/* $OpenBSD: moduli.c,v 1.20 2007/02/24 03:30:11 ray 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 <sys/types.h>
#include <openssl/bn.h>
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <stdarg.h>
#include <time.h>
#include "xmalloc.h"
#include "log.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_SCHNORR (3)
#define QTYPE_SOPHIE_GERMAIN (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)
/*
* Using virtual memory can cause thrashing. This should be the largest
* number that is supported without a large amount of disk activity --
* that would increase the run time from hours to days or weeks!
*/
#define LARGE_MINIMUM (8UL) /* megabytes */
/*
* Do not increase this number beyond the unsigned integer bit size.
* Due to a multiple of 4, it must be LESS than 128 (yielding 2**30 bits).
*/
#define LARGE_MAXIMUM (127UL) /* megabytes */
/*
* 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
*/
/* Minimum number of primality tests to perform */
#define TRIAL_MINIMUM (4)
/*
* 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;
int gen_candidates(FILE *, u_int32_t, u_int32_t, BIGNUM *);
int prime_test(FILE *, FILE *, u_int32_t, u_int32_t);
/*
* 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-Germain 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, u_int32_t memory, u_int32_t 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;
u_int32_t i;
int ret = 0;
largememory = memory;
if (memory != 0 &&
(memory < LARGE_MINIMUM || memory > LARGE_MAXIMUM)) {
error("Invalid memory amount (min %ld, max %ld)",
LARGE_MINIMUM, LARGE_MAXIMUM);
return (-1);
}
/*
* 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 = xcalloc(tinywords, sizeof(u_int32_t));
tinybits = tinywords << SHIFT_WORD;
SmallSieve = xcalloc(smallwords, sizeof(u_int32_t));
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;
if ((q = BN_new()) == NULL)
fatal("BN_new failed");
/*
* Generate random starting point for subprime search, or use
* specified parameter.
*/
if ((largebase = BN_new()) == NULL)
fatal("BN_new failed");
if (start == NULL) {
if (BN_rand(largebase, power, 1, 1) == 0)
fatal("BN_rand failed");
} else {
if (BN_copy(largebase, start) == NULL)
fatal("BN_copy: failed");
}
/* ensure odd */
if (BN_set_bit(largebase, 0) == 0)
fatal("BN_set_bit: failed");
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);
if (BN_set_word(q, 2 * j) == 0)
fatal("BN_set_word failed");
if (BN_add(q, q, largebase) == 0)
fatal("BN_add failed");
if (qfileout(out, QTYPE_SOPHIE_GERMAIN, 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;
if (trials < TRIAL_MINIMUM) {
error("Minimum primality trials is %d", TRIAL_MINIMUM);
return (-1);
}
time(&time_start);
if ((p = BN_new()) == NULL)
fatal("BN_new failed");
if ((q = BN_new()) == NULL)
fatal("BN_new failed");
if ((ctx = BN_CTX_new()) == NULL)
fatal("BN_CTX_new failed");
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 + 1, in) != NULL) {
count_in++;
if (strlen(lp) < 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_GERMAIN:
debug2("%10u: (%u) Sophie-Germain", count_in, in_type);
a = q;
if (BN_hex2bn(&a, cp) == 0)
fatal("BN_hex2bn failed");
/* p = 2*q + 1 */
if (BN_lshift(p, q, 1) == 0)
fatal("BN_lshift failed");
if (BN_add_word(p, 1) == 0)
fatal("BN_add_word failed");
in_size += 1;
generator_known = 0;
break;
case QTYPE_UNSTRUCTURED:
case QTYPE_SAFE:
case QTYPE_SCHNORR:
case QTYPE_STRONG:
case QTYPE_UNKNOWN:
debug2("%10u: (%u)", count_in, in_type);
a = p;
if (BN_hex2bn(&a, cp) == 0)
fatal("BN_hex2bn failed");
/* q = (p-1) / 2 */
if (BN_rshift(q, p, 1) == 0)
fatal("BN_rshift failed");
break;
default:
debug2("Unknown prime type");
break;
}
/*
* due to earlier inconsistencies in interpretation, check
* the proposed bit size.
*/
if ((u_int32_t)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);
}
|
