#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include <time.h>
#include <gmp.h>
#include "ptypes.h"
#include "gmp_main.h"
#include "prime_iterator.h"
#include "bls75.h"
#include "ecpp.h"
#include "utility.h"
#include "factor.h"
#define AKS_VARIANT_V6 1 /* The V6 paper with Lenstra impr */
#define AKS_VARIANT_BORNEMANN 2 /* Based on Folkmar Bornemann's impl */
#define AKS_VARIANT_BERNEXAMPLE 3 /* AKS-Bernstein-Morain */
#define AKS_VARIANT AKS_VARIANT_BORNEMANN
static mpz_t _bgcd;
static mpz_t _bgcd2;
static mpz_t _bgcd3;
#define BGCD_PRIMES 168
#define BGCD_LASTPRIME 997
#define BGCD_NEXTPRIME 1009
#define BGCD2_PRIMES 1229
#define BGCD2_NEXTPRIME 10007
#define BGCD3_PRIMES 4203
#define BGCD3_NEXTPRIME 40009
#define TSTAVAL(arr, val) (arr[(val) >> 6] & (1U << (((val)>>1) & 0x1F)))
#define SETAVAL(arr, val) arr[(val) >> 6] |= 1U << (((val)>>1) & 0x1F)
void _GMP_init(void)
{
/* We should not use this random number system for crypto, so
* using this lousy seed is ok. We just would like something a
* bit different every run. Using Perl_seed(aTHX) would be better. */
unsigned long seed = time(NULL);
init_randstate(seed);
prime_iterator_global_startup();
mpz_init(_bgcd);
_GMP_pn_primorial(_bgcd, BGCD_PRIMES); /* mpz_primorial_ui(_bgcd, 1000) */
mpz_init_set_ui(_bgcd2, 0);
mpz_init_set_ui(_bgcd3, 0);
_init_factor();
}
void _GMP_destroy(void)
{
prime_iterator_global_shutdown();
clear_randstate();
mpz_clear(_bgcd);
mpz_clear(_bgcd2);
mpz_clear(_bgcd3);
destroy_ecpp_gcds();
}
static const unsigned char next_wheel[30] =
{1,7,7,7,7,7,7,11,11,11,11,13,13,17,17,17,17,19,19,23,23,23,23,29,29,29,29,29,29,1};
static const unsigned char prev_wheel[30] =
{29,29,1,1,1,1,1,1,7,7,7,7,11,11,13,13,13,13,17,17,19,19,19,19,23,23,23,23,23,23};
static const unsigned char wheel_advance[30] =
{1,6,5,4,3,2,1,4,3,2,1,2,1,4,3,2,1,2,1,4,3,2,1,6,5,4,3,2,1,2};
static const unsigned char wheel_retreat[30] =
{1,2,1,2,3,4,5,6,1,2,3,4,1,2,1,2,3,4,1,2,1,2,3,4,1,2,3,4,5,6};
static INLINE int _GMP_miller_rabin_ui(mpz_t n, UV base)
{
int rval;
mpz_t a;
mpz_init_set_ui(a, base);
rval = _GMP_miller_rabin(n, a);
mpz_clear(a);
return rval;
}
int _GMP_miller_rabin_random(mpz_t n, UV numbases, char* seedstr)
{
gmp_randstate_t* p_randstate = get_randstate();
mpz_t t, base;
UV i;
if (numbases == 0) return 1;
if (mpz_cmp_ui(n, 100) < 0) /* tiny n */
return (_GMP_is_prob_prime(n) > 0);
mpz_init(base); mpz_init(t);
if (seedstr != 0) { /* Set the RNG seed if they gave us a seed */
mpz_set_str(t, seedstr, 0);
gmp_randseed(*p_randstate, t);
}
mpz_sub_ui(t, n, 3);
for (i = 0; i < numbases; i++) {
mpz_urandomm(base, *p_randstate, t); /* base = 0 .. (n-3)-1 */
mpz_add_ui(base, base, 2); /* base = 2 .. n-2 */
if (_GMP_miller_rabin(n, base) == 0)
break;
}
mpz_clear(base); mpz_clear(t);
return (i >= numbases);
}
int _GMP_miller_rabin(mpz_t n, mpz_t a)
{
mpz_t nminus1, d, x;
UV s, r;
int rval;
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
if (mpz_even_p(n)) return 0; /* multiple of 2 is composite */
}
if (mpz_cmp_ui(a, 1) <= 0)
croak("Base %ld is invalid", mpz_get_si(a));
mpz_init_set(nminus1, n);
mpz_sub_ui(nminus1, nminus1, 1);
mpz_init_set(x, a);
/* Handle large and small bases. Use x so we don't modify their input a. */
if (mpz_cmp(x, n) >= 0)
mpz_mod(x, x, n);
if ( (mpz_cmp_ui(x, 1) <= 0) || (mpz_cmp(x, nminus1) >= 0) ) {
mpz_clear(nminus1);
mpz_clear(x);
return 1;
}
mpz_init_set(d, nminus1);
s = mpz_scan1(d, 0);
mpz_tdiv_q_2exp(d, d, s);
mpz_powm(x, x, d, n);
mpz_clear(d); /* done with a and d */
rval = 0;
if (!mpz_cmp_ui(x, 1) || !mpz_cmp(x, nminus1)) {
rval = 1;
} else {
for (r = 1; r < s; r++) {
mpz_powm_ui(x, x, 2, n);
if (!mpz_cmp_ui(x, 1)) {
break;
}
if (!mpz_cmp(x, nminus1)) {
rval = 1;
break;
}
}
}
mpz_clear(nminus1); mpz_clear(x);
return rval;
}
/* Returns Lucas sequence U_k mod n and V_k mod n defined by P,Q */
void _GMP_lucas_seq(mpz_t U, mpz_t V, mpz_t n, IV P, IV Q, mpz_t k,
mpz_t Qk, mpz_t t)
{
UV b = mpz_sizeinbase(k, 2);
IV D = P*P - 4*Q;
if (mpz_cmp_ui(n, 2) < 0) croak("Lucas sequence modulus n must be > 1");
MPUassert( mpz_cmp_ui(k, 0) >= 0, "lucas_seq: k is negative" );
MPUassert( mpz_cmp_si(n,(P>=0) ? P : -P) > 0, "lucas_seq: P is out of range");
MPUassert( mpz_cmp_si(n,(Q>=0) ? Q : -Q) > 0, "lucas_seq: Q is out of range");
MPUassert( D != 0, "lucas_seq: D is zero" );
if (mpz_cmp_ui(k, 0) <= 0) {
mpz_set_ui(U, 0);
mpz_set_ui(V, 2);
return;
}
mpz_set_ui(U, 1);
mpz_set_si(V, P);
mpz_set_si(Qk, Q);
if (Q == 1) {
/* Use the fast V method if possible. Much faster with small n. */
mpz_set_si(t, P*P-4);
if (P > 2 && mpz_invert(t, t, n)) {
/* Compute V_k and V_{k+1}, then computer U_k from them. */
mpz_set_si(V, P);
mpz_set_si(U, P*P-2);
while (b > 1) {
b--;
if (mpz_tstbit(k, b-1)) {
mpz_mul(V, V, U); mpz_sub_ui(V, V, P); mpz_mod(V, V, n);
mpz_mul(U, U, U); mpz_sub_ui(U, U, 2); mpz_mod(U, U, n);
} else {
mpz_mul(U, V, U); mpz_sub_ui(U, U, P); mpz_mod(U, U, n);
mpz_mul(V, V, V); mpz_sub_ui(V, V, 2); mpz_mod(V, V, n);
}
}
mpz_mul_ui(U, U, 2);
mpz_submul_ui(U, V, P);
mpz_mul(U, U, t);
} else {
/* Fast computation of U_k and V_k, specific to Q = 1 */
while (b > 1) {
mpz_mulmod(U, U, V, n, t); /* U2k = Uk * Vk */
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
mpz_mod(V, V, n); /* V2k = Vk^2 - 2 Q^k */
b--;
if (mpz_tstbit(k, b-1)) {
mpz_mul_si(t, U, D);
/* U: U2k+1 = (P*U2k + V2k)/2 */
mpz_mul_si(U, U, P);
mpz_add(U, U, V);
if (mpz_odd_p(U)) mpz_add(U, U, n);
mpz_fdiv_q_2exp(U, U, 1);
/* V: V2k+1 = (D*U2k + P*V2k)/2 */
mpz_mul_si(V, V, P);
mpz_add(V, V, t);
if (mpz_odd_p(V)) mpz_add(V, V, n);
mpz_fdiv_q_2exp(V, V, 1);
}
}
}
} else {
while (b > 1) {
mpz_mulmod(U, U, V, n, t); /* U2k = Uk * Vk */
mpz_mul(V, V, V);
mpz_submul_ui(V, Qk, 2);
mpz_mod(V, V, n); /* V2k = Vk^2 - 2 Q^k */
mpz_mul(Qk, Qk, Qk); /* Q2k = Qk^2 */
b--;
if (mpz_tstbit(k, b-1)) {
mpz_mul_si(t, U, D);
/* U: U2k+1 = (P*U2k + V2k)/2 */
mpz_mul_si(U, U, P);
mpz_add(U, U, V);
if (mpz_odd_p(U)) mpz_add(U, U, n);
mpz_fdiv_q_2exp(U, U, 1);
/* V: V2k+1 = (D*U2k + P*V2k)/2 */
mpz_mul_si(V, V, P);
mpz_add(V, V, t);
if (mpz_odd_p(V)) mpz_add(V, V, n);
mpz_fdiv_q_2exp(V, V, 1);
mpz_mul_si(Qk, Qk, Q);
}
mpz_mod(Qk, Qk, n);
}
}
mpz_mod(U, U, n);
mpz_mod(V, V, n);
}
int lucas_lehmer(UV p)
{
UV k, tlim;
int res;
mpz_t V, mp, t;
if (p == 2) return 1;
if (!(p&1)) return 0;
mpz_init_set_ui(t, p);
if (!_GMP_is_prob_prime(t)) /* p must be prime */
{ mpz_clear(t); return 0; }
if (p < 23)
{ mpz_clear(t); return (p != 11); }
/* If p=3 mod 4 and p,2p+1 both prime, then 2p+1 | 2^p-1. Cheap test. */
if (p > 3 && p % 4 == 3) {
mpz_mul_ui(t, t, 2);
mpz_add_ui(t, t, 1);
if (_GMP_is_prob_prime(t))
{ mpz_clear(t); return 0; }
}
mpz_init(mp);
mpz_setbit(mp, p);
mpz_sub_ui(mp, mp, 1);
/* Do a little trial division first. Saves quite a bit of time. */
tlim = (p < 1500) ? p/2 : (p < 5000) ? p : 2*p;
if (tlim > UV_MAX/(2*p)) tlim = UV_MAX/(2*p);
for (k = 1; k < tlim; k++) {
UV q = 2*p*k+1;
if ((q%8==1 || q%8==7) && mpz_divisible_ui_p(mp, q))
{ mpz_clear(mp); mpz_clear(t); return 0; }
}
/* We could do some specialized p+1 factoring here. */
mpz_init_set_ui(V, 4);
for (k = 3; k <= p; k++) {
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
/* mpz_mod(V, V, mp) but more efficiently done given mod 2^p-1 */
if (mpz_sgn(V) < 0) mpz_add(V, V, mp);
/* while (n > mp) { n = (n >> p) + (n & mp) } if (n==mp) n=0 */
/* but in this case we can have at most one loop plus a carry */
mpz_tdiv_r_2exp(t, V, p);
mpz_tdiv_q_2exp(V, V, p);
mpz_add(V, V, t);
while (mpz_cmp(V, mp) >= 0) mpz_sub(V, V, mp);
}
res = !mpz_sgn(V);
mpz_clear(t); mpz_clear(mp); mpz_clear(V);
return res;
}
/* Returns: -1 unknown, 0 composite, 2 definitely prime */
int llr(mpz_t N)
{
mpz_t v, k, V;
UV i, n;
int res = -1;
