/*========================================================================
Compute Factor Base:
Function: Computes primes p up to B for which n is a square mod p,
allocates memory and stores them in an array pointed to by factorBase.
Additionally allocates and computes the primeSizes array.
Returns: number of primes actually in the factor base
========================================================================*/
static void computeFactorBase(mpz_t n, unsigned long B,unsigned long multiplier)
{
UV p;
UV primesinbase = 0;
PRIME_ITERATOR(iter);
if (factorBase) { Safefree(factorBase); factorBase = 0; }
New(0, factorBase, B, unsigned int);
factorBase[primesinbase++] = multiplier;
if (multiplier != 2)
factorBase[primesinbase++] = 2;
prime_iterator_setprime(&iter, 3);
for (p = 3; primesinbase < B; p = prime_iterator_next(&iter)) {
if (mpz_kronecker_ui(n, p) == 1)
factorBase[primesinbase++] = p;
}
prime_iterator_destroy(&iter);
#ifdef LARGESTP
gmp_printf("Largest prime less than %Zd\n",p);
#endif
/* Allocate and compute the number of bits required to store each prime */
New(0, primeSizes, B, unsigned char);
for (p = 0; p < B; p++)
primeSizes[p] =
(unsigned char) floor( log(factorBase[p]) / log(2.0) - SIZE_FUDGE + 0.5 );
}
static unsigned long knuthSchroeppel(mpz_t n, unsigned long numPrimes)
{
unsigned int i, j, best_mult, knmod8;
unsigned int maxprimes = (2*numPrimes <= 1000) ? 2*numPrimes : 1000;
float best_score, contrib;
float scores[NUMMULTS];
mpz_t temp;
mpz_init(temp);
for (i = 0; i < NUMMULTS; i++) {
scores[i] = 0.5 * logf((float)multipliers[i]);
mpz_mul_ui(temp, n, multipliers[i]);
knmod8 = mpz_mod_ui(temp, temp, 8);
switch (knmod8) {
case 1: scores[i] -= 2 * M_LN2; break;
case 5: scores[i] -= M_LN2; break;
case 3:
case 7: scores[i] -= 0.5 * M_LN2; break;
default: break;
}
}
{
unsigned long prime, modp, knmodp;
PRIME_ITERATOR(iter);
for (i = 1; i < maxprimes; i++) {
prime = prime_iterator_next(&iter);
modp = mpz_mod_ui(temp, n, prime);
contrib = logf((float)prime) / (float)(prime-1);
for (j = 0; j < NUMMULTS; j++) {
knmodp = (modp * multipliers[j]) % prime;
if (knmodp == 0) {
scores[j] -= contrib;
} else {
mpz_set_ui(temp, knmodp);
if (mpz_kronecker_ui(temp, prime) == 1)
scores[j] -= 2*contrib;
}
}
}
prime_iterator_destroy(&iter);
}
mpz_clear(temp);
best_score = 1000.0;
best_mult = 1;
for (i = 0; i < NUMMULTS; i++) {
float score = scores[i];
if (score < best_score) {
best_score = score;
best_mult = multipliers[i];
}
}
/* gmp_printf("%Zd mult %lu\n", n, best_mult); */
return best_mult;
}
double
qcn_estimate (mpz_t d)
{
#define P_LIMIT 132000
double h;
unsigned long p;
/* p=2 */
h = sqrt (-mpz_get_d (d)) / M_PI
* 2.0 / (2.0 - mpz_kronecker_ui (d, 2));
if (mpz_cmp_si (d, -3) == 0) h *= 3;
else if (mpz_cmp_si (d, -4) == 0) h *= 2;
for (p = 3; p < P_LIMIT; p += 2)
if (prime_p (p))
h *= (double) p / (double) (p - mpz_kronecker_ui (d, p));
return h;
}
/* evaluate the Knuth-Schroeppel function, cf Robert D. Silverman,
"The Multiple Polynomial Quadratic Sieve", Math. of Comp. volume 48,
number 177, 1987, page 335 */
unsigned long
find_multiplier (mpz_t N, double B)
{
unsigned long k, bestk = 1;
double p, f, g, maxf = 0.0;
mpz_t kN;
mpz_init (kN);
for (k = 1; k < 100; k = nextprime (k))
{
mpz_mul_ui (kN, N, k);
/* FIXME: Silverman writes "if N = 1 mod 8" but isn't it kN instead? */
if (mpz_kronecker_ui (kN, 2) == 1 && mpz_fdiv_ui (kN, 8) == 1)
f = 2.0 * log (2.0);
else
f = 0.0;
for (p = getprime (2.0); p <= B; p = getprime (p))
{
if (mpz_kronecker_ui (kN, (unsigned long) p) == 1)
{
g = ((k % (unsigned long) p) == 0) ? (1.0 / p) : (2.0 / p);
f += g * log (p);
}
}
f -= 0.5 * log ((double) k);
if (f > maxf)
{
maxf = f;
bestk = k;
}
getprime (0.0); /* free prime buffer */
}
mpz_clear (kN);
return bestk;
}
static int ec_order(mpz_t order, mpz_t a, mpz_t b, unsigned long n)
{
/* use the inefficient naive approach to count the points of an elliptic
* curve E(a, b) over finite field GF(n):
* |(E(a, b))/GF(n)| =
* n + 1 + sum (x in GF(n)) (jacobi_symbol((x^3 + a*x + b), n))
*/
unsigned long i;
mpz_t tmp;
int order_exists;
if (!(n & 1))
{
order_exists = 0;
}
else
{
mpz_init(tmp);
mpz_set_ui(order, n);
mpz_add_ui(order, order, 1);
for (i = 0; i < n; ++i)
{
mpz_set_ui(tmp, i);
mpz_mul_ui(tmp, tmp, i);
mpz_mul_ui(tmp, tmp, i);
mpz_addmul_ui(tmp, a, i);
mpz_add(tmp, tmp, b);
mpz_set_si(tmp, mpz_kronecker_ui(tmp, n));
mpz_add(order, order, tmp);
}
order_exists = 1;
mpz_clear(tmp);
}
return order_exists;
}