| /* |
| * Sophie Germain prime generation using the bignum library and sieving. |
| * |
| * Copyright (c) 1995 Colin Plumb. All rights reserved. |
| * For licensing and other legal details, see the file legal.c. |
| */ |
| #ifndef HAVE_CONFIG_H |
| #define HAVE_CONFIG_H 0 |
| #endif |
| #if HAVE_CONFIG_H |
| #include "bnconfig.h" |
| #endif |
| |
| /* |
| * Some compilers complain about #if FOO if FOO isn't defined, |
| * so do the ANSI-mandated thing explicitly... |
| */ |
| #ifndef NO_ASSERT_H |
| #define NO_ASSERT_H 0 |
| #endif |
| #if !NO_ASSERT_H |
| #include <assert.h> |
| #else |
| #define assert(x) (void)0 |
| #endif |
| |
| #define BNDEBUG 1 |
| #ifndef BNDEBUG |
| #define BNDEBUG 0 |
| #endif |
| #if BNDEBUG |
| #include <stdio.h> |
| #endif |
| |
| #include "bn.h" |
| #include "germain.h" |
| #include "jacobi.h" |
| #include "lbnmem.h" /* For lbnMemWipe */ |
| #include "sieve.h" |
| |
| #include "kludge.h" |
| |
| /* Size of the sieve area (can be up to 65536/8 = 8192) */ |
| #define SIEVE 8192 |
| |
| static unsigned const confirm[] = {2, 3, 5, 7, 11, 13, 17}; |
| #define CONFIRMTESTS (sizeof(confirm)/sizeof(*confirm)) |
| |
| #if BNDEBUG |
| /* |
| * For sanity checking the sieve, we check for small divisors of the numbers |
| * we get back. This takes "rem", a partially reduced form of the prime, |
| * "div" a divisor to check for, and "order", a parameter of the "order" |
| * of Sophie Germain primes (0 = normal primes, 1 = Sophie Germain primes, |
| * 2 = 4*p+3 is also prime, etc.) and does the check. It just complains |
| * to stdout if the check fails. |
| */ |
| static void |
| germainSanity(unsigned rem, unsigned div, unsigned order) |
| { |
| unsigned mul = 1; |
| |
| rem %= div; |
| if (!rem) |
| printf("bn div by %u!\n", div); |
| while (order--) { |
| rem += rem+1; |
| if (rem >= div) |
| rem -= div; |
| mul += mul; |
| if (!rem) |
| printf("%u*bn+%u div by %u!\n", mul, mul-1, div); |
| } |
| } |
| #endif /* BNDEBUG */ |
| |
| /* |
| * Helper function that does the slow primality test. |
| * bn is the input bignum; a, e and bn2 are temporary buffers that are |
| * allocated by the caller to save overhead. bn2 is filled with |
| * a copy of 2^order*bn+2^order-1 if bn is found to be prime. |
| * |
| * Returns 0 if both bn and bn2 are 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. Prints a '+' or '-' on the |
| * given FILE (if any) for each test that is passed by bn, and a '*' |
| * for each test that is passed by bn2. |
| * |
| * The testing consists of strong pseudoprimality tests, to the bases given |
| * in the confirm[] array above. (Also called Miller-Rabin, although that's |
| * not technically correct if we're using fixed bases.) 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. |
| * |
| * 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 |
| germainPrimeTest(struct BigNum const *bn, struct BigNum *bn2, struct BigNum *e, |
| struct BigNum *a, unsigned order, int (*f)(void *arg, int c), void *arg) |
| { |
| int err; |
| unsigned i; |
| int j; |
| unsigned k, l, n; |
| |
| #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 = bnLSWord(bn); |
| if (!(i % 2)) printf("bn div by 2!"); |
| i = bnModQ(bn, 51051); /* 51051 = 3 * 7 * 11 * 13 * 17 */ |
| germainSanity(i, 3, order); |
| germainSanity(i, 7, order); |
| germainSanity(i, 11, order); |
| germainSanity(i, 13, order); |
| germainSanity(i, 17, order); |
| i = bnModQ(bn, 63365); /* 63365 = 5 * 19 * 23 * 29 */ |
| germainSanity(i, 5, order); |
| germainSanity(i, 19, order); |
| germainSanity(i, 23, order); |
| germainSanity(i, 29, order); |
| i = bnModQ(bn, 47027); /* 47027 = 31 * 37 * 41 */ |