if (mpz_cmp_ui(N,100) <= 0) return (_GMP_is_prob_prime(N) ? 2 : 0);
if (mpz_even_p(N) || mpz_divisible_ui_p(N, 3)) return 0;
mpz_init(v); mpz_init(k);
mpz_add_ui(v, N, 1);
n = mpz_scan1(v, 0);
mpz_tdiv_q_2exp(k, v, n);
/* N = k * 2^n - 1 */
if (mpz_cmp_ui(k,1) == 0) {
res = lucas_lehmer(n) ? 2 : 0;
goto DONE_LLR;
}
if (mpz_sizeinbase(k,2) > n)
goto DONE_LLR;
mpz_init(V);
if (!mpz_divisible_ui_p(k, 3)) { /* Select V for 3 not divis k */
mpz_t U, Qk, t;
mpz_init(U); mpz_init(Qk); mpz_init(t);
_GMP_lucas_seq(U, V, N, 4, 1, k, Qk, t);
mpz_clear(t); mpz_clear(Qk); mpz_clear(U);
} else if ((n % 4 == 0 || n % 4 == 3) && mpz_cmp_ui(k,3)==0) {
mpz_set_ui(V, 5778);
} else {
/* TODO: No logic for the k | 3 case yet. */
mpz_clear(V);
goto DONE_LLR;
}
for (i = 3; i <= n; i++) {
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
mpz_mod(V, V, N);
}
res = mpz_sgn(V) ? 0 : 2;
mpz_clear(V);
DONE_LLR:
if (res != -1 && get_verbose_level() > 1) printf("N shown %s with LLR\n", res ? "prime" : "composite");
mpz_clear(k); mpz_clear(v);
return res;
}
static int lucas_selfridge_params(IV* P, IV* Q, mpz_t n, mpz_t t)
{
IV D = 5;
UV Dui = (UV) D;
while (1) {
UV gcd = mpz_gcd_ui(NULL, n, Dui);
if ((gcd > 1) && mpz_cmp_ui(n, gcd) != 0)
return 0;
mpz_set_si(t, D);
if (mpz_jacobi(t, n) == -1)
break;
if (Dui == 21 && mpz_perfect_square_p(n))
return 0;
Dui += 2;
D = (D > 0) ? -Dui : Dui;
if (Dui > 1000000)
croak("lucas_selfridge_params: D exceeded 1e6");
}
if (P) *P = 1;
if (Q) *Q = (1 - D) / 4;
return 1;
}
static int lucas_extrastrong_params(IV* P, IV* Q, mpz_t n, mpz_t t, UV inc)
{
UV tP = 3;
if (inc < 1 || inc > 256)
croak("Invalid lucas parameter increment: %"UVuf"\n", inc);
while (1) {
UV D = tP*tP - 4;
UV gcd = mpz_gcd_ui(NULL, n, D);
if (gcd > 1 && mpz_cmp_ui(n, gcd) != 0)
return 0;
mpz_set_ui(t, D);
if (mpz_jacobi(t, n) == -1)
break;
if (tP == (3+20*inc) && mpz_perfect_square_p(n))
return 0;
tP += inc;
if (tP > 65535)
croak("lucas_extrastrong_params: P exceeded 65535");
}
if (P) *P = (IV)tP;
if (Q) *Q = 1;
return 1;
}
/* This code was verified against Feitsma's psps-below-2-to-64.txt file.
* is_strong_pseudoprime reduced it from 118,968,378 to 31,894,014.
* all three variations of the Lucas test reduce it to 0.
* The test suite should check that they generate the correct pseudoprimes.
*
* The standard and strong versions use the method A (Selfridge) parameters,
* while the extra strong version uses Baillie's parameters from OEIS A217719.
*
* Using the strong version, we can implement the strong BPSW test as
* specified by Baillie and Wagstaff, 1980, page 1401.
*
* Testing on my x86_64 machine, the strong Lucas code is over 35% faster than
* T.R. Nicely's implementation, and over 40% faster than David Cleaver's.
*/
int _GMP_is_lucas_pseudoprime(mpz_t n, int strength)
{
mpz_t d, U, V, Qk, t;
IV P, Q;
UV s = 0;
int rval;
int _verbose = get_verbose_level();
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
if (mpz_even_p(n)) return 0; /* multiple of 2 is composite */
}
mpz_init(t);
rval = (strength < 2) ? lucas_selfridge_params(&P, &Q, n, t)
: lucas_extrastrong_params(&P, &Q, n, t, 1);
if (!rval) {
mpz_clear(t);
return 0;
}
if (_verbose>3) gmp_printf("N: %Zd D: %ld P: %lu Q: %ld\n", n, P*P-4*Q, P, Q);
mpz_init(U); mpz_init(V); mpz_init(Qk);
mpz_init_set(d, n);
mpz_add_ui(d, d, 1);
if (strength > 0) {
s = mpz_scan1(d, 0);
mpz_tdiv_q_2exp(d, d, s);
}
_GMP_lucas_seq(U, V, n, P, Q, d, Qk, t);
mpz_clear(d);
rval = 0;
if (strength == 0) {
/* Standard checks U_{n+1} = 0 mod n. */
rval = (mpz_sgn(U) == 0);
} else if (strength == 1) {
if (mpz_sgn(U) == 0) {
rval = 1;
} else {
while (s--) {
if (mpz_sgn(V) == 0) {
rval = 1;
break;
}
if (s) {
mpz_mul(V, V, V);
mpz_submul_ui(V, Qk, 2);
mpz_mod(V, V, n);
mpz_mulmod(Qk, Qk, Qk, n, t);
}
}
}
} else {
mpz_sub_ui(t, n, 2);
if ( mpz_sgn(U) == 0 && (mpz_cmp_ui(V, 2) == 0 || mpz_cmp(V, t) == 0) ) {
rval = 1;
} else {
s--; /* The extra strong test tests r < s-1 instead of r < s */
while (s--) {
if (mpz_sgn(V) == 0) {
rval = 1;
break;
}
if (s) {
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
mpz_mod(V, V, n);
}
}
}
}
mpz_clear(Qk); mpz_clear(V); mpz_clear(U); mpz_clear(t);
return rval;
}
/* Pari's clever method. It's an extra-strong Lucas test, but without
* computing U_d. This makes it faster, but yields more pseudoprimes.
*
* increment: 1 for Baillie OEIS, 2 for Pari.
*
* With increment = 1, these results will be a subset of the extra-strong
* Lucas pseudoprimes. With increment = 2, we produce Pari's results (we've
* added the necessary GCD with D so we produce somewhat fewer).
*/
int _GMP_is_almost_extra_strong_lucas_pseudoprime(mpz_t n, UV increment)
{
mpz_t d, V, W, t;
UV P, s;
int rval;
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
if (mpz_even_p(n)) return 0; /* multiple of 2 is composite */
}
mpz_init(t);
{
IV PP;
if (! lucas_extrastrong_params(&PP, 0, n, t, increment) ) {
mpz_clear(t);
return 0;
}
P = (UV) PP;
}
mpz_init(d);
mpz_add_ui(d, n, 1);
s = mpz_scan1(d, 0);
mpz_tdiv_q_2exp(d, d, s);
/* Calculate V_d */
{
UV b = mpz_sizeinbase(d, 2);
mpz_init_set_ui(V, P);
mpz_init_set_ui(W, P*P-2); /* V = V_{k}, W = V_{k+1} */
while (b > 1) {
b--;
if (mpz_tstbit(d, b-1)) {
mpz_mul(V, V, W);
mpz_sub_ui(V, V, P);
mpz_mul(W, W, W);
mpz_sub_ui(W, W, 2);
} else {
mpz_mul(W, V, W);
mpz_sub_ui(W, W, P);
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
}
mpz_mod(V, V, n);
mpz_mod(W, W, n);
}
mpz_clear(W);
}
mpz_clear(d);
rval = 0;
mpz_sub_ui(t, n, 2);
if ( mpz_cmp_ui(V, 2) == 0 || mpz_cmp(V, t) == 0 ) {
rval = 1;
} else {
s--; /* The extra strong test tests r < s-1 instead of r < s */
while (s--) {
if (mpz_sgn(V) == 0) {
rval = 1;
break;
}
if (s) {
mpz_mul(V, V, V);
mpz_sub_ui(V, V, 2);
mpz_mod(V, V, n);
}
}
}
mpz_clear(V); mpz_clear(t);
return rval;
}
static void mat_mulmod_3x3(mpz_t* a, mpz_t* b, mpz_t n, mpz_t* t, mpz_t t2) {
int i, row, col;
for (row = 0; row < 3; row++) {
for (col = 0; col < 3; col++) {
mpz_mul(t[3*row+col], a[3*row+0], b[0+col]);
mpz_mul(t2, a[3*row+1], b[3+col]);
mpz_add(t[3*row+col], t[3*row+col], t2);
mpz_mul(t2, a[3*row+2], b[6+col]);
mpz_add(t[3*row+col], t[3*row+col], t2);
}
}
for (i = 0; i < 9; i++) mpz_mod(a[i], t[i], n);
}
static void mat_powmod_3x3(mpz_t* m, mpz_t kin, mpz_t n) {
mpz_t k, t2, t[9], res[9];
int i;
mpz_init_set(k, kin);
mpz_init(t2);
for (i = 0; i < 9; i++) { mpz_init(t[i]); mpz_init(res[i]); }
mpz_set_ui(res[0],1); mpz_set_ui(res[4],1); mpz_set_ui(res[8],1);
while (mpz_sgn(k)) {
if (mpz_odd_p(k)) mat_mulmod_3x3(res, m, n, t, t2);
mpz_fdiv_q_2exp(k, k, 1);
if (mpz_sgn(k)) mat_mulmod_3x3(m, m, n, t, t2);
}
for (i = 0; i < 9; i++)
{ mpz_set(m[i],res[i]); mpz_clear(res[i]); mpz_clear(t[i]); }
mpz_clear(t2);
mpz_clear(k);
}
int is_perrin_pseudoprime(mpz_t n)
{
int P[9] = {0,1,0, 0,0,1, 1,1,0};
mpz_t m[9];
int i, rval;
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
}
for (i = 0; i < 9; i++) mpz_init_set_ui(m[i], P[i]);
mat_powmod_3x3(m, n, n);
mpz_add(m[1], m[0], m[4]);
mpz_add(m[2], m[1], m[8]);
mpz_mod(m[0], m[2], n);
rval = mpz_sgn(m[0]) ? 0 : 1;
for (i = 0; i < 9; i++) mpz_clear(m[i]);
return rval;
}
int is_frobenius_pseudoprime(mpz_t n, IV P, IV Q)
{
mpz_t t, Vcomp, d, U, V, Qk;
IV D;
int k = 0;
int rval;
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
if (mpz_even_p(n)) return 0; /* multiple of 2 is composite */
if (mpz_perfect_square_p(n)) return 0;
}
mpz_init(t);
if (P == 0 && Q == 0) {
P = 1; Q = 2;
while (k != -1) {
P += 2;
if (P == 3) P = 5; /* P=3,Q=2 -> D=9-8=1 => k=1, so skip */
D = P*P-4*Q;
if (mpz_cmp_ui(n, P >= 0 ? P : -P) <= 0) break;
if (mpz_cmp_ui(n, D >= 0 ? D : -D) <= 0) break;
mpz_set_si(t, D);
k = mpz_jacobi(t, n);
if (k != 1) break;
}
} else {
D = P*P-4*Q;
mpz_set_si(t, D);
if (mpz_perfect_square_p(t))
croak("Frobenius invalid P,Q: (%"IVdf",%"IVdf")", P, Q);
