| /* |
| * 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 |
| |
| #include <stdarg.h> /* We just can't live without this... */ |
| |
| #ifndef BNDEBUG |
| #define BNDEBUG 1 |
| #endif |
| #if BNDEBUG |
| #include <stdio.h> |
| #endif |
| |
| #include "bn.h" |
| #include "lbnmem.h" |
| #include "prime.h" |
| #include "sieve.h" |
| |
| #include "kludge.h" |
| |
| /* Size of the shuffle table */ |
| #define SHUFFLE 256 |
| /* Size of the sieve area */ |
| #define SIEVE 32768u/16 |
| |
| /* Confirmation tests. The first one *must* be 2 */ |
| static unsigned const confirm[] = {2, 3, 5, 7, 11, 13, 17}; |
| #define CONFIRMTESTS (sizeof(confirm)/sizeof(*confirm)) |
| |
| /* |
| * 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, returns the number of modular exponentiations performed. |
| * Calls the given progress function with a '*' for each primality test |
| * that is passed. |
| * |
| * 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 |
| primeTest(struct BigNum const *bn, struct BigNum *e, struct BigNum *a, |
| int (*f)(void *arg, int c), void *arg) |
| { |
| unsigned i, j; |
| unsigned k, l; |
| int err; |
| |
| #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 */ |
| if (!(i % 3)) printf("bn div by 3!"); |
| if (!(i % 7)) printf("bn div by 7!"); |
| if (!(i % 11)) printf("bn div by 11!"); |
| if (!(i % 13)) printf("bn div by 13!"); |
| if (!(i % 17)) printf("bn div by 17!"); |
| i = bnModQ(bn, 63365); /* 63365 = 5 * 19 * 23 * 29 */ |
| if (!(i % 5)) printf("bn div by 5!"); |
| if (!(i % 19)) printf("bn div by 19!"); |
| if (!(i % 23)) printf("bn div by 23!"); |
| if (!(i % 29)) printf("bn div by 29!"); |
| i = bnModQ(bn, 47027); /* 47027 = 31 * 37 * 41 */ |
| if (!(i % 31)) printf("bn div by 31!"); |
| if (!(i % 37)) printf("bn div by 37!"); |
| if (!(i % 41)) printf("bn div by 41!"); |
| #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 */ |
| |
| /* |
| * Now, e = (bn-1)/2^k is odd. k >= 1, and has a given value |
| * with probability 2^-k, so its expected value is 2. |
| * j = 1 in the usual case when the previous test was as good as |
| * a strong prime test, but 1/8 of the time, j = 0 because |
| * the strong prime test to the base 2 needs to be re-done. |
| */ |
| for (i = j; i < CONFIRMTESTS; i++) { |
| if (f && (err = f(arg, '*')) < 0) |
| return err; |
| (void)bnSetQ(a, confirm[i]); |
| if (bnExpMod(a, a, e, bn) < 0) |
| return -1; |
| if (bnBits(a) == 1) |
| continue; /* Passed this test */ |
| |
| l = k; |
| for (;;) { |
| if (bnAddQ(a, 1) < 0) |
| return -1; |
| if (bnCmp(a, bn) == 0) /* Was result bn-1? */ |
| break; /* Prime */ |
| if (!--l) /* Reached end, not -1? luck? */ |
| return i+2-j; /* Failed, not prime */ |
| /* This portion is executed, on average, once. */ |
| (void)bnSubQ(a, 1); /* Put a back where it was. */ |
| if (bnSquare(a, a) < 0 || bnMod(a, a, bn) < 0) |
| return -1; |
| if (bnBits(a) == 1) |
| return i+2-j; /* Failed, not prime */ |
| } |
| /* It worked (to the base confirm[i]) */ |
| } |
| |
| /* Yes, we've decided that it's prime. */ |
| 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 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); |
| } |
| |
| static int |
| bnSubMult(struct BigNum *bn, unsigned x, unsigned y) |
| { |
| unsigned long z = (unsigned long)x * y; |
| |
| while (z > 65535) { |
| if (bnSubQ(bn, 65535) < 0) |
| return -1; |
| z -= 65535; |
| } |
| return bnSubQ(bn, (unsigned)z); |
| } |
| |
| /* |
| * Modifies the bignum to return a nearby (slightly larger) number which |
| * is a probable prime. Returns >=0 on success or -1 on failure (out of |
| * memory). The return value is the number of unsuccessful modular |
| * exponentiations performed. This never gives up searching. |
| * |
| * All other arguments are optional. They may be NULL. They are: |
| * |
| * unsigned (*rand)(unsigned limit) |
| * For better distributed numbers, supply a non-null pointer to a |
| * function which returns a random x, 0 <= x < limit. (It may make it |
| * simpler to know that 0 < limit <= SHUFFLE, so you need at most a byte.) |
| * The program generates a large window of sieve data and then does |
| * pseudoprimality tests on the data. If a rand function is supplied, |