| germainSanity(i, 31, order); |
| germainSanity(i, 37, order); |
| germainSanity(i, 41, order); |
| #endif |
| /* |
| * First, check whether bn is prime. This uses a fast primality |
| * test which usually obviates the need to do one of the |
| * confirmation tests later. See prime.c for a full explanation. |
| * We check bn first because it's one bit smaller, saving one |
| * modular squaring, and because we might be able to save another |
| * when testing it. (1/4 of the time.) A small speed hack, |
| * but finding big Sophie Germain primes is *slow*. |
| */ |
| 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 check higher-order forms bn2 = 2*bn+1, 4*bn+3, |
| * etc. Since bn2 == 3 mod 4, a strong pseudoprimality test boils |
| * down to looking at a^((bn2-1)/2) mod bn and seeing if it's +/-1. |
| * (+1 if bn2 is == 7 mod 8, -1 if it's == 3) |
| * Of course, that exponent is just the previous bn2 or bn... |
| */ |
| if (bnCopy(bn2, bn) < 0) |
| return -1; |
| for (n = 0; n < order; n++) { |
| /* |
| * Print a success indicator: the sign of Jacobi(2,bn2), |
| * which is available to us in l. bn2 = 2*bn + 1. Since bn |
| * is odd, bn2 must be == 3 mod 4, so the options modulo 8 |
| * are 3 and 7. 3 if l == 1 mod 4, 7 if l == 3 mod 4. |
| * The sign of the Jacobi symbol is - and + for these cases, |
| * respectively. |
| */ |
| if (f && (err = f(arg, "-+"[(l >> 1) & 1])) < 0) |
| return err; |
| /* Exponent is previous bn2 */ |
| if (bnCopy(e, bn2) < 0 || bnLShift(bn2, 1) < 0) |
| return -1; |
| (void)bnAddQ(bn2, 1); /* Can't overflow */ |
| if (bnTwoExpMod(a, e, bn2) < 0) |
| return -1; |
| if (n | l) { /* Expect + */ |
| if (bnBits(a) != 1) |
| return 2+n; /* Not prime */ |
| } else { |
| if (bnAddQ(a, 1) < 0) |
| return -1; |
| if (bnCmp(a, bn2) != 0) |
| return 2+n; /* Not prime */ |
| } |
| l = bnLSWord(bn2); |
| } |
| |
| /* Final success indicator - it's in the bag. */ |
| if (f && (err = f(arg, '*')) < 0) |
| return err; |
| |
| /* |
| * Success! We have found a prime! Now go on to confirmation |
| * tests... k is an amount by which we know it's safe to shift |
| * down e. j = 1 unless the test to the base 2 could stand to be |
| * re-done (it wasn't *quite* a strong test), in which case it's 0. |
| * |
| * Here, we do the full strong pseudoprimality test. This proves |
| * that a number is composite, or says that it's probably prime. |
| * |
| * For the given base a, find bn-1 = 2^k * e, then find |
| * x == a^e (mod bn). |
| * If x == +1 -> strong pseudoprime to base a |
| * Otherwise, repeat k times: |
| * If x == -1, -> strong pseudoprime |
| * x = x^2 (mod bn) |
| * If x = +1 -> composite |
| * If we reach the end of the iteration and x is *not* +1, at the |
| * end, it is composite. But it's also composite if the result |
| * *is* +1. Which means that the squaring actually only has to |
| * proceed k-1 times. If x is not -1 by then, it's composite |
| * no matter what the result of the squaring is. |
| * |
| * For the multiples 2*bn+1, 4*bn+3, etc. then k = 1 (and e is |
| * the previous multiple of bn) so the squaring loop is never |
| * actually executed at all. |
| */ |
| for (i = j; i < CONFIRMTESTS; i++) { |
| if (bnCopy(e, bn) < 0) |
| return -1; |
| bnRShift(e, k); |
| k += bnMakeOdd(e); |
| (void)bnSetQ(a, confirm[i]); |
| if (bnExpMod(a, a, e, bn) < 0) |
| return -1; |
| |
| if (bnBits(a) != 1) { |
| l = k; |
| for (;;) { |
| if (bnAddQ(a, 1) < 0) |
| return -1; |
| if (bnCmp(a, bn) == 0) /* Was result bn-1? */ |
| break; /* Prime */ |
| if (!--l) |
| return (1+order)*i+2; /* Fail */ |
| /* This part is executed once, on average. */ |