k = mpz_jacobi(t, n);
}
if (k == 0) { mpz_clear(t); return 0; }
{
UV Pu = P >= 0 ? P : -P;
UV Qu = Q >= 0 ? Q : -Q;
UV Du = D >= 0 ? D : -D;
if (mpz_cmp_ui(n, Pu) <= 0 || mpz_cmp_ui(n, Qu) <= 0 || mpz_cmp_ui(n, Du) <= 0) {
mpz_clear(t);
return _GMP_trial_factor(n, 2, Du+Pu+Qu) ? 0 : 1;
}
if (mpz_gcd_ui(NULL, n, Du*Pu*Qu) > 1) {
mpz_clear(t);
return 0;
}
}
mpz_init(Vcomp);
if (k == 1) {
mpz_set_si(Vcomp, 2);
} else {
mpz_set_si(Vcomp, Q);
mpz_mul_ui(Vcomp, Vcomp, 2);
mpz_mod(Vcomp, Vcomp, n);
}
mpz_init(U); mpz_init(V); mpz_init(Qk); mpz_init(d);
if (k == 1) mpz_sub_ui(d, n, 1);
else mpz_add_ui(d, n, 1);
_GMP_lucas_seq(U, V, n, P, Q, d, Qk, t);
rval = ( mpz_sgn(U) == 0 && mpz_cmp(V, Vcomp) == 0 );
mpz_clear(d); mpz_clear(Qk); mpz_clear(V); mpz_clear(U);
mpz_clear(Vcomp); mpz_clear(t);
return rval;
}
/* New code based on draft paper */
int _GMP_is_frobenius_underwood_pseudoprime(mpz_t n)
{
mpz_t temp1, temp2, n_plus_1, s, t;
unsigned long a, ap2, len;
int bit, j, rval = 0;
int _verbose = get_verbose_level();
{
int cmpr = mpz_cmp_ui(n, 2);
if (cmpr == 0) return 1; /* 2 is prime */
if (cmpr < 0) return 0; /* below 2 is composite */
if (mpz_even_p(n)) return 0; /* multiple of 2 is composite */
}
mpz_init(temp1);
for (a = 0; a < 1000000; a++) {
if (a==2 || a==4 || a==7 || a==8 || a==10 || a==14 || a==16 || a==18)
continue;
mpz_set_si(temp1, (long)(a*a) - 4);
j = mpz_jacobi(temp1, n);
if (j == -1) break;
if (j == 0 || (a == 20 && mpz_perfect_square_p(n)))
{ mpz_clear(temp1); return 0; }
}
if (a >= 1000000)
{ mpz_clear(temp1); croak("FU test failure, unable to find suitable a"); }
if (mpz_gcd_ui(NULL, n, (a+4)*(2*a+5)) != 1)
{ mpz_clear(temp1); return 0; }
mpz_init(temp2); mpz_init(n_plus_1),
ap2 = a+2;
mpz_add_ui(n_plus_1, n, 1);
len = mpz_sizeinbase(n_plus_1, 2);
mpz_init_set_ui(s, 1);
mpz_init_set_ui(t, 2);
for (bit = len-2; bit >= 0; bit--) {
mpz_add(temp2, t, t);
if (a != 0) {
mpz_mul_ui(temp1, s, a);
mpz_add(temp2, temp1, temp2);
}
mpz_mul(temp1, temp2, s);
mpz_sub(temp2, t, s);
mpz_add(s, s, t);
mpz_mul(t, s, temp2);
mpz_mod(t, t, n);
mpz_mod(s, temp1, n);
if ( mpz_tstbit(n_plus_1, bit) ) {
if (a == 0) mpz_add(temp1, s, s);
else mpz_mul_ui(temp1, s, ap2);
mpz_add(temp1, temp1, t);
mpz_add(temp2, t, t);
mpz_sub(t, temp2, s);
mpz_set(s, temp1);
}
}
/* n+1 always has an even last bit, so s and t always modded */
mpz_set_ui(temp1, 2*a+5);
mpz_mod(temp1, temp1, n);
if (mpz_cmp_ui(s, 0) == 0 && mpz_cmp(t, temp1) == 0)
rval = 1;
if (_verbose>1) gmp_printf("%Zd is %s with a = %lu\n", n, (rval) ? "probably prime" : "composite", a);
mpz_clear(temp1); mpz_clear(temp2); mpz_clear(n_plus_1);
mpz_clear(s); mpz_clear(t);
return rval;
}
UV _GMP_trial_factor(mpz_t n, UV from_n, UV to_n)
{
size_t log2n = mpz_sizeinbase(n, 2);
UV p = 0;
PRIME_ITERATOR(iter);
if (mpz_cmp_ui(n, 6) < 0) {
unsigned long un = mpz_get_ui(n);
if (un == 1) p = 1;
else if (un == 4 && from_n <= 2 && to_n >= 2) p = 2;
prime_iterator_destroy(&iter);
return p;
}
if (from_n <= 2 && to_n >= 2 && mpz_even_p(n)) p = 2;
else if (from_n <= 3 && to_n >= 3 && mpz_divisible_ui_p(n, 3)) p = 3;
else if (from_n <= 5 && to_n >= 5 && mpz_divisible_ui_p(n, 5)) p = 5;
if (p != 0) {
prime_iterator_destroy(&iter);
return p;
}
if (from_n < 7)
from_n = 7;
if (from_n > to_n)
{ prime_iterator_destroy(&iter); return 0; }
/* p will be the next prime >= from_n */
prime_iterator_setprime(&iter, from_n-1);
p = prime_iterator_next(&iter);
/* All native math if n fits in an unsigned long */
if (log2n <= sizeof(unsigned long)*8) {
unsigned long un = mpz_get_ui(n);
unsigned long sqrtn = (unsigned long) sqrt((double)un);
/* Be extra careful here, as we are using unsigned long, which may not
* match a UV. But GMP's ui is 'unsigned long' so that's what we have
* to deal with. We want to make sure we get the correct integer sqrt,
* but also watch out for overflow. */
while (sqrtn*sqrtn > un) sqrtn--;
while ( (sqrtn+1)*(sqrtn+1) <= un
&& sqrtn < (1UL << 4*sizeof(unsigned long)) )
sqrtn++;
if (to_n > sqrtn)
to_n = sqrtn;
while (p <= to_n) {
if ((un % p) == 0)
break;
p = prime_iterator_next(&iter);
}
prime_iterator_destroy(&iter);
return (p <= to_n) ? p : 0;
}
/* For "small" numbers, this simple method is best. */
{
UV small_to = (log2n < 3000) ? to_n : 30000;
while (p <= small_to) {
if (mpz_divisible_ui_p(n, p))
break;
p = prime_iterator_next(&iter);
}
if (p <= small_to || p > to_n) {
prime_iterator_destroy(&iter);
return (p <= small_to) ? p : 0;
}
}
/* Simple treesieve.
* This is much faster than simple divisibility for really big numbers.
* Credit to Jens K Andersen for writing up the generic algorithm.
*
* This will search until the first group element is > to_n, which means
* we will search a bit farther than to_n.
*/
{
UV found = 0;
unsigned long* xn; /* leaves */
mpz_t* xtree[16+1]; /* the tree (maxdepth = 16) */
mpz_t* xtemp;
unsigned int i, j, d, depth, leafsize, nleaves;
/* Decide on the tree depth (3-16) and number of leaves (10-31) */
{
unsigned int dp = log2n >> 10;
depth = 0;
while (dp >>= 1) depth++;
if (depth < 3) depth = 3;
if (depth > 16) depth = 16;
}
leafsize = log2n / (1U << depth) / 68;
nleaves = 1 << depth;
/* printf("log2n %lu depth %u leafsize %u nleaves %u\n",log2n,depth,leafsize,nleaves); */
New(0, xn, nleaves * leafsize, unsigned long);
for (d = 0; d <= depth; d++) {
unsigned int nodes = 1 << (depth - d);
New(0, xtree[d], nodes, mpz_t);
for (j = 0; j < nodes; j++)
mpz_init(xtree[d][j]);
}
xtemp = xtree[1]; /* implies mindepth = 3 */
while (!found && p <= to_n) {
/* Create nleaves x[0] values, each the product of leafsize primes */
for (i = 0; i < nleaves; i++) {
for (j = 0; j < 4; j++) /* Create 4 sub-products */
mpz_set_ui(xtemp[j], 1);
for (j = 0; j < leafsize; j++) {
xn[i*leafsize+j] = p;
mpz_mul_ui(xtemp[j&3], xtemp[j&3], p);
p = prime_iterator_next(&iter);
}
mpz_mul(xtemp[0], xtemp[0], xtemp[1]); /* Combine for final product*/
mpz_mul(xtemp[2], xtemp[2], xtemp[3]);
mpz_mul(xtree[0][i], xtemp[0], xtemp[2]);
}
/* Multiply product tree, xtree[depth][0] has nleaves*leafsize product */
for (d = 1; d <= depth; d++)
for (i = 0; i < (1U << (depth-d)); i++)
mpz_mul(xtree[d][i], xtree[d-1][2*i], xtree[d-1][2*i+1]);
/* Go backwards replacing the products with remainders */
mpz_tdiv_r(xtree[depth][0], n, xtree[depth][0]);
for (d = 1; d <= depth; d++)
for (i = 0; i < (1U << d); i++)
mpz_tdiv_r(xtree[depth-d][i], xtree[depth-d+1][i>>1], xtree[depth-d][i]);
/* Search each leaf for divisors */
for (i = 0; !found && i < nleaves; i++)
for (j = 0; j < leafsize; j++)
if (mpz_divisible_ui_p(xtree[0][i], xn[i*leafsize+j]))
{ found = xn[i*leafsize+j]; break; }
}
p = found;
for (d = 0; d <= depth; d++) {
unsigned int nodes = 1U << (depth - d);
for (j = 0; j < nodes; j++)
mpz_clear(xtree[d][j]);
Safefree(xtree[d]);
}
Safefree(xn);
if (p > 0 && !mpz_divisible_ui_p(n, p))
croak("incorrect trial factor\n");
}
prime_iterator_destroy(&iter);
return p;
}
/*
* is_prob_prime BPSW -- fast, no known counterexamples
* is_prime is_prob_prime + a little extra
* is_provable_prime really prove it, which could take a very long time
*
* They're all identical for numbers <= 2^64.
*
* The extra M-R tests in is_prime start actually costing something after
* 1000 bits or so. Primality proving will get quite a bit slower as the
* number of bits increases.
*
* What are these extra M-R tests getting us? The primary reference is
* Damgård, Landrock, and Pomerance, 1993. From Rabin-Monier, with random
* bases we have p <= 4^-t. So one extra test gives us p = 0.25, and four
* tests gives us p = 0.00390625. But for larger k (bits in n) this is very
* conservative. Since the value has passed BPSW, we are only interested in
* k > 64. See the calculate-mr-probs.pl script in the xt/ directory.