| * the candidates which survive sieving are shuffled with a window of |
| * size SHUFFLE before testing to increase the uniformity of the prime |
| * selection. This isn't perfect, but it reduces the correlation between |
| * the size of the prime-free gap before a prime and the probability |
| * that that prime will be found by a sequential search. |
| * |
| * If rand is NULL, sequential search is used. If you want sequential |
| * search, note that the search begins with the given number; if you're |
| * trying to generate consecutive primes, you must increment the previous |
| * one by two before calling this again. |
| * |
| * int (*f)(void *arg, int c), void *arg |
| * The function f argument, if non-NULL, is called with progress indicator |
| * characters for printing. A dot (.) is written every time a primality test |
| * is failed, a star (*) every time one is passed, and a slash (/) in the |
| * (very rare) case that the sieve was emptied without finding a prime |
| * and is being refilled. f is also passed the void *arg argument for |
| * private context storage. If f returns < 0, the test aborts and returns |
| * that value immediately. (bn is set to the last value tested, so you |
| * can increment bn and continue.) |
| * |
| * The "exponent" argument, and following unsigned numbers, are exponents |
| * for which an inverse is desired, modulo p. For a d to exist such that |
| * (x^e)^d == x (mod p), then d*e == 1 (mod p-1), so gcd(e,p-1) must be 1. |
| * The prime returned is constrained to not be congruent to 1 modulo |
| * any of the zero-terminated list of 16-bit numbers. Note that this list |
| * should contain all the small prime factors of e. (You'll have to test |
| * for large prime factors of e elsewhere, but the chances of needing to |
| * generate another prime are low.) |
| * |
| * The list is terminated by a 0, and may be empty. |
| */ |
| int |
| primeGen(struct BigNum *bn, unsigned (*rand)(unsigned), |
| int (*f)(void *arg, int c), void *arg, unsigned exponent, ...) |
| { |
| int retval; |
| int modexps = 0; |
| unsigned short offsets[SHUFFLE]; |
| unsigned i, j; |
| unsigned p, q, prev; |
| struct BigNum a, e; |
| #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); |
| |
| #if 0 /* Self-test (not used for production) */ |
| { |
| struct BigNum t; |
| static unsigned char const prime1[] = {5}; |
| static unsigned char const prime2[] = {7}; |
| static unsigned char const prime3[] = {11}; |
| static unsigned char const prime4[] = {1, 1}; /* 257 */ |
| static unsigned char const prime5[] = {0xFF, 0xF1}; /* 65521 */ |
| static unsigned char const prime6[] = {1, 0, 1}; /* 65537 */ |
| static unsigned char const prime7[] = {1, 0, 3}; /* 65539 */ |
| /* A small prime: 1234567891 */ |
| static unsigned char const prime8[] = {0x49, 0x96, 0x02, 0xD3}; |
| /* A slightly larger prime: 12345678901234567891 */ |
| static unsigned char const prime9[] = { |
| 0xAB, 0x54, 0xA9, 0x8C, 0xEB, 0x1F, 0x0A, 0xD3 }; |
| /* |
| * No, 123456789012345678901234567891 isn't prime; it's just a |
| * lucky, easy-to-remember conicidence. (You have to go to |
| * ...4567907 for a prime.) |
| */ |
| static struct { |
| unsigned char const *prime; |
| unsigned size; |
| } const primelist[] = { |
| { prime1, sizeof(prime1) }, |
| { prime2, sizeof(prime2) }, |
| { prime3, sizeof(prime3) }, |
| { prime4, sizeof(prime4) }, |
| { prime5, sizeof(prime5) }, |
| { prime6, sizeof(prime6) }, |
| { prime7, sizeof(prime7) }, |
| { prime8, sizeof(prime8) }, |
| { prime9, sizeof(prime9) } }; |
| |
| bnBegin(&t); |
| |
| for (i = 0; i < sizeof(primelist)/sizeof(primelist[0]); i++) { |
| bnInsertBytes(&t, primelist[i].prime, 0, |
| primelist[i].size); |
| bnCopy(&e, &t); |
| (void)bnSubQ(&e, 1); |
| bnTwoExpMod(&a, &e, &t); |
| p = bnBits(&a); |
| if (p != 1) { |
| printf( |
| "Bug: Fermat(2) %u-bit output (1 expected)\n", p); |
| fputs("Prime = 0x", stdout); |
| for (j = 0; j < primelist[i].size; j++) |
| printf("%02X", primelist[i].prime[j]); |
| putchar('\n'); |
| } |
| bnSetQ(&a, 3); |
| bnExpMod(&a, &a, &e, &t); |
| p = bnBits(&a); |
| if (p != 1) { |
| printf( |
| "Bug: Fermat(3) %u-bit output (1 expected)\n", p); |
| fputs("Prime = 0x", stdout); |
| for (j = 0; j < primelist[i].size; j++) |
