int _XS_is_aks_prime(UV n)
{
UV sqrtn, limit, r, rlimit, a;
double log2n;
int verbose = _XS_get_verbose();
if (n < 2)
return 0;
if (n == 2)
return 1;
if (is_perfect_power(n))
return 0;
sqrtn = sqrt(n);
log2n = log(n) / log(2); /* C99 has a log2() function */
limit = (UV) floor(log2n * log2n);
if (verbose) { printf("# aks limit is %lu\n", (unsigned long) limit); }
for (r = 2; r < n; r++) {
if ((n % r) == 0)
return 0;
#if SQRTN_SHORTCUT
if (r > sqrtn)
return 1;
#endif
if (order(r, n, limit) > limit)
break;
}
if (r >= n)
return 1;
rlimit = (UV) floor(sqrt(r-1) * log2n);
if (verbose) { printf("# aks r = %lu rlimit = %lu\n", (unsigned long) r, (unsigned long) rlimit); }
for (a = 1; a <= rlimit; a++) {
if (! test_anr(a, n, r) )
return 0;
if (verbose>1) { printf("."); fflush(stdout); }
}
if (verbose>1) { printf("\n"); }
return 1;
}
int is_aks_prime(mpz_t n)
{
mpz_t *px, *py;
int retval;
UV i, s, r, a;
UV starta = 1;
int _verbose = get_verbose_level();
if (mpz_cmp_ui(n, 4) < 0)
return (mpz_cmp_ui(n, 1) <= 0) ? 0 : 1;
/* Just for performance: check small divisors: 2*3*5*7*11*13*17*19*23 */
if (mpz_gcd_ui(0, n, 223092870UL) != 1 && mpz_cmp_ui(n, 23) > 0)
return 0;
if (mpz_perfect_power_p(n))
return 0;
#if AKS_VARIANT == AKS_VARIANT_V6 /* From the V6 AKS paper */
{
mpz_t sqrtn, t;
double log2n;
UV limit, startr;
PRIME_ITERATOR(iter);
mpz_init(sqrtn);
mpz_sqrt(sqrtn, n);
log2n = mpz_log2(n);
limit = (UV) floor( log2n * log2n );
if (_verbose>1) gmp_printf("# AKS checking order_r(%Zd) to %"UVuf"\n", n, (unsigned long) limit);
/* Using a native r limits us to ~2000 digits in the worst case (r ~ log^5n)
* but would typically work for 100,000+ digits (r ~ log^3n). This code is
* far too slow to matter either way. Composite r is ok here, but it will
* always end up prime, so save time and just check primes. */
retval = 0;
/* Start order search at a good spot. Idea from Nemana and Venkaiah. */
startr = (mpz_sizeinbase(n,2)-1) * (mpz_sizeinbase(n,2)-1);
startr = (startr < 1002) ? 2 : startr - 100;
for (r = 2; /* */; r = prime_iterator_next(&iter)) {
if (mpz_divisible_ui_p(n, r) ) /* r divides n. composite. */
{ retval = 0; break; }
if (mpz_cmp_ui(sqrtn, r) <= 0) /* no r <= sqrtn divides n. prime. */
{ retval = 1; break; }
if (r < startr) continue;
if (mpz_order_ui(r, n, limit) > limit)
{ retval = 2; break; }
}
prime_iterator_destroy(&iter);
mpz_clear(sqrtn);
if (retval != 2) return retval;
/* Since r is prime, phi(r) = r-1. */
s = (UV) floor( sqrt(r-1) * log2n );
}
#elif AKS_VARIANT == AKS_VARIANT_BORNEMANN /* Bernstein + Voloch */
{
UV slim;
double c2, x;
/* small t = few iters of big poly. big t = many iters of small poly */
double const t = (mpz_sizeinbase(n, 2) <= 64) ? 32 : 40;
double const t1 = (1.0/((t+1)*log(t+1)-t*log(t)));
double const dlogn = mpz_logn(n);
mpz_t tmp;
PRIME_ITERATOR(iter);
mpz_init(tmp);
prime_iterator_setprime(&iter, (UV) (t1*t1 * dlogn*dlogn) );