| (void)bnSubQ(a, 1); /* Restore a */ |
| if (bnSquare(a, a) < 0 || bnMod(a, a, bn) < 0) |
| return -1; |
| if (bnBits(a) == 1) |
| return (1+order)*i+1; /* Fail */ |
| } |
| } |
| |
| if (bnCopy(bn2, bn) < 0) |
| return -1; |
| |
| /* Only do the following if we're not re-doing base 2 */ |
| if (i) for (n = 0; n < order; n++) { |
| if (bnCopy(e, bn2) < 0 || bnLShift(bn2, 1) < 0) |
| return -1; |
| (void)bnAddQ(bn2, 1); |
| |
| /* Print success indicator for previous test */ |
| j = bnJacobiQ(confirm[i], bn2); |
| if (f && (err = f(arg, j < 0 ? '-' : '+')) < 0) |
| return err; |
| |
| /* Check that p^e == Jacobi(p,bn2) (mod bn2) */ |
| (void)bnSetQ(a, confirm[i]); |
| if (bnExpMod(a, a, e, bn2) < 0) |
| return -1; |
| /* |
| * FIXME: Actually, we don't need to compute the |
| * Jacobi symbol externally... it never happens that |
| * a = +/-1 but it's the wrong one. So we can just |
| * look at a and use its sign. Find a proof somewhere. |
| */ |
| if (j < 0) { |
| /* Not a Q.R., should have a = bn2-1 */ |
| if (bnAddQ(a, 1) < 0) |
| return -1; |
| if (bnCmp(a, bn2) != 0) /* Was result bn2-1? */ |
| return (1+order)*i+n+2; /* Fail */ |
| } else { |
| /* Quadratic residue, should have a = 1 */ |
| if (bnBits(a) != 1) |
| return (1+order)*i+n+2; /* Fail */ |
| } |
| } |
| /* Final success indicator for the base confirm[i]. */ |
| if (f && (err = f(arg, '*')) < 0) |
| return err; |
| } |
| |
| return 0; /* Prime! */ |
| } |
| |
| /* |
| * Add x*y to bn, which is usually (but not always) < 65536. |
| * Do it in a simple linear manner. |
| */ |
| static int |
| bnAddMult(struct BigNum *bn, unsigned long x, unsigned y) |
| { |
| unsigned long z = (unsigned long)x * y; |
| |
| while (z > 65535) { |
| if (bnAddQ(bn, 65535) < 0) |
| return -1; |
| z -= 65535; |
| } |
| return bnAddQ(bn, (unsigned)z); |
| } |
| |
| /* |
| * Modifies the bignum to return the next Sophie Germain prime >= the |
| * input value. Sohpie Germain primes are number such that p is |
| * prime and 2*p+1 is also prime. |
| * |
| * This is actually parameterized: it generates primes p such that "order" |
| * multiples-plus-two are also prime, 2*p+1, 2*(2*p+1)+1 = 4*p+3, etc. |
| * |
| * Returns >=0 on success or -1 on failure (out of memory). On success, |
| * the return value is the number of modular exponentiations performed |
| * (excluding the final confirmations). This never gives up searching. |
| * |
| * The FILE *f argument, if non-NULL, has progress indicators written |
| * to it. A dot (.) is written every time a primeality test is failed, |
| * a plus (+) or minus (-) when the smaller prime of the pair passes a |
| * test, and a star (*) when the larger one does. Finally, a slash (/) |
| * is printed when the sieve was emptied without finding a prime and is |
| * being refilled. |
| * |
| * Apologies to structured programmers for all the GOTOs. |
| */ |
| int |
| germainPrimeGen(struct BigNum *bn, unsigned order, |
| int (*f)(void *arg, int c), void *arg) |
| { |
| int retval; |
| unsigned p, prev; |
| unsigned inc; |
| struct BigNum a, e, bn2; |
| int modexps = 0; |
| #ifdef MSDOS |
| unsigned char *sieve; |
| #else |
| unsigned char sieve[SIEVE]; |
| #endif |
| |
| #ifdef MSDOS |
| sieve = lbnMemAlloc(SIEVE); |
| if (!sieve) |
| return -1; |
| #endif |
| |
| bnBegin(&a); |
| bnBegin(&e); |
| bnBegin(&bn2); |
| |
| /* |
| * Obviously, the prime we find must be odd. Further, if 2*p+1 |
| * is also to be prime (order > 0) then p != 1 (mod 3), lest |
| * 2*p+1 == 3 (mod 3). Added to p != 3 (mod 3), p == 2 (mod 3) |
| * and p == 5 (mod 6). |
| * If order > 2 and we care about 4*p+3 and 8*p+7, then similarly |