| printf("%02X", primelist[i].prime[j]); |
| putchar('\n'); |
| } |
| } |
| |
| bnEnd(&t); |
| } |
| #endif |
| |
| /* First, make sure that bn is odd. */ |
| if ((bnLSWord(bn) & 1) == 0) |
| (void)bnAddQ(bn, 1); |
| |
| retry: |
| /* Then build a sieve starting at bn. */ |
| sieveBuild(sieve, SIEVE, bn, 2, 0); |
| |
| /* Do the extra exponent sieving */ |
| if (exponent) { |
| va_list ap; |
| unsigned t = exponent; |
| |
| va_start(ap, exponent); |
| |
| do { |
| /* The exponent had better be odd! */ |
| assert(t & 1); |
| |
| i = bnModQ(bn, t); |
| /* Find 1-i */ |
| if (i == 0) |
| i = 1; |
| else if (--i) |
| i = t - i; |
| |
| /* Divide by 2, modulo the exponent */ |
| i = (i & 1) ? i/2 + t/2 + 1 : i/2; |
| |
| /* Remove all following multiples from the sieve. */ |
| sieveSingle(sieve, SIEVE, i, t); |
| |
| /* Get the next exponent value */ |
| t = va_arg(ap, unsigned); |
| } while (t); |
| |
| va_end(ap); |
| } |
| |
| /* Fill up the offsets array with the first SHUFFLE candidates */ |
| i = p = 0; |
| /* Get first prime */ |
| if (sieve[0] & 1 || (p = sieveSearch(sieve, SIEVE, p)) != 0) { |
| offsets[i++] = p; |
| p = sieveSearch(sieve, SIEVE, p); |
| } |
| /* |
| * Okay, from this point onwards, p is always the next entry |
| * from the sieve, that has not been added to the shuffle table, |
| * and is 0 iff the sieve has been exhausted. |
| * |
| * If we want to shuffle, then fill the shuffle table until the |
| * sieve is exhausted or the table is full. |
| */ |
| if (rand && p) { |
| do { |
| offsets[i++] = p; |
| p = sieveSearch(sieve, SIEVE, p); |
| } while (p && i < SHUFFLE); |
| } |
| |
| /* Choose a random candidate for experimentation */ |
| prev = 0; |
| while (i) { |
| /* Pick a random entry from the shuffle table */ |
| j = rand ? rand(i) : 0; |
| q = offsets[j]; /* The entry to use */ |
| |
| /* Replace the entry with some more data, if possible */ |
| if (p) { |
| offsets[j] = p; |
| p = sieveSearch(sieve, SIEVE, p); |
| } else { |
| offsets[j] = offsets[--i]; |
| offsets[i] = 0; |
| } |
| |
| /* Adjust bn to have the right value */ |
| if ((q > prev ? bnAddMult(bn, q-prev, 2) |
| : bnSubMult(bn, prev-q, 2)) < 0) |
| goto failed; |
| prev = q; |
| |
| /* Now do the Fermat tests */ |
| retval = primeTest(bn, &e, &a, f, arg); |
| if (retval <= 0) |
| goto done; /* Success or error */ |
| modexps += retval; |
| if (f && (retval = f(arg, '.')) < 0) |
| goto done; |
| } |
| |
| /* Ran out of sieve space - increase bn and keep trying. */ |
| if (bnAddMult(bn, SIEVE*8-prev, 2) < 0) |
| goto failed; |
| if (f && (retval = f(arg, '/')) < 0) |
| goto done; |
| goto retry; |
| |
| failed: |
| retval = -1; |
| done: |
| bnEnd(&e); |
| bnEnd(&a); |
| lbnMemWipe(offsets, sizeof(offsets)); |
| #ifdef MSDOS |
| lbnMemFree(sieve, SIEVE); |
| #else |
| lbnMemWipe(sieve, sizeof(sieve)); |
| #endif |
| |
| return retval < 0 ? retval : modexps + CONFIRMTESTS; |
| } |
| |
| /* |
| * Similar, but searches forward from the given starting value in steps of |
| * "step" rather than 1. The step size must be even, and bn must be odd. |
| * Among other possibilities, this can be used to generate "strong" |
| * primes, where p-1 has a large prime factor. |
| */ |
| int |
| primeGenStrong(struct BigNum *bn, struct BigNum const *step, |
| int (*f)(void *arg, int c), void *arg) |
| { |
| int retval; |
| unsigned p, prev; |
| struct BigNum a, e; |
| int modexps = 0; |
| #ifdef MSDOS |
| unsigned char *sieve; |
| #else |
| unsigned char sieve[SIEVE]; |
| #endif |
| |
| #ifdef MSDOS |
| sieve = lbnMemAlloc(SIEVE); |
| if (!sieve) |
| return -1; |
| #endif |
| |
| /* Step must be even and bn must be odd */ |
| assert((bnLSWord(step) & 1) == 0); |
| assert((bnLSWord(bn) & 1) == 1); |
| |
| bnBegin(&a); |
| bnBegin(&e); |
| |
| for (;;) { |
| if (sieveBuildBig(sieve, SIEVE, bn, step, 0) < 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; |
| |
| retval = primeTest(bn, &e, &a, f, arg); |
| if (retval <= 0) |
| goto done; /* Success! */ |
| 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(&e); |
| bnEnd(&a); |
| #ifdef MSDOS |
| lbnMemFree(sieve, SIEVE); |
| #else |
| lbnMemWipe(sieve, sizeof(sieve)); |
| #endif |
| return retval < 0 ? retval : modexps + CONFIRMTESTS; |
| } |