r = prime_iterator_next(&iter);
while (!is_primitive_root_uiprime(n,r))
r = prime_iterator_next(&iter);
prime_iterator_destroy(&iter);
slim = (UV) (2*t*(r-1));
c2 = dlogn * floor(sqrt(r));
{ /* Binary search for first s in [1,slim] where x >= 0 */
UV bi = 1;
UV bj = slim;
while (bi < bj) {
s = bi + (bj-bi)/2;
mpz_bin_uiui(tmp, r+s-1, s);
x = mpz_logn(tmp) / c2 - 1.0;
if (x < 0) bi = s+1;
else bj = s;
}
s = bi-1;
}
s = (s+3) >> 1;
/* Bornemann checks factors up to (s-1)^2, we check to max(r,s) */
/* slim = (s-1)*(s-1); */
slim = (r > s) ? r : s;
if (_verbose > 1) printf("# aks trial to %"UVuf"\n", slim);
if (_GMP_trial_factor(n, 2, slim) > 1)
{ mpz_clear(tmp); return 0; }
mpz_sqrt(tmp, n);
if (mpz_cmp_ui(tmp, slim) <= 0)
{ mpz_clear(tmp); return 1; }
mpz_clear(tmp);
}
#elif AKS_VARIANT == AKS_VARIANT_BERN21
{ /* Bernstein 2003, theorem 2.1 (simplified) */
UV q;
double slim, scmp, x;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
r = s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r = prime_iterator_next(&iter);
q = largest_factor(r-1);
mpz_set_ui(t, r);
mpz_powm_ui(t, n, (r-1)/q, t);
if (mpz_cmp_ui(t, 1) <= 0) continue;
scmp = 2 * floor(sqrt(r)) * mpz_log2(n);
slim = 20 * (r-1);
/* Check viability */
mpz_bin_uiui(t, q+slim-1, slim); if (mpz_log2(t) < scmp) continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
if (mpz_log2(t) > scmp) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN22
{ /* Bernstein 2003, theorem 2.2 (simplified) */
UV q;
double slim, scmp, x;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
r = s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r = prime_iterator_next(&iter);
if (!is_primitive_root_uiprime(n,r)) continue;
q = r-1; /* Since r is prime, phi(r) = r-1 */
scmp = 2 * floor(sqrt(r-1)) * mpz_log2(n);
slim = 20 * (r-1);
/* Check viability */
mpz_bin_uiui(t, q+slim-1, slim); if (mpz_log2(t) < scmp) continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
if (mpz_log2(t) > scmp) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN23
{ /* Bernstein 2003, theorem 2.3 (simplified) */
UV q, d, limit;
double slim, scmp, sbin, x, log2n;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
log2n = mpz_log2(n);
limit = (UV) floor( log2n * log2n );
r = 2;
s = 0;
while (1) {
/* todo: Check r|n and r >= sqrt(n) here instead of waiting */
if (mpz_cmp_ui(n, r) <= 0) break;
r++;
UV gcd = mpz_gcd_ui(NULL, n, r);
if (gcd != 1) { mpz_clear(t); mpz_clear(t2); return 0; }
UV v = mpz_order_ui(r, n, limit);
if (v >= limit) continue;
mpz_set_ui(t2, r);
totient(t, t2);
q = mpz_get_ui(t);
UV phiv = q/v;
/* printf("phi(%lu)/v = %lu/%lu = %lu\n", r, q, v, phiv); */
/* This is extremely inefficient. */
/* Choose an s value we'd be happy with */
slim = 20 * (r-1);
/* Quick check to see if it could work with s=slim, d=1 */
mpz_bin_uiui(t, q+slim-1, slim);
sbin = mpz_log2(t);
