moduli.c - security/openssh-library - Rivoreo Source Code Repositories

 /* $OpenBSD: moduli.c,v 1.1 2003/07/28 09:49:56 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>


 /*
  * Debugging defines
  */

 /* define DEBUG_LARGE 1 */
 /* define DEBUG_SMALL 1 */
 /* define DEBUG_TEST  1 */

 /*
  * 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;

 	debug2("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;
 		default:
 			debug2("%10u: (%u)", count_in, in_type);
 			a = p;
 			BN_hex2bn(&a, cp);
 			/* q = (p-1) / 2 */
 			BN_rshift(q, p, 1);
 			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;
 		}

 		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) {
 			debug2("%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)) {
 			debug2("%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);
 }
	/* $OpenBSD: moduli.c,v 1.1 2003/07/28 09:49:56 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>


	/*
	* Debugging defines
	*/

	/* define DEBUG_LARGE 1 */
	/* define DEBUG_SMALL 1 */
	/* define DEBUG_TEST 1 */

	/*
	* 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 232, as 655372 > 2*32-1. /
	#define TINY_NUMBER (1UL<<16)

	/* Ensure enough bit space for testing 2q. /
	#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 230 in 216 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;

	debug2("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 2s /
	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 4s /
	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; /* 2i+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; /* 2i+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 2t /
	for (s /= 2; s < smallbits; s += t)
	BIT_SET(SmallSieve, s);
	}

	/*
	* SmallSieve
	*/
	for (i = 0; i < smallbits; i++) {
	if (BIT_TEST(SmallSieve, i))
	continue; /* 2i+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 = 2q + 1 /
	BN_lshift(p, q, 1);
	BN_add_word(p, 1);
	in_size += 1;
	generator_known = 0;
	break;
	default:
	debug2("%10u: (%u)", count_in, in_type);
	a = p;
	BN_hex2bn(&a, cp);
	/* q = (p-1) / 2 */
	BN_rshift(q, p, 1);
	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;
	}

	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) {
	debug2("%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)) {
	debug2("%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);
	}