bndsaprime.c

著名的加密软件的应用于电子邮件中

/*
 * bndsaprime.c - Digital Signature Algorithm prime generation using the
 * bignum library and sieving.
 *
 * Copyright (C) 1995-1997 Pretty Good Privacy, Inc. All rights reserved.
 *
 * Written by Colin Plumb.
 *
 * $Id: bndsaprime.c,v 1.2.2.1 1997/06/07 09:49:27 mhw Exp $
 */
#include <stdarg.h>
#include <string.h>
#if BNDEBUG
#include <stdio.h>
#endif
#include "bnkludge.h"

#include "bn.h"
#include "bnimem.h"	 /* For bniMemWipe */
#include "bndsaprime.h"
#include "bnsieve.h"
#include "pgpDebug.h"

/* Size of the sieve area (can be up to 65536/8 = 8192) */
#define SIEVE 1024

/*
* Helper function that does the slow primality test.
* bn is the input bignum; a and e are temporary buffers that are
* allocated by the caller to save overhead.
*
* Returns 0 if prime, >0 if not prime, and -1 on error (out of memory).
* If not prime, the return value is the number of modular exponentiations
* performed. Calls the progress function with progress indicators as
* tests are passed.
*
* Currently, the testing performs Euler tests, to the bases 2, 3, 5, 7,
* 11, 13 and 17. (An Euler test is a slightly strengthened Fermat test;
* Euler pseudoprimes are a subset of Fermat pseudoprimes.) Some people
* worry that this might not be enough. Number theorists may wish to
* generate primality proofs, but for random inputs, this returns non-primes
* with a probability which is quite negligible, which is good enough.
* NOTE that for NON-random inputs, there are indeed numbers which are
* strong pseudoprimes to a lot of bases. Up to 1/4 of all bases 0 < b < n.
*
* It has been proved (see Carl Pomerance, "On the Distribution of
* Pseudoprimes", Math. Comp. v.37 (1981) pp. 587-593) that the number
* of pseudoprimes (composite numbers that pass a Fermat test to the
* base 2) less than x is bounded by:
* exp(ln(x)^(5/14)) <= P_2(x)	### CHECK THIS FORMULA - it looks wrong! ###
* P_2(x) <= x * exp(-1/2 * ln(x) * ln(ln(ln(x))) / ln(ln(x))).
* Thus, the local density of Pseudoprimes near x is at most
* exp(-1/2 * ln(x) * ln(ln(ln(x))) / ln(ln(x))), and at least
* exp(ln(x)^(5/14) - ln(x)). Here are some values of this function
* for various k-bit numbers x = 2^k:
* Bits	Density <=	Bit equivalent	Density >=	Bit equivalent
* 128	3.577869e-07	21.414396	4.202213e-37	 120.840190
* 192	4.175629e-10	31.157288	4.936250e-56	 183.724558
	* 256 5.804314e-13		40.647940		4.977813e-75		246.829095
	* 384 1.578039e-18		59.136573		3.938861e-113		373.400096
	* 512 5.858255e-24		77.175803		2.563353e-151		500.253110
	* 768 1.489276e-34		112.370944		7.872825e-228		754.422724
	* 1024 6.633188e-45		146.757062		1.882404e-304		1008.953565
*
* As you can see, there's quite a bit of slop between these estimates.
* In fact, the density of pseudoprimes is conjectured to be closer
* to the square of that upper bound. E.g. the density of pseudoprimes
* of size 256 is around 3 * 10^-27. The density of primes is very
* high, from 0.005636 at 256 bits to 0.001409 at 1024 bits, i.e.
* more than 10^-3.
*
* For those people used to cryptographic levels of security where the
* 56 bits of DES key space is too small because it's exhaustible with
* custom hardware searching engines, note that you are not generating
* 50,000,000 primes per second on each of 56,000 custom hardware chips
* for several hours. The chances that another Dinosaur Killer asteroid
* will land today is about 10^-11 or 2^-36, so it would be better to
* spend your time worrying about *that*. Well, okay, there should be
* some derating for the chance that astronomers haven't seen it yet, but
* I think you get the idea. For a good feel about the probability of
* various events, I have heard that a good book is by E'mile Borel,
* "Les Probabilite's et la vie". (The 's are accents, not apostrophes.)
*
* For more on the subject, try "Finding Four Million Large Random Primes",
* by Ronald Rivest, in Advancess in Cryptology: Proceedings of Crypto '90.
* He used a small-divisor test, then a Fermat test to the base 2, and then
* 8 iterations of a Miller-Rabin test. About 718 million random 256-bit
* integers were generated, 43,741,404 passed the small divisor test,
* 4,058,000 passed the Fermat test, and all 4,058,000 passed all 8
* iterations of the Miller-Rabin test, proving their primality beyond most
* reasonable doubts.
*
* If the probability of getting a pseudoprime is some small p, then
* the probability of not getting it in t trials is (1-p)^t. Remember
* that, for small p, (1-p)^(1/p) ~ 1/e, the base of natural logarithms.
* (This is more commonly expressed as e = lim_{x\to\infty} (1+1/x)^x.)
* Thus, (1-p)^t ~ e^(-p*t) = exp(-p*t). So the odds of being able to
* do this many tests without seeing a pseudoprime if you assume that
* p = 10^-6 (one in a million) is one in 57.86. If you assume that
* p = 2*10^-6, it's one in 3347.6. So it's implausible that the
* density of pseudoprimes is much more than one millionth the density
* of primes.
*
* He also gives a theoretical argument that the chance of finding a
* 256-bit non-prime which satisfies one Fermat test to the base 2 is less
* than 10^-22. The small divisor test improves this number, and if the
* numbers are 512 bits (as needed for a 1024-bit key) the odds of failure
* shrink to about 10^-44. Thus, he concludes, for practical purposes
* *one* Fermat test to the base 2 is sufficient.
*/
static int
primeTest(struct BigNum const *bn, struct BigNum *e, struct BigNum *a,
	int (*f)(void *arg, int c), void *arg)
{
	int err;
	unsigned i, j;
 unsigned k, l;
	static unsigned const primes[] = {2, 3, 5, 7, 11, 13, 17};
#define CONFIRMTESTS 7