* For a 256-bit input, we get:
* 1 test: p < 0.000244140625
* 2 tests: p < 0.00000000441533
* 3 tests: p < 0.0000000000062550875
* 4 tests: p < 0.000000000000028421709
*
* It's even more extreme as k goes higher. Also recall that this is the
* probability once we've somehow found a BPSW pseudoprime.
*/
static int primality_pretest(mpz_t n)
{
/* If less than 1009, make trial factor handle it. */
if (mpz_cmp_ui(n, BGCD_NEXTPRIME) < 0)
return _GMP_trial_factor(n, 2, BGCD_LASTPRIME) ? 0 : 2;
/* Check for tiny divisors */
if (mpz_even_p(n)) return 0;
if (sizeof(unsigned long) < 8) {
if (mpz_gcd_ui(NULL, n, 3234846615UL) != 1) return 0; /* 3-29 */
} else {
if (mpz_gcd_ui(NULL, n, 4127218095UL*3948078067UL)!=1) return 0;/* 3-53 */
if (mpz_gcd_ui(NULL, n, 4269855901UL*1673450759UL)!=1) return 0;/* 59-101 */
}
{
UV log2n = mpz_sizeinbase(n,2);
mpz_t t;
mpz_init(t);
/* Do a GCD with all primes < 1009 */
mpz_gcd(t, n, _bgcd);
if (mpz_cmp_ui(t, 1))
{ mpz_clear(t); return 0; }
/* No divisors under 1009 */
if (mpz_cmp_ui(n, BGCD_NEXTPRIME*BGCD_NEXTPRIME) < 0)
{ mpz_clear(t); return 2; }
/* If we're reasonably large, do a gcd with more primes */
if (log2n > 700) {
if (mpz_sgn(_bgcd3) == 0) {
_GMP_pn_primorial(_bgcd3, BGCD3_PRIMES);
mpz_divexact(_bgcd3, _bgcd3, _bgcd);
}
mpz_gcd(t, n, _bgcd3);
if (mpz_cmp_ui(t, 1))
{ mpz_clear(t); return 0; }
} else if (log2n > 300) {
if (mpz_sgn(_bgcd2) == 0) {
_GMP_pn_primorial(_bgcd2, BGCD2_PRIMES);
mpz_divexact(_bgcd2, _bgcd2, _bgcd);
}
mpz_gcd(t, n, _bgcd2);
if (mpz_cmp_ui(t, 1))
{ mpz_clear(t); return 0; }
}
mpz_clear(t);
/* Do more trial division if we think we should.
* According to Menezes (section 4.45) as well as Park (ISPEC 2005),
* we want to select a trial limit B such that B = E/D where E is the
* time for our primality test (one M-R test) and D is the time for
* one trial division. Example times on my machine came out to
* log2n = 840375, E= 6514005000 uS, D=1.45 uS, E/D = 0.006 * log2n
* log2n = 465618, E= 1815000000 uS, D=1.05 uS, E/D = 0.008 * log2n
* log2n = 199353, E= 287282000 uS, D=0.70 uS, E/D = 0.01 * log2n
* log2n = 99678, E= 56956000 uS, D=0.55 uS, E/D = 0.01 * log2n
* log2n = 33412, E= 4289000 uS, D=0.30 uS, E/D = 0.013 * log2n
* log2n = 13484, E= 470000 uS, D=0.21 uS, E/D = 0.012 * log2n
* Our trial division could also be further improved for large inputs.
*/
if (log2n > 16000) {
double dB = (double)log2n * (double)log2n * 0.005;
if (BITS_PER_WORD == 32 && dB > 4200000000.0) dB = 4200000000.0;
if (_GMP_trial_factor(n, BGCD3_NEXTPRIME, (UV)dB)) return 0;
} else if (log2n > 4000) {
if (_GMP_trial_factor(n, BGCD3_NEXTPRIME, 80*log2n)) return 0;
} else if (log2n > 1600) {
if (_GMP_trial_factor(n, BGCD3_NEXTPRIME, 30*log2n)) return 0;
}
}
return 1;
}
int _GMP_BPSW(mpz_t n)
{
if (mpz_cmp_ui(n, 4) < 0)
return (mpz_cmp_ui(n, 1) <= 0) ? 0 : 1;
if (_GMP_miller_rabin_ui(n, 2) == 0) /* Miller Rabin with base 2 */
return 0;
if (_GMP_is_lucas_pseudoprime(n, 2 /*extra strong*/) == 0)
return 0;
if (mpz_sizeinbase(n, 2) <= 64) /* BPSW is deterministic below 2^64 */
return 2;
return 1;
}
int _GMP_is_prob_prime(mpz_t n)
{
/* Step 1: Look for small divisors. This is done purely for performance.
* It is *not* a requirement for the BPSW test. */
int res = primality_pretest(n);
if (res != 1) return res;
/* We'd like to do the LLR test here, but it screws with certificates. */
/* Step 2: The BPSW test. spsp base 2 and slpsp. */
return _GMP_BPSW(n);
}
int _GMP_is_prime(mpz_t n)
{
UV nbits;
/* Similar to is_prob_prime, but put LLR before BPSW, then do more testing */
/* First, simple pretesting */
int prob_prime = primality_pretest(n);
if (prob_prime != 1) return prob_prime;
/* If the number is of form N=k*2^n-1 and we have a fast proof, do it. */
prob_prime = llr(n);
if (prob_prime == 0 || prob_prime == 2) return prob_prime;
/* Start with BPSW */
prob_prime = _GMP_BPSW(n);
nbits = mpz_sizeinbase(n, 2);
/* n has passed the ES BPSW test, making it quite unlikely it is a
* composite (and it cannot be if n < 2^64). */
/* For small numbers, try a quick BLS75 n-1 proof. */
if (prob_prime == 1 && nbits <= 200)
prob_prime = _GMP_primality_bls_nm1(n, 1 /* effort */, 0 /* proof */);
/* If prob_prime is still 1, let's run some extra tests. We could run
* a Frobenius test or some random-base M-R tests. The FU test is
* attractive as it does not overlap with the BPSW test and gives very
* strong assurances. However for small sizes it can take 6x more time
* than a random-base MR test (this narrows to ~2.5x at large sizes). It
* also has the advantage of being deterministic.
*
* For performance reasons, we'll use random-base MRs.
* Assuming we are choosing uniformly random bases and the caller cannot
* predict our random numbers (not guaranteed), then there is less than
* a 1 in 595,000 chance that a composite will pass the extra tests.
*/
if (prob_prime == 1) {
UV ntests;
if (nbits < 80) ntests = 5; /* p < .00000168 */
else if (nbits < 105) ntests = 4; /* p < .00000156 */
else if (nbits < 160) ntests = 3; /* p < .00000164 */
else if (nbits < 413) ntests = 2; /* p < .00000156 */
else ntests = 1; /* p < .00000159 */
prob_prime = _GMP_miller_rabin_random(n, ntests, 0);
/* prob_prime = _GMP_is_frobenius_underwood_pseudoprime(n); */
}
/* Using Damgård, Landrock, and Pomerance, we get upper bounds:
* k <= 64 p = 0
* k < 80 p < 7.53e-08
* k < 105 p < 3.97e-08
* k < 160 p < 1.68e-08
* k >= 160 p < 2.57e-09
* This (1) pretends the SPSP-2 is included in the random tests, (2) doesn't
* take into account the trial division, (3) ignores the observed
* SPSP-2 / Strong Lucas anti-correlation that makes the BPSW test so
* useful. Hence these numbers are still extremely conservative.
*
* For an idea of how conservative we are being, we have now exceeded the
* tests used in Mathematica, Maple, Pari, and SAGE. Note: Pari pre-2.3
* used just M-R tests. Pari 2.3+ uses BPSW with no extra M-R checks for
* is_pseudoprime, but isprime uses APRCL which, being a proof, could only
* output a pseudoprime through a coding error.
*/
return prob_prime;
}
int _GMP_is_provable_prime(mpz_t n, char** prooftext)
{
int prob_prime = _GMP_is_prob_prime(n);
/* Try LLR test if they don't need a proof certificate. */
if (prooftext == 0) {
int res = llr(n);
if (res == 0 || res == 2) return res;
}
/* Run one more M-R test, just in case. */
if (prob_prime == 1)
prob_prime = _GMP_miller_rabin_random(n, 1, 0);
/* We can choose a primality proving algorithm:
* AKS _GMP_is_aks_prime really slow, don't bother
* N-1 _GMP_primality_bls_nm1 small or special numbers
* ECPP _GMP_ecpp fastest in general
*/
/* Give n-1 a small go */
if (prob_prime == 1)
prob_prime = _GMP_primality_bls_nm1(n, 2, prooftext);
/* ECPP */
if (prob_prime == 1)
prob_prime = _GMP_ecpp(n, prooftext);
return prob_prime;
}
/*****************************************************************************/
/* AKS. This implementation is quite slow, but useful to have. */
static int test_anr(UV a, mpz_t n, UV r, mpz_t* px, mpz_t* py)
{
int retval = 1;
UV i, n_mod_r;
mpz_t t;
for (i = 0; i < r; i++)
mpz_set_ui(px[i], 0);
a %= r;
mpz_set_ui(px[0], a);
mpz_set_ui(px[1], 1);
poly_mod_pow(py, px, n, r, n);
mpz_init(t);
n_mod_r = mpz_mod_ui(t, n, r);
if (n_mod_r >= r) croak("n mod r >= r ?!");
mpz_sub_ui(t, py[n_mod_r], 1);
mpz_mod(py[n_mod_r], t, n);
mpz_sub_ui(t, py[0], a);
mpz_mod(py[0], t, n);
mpz_clear(t);
for (i = 0; i < r; i++)
if (mpz_sgn(py[i]))
retval = 0;
return retval;
}
#if AKS_VARIANT == AKS_VARIANT_BORNEMANN
static int is_primitive_root(mpz_t n, UV r)
{
mpz_t m, modr;
UV p, rm1 = r-1;
UV lim = (UV) (sqrt(rm1) + 0.00001);
UV ret = 0;
mpz_init(m);
mpz_init_set_ui(modr, r);
if ((rm1 % 2) == 0) {
mpz_powm_ui(m, n, (rm1/2), modr);
if (!mpz_cmp_ui(m,1))
goto END_PRIMROOT;
}
for (p = 3; p <= lim; p += 2) {
if ((rm1 % p) == 0) {
mpz_powm_ui(m, n, (rm1/p), modr);
if (!mpz_cmp_ui(m,1))
goto END_PRIMROOT;
}
}
ret = 1;
END_PRIMROOT:
mpz_clear(m);
mpz_clear(modr);
return ret;
}
static double mpz_logn(mpz_t n)
{
long exp;
double logn = mpz_get_d_2exp(&exp, n);
logn = log(logn) + (log(2) * exp);
return logn;
}
#endif
#if AKS_VARIANT == AKS_VARIANT_BERNEXAMPLE
static UV largest_factor(UV n) {
UV p = 2;
PRIME_ITERATOR(iter);
while (n >= p*p && !prime_iterator_isprime(&iter, n)) {
while ( (n % p) == 0 && n >= p*p ) { n /= p; }
p = prime_iterator_next(&iter);
}
prime_iterator_destroy(&iter);
return n;
}
#endif
int _GMP_is_aks_prime(mpz_t n)
{
mpz_t *px, *py;
int retval;
UV i, s, r, a;
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;
PRIME_ITERATOR(iter);
mpz_init(sqrtn);
mpz_sqrt(sqrtn, n);
/* limit should be floor( log2(n) ** 2 ). The simple GMP solution is
* to get ceil(log2(n)) via mpz_sizeinbase(n,2) and square, but that
* overcalculates by a fair amount. We'll calculate float log2n as:
* ceil(log2(n**k)) / k [mpz_sizeinbase(n,2) <=> ceil(log2(n))]
* which gives us a value that slightly overestimates log2(n).