if (sbin < 2*floor(sqrt(q))*log2n)
continue;
for (s = 2; s < slim; s++) {
mpz_bin_uiui(t, q+s-1, s);
sbin = mpz_log2(t);
if (sbin < 2*floor(sqrt(q))*log2n) continue; /* d=1 */
/* Check each number dividing phi(r)/v */
for (d = 2; d < phiv; d++) {
if ((phiv % d) != 0) continue;
scmp = 2 * d * floor(sqrt(q/d)) * log2n;
if (sbin < scmp) break;
}
/* if we did not exit early, this s worked for each d. This s wins. */
if (d >= phiv) break;
}
if (s < slim) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#elif AKS_VARIANT == AKS_VARIANT_BERN41
{
double const log2n = mpz_log2(n);
/* Tuning: Initial 'r' selection */
double const r0 = 0.008 * log2n * log2n;
/* Tuning: Try a larger 'r' if 's' looks very large */
UV const rmult = 8;
UV slim;
mpz_t tmp, tmp2;
PRIME_ITERATOR(iter);
mpz_init(tmp); mpz_init(tmp2);
/* r has to be at least 3. */
prime_iterator_setprime(&iter, (r0 < 2) ? 2 : (UV) r0);
r = prime_iterator_next(&iter);
/* r must be a primitive root. For performance, skip if s looks too big. */
while ( !is_primitive_root_uiprime(n, r) ||
!bern41_acceptable(n, r, rmult*(r-1), tmp, tmp2) )
r = prime_iterator_next(&iter);
prime_iterator_destroy(&iter);
{ /* Binary search for first s in [1,lim] where conditions met */
UV bi = 1;
UV bj = rmult * (r-1);
while (bi < bj) {
s = bi + (bj-bi)/2;
if (!bern41_acceptable(n,r,s,tmp,tmp2)) bi = s+1;
else bj = s;
}
s = bj;
/* Our S goes from 2 to s+1. */
starta = 2;
s = s+1;
}
/* printf("chose r=%lu s=%lu d = %lu i = %lu j = %lu\n", r, s, d, i, j); */
/* Check divisibility to s(s-1) to cover both gcd conditions */
slim = s * (s-1);
if (_verbose > 1) printf("# aks trial to %"UVuf"\n", slim);
if (_GMP_trial_factor(n, 2, slim) > 1)
{ mpz_clear(tmp); mpz_clear(tmp2); return 0; }
/* If we checked divisibility to sqrt(n), then it is prime. */
mpz_sqrt(tmp, n);
if (mpz_cmp_ui(tmp, slim) <= 0)
{ mpz_clear(tmp); mpz_clear(tmp2); return 1; }
/* Check b^(n-1) = 1 mod n for b in [2..s] */
if (_verbose > 1) printf("# aks checking fermat to %"UVuf"\n", s);
mpz_sub_ui(tmp2, n, 1);
for (i = 2; i <= s; i++) {
mpz_set_ui(tmp, i);
mpz_powm(tmp, tmp, tmp2, n);
if (mpz_cmp_ui(tmp, 1) != 0)
{ mpz_clear(tmp); mpz_clear(tmp2); return 0; }
}
mpz_clear(tmp); mpz_clear(tmp2);
}
#endif
if (_verbose) gmp_printf("# AKS %Zd. r = %"UVuf" s = %"UVuf"\n", n, (unsigned long) r, (unsigned long) s);
/* Create the three polynomials we will use */
New(0, px, r, mpz_t);
New(0, py, r, mpz_t);
if ( !px || !py )
croak("allocation failure\n");
for (i = 0; i < r; i++) {
mpz_init(px[i]);
mpz_init(py[i]);
}
retval = 1;
for (a = starta; a <= s; a++) {
retval = test_anr(a, n, r, px, py);
if (!retval) break;
if (_verbose>1) { printf("."); fflush(stdout); }
}
if (_verbose>1) { printf("\n"); fflush(stdout); };
/* Free the polynomials */
for (i = 0; i < r; i++) {
mpz_clear(px[i]);
mpz_clear(py[i]);
}
Safefree(px);
Safefree(py);
return retval;
}