#if BNDEBUG	 /* Debugging */
			/*
			* This is debugging code to test the sieving stage.
			* If the sieving is wrong, it will let past numbers with
			* small divisors. The prime test here will still work, and
			* weed them out, but you'll be doing a lot more slow tests,
			* and presumably excluding from consideration some other numbers
			* which might be prime. This check just verifies that none
			* of the candidates have any small divisors. If this
			* code is enabled and never triggers, you can feel quite
			* confident that the sieving is doing its job.
*/
i = bnModQ(bn, 30030);	/* 30030 = 2 * 3 * 5 * 7 * 11 * 13 */
if (!(i % 2)) printf("bn div by 2!");
if (!(i % 3)) printf("bn div by 3!");
if (!(i % 5)) printf("bn div by 5!");
if (!(i % 7)) printf("bn div by 7!");
if (!(i % 11)) printf("bn div by 11!");
if (!(i % 13)) printf("bn div by 13!");
i = bnModQ(bn, 7429);	/* 7429 = 17 * 19 * 23 */
if (!(i % 17)) printf("bn div by 17!");
if (!(i % 19)) printf("bn div by 19!");
if (!(i % 23)) printf("bn div by 23!");
i = bnModQ(bn, 33263);	/* 33263 = 29 * 31 * 37 */
if (!(i % 29)) printf("bn div by 29!");
if (!(i % 31)) printf("bn div by 31!");
if (!(i % 37)) printf("bn div by 37!");
#endif