*/
mpz_init(t);
mpz_pow_ui(t, n, 32);
log2n = ((double) mpz_sizeinbase(t, 2) + 0.000001) / 32.0;
limit = (UV) floor( log2n * log2n );
mpz_clear(t);
if (_verbose>1) gmp_printf("# AKS checking order_r(%Zd) to %lu\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;
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 (mpz_order_ui(r, n, limit) > limit)
{ retval = 2; break; }
}
prime_iterator_destroy(&iter);
mpz_clear(sqrtn);
if (retval != 2) return retval;
/* Bernstein 2002 suggests we could use:
* s = (UV) floor( ceil(sqrt(((double)(r-1))/3.0)) * log2n );
* here, by Minkowski's theorem. r-1 is really phi(r). */
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(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 i = 1;
UV j = slim;
while (i < j) {
s = i + (j-i)/2;
mpz_bin_uiui(tmp, r+s-1, s);
x = mpz_logn(tmp) / c2 - 1.0;
if (x < 0) i = s+1;
else j = s;
}
s = i-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 %lu\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_BERNEXAMPLE
{
double slim, x;
mpz_t t, t2;
PRIME_ITERATOR(iter);
mpz_init(t); mpz_init(t2);
r = 0;
s = 0;
while (1) {
UV q, tmp;
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);
/* gmp_printf("r %lu q %lu t %Zd\n", r, q, t); */
if (mpz_cmp_ui(t, 1) <= 0) continue;
/* printf("trying r=%lu\n", r); */
s = 2;
slim = 2 * floor(sqrt(r)) * log(mpz_get_d(n));
while (1) {
mpz_bin_uiui(t, q+s-1, s);
x = log(mpz_get_d(t)) / slim - 1.0;
if (x > 0) break;
s++;
if (s > 1000000) { s=0; break; }
}
if (s != 0) break;
}
mpz_clear(t); mpz_clear(t2);
prime_iterator_destroy(&iter);
if (_GMP_trial_factor(n, 2, s) > 1)
return 0;
}
#endif
if (_verbose) gmp_printf("# AKS %Zd. r = %lu s = %lu\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 = 1; a <= s; a++) {
if (! test_anr(a, n, r, px, py) ) {
retval = 0;
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;
}
/*****************************************************************************/
/* Controls how many numbers to sieve. Little time impact. */
#define NPS_MERIT 30.0
/* Controls how many primes to use. Big time impact. */
#define NPS_DEPTH (log2n > 200000 ? 4200000000UL : log2n * (log2n/10))
static void next_prime_with_sieve(mpz_t n) {
uint32_t* comp;
mpz_t t, base;
UV i;
UV log2n = mpz_sizeinbase(n, 2);
UV width = (UV) (NPS_MERIT/1.4427 * (double)log2n + 0.5);
UV depth = NPS_DEPTH;
if (width & 1) width++; /* Make width even */
mpz_add_ui(n, n, mpz_even_p(n) ? 1 : 2); /* Set n to next odd */
mpz_init(t); mpz_init(base);
while (1) {
mpz_set(base, n);
comp = partial_sieve(base, width, depth); /* sieve range to depth */
for (i = 1; i <= width; i += 2) {
if (!TSTAVAL(comp, i)) {
mpz_add_ui(t, base, i); /* We found a candidate */
if (_GMP_BPSW(t)) {
mpz_set(n, t);
mpz_clear(t); mpz_clear(base);
Safefree(comp);
return;
}
}
}
Safefree(comp); /* A huge gap found, so sieve another range */
mpz_add_ui(n, n, width);
}
}
static void prev_prime_with_sieve(mpz_t n) {
uint32_t* comp;
mpz_t t, base;
UV i, j;
UV log2n = mpz_sizeinbase(n, 2);
UV width = (UV) (NPS_MERIT/1.4427 * (double)log2n + 0.5);
UV depth = NPS_DEPTH;
mpz_sub_ui(n, n, mpz_even_p(n) ? 1 : 2); /* Set n to prev odd */
width = 64 * ((width+63)/64); /* Round up to next 64 */
mpz_init(t); mpz_init(base);
while (1) {
mpz_sub_ui(base, n, width-2);
/* gmp_printf("sieve from %Zd to %Zd width %lu\n", base, n, width); */
comp = partial_sieve(base, width, depth); /* sieve range to depth */
/* if (mpz_odd_p(base)) croak("base off after partial");
if (width & 1) croak("width is odd after partial"); */
for (j = 1; j < width; j += 2) {
i = width - j;
if (!TSTAVAL(comp, i)) {
mpz_add_ui(t, base, i); /* We found a candidate */
if (_GMP_BPSW(t)) {
mpz_set(n, t);
mpz_clear(t); mpz_clear(base);
Safefree(comp);
return;
}
}
}
Safefree(comp); /* A huge gap found, so sieve another range */
mpz_sub_ui(n, n, width);
}
}
/* Modifies argument */
void _GMP_next_prime(mpz_t n)
{
if (mpz_cmp_ui(n, 29) < 0) { /* small inputs */
UV m = mpz_get_ui(n);
m = (m < 2) ? 2 : (m < 3) ? 3 : (m < 5) ? 5 : next_wheel[m];
mpz_set_ui(n, m);
} else if (mpz_sizeinbase(n,2) > 120) {
next_prime_with_sieve(n);
} else {
UV m23 = mpz_fdiv_ui(n, 223092870UL); /* 2*3*5*7*11*13*17*19*23 */
UV m = m23 % 30;
do {
UV skip = wheel_advance[m];
mpz_add_ui(n, n, skip);
m23 += skip;
m = next_wheel[m];
} while ( !(m23% 7) || !(m23%11) || !(m23%13) || !(m23%17) ||
!(m23%19) || !(m23%23) || !_GMP_is_prob_prime(n) );
}
}
/* Modifies argument */
void _GMP_prev_prime(mpz_t n)
{
UV m, m23;
if (mpz_cmp_ui(n, 29) <= 0) { /* small inputs */
m = mpz_get_ui(n);
m = (m < 3) ? 0 : (m < 4) ? 2 : (m < 6) ? 3 : (m < 8) ? 5 : prev_wheel[m];
mpz_set_ui(n, m);
} else if (mpz_sizeinbase(n,2) > 200) {
prev_prime_with_sieve(n);
} else {
m23 = mpz_fdiv_ui(n, 223092870UL); /* 2*3*5*7*11*13*17*19*23 */
m = m23 % 30;
m23 += 223092870UL; /* No need to re-mod inside the loop */
do {
UV skip = wheel_retreat[m];
mpz_sub_ui(n, n, skip);
m23 -= skip;
m = prev_wheel[m];
} while ( !(m23% 7) || !(m23%11) || !(m23%13) || !(m23%17) ||
!(m23%19) || !(m23%23) || !_GMP_is_prob_prime(n) );
}
}
#define LAST_DOUBLE_PROD \
((BITS_PER_WORD == 32) ? UVCONST(65521) : UVCONST(4294967291))
void _GMP_pn_primorial(mpz_t prim, UV n)
{
UV p = 2;
PRIME_ITERATOR(iter);
if (n < 800) { /* Don't go above 6500 to prevent overflow below */
/* Simple linear multiplication, two at a time */
mpz_set_ui(prim, 1);
while (n-- > 0) {
if (n > 0) { p *= prime_iterator_next(&iter); n--; }
mpz_mul_ui(prim, prim, p);
p = prime_iterator_next(&iter);
}
} else {
/* Shallow product tree, ~10x faster for large values */
mpz_t t[16];
UV i;
for (i = 0; i < 16; i++) mpz_init_set_ui(t[i], 1);
i = 0;
while (n-- > 0) {
if (p <= LAST_DOUBLE_PROD && n > 0)
{ p *= prime_iterator_next(&iter); n--; }
mpz_mul_ui(t[i&15], t[i&15], p);
p = prime_iterator_next(&iter);
i++;
}
mpz_product(t, 0, 16-1);
mpz_set(prim, t[0]);
for (i = 0; i < 16; i++) mpz_clear(t[i]);
}
prime_iterator_destroy(&iter);
}
void _GMP_primorial(mpz_t prim, mpz_t n)
{
UV p = 2;
PRIME_ITERATOR(iter);
if (mpz_cmp_ui(n, 1000) < 0) {
/* Simple linear multiplication, one at a time */
mpz_set_ui(prim, 1);
while (mpz_cmp_ui(n, p) >= 0) {
mpz_mul_ui(prim, prim, p);
p = prime_iterator_next(&iter);
}
} else {
/* Shallow product tree, ~10x faster for large values */
mpz_t t[16];
UV i;
for (i = 0; i < 16; i++) mpz_init_set_ui(t[i], 1);
i = 0;
while (mpz_cmp_ui(n, p) >= 0) {
mpz_mul_ui(t[i&15], t[i&15], p);
p = prime_iterator_next(&iter);
i++;
}
mpz_product(t, 0, 16-1);
mpz_set(prim, t[0]);
for (i = 0; i < 16; i++) mpz_clear(t[i]);
}
prime_iterator_destroy(&iter);
}
/* Luschny's version of the "Brent-Harvey" method */
void bernfrac(mpz_t num, mpz_t den, mpz_t zn)
{
UV k, j, n = mpz_get_ui(zn);
mpz_t* T;
mpz_t t;
if (n == 0) { mpz_set_ui(num, 1); mpz_set_ui(den, 1); return; }
else if (n == 1) { mpz_set_ui(num, 1); mpz_set_ui(den, 2); return; }
else if (n & 1) { mpz_set_ui(num, 0); mpz_set_ui(den, 1); return; }
n >>= 1;
New(0, T, n+1, mpz_t);
for (k = 1; k <= n; k++) mpz_init(T[k]);
mpz_set_ui(T[1], 1);
mpz_init(t);
for (k = 2; k <= n; k++)
mpz_mul_ui(T[k], T[k-1], k-1);
for (k = 2; k <= n; k++) {
for (j = k; j <= n; j++) {
mpz_mul_ui(t, T[j], j-k+2);
mpz_mul_ui(T[j], T[j-1], j-k);
mpz_add(T[j], T[j], t);
}
}
mpz_mul_ui(num, T[n], n);
mpz_mul_si(num, num, (n & 1) ? 2 : -2);
mpz_set_ui(t, 1);
mpz_mul_2exp(den, t, 2*n); /* den = U = 1 << 2n */
mpz_sub_ui(t, den, 1); /* t = U-1 */
mpz_mul(den, den, t); /* den = U*(U-1) */
mpz_gcd(t, num, den);
mpz_divexact(num, num, t);
mpz_divexact(den, den, t);
mpz_clear(t);
for (k = 1; k <= n; k++) mpz_clear(T[k]);
Safefree(T);
}
