on the first Rabin-Miller iteration so it
		 * doesn't hurt to specify a high iteration count.
		 */
		if (!BN_is_prime(p, trials, NULL, ctx, NULL)) {
			debug("%10u: p is not prime", count_in);
			continue;
		}
		debug("%10u: p is almost certainly prime", count_in);

		/* recheck q more rigorously */
		if (!BN_is_prime(q, trials - 1, NULL, ctx, NULL)) {
			debug("%10u: q is not prime", count_in);
			continue;
		}
		debug("%10u: q is almost certainly prime", count_in);

		if (qfileout(out, QTYPE_SAFE, (in_tests | QTEST_MILLER_RABIN),
		    in_tries, in_size, generator_known, p)) {
			res = -1;
			break;
		}

		count_out++;
	}

	time(&time_stop);
	xfree(lp);
	BN_free(p);
	BN_free(q);
	BN_CTX_free(ctx);

	logit("%.24s Found %u safe primes of %u candidates in %ld seconds",
	    ctime(&time_stop), count_out, count_possible,
	    (long) (time_stop - time_start));

	return (res);
}