/*
* Now, check that bn is prime. If it passes to the base 2,
* it's prime beyond all reasonable doubt, and everything else
* is just gravy, but it gives people warm fuzzies to do it.
*
* This starts with verifying Euler's criterion for a base of 2.
* This is the fastest pseudoprimality test that I know of,
* saving a modular squaring over a Fermat test, as well as
* being stronger. 7/8 of the time, it's as strong as a strong
* pseudoprimality test, too. (The exception being when bn ==
* 1 mod 8 and 2 is a quartic residue, i.e. bn is of the form
* a^2 + (8*b)^2.) The precise series of tricks used here is
* not documented anywhere, so here's an explanation.
* Euler's criterion states that if p is prime then a^((p-1)/2)
* is congruent to Jacobi(a,p), modulo p. Jacobi(a,p) is
* a function which is +1 if a is a square modulo p, and -1 if
* it is not. For a = 2, this is particularly simple. It's
* +1 if p == +/-1 (mod 8), and -1 if m == +/-3 (mod 8).
* If p == 3 mod 4, then all a strong test does is compute
* 2^((p-1)/2). and see if it's +1 or -1. (Euler's criterion
* says *which* it should be.) If p == 5 (mod 8), then
* 2^((p-1)/2) is -1, so the initial step in a strong test,
* looking at 2^((p-1)/4), is wasted - you're not going to
* find a +/-1 before then if it *is* prime, and it shouldn't
* have either of those values if it isn't. So don't bother.
			*
			* The remaining case is p == 1 (mod 8). In this case, we
			* expect 2^((p-1)/2) == 1 (mod p), so we expect that the
			* square root of this, 2^((p-1)/4), will be +/-1 (mod p).
			* Evaluating this saves us a modular squaring 1/4 of the time.
			* If it's -1, a strong pseudoprimality test would call p
			* prime as well. Only if the result is +1, indicating that
			* 2 is not only a quadratic residue, but a quartic one as well,
			* does a strong pseudoprimality test verify more things than
			* this test does. Good enough.
			*
			* We could back that down another step, looking at 2^((p-1)/8)
			* if there was a cheap way to determine if 2 were expected to
			* be a quartic residue or not. Dirichlet proved that 2 is
			* a quartic residue iff p is of the form a^2 + (8*b^2).
			* All primes == 1 (mod 4) can be expressed as a^2 + (2*b)^2,
			* but I see no cheap way to evaluate this condition.
			*/
			if (bnCopy(e, bn) < 0)
				 return -1;
			(void)bnSubQ(e, 1);
			l = bnLSWord(e);

			j = 1;	/* Where to start in prime array for strong prime tests */

			if (l & 7) {
					bnRShift(e, 1);
					if (bnTwoExpMod(a, e, bn) < 0)
						return -1;
					if ((l & 7) == 6) {
						/* bn == 7 mod 8, expect +1 */
						if (bnBits(a) != 1)
							 return 1;	/* Not prime */
						k = 1;
					} else {
							/* bn == 3 or 5 mod 8, expect -1 == bn-1 */
							if (bnAddQ(a, 1) < 0)
								 return -1;
							if (bnCmp(a, bn) != 0)
								 return 1;		/* Not prime */
							k = 1;
							if (l & 4) {
									/* bn == 5 mod 8, make odd for strong tests */
									bnRShift(e, 1);
									k = 2;
							}
					}
			} else {
					/* bn == 1 mod 8, expect 2^((bn-1)/4) == +/-1 mod bn */
					bnRShift(e, 2);
					if (bnTwoExpMod(a, e, bn) < 0)
						return -1;
					if (bnBits(a) == 1) {
						j = 0;	/* Re-do strong prime test to base 2 */
					} else {
						if (bnAddQ(a, 1) < 0)
							 return -1;
						if (bnCmp(a, bn) != 0)
			return 1;	 /* Not prime */
	}
	k = 2 + bnMakeOdd(e);
}
/* It's prime! Now go on to confirmation tests */