void stirling(mpz_t r, unsigned long n, unsigned long m, UV type)
{
mpz_t t, t2;
unsigned long j;
if (type < 1 || type > 3) croak("stirling type must be 1, 2, or 3");
if (n == m) {
mpz_set_ui(r, 1);
} else if (n == 0 || m == 0 || m > n) {
mpz_set_ui(r,0);
} else if (m == 1) {
switch (type) {
case 1: mpz_fac_ui(r, n-1); if (!(n&1)) mpz_neg(r, r); break;
case 2: mpz_set_ui(r, 1); break;
case 3:
default: mpz_fac_ui(r, n); break;
}
} else {
mpz_init(t); mpz_init(t2);
mpz_set_ui(r,0);
if (type == 3) {
mpz_bin_uiui(t, n-1, m-1);
mpz_fac_ui(t2, n);
mpz_mul(r, t, t2);
mpz_fac_ui(t2, m);
mpz_divexact(r, r, t2);
} else if (type == 2) {
for (j = 1; j <= m; j++) {
mpz_bin_uiui(t, m, j);
mpz_ui_pow_ui(t2, j, n);
mpz_mul(t, t, t2);
if ((m-j) & 1) mpz_sub(r, r, t);
else mpz_add(r, r, t);
}
mpz_fac_ui(t, m);
mpz_divexact(r, r, t);
} else {
for (j = 1; j <= n-m; j++) {
mpz_bin_uiui(t, n+j-1, n+j-m);
mpz_bin_uiui(t2, n+n-m, n-j-m);
mpz_mul(t, t, t2);
stirling(t2, n+j-m, j, 2);
mpz_mul(t, t, t2);
if (j & 1) mpz_sub(r, r, t);
else mpz_add(r, r, t);
}
}
mpz_clear(t2); mpz_clear(t);
}
}
/* Goetgheluck method. Also thanks to Peter Luschny */
void binomial(mpz_t r, UV n, UV k)
{
UV fi, nk, sqrtn, piN, prime, i, j;
UV* primes;
mpz_t* mprimes;
if (k > n) { mpz_set_ui(r, 0); return; }
if (k == 0 || k == n) { mpz_set_ui(r, 1); return; }
if (k > n/2) k = n-k;
sqrtn = (UV) (sqrt((double)n));
fi = 0;
nk = n-k;
primes = sieve_to_n(n, &piN);
#define PUSHP(p) \
do { \
if ((j++ % 8) == 0) mpz_init_set_ui(mprimes[fi++], p); \
else mpz_mul_ui(mprimes[fi-1], mprimes[fi-1], p); \
} while (0)
New(0, mprimes, (piN+7)/8, mpz_t);
for (i = 0, j = 0; i < piN; i++) {
prime = primes[i];
if (prime > nk) {
PUSHP(prime);
} else if (prime > n/2) {
/* nothing */
} else if (prime > sqrtn) {
if (n % prime < k % prime)
PUSHP(prime);
} else {
UV N = n, K = k, p = 1, r = 0;
while (N > 0) {
r = (N % prime) < (K % prime + r) ? 1 : 0;
if (r == 1) p *= prime;
N /= prime;
K /= prime;
}
if (p > 1)
PUSHP(p);
}
}
Safefree(primes);
mpz_product(mprimes, 0, fi-1);
mpz_set(r, mprimes[0]);
for (i = 0; i < fi; i++)
mpz_clear(mprimes[i]);
Safefree(mprimes);
}
#define TEST_FOR_2357(n, f) \
{ \
if (mpz_divisible_ui_p(n, 2)) { mpz_set_ui(f, 2); return 1; } \
if (mpz_divisible_ui_p(n, 3)) { mpz_set_ui(f, 3); return 1; } \
if (mpz_divisible_ui_p(n, 5)) { mpz_set_ui(f, 5); return 1; } \
if (mpz_divisible_ui_p(n, 7)) { mpz_set_ui(f, 7); return 1; } \
if (mpz_cmp_ui(n, 121) < 0) { return 0; } \
}
int _GMP_prho_factor(mpz_t n, mpz_t f, UV a, UV rounds)
{
mpz_t U, V, oldU, oldV, m;
int i;
const UV inner = 256;
TEST_FOR_2357(n, f);
rounds = (rounds + inner - 1) / inner;
mpz_init_set_ui(U, 7);
mpz_init_set_ui(V, 7);
mpz_init(m);
mpz_init(oldU);
mpz_init(oldV);
while (rounds-- > 0) {
mpz_set_ui(m, 1); mpz_set(oldU, U); mpz_set(oldV, V);
for (i = 0; i < (int)inner; i++) {
mpz_mul(U, U, U); mpz_add_ui(U, U, a); mpz_tdiv_r(U, U, n);
mpz_mul(V, V, V); mpz_add_ui(V, V, a); mpz_tdiv_r(V, V, n);
mpz_mul(V, V, V); mpz_add_ui(V, V, a); mpz_tdiv_r(V, V, n);
if (mpz_cmp(U, V) >= 0) mpz_sub(f, U, V);
else mpz_sub(f, V, U);
mpz_mul(m, m, f);
mpz_tdiv_r(m, m, n);
}
mpz_gcd(f, m, n);
if (!mpz_cmp_ui(f, 1))
continue;
if (!mpz_cmp(f, n)) {
/* f == n, so we have to back up to see what factor got found */
mpz_set(U, oldU); mpz_set(V, oldV);
i = inner;
do {
mpz_mul(U, U, U); mpz_add_ui(U, U, a); mpz_tdiv_r(U, U, n);
mpz_mul(V, V, V); mpz_add_ui(V, V, a); mpz_tdiv_r(V, V, n);
mpz_mul(V, V, V); mpz_add_ui(V, V, a); mpz_tdiv_r(V, V, n);
if (mpz_cmp(U, V) >= 0) mpz_sub(f, U, V);
else mpz_sub(f, V, U);
mpz_gcd(f, f, n);
} while (!mpz_cmp_ui(f, 1) && i-- != 0);
if ( (!mpz_cmp_ui(f, 1)) || (!mpz_cmp(f, n)) ) break;
}
mpz_clear(U); mpz_clear(V); mpz_clear(m); mpz_clear(oldU); mpz_clear(oldV);
return 1;
}
mpz_clear(U); mpz_clear(V); mpz_clear(m); mpz_clear(oldU); mpz_clear(oldV);
mpz_set(f, n);
return 0;
}
int _GMP_pbrent_factor(mpz_t n, mpz_t f, UV a, UV rounds)
{
mpz_t Xi, Xm, saveXi, m, t;
UV i, r;
const UV inner = 256;
TEST_FOR_2357(n, f);
mpz_init_set_ui(Xi, 2);
mpz_init_set_ui(Xm, 2);
mpz_init(m);
mpz_init(t);
mpz_init(saveXi);
r = 1;
while (rounds > 0) {
UV rleft = (r > rounds) ? rounds : r;
while (rleft > 0) { /* Do rleft rounds, inner at a time */
UV dorounds = (rleft > inner) ? inner : rleft;
mpz_set_ui(m, 1);
mpz_set(saveXi, Xi);
for (i = 0; i < dorounds; i++) {
mpz_mul(t, Xi, Xi); mpz_add_ui(t, t, a); mpz_tdiv_r(Xi, t, n);
if (mpz_cmp(Xi, Xm) >= 0) mpz_sub(f, Xi, Xm);
else mpz_sub(f, Xm, Xi);
mpz_mul(t, m, f);
mpz_tdiv_r(m, t, n);
}
rleft -= dorounds;
rounds -= dorounds;
mpz_gcd(f, m, n);
if (mpz_cmp_ui(f, 1) != 0)
break;
}
if (!mpz_cmp_ui(f, 1)) {
r *= 2;
mpz_set(Xm, Xi);
continue;
}
if (!mpz_cmp(f, n)) {
/* f == n, so we have to back up to see what factor got found */
mpz_set(Xi, saveXi);
do {
mpz_mul(t, Xi, Xi); mpz_add_ui(t, t, a); mpz_tdiv_r(Xi, t, n);
if (mpz_cmp(Xi, Xm) >= 0) mpz_sub(f, Xi, Xm);
else mpz_sub(f, Xm, Xi);
mpz_gcd(f, f, n);
} while (!mpz_cmp_ui(f, 1) && r-- != 0);
if ( (!mpz_cmp_ui(f, 1)) || (!mpz_cmp(f, n)) ) break;
}
mpz_clear(Xi); mpz_clear(Xm); mpz_clear(m); mpz_clear(saveXi); mpz_clear(t);
return 1;
}
mpz_clear(Xi); mpz_clear(Xm); mpz_clear(m); mpz_clear(saveXi); mpz_clear(t);
mpz_set(f, n);
return 0;
}
void _GMP_lcm_of_consecutive_integers(UV B, mpz_t m)
{
UV i, p, p_power, pmin;
mpz_t t[8];
PRIME_ITERATOR(iter);
/* Create eight sub-products to combine at the end. */
for (i = 0; i < 8; i++) mpz_init_set_ui(t[i], 1);
i = 0;
/* For each prime, multiply m by p^floor(log B / log p), which means
* raise p to the largest power e such that p^e <= B.
*/
if (B >= 2) {
p_power = 2;
while (p_power <= B/2)
p_power *= 2;
mpz_mul_ui(t[i&7], t[i&7], p_power); i++;
}
p = prime_iterator_next(&iter);
while (p <= B) {
pmin = B/p;
if (p > pmin)
break;
p_power = p*p;
while (p_power <= pmin)
p_power *= p;
mpz_mul_ui(t[i&7], t[i&7], p_power); i++;
p = prime_iterator_next(&iter);
}
while (p <= B) {
mpz_mul_ui(t[i&7], t[i&7], p); i++;
p = prime_iterator_next(&iter);
}
/* combine the eight sub-products */
for (i = 0; i < 4; i++) mpz_mul(t[i], t[2*i], t[2*i+1]);
for (i = 0; i < 2; i++) mpz_mul(t[i], t[2*i], t[2*i+1]);
mpz_mul(m, t[0], t[1]);
for (i = 0; i < 8; i++) mpz_clear(t[i]);
prime_iterator_destroy(&iter);
}
int _GMP_pminus1_factor(mpz_t n, mpz_t f, UV B1, UV B2)
{
mpz_t a, savea, t;
UV q, saveq, j, sqrtB1;
int _verbose = get_verbose_level();
PRIME_ITERATOR(iter);
TEST_FOR_2357(n, f);
if (B1 < 7) return 0;
mpz_init(a);
mpz_init(savea);
mpz_init(t);
if (_verbose>2) gmp_printf("# p-1 trying %Zd (B1=%lu B2=%lu)\n", n, (unsigned long)B1, (unsigned long)B2);
/* STAGE 1
* Montgomery 1987 p249-250 and Brent 1990 p5 both indicate we can calculate
* a^m mod n where m is the lcm of the integers to B1. This can be done
* using either
* m = calc_lcm(B), b = a^m mod n
* or
* calculate_b_lcm(b, B1, a, n);
*
* The first means raising a to a huge power then doing the mod, which is
* inefficient and can be _very_ slow on some machines. The latter does
* one powmod for each prime power, which works pretty well. Yet another
* way to handle this is to loop over each prime p below B1, calculating
* a = a^(p^e) mod n, where e is the largest e such that p^e <= B1.
* My experience with GMP is that this last method is faster with large B1,
* sometimes a lot faster.
*
* One thing that can speed things up quite a bit is not running the GCD
* on every step. However with small factors this means we can easily end
* up with multiple factors between GCDs, so we allow backtracking. This
* could also be added to stage 2, but it's far less likely to happen there.