| * p == 4 (mod 5), so p == 29 (mod 30). |
| * So pick the step size for searching based on the order |
| * and increse bn until it's == -1 (mod inc). |
| * |
| * mod 7 doesn't have a unique value for p because 2 -> 5 -> 4 -> 2, |
| * nor does mod 11, and I don't want to think about things past |
| * that. The required order would be impractically high, in any case. |
| */ |
| inc = order ? ((order > 2) ? 30 : 6) : 2; |
| if (bnAddQ(bn, inc-1 - bnModQ(bn, inc)) < 0) |
| goto failed; |
| |
| for (;;) { |
| if (sieveBuild(sieve, SIEVE, bn, inc, order) < 0) |
| goto failed; |
| |
| p = prev = 0; |
| if (sieve[0] & 1 || (p = sieveSearch(sieve, SIEVE, p)) != 0) { |
| do { |
| /* Adjust bn to have the right value. */ |
| assert(p >= prev); |
| if (bnAddMult(bn, p-prev, inc) < 0) |
| goto failed; |
| prev = p; |
| |
| /* Okay, do the strong tests. */ |
| retval = germainPrimeTest(bn, &bn2, &e, &a, |
| order, f, arg); |
| if (retval <= 0) |
| goto done; |
| modexps += retval; |
| if (f && (retval = f(arg, '.')) < 0) |
| goto done; |
| |
| /* And try again */ |
| p = sieveSearch(sieve, SIEVE, p); |
| } while (p); |
| } |
| |
| /* Ran out of sieve space - increase bn and keep trying. */ |
| if (bnAddMult(bn, (unsigned long)SIEVE*8-prev, inc) < 0) |
| goto failed; |
| if (f && (retval = f(arg, '/')) < 0) |
| goto done; |
| } /* for (;;) */ |
| |
| failed: |
| retval = -1; |
| done: |
| bnEnd(&bn2); |
| bnEnd(&e); |
| bnEnd(&a); |
| #ifdef MSDOS |
| lbnMemFree(sieve, SIEVE); |
| #else |
| lbnMemWipe(sieve, sizeof(sieve)); |
| #endif |
| return retval < 0 ? retval : modexps+(order+1)*CONFIRMTESTS; |
| } |
| |
| int |
| germainPrimeGenStrong(struct BigNum *bn, struct BigNum const *step, |
| unsigned order, int (*f)(void *arg, int c), void *arg) |
| { |
| int retval; |
| unsigned p, prev; |
| struct BigNum a, e, bn2; |
| int modexps = 0; |
| #ifdef MSDOS |
| unsigned char *sieve; |
| #else |
| unsigned char sieve[SIEVE]; |
| #endif |
| |
| #ifdef MSDOS |
| sieve = lbnMemAlloc(SIEVE); |
| if (!sieve) |
| return -1; |
| #endif |
| bnBegin(&a); |
| bnBegin(&e); |
| bnBegin(&bn2); |
| |
| for (;;) { |
| if (sieveBuildBig(sieve, SIEVE, bn, step, order) < 0) |
| goto failed; |
| |
| p = prev = 0; |
| if (sieve[0] & 1 || (p = sieveSearch(sieve, SIEVE, p)) != 0) { |
| do { |
| /* |
| * Adjust bn to have the right value, |
| * adding (p-prev) * 2*step. |
| */ |
| assert(p >= prev); |
| /* Compute delta into a */ |
| if (bnMulQ(&a, step, p-prev) < 0) |
| goto failed; |
| if (bnAdd(bn, &a) < 0) |
| goto failed; |
| prev = p; |
| |
| /* Okay, do the strong tests. */ |
| retval = germainPrimeTest(bn, &bn2, &e, &a, |
| order, f, arg); |
| if (retval <= 0) |
| goto done; |
| modexps += retval; |
| if (f && (retval = f(arg, '.')) < 0) |
| goto done; |
| |
| /* And try again */ |
| p = sieveSearch(sieve, SIEVE, p); |
| } while (p); |
| } |
| |
| /* Ran out of sieve space - increase bn and keep trying. */ |
| #if SIEVE*8 == 65536 |
| /* Corner case that will never actually happen */ |
| if (!prev) { |
| if (bnAdd(bn, step) < 0) |
| goto failed; |
| p = 65535; |
| } else { |
| p = (unsigned)(SIEVE*8 - prev); |
| } |
| #else |
| p = SIEVE*8 - prev; |
| #endif |
| if (bnMulQ(&a, step, p) < 0 || bnAdd(bn, &a) < 0) |
| goto failed; |
| if (f && (retval = f(arg, '/')) < 0) |
| goto done; |
| } /* for (;;) */ |
| |
| failed: |
| retval = -1; |
| done: |
| bnEnd(&bn2); |
| bnEnd(&e); |
| bnEnd(&a); |
| #ifdef MSDOS |
| lbnMemFree(sieve, SIEVE); |
| #else |
| lbnMemWipe(sieve, sizeof(sieve)); |
| #endif |
| return retval < 0 ? retval : modexps+(order+1)*CONFIRMTESTS; |
| } |