*/
j = 15;
mpz_set_ui(a, 2);
mpz_set_ui(savea, 2);
saveq = 2;
/* We could wrap this in a loop trying a few different a values, in case
* the current one ended up going to 0. */
q = 2;
mpz_set_ui(t, 1);
sqrtB1 = (UV) sqrt(B1);
while (q <= B1) {
UV k = q;
if (q <= sqrtB1) {
UV kmin = B1/q;
while (k <= kmin)
k *= q;
}
mpz_mul_ui(t, t, k); /* Accumulate powers for a */
if ( (j++ % 32) == 0) {
mpz_powm(a, a, t, n); /* a=a^(k1*k2*k3*...) mod n */
if (mpz_sgn(a)) mpz_sub_ui(t, a, 1);
else mpz_sub_ui(t, n, 1);
mpz_gcd(f, t, n); /* f = gcd(a-1, n) */
mpz_set_ui(t, 1);
if (mpz_cmp(f, n) == 0)
break;
if (mpz_cmp_ui(f, 1) != 0)
goto end_success;
saveq = q;
mpz_set(savea, a);
}
q = prime_iterator_next(&iter);
}
mpz_powm(a, a, t, n);
if (mpz_sgn(a)) mpz_sub_ui(t, a, 1);
else mpz_sub_ui(t, n, 1);
mpz_gcd(f, t, n);
if (mpz_cmp(f, n) == 0) {
/* We found multiple factors. Loop one at a time. */
prime_iterator_setprime(&iter, saveq);
mpz_set(a, savea);
for (q = saveq; q <= B1; q = prime_iterator_next(&iter)) {
UV k = q;
if (q <= sqrtB1) {
UV kmin = B1/q;
while (k <= kmin)
k *= q;
}
mpz_powm_ui(a, a, k, n );
mpz_sub_ui(t, a, 1);
mpz_gcd(f, t, n);
if (mpz_cmp(f, n) == 0)
goto end_fail;
if (mpz_cmp_ui(f, 1) != 0)
goto end_success;
}
}
if ( (mpz_cmp_ui(f, 1) != 0) && (mpz_cmp(f, n) != 0) )
goto end_success;
/* STAGE 2
* This is the standard continuation which replaces the powmods in stage 1
* with two mulmods, with a GCD every 64 primes (no backtracking).
* This is quite a bit faster than stage 1.
* See Montgomery 1987, p250-253 for possible optimizations.
* We quickly precalculate a few of the prime gaps, and lazily cache others
* up to a gap of 222. That's enough for a B2 value of 189 million. We
* still work above that, we just won't cache the value for big gaps.
*/
if (B2 > B1) {
mpz_t b, bm, bmdiff;
mpz_t precomp_bm[111];
int is_precomp[111] = {0};
UV* primes = 0;
UV sp = 1;
mpz_init(bmdiff);
mpz_init_set(bm, a);
mpz_init_set_ui(b, 1);
/* Set the first 20 differences */
mpz_powm_ui(bmdiff, bm, 2, n);
mpz_init_set(precomp_bm[0], bmdiff);
is_precomp[0] = 1;
for (j = 1; j < 22; j++) {
mpz_mul(bmdiff, bmdiff, bm);
mpz_mul(bmdiff, bmdiff, bm);
mpz_tdiv_r(bmdiff, bmdiff, n);
mpz_init_set(precomp_bm[j], bmdiff);
is_precomp[j] = 1;
}
mpz_powm_ui(a, a, q, n );
if (B2 < 10000000) {
/* grab all the primes at once. Hack around non-perfect iterator. */
primes = sieve_to_n(B2+300, 0);
for (sp = B1>>4; primes[sp] <= q; sp++) ;
/* q is primes <= B1, primes[sp] is the next prime */
}
j = 31;
while (q <= B2) {
UV lastq, qdiff;
lastq = q;
q = primes ? primes[sp++] : prime_iterator_next(&iter);
qdiff = (q - lastq) / 2 - 1;
if (qdiff < 111 && is_precomp[qdiff]) {
mpz_mul(t, a, precomp_bm[qdiff]);
} else if (qdiff < 111) {
mpz_powm_ui(bmdiff, bm, q-lastq, n);
mpz_init_set(precomp_bm[qdiff], bmdiff);
is_precomp[qdiff] = 1;
mpz_mul(t, a, bmdiff);
} else {
mpz_powm_ui(bmdiff, bm, q-lastq, n); /* Big gap */
mpz_mul(t, a, bmdiff);
}
mpz_tdiv_r(a, t, n);
if (mpz_sgn(a)) mpz_sub_ui(t, a, 1);
else mpz_sub_ui(t, n, 1);
mpz_mul(b, b, t);
if ((j % 2) == 0) /* put off mods a little */
mpz_tdiv_r(b, b, n);
if ( (j++ % 64) == 0) { /* GCD every so often */
mpz_gcd(f, b, n);
if ( (mpz_cmp_ui(f, 1) != 0) && (mpz_cmp(f, n) != 0) )
break;
}
}
mpz_gcd(f, b, n);
mpz_clear(b);
mpz_clear(bm);
mpz_clear(bmdiff);
for (j = 0; j < 111; j++) {
if (is_precomp[j])
mpz_clear(precomp_bm[j]);
}
if (primes != 0) Safefree(primes);
if ( (mpz_cmp_ui(f, 1) != 0) && (mpz_cmp(f, n) != 0) )
goto end_success;
}
end_fail:
mpz_set(f,n);
end_success:
prime_iterator_destroy(&iter);
mpz_clear(a);
mpz_clear(savea);
mpz_clear(t);
if ( (mpz_cmp_ui(f, 1) != 0) && (mpz_cmp(f, n) != 0) ) {
if (_verbose>2) gmp_printf("# p-1: %Zd\n", f);
return 1;
}
if (_verbose>2) gmp_printf("# p-1: no factor\n");
mpz_set(f, n);
return 0;
}
static void pp1_pow(mpz_t X, mpz_t Y, unsigned long exp, mpz_t n)
{
mpz_t x0;
unsigned long bit;
{
unsigned long v = exp;
unsigned long b = 1;
while (v >>= 1) b++;
bit = 1UL << (b-2);
}
mpz_init_set(x0, X);
mpz_mul(Y, X, X);
mpz_sub_ui(Y, Y, 2);
mpz_tdiv_r(Y, Y, n);
while (bit) {
if ( exp & bit ) {
mpz_mul(X, X, Y);
mpz_sub(X, X, x0);
mpz_mul(Y, Y, Y);
mpz_sub_ui(Y, Y, 2);
} else {
mpz_mul(Y, X, Y);
mpz_sub(Y, Y, x0);
mpz_mul(X, X, X);
mpz_sub_ui(X, X, 2);
}
mpz_mod(X, X, n);
mpz_mod(Y, Y, n);
bit >>= 1;
}
mpz_clear(x0);
}
int _GMP_pplus1_factor(mpz_t n, mpz_t f, UV P0, UV B1, UV B2)
{
UV j, q, saveq, sqrtB1;
mpz_t X, Y, saveX;
PRIME_ITERATOR(iter);
TEST_FOR_2357(n, f);
if (B1 < 7) return 0;
mpz_init_set_ui(X, P0);
mpz_init(Y);
mpz_init(saveX);
/* Montgomery 1987 */
if (P0 == 0) {
mpz_set_ui(X, 7);
if (mpz_invert(X, X, n)) {
mpz_mul_ui(X, X, 2);
mpz_mod(X, X, n);
} else
P0 = 1;
}
if (P0 == 1) {
mpz_set_ui(X, 5);
if (mpz_invert(X, X, n)) {
mpz_mul_ui(X, X, 6);
mpz_mod(X, X, n);
} else
P0 = 2;
}
if (P0 == 2) {
mpz_set_ui(X, 11);
if (mpz_invert(X, X, n)) {
mpz_mul_ui(X, X, 23);
mpz_mod(X, X, n);
}
}
sqrtB1 = (UV) sqrt(B1);
j = 8;
q = 2;
saveq = q;
mpz_set(saveX, X);
while (q <= B1) {
UV k = q;
if (q <= sqrtB1) {
UV kmin = B1/q;
while (k <= kmin)
k *= q;
}
pp1_pow(X, Y, k, n);
if ( (j++ % 16) == 0) {
mpz_sub_ui(f, X, 2);
if (mpz_sgn(f) == 0) break;
mpz_gcd(f, f, n);
if (mpz_cmp(f, n) == 0) break;
if (mpz_cmp_ui(f, 1) > 0) goto end_success;
saveq = q;
mpz_set(saveX, X);
}
q = prime_iterator_next(&iter);
}
mpz_sub_ui(f, X, 2);
mpz_gcd(f, f, n);
if (mpz_cmp_ui(X, 2) == 0 || mpz_cmp(f, n) == 0) {
/* Backtrack */
prime_iterator_setprime(&iter, saveq);
mpz_set(X, saveX);
for (q = saveq; q <= B1; q = prime_iterator_next(&iter)) {
UV k = q;
if (q <= sqrtB1) {
UV kmin = B1/q;
while (k <= kmin)
k *= q;
}
pp1_pow(X, Y, k, n);
mpz_sub_ui(f, X, 2);
if (mpz_sgn(f) == 0) goto end_fail;
mpz_gcd(f, f, n);
if (mpz_cmp(f, n) == 0) break;
if (mpz_cmp_ui(f, 1) > 0) goto end_success;
}
}
if ( (mpz_cmp_ui(f, 1) > 0) && (mpz_cmp(f, n) != 0) )
goto end_success;
/* TODO: stage 2 */
end_fail:
mpz_set(f,n);
end_success:
prime_iterator_destroy(&iter);
mpz_clear(X); mpz_clear(Y); mpz_clear(saveX);
return (mpz_cmp_ui(f, 1) != 0) && (mpz_cmp(f, n) != 0);
}
int _GMP_holf_factor(mpz_t n, mpz_t f, UV rounds)
{
mpz_t s, m;
UV i;
#define PREMULT 480 /* 1 2 6 12 480 151200 */
TEST_FOR_2357(n, f);
if (mpz_perfect_square_p(n)) {
mpz_sqrt(f, n);
return 1;
}
mpz_mul_ui(n, n, PREMULT);
mpz_init(s);
mpz_init(m);
for (i = 1; i <= rounds; i++) {
mpz_mul_ui(f, n, i); /* f = n*i */
if (mpz_perfect_square_p(f)) {
/* s^2 = n*i, so m = s^2 mod n = 0. Hence f = GCD(n, s) = GCD(n, n*i) */
mpz_divexact_ui(n, n, PREMULT);
mpz_gcd(f, f, n);
mpz_clear(s); mpz_clear(m);
if (mpz_cmp(f, n) == 0) return 0;
return 1;
}
mpz_sqrt(s, f);
mpz_add_ui(s, s, 1); /* s = ceil(sqrt(n*i)) */
mpz_mul(m, s, s);
mpz_sub(m, m, f); /* m = s^2 mod n = s^2 - n*i */
if (mpz_perfect_square_p(m)) {
mpz_divexact_ui(n, n, PREMULT);
mpz_sqrt(f, m);
mpz_sub(s, s, f);
mpz_gcd(f, s, n);
mpz_clear(s); mpz_clear(m);
return (mpz_cmp_ui(f, 1) > 0);
}
}
mpz_divexact_ui(n, n, PREMULT);
mpz_set(f, n);
mpz_clear(s); mpz_clear(m);
return 0;
}
/*----------------------------------------------------------------------
* GMP version of Ben Buhrow's public domain 9/24/09 implementation.
* It uses ideas and code from Jason Papadopoulos, Scott Contini, and
* Tom St. Denis. Also see the papers of Stephen McMath, Daniel Shanks,
* and Jason Gower. Gower and Wagstaff is particularly useful:
* http://homes.cerias.purdue.edu/~ssw/squfof.pdf
*--------------------------------------------------------------------*/
static int shanks_mult(mpz_t n, mpz_t f)
{
/*
* use shanks SQUFOF to factor N.
*
* return 0 if no factor found, 1 if found with factor in f1.
*
* Input should have gone through trial division to 5.
*/
int result = 0;
unsigned long j=0;
mpz_t b0, bn, imax, tmp, Q0, Qn, P, i, t1, t2, S, Ro, So, bbn;
if (mpz_cmp_ui(n, 3) <= 0)
return 0;
if (mpz_perfect_square_p(n)) {
mpz_sqrt(f, n);
return 1;
}
mpz_init(b0);
mpz_init(bn);
mpz_init(imax);
mpz_init(tmp);
mpz_init(Q0);
mpz_init(Qn);
mpz_init(P);
mpz_init(i);
mpz_init(t1);
mpz_init(t2);
mpz_init(S);
mpz_init(Ro);
mpz_init(So);
mpz_init(bbn);
mpz_mod_ui(t1, n, 4);
if (mpz_cmp_ui(t1, 3))
croak("Incorrect call to shanks\n");
mpz_sqrt(b0, n);
mpz_sqrt(tmp, b0);
mpz_mul_ui(imax, tmp, 3);
/* set up recurrence */
mpz_set_ui(Q0, 1);
mpz_set(P, b0);
mpz_mul(tmp, b0, b0);
mpz_sub(Qn, n, tmp);
mpz_add(tmp, b0, P);
mpz_tdiv_q(bn, tmp, Qn);
mpz_set_ui(i, 0);
while (1) {
j=0;
while (1) {
mpz_set(t1, P); /* hold Pn for this iteration */
mpz_mul(tmp, bn, Qn);
mpz_sub(P, tmp, P);
mpz_set(t2, Qn); /* hold Qn for this iteration */
mpz_sub(tmp, t1, P);
mpz_mul(tmp, tmp, bn);
mpz_add(Qn, Q0, tmp);
mpz_set(Q0, t2); /* remember last Q */
mpz_add(tmp, b0, P);
mpz_tdiv_q(bn, tmp, Qn);
if (mpz_even_p(i)) {
if (mpz_perfect_square_p(Qn)) {
mpz_add_ui(i, i, 1);
break;
}
}
mpz_add_ui(i, i, 1);
if (mpz_cmp(i, imax) >= 0) {
result = 0;
goto end;
}
}
/* reduce to G0 */
mpz_sqrt(S, Qn);
mpz_sub(tmp, b0, P);
mpz_tdiv_q(tmp, tmp, S);
mpz_mul(tmp, S, tmp);
mpz_add(Ro, P, tmp);
mpz_mul(tmp, Ro, Ro);
mpz_sub(tmp, n, tmp);
mpz_tdiv_q(So, tmp, S);
mpz_add(tmp, b0, Ro);
mpz_tdiv_q(bbn, tmp, So);
/* search for symmetry point */
while (1) {
mpz_set(t1, Ro); /* hold Ro for this iteration */
mpz_mul(tmp, bbn, So);
mpz_sub(Ro, tmp, Ro);
mpz_set(t2, So); /* hold So for this iteration */
mpz_sub(tmp, t1, Ro);
mpz_mul(tmp, bbn, tmp);
mpz_add(So, S, tmp);
mpz_set(S, t2); /* remember last S */
mpz_add(tmp, b0, Ro);
mpz_tdiv_q(bbn, tmp, So);
/* check for symmetry point */
if (mpz_cmp(Ro, t1) == 0)
break;
/* this gets stuck very rarely, but it does happen. */
if (++j > 1000000000)
{
result = -1;
goto end;
}
}
mpz_gcd(t1, Ro, n);
if (mpz_cmp_ui(t1, 1) > 0) {
mpz_set(f, t1);
/* gmp_printf("GMP SQUFOF found factor after %Zd/%lu rounds: %Zd\n", i, j, f); */
result = 1;
goto end;
}
}
end:
mpz_clear(b0);
mpz_clear(bn);
mpz_clear(imax);
mpz_clear(tmp);
mpz_clear(Q0);
mpz_clear(Qn);
mpz_clear(P);
mpz_clear(i);
mpz_clear(t1);
mpz_clear(t2);
mpz_clear(S);
mpz_clear(Ro);
mpz_clear(So);
mpz_clear(bbn);
return result;
}
int _GMP_squfof_factor(mpz_t n, mpz_t f, UV rounds)
{
const UV multipliers[] = {
3*5*7*11, 3*5*7, 3*5*11, 3*5, 3*7*11, 3*7, 5*7*11, 5*7,
3*11, 3, 5*11, 5, 7*11, 7, 11, 1 };
const size_t sz_mul = sizeof(multipliers)/sizeof(multipliers[0]);
size_t i;
int result;
UV nmod4;
mpz_t nm, t;
TEST_FOR_2357(n, f);
mpz_init(nm);
mpz_init(t);
mpz_set_ui(f, 1);
nmod4 = mpz_tdiv_r_ui(nm, n, 4);
for (i = 0; i < sz_mul; i++) {
UV mult = multipliers[i];
/* Only use multipliers where n*m = 3 mod 4 */
if (nmod4 == (mult % 4))
continue;
/* Only run when 64*m^3 < n */
mpz_set_ui(t, mult);
mpz_pow_ui(t, t, 3);
mpz_mul_ui(t, t, 64);
if (mpz_cmp(t, n) >= 0)
continue;
/* Run with this multiplier */
mpz_mul_ui(nm, n, mult);
result = shanks_mult(nm, f);
if (result == -1)
break;
if ( (result == 1) && (mpz_cmp_ui(f, mult) != 0) ) {
unsigned long gcdf = mpz_gcd_ui(NULL, f, mult);
mpz_divexact_ui(f, f, gcdf);
if (mpz_cmp_ui(f, 1) > 0)
break;
}
}
mpz_clear(t);
mpz_clear(nm);
return (mpz_cmp_ui(f, 1) > 0);
}
/* See if n is a perfect power */
UV power_factor(mpz_t n, mpz_t f)
{
if (mpz_cmp_ui(n, 1) <= 0) return 0;
if (mpz_perfect_power_p(n)) {
UV k;
mpz_set_ui(f, 1);
for (k = 2; mpz_sgn(f); k++) {
if (mpz_root(f, n, k)) {
if (mpz_perfect_power_p(f)) { /* If root is a power, recurse */
mpz_t nf;
mpz_init_set(nf, f);
k *= power_factor(nf, f);
mpz_clear(nf);
}
return k;
}
}
/* GMP says it's a perfect power, but we couldn't find an integer root? */
}
return 0;
}
/* a=0, return power. a>1, return bool if an a-th power */
UV is_power(mpz_t n, UV a)
{
if (mpz_cmp_ui(n,3) <= 0)
return 0;
else if (a == 1)
return 1;
else if (a == 2)
return mpz_perfect_square_p(n);
else {
UV result;
mpz_t t;
mpz_init(t);
result = (a == 0) ? power_factor(n, t) : (UV)mpz_root(t, n, a);
mpz_clear(t);
return result;
}
}
void exp_mangoldt(mpz_t res, mpz_t n)
{
UV k;
mpz_set_ui(res, 1);
if (mpz_cmp_ui(n, 1) <= 0)
return;
k = mpz_scan1(n, 0);
if (k > 0) {
if (k+1 == mpz_sizeinbase(n, 2))
mpz_set_ui(res, 2);
return;
}
if (_GMP_is_prob_prime(n)) {
mpz_set(res, n);
return;
}
k = power_factor(n, res);
if (k > 1 && _GMP_is_prob_prime(res))
return;
mpz_set_ui(res, 1);
}
uint32_t* partial_sieve(mpz_t start, UV length, UV maxprime)
{
UV p, m, pos;
uint32_t* comp;
mpz_t t;
PRIME_ITERATOR(iter);
/* mpz_init(t);
mpz_add_ui(t, start, (length & 1) ? length-1 : length-2);
gmp_printf("partial sieve start %Zd length %lu mark %Zd to %Zd\n", start, length, start, t); */
MPUassert(mpz_odd_p(start), "partial sieve given even start");
MPUassert(length > 0, "partial sieve given zero length");
mpz_sub_ui(start, start, 1);
if (length & 1) length++;
Newz(0, comp, (length+63)/64, uint32_t);
mpz_init(t);
p = prime_iterator_next(&iter);
for (p = 3; p <= maxprime; p = prime_iterator_next(&iter)) {
m = mpz_mod_ui(t, start, p);
pos = (m == 0) ? 0 : p-m; /* First multiple of p after start */
if (!(pos & 1)) pos += p; /* Make sure it is odd. */
while (pos < length) {
SETAVAL(comp, pos);
pos += 2*p;
}
}
mpz_clear(t);
prime_iterator_destroy(&iter);
return comp;
}
char* pidigits(UV n) {
char* out;
New(0, out, n+4, char);
out[0] = '3'; out[1] = '\0';
if (n <= 1)
return out;
#if 0
/* Spigot method.
* ~40x slower than the Machin formulas, 2x slower than spigot in plain C */
{
mpz_t t1, t2, acc, den, num;
UV i, k, d, d2;
mpz_init(t1); mpz_init(t2);
mpz_init_set_ui(acc, 0);
mpz_init_set_ui(den, 1);
mpz_init_set_ui(num, 1);
n++; /* rounding */
for (i = k = 0; i < n; ) {
{
UV k2 = ++k * 2 + 1;
mpz_mul_2exp(t1, num, 1);
mpz_add(acc, acc, t1);
mpz_mul_ui(acc, acc, k2);
mpz_mul_ui(den, den, k2);
mpz_mul_ui(num, num, k);
}
if (mpz_cmp(num, acc) > 0) continue;
{
mpz_mul_ui(t1, num, 3);
mpz_add(t2, t1, acc);
mpz_tdiv_q(t1, t2, den);
d = mpz_get_ui(t1);
}
{
mpz_mul_ui(t1, num, 4);
mpz_add(t2, t1, acc);
mpz_tdiv_q(t1, t2, den);
d2 = mpz_get_ui(t1);
}
if (d != d2) continue;
out[++i] = '0' + d;
{
mpz_submul_ui(acc, den, d);
mpz_mul_ui(acc, acc, 10);
mpz_mul_ui(num, num, 10);
}
}
mpz_clear(num); mpz_clear(den); mpz_clear(acc); mpz_clear(t2);mpz_clear(t1);
if (out[n] >= '5') out[n-1]++; /* Round */
for (i = n-1; out[i] == '9'+1; i--) /* Keep rounding */
{ out[i] = '0'; out[i-1]++; }
n--; /* Undo the extra digit we used for rounding */
out[1] = '.';
out[n+1] = '\0';
}
#elif 0
/* https://en.wikipedia.org/wiki/Machin-like_formula
* Thanks to Ledrug from RosettaCode for the simple code for base 10.
* Pretty fast, but growth is a lot slower than AGM. */
{
mpz_t t1, t2, term1, term2, pows;
UV i, k;
mpz_init(t1); mpz_init(t2); mpz_init(term1); mpz_init(term2); mpz_init(pows);
n++; /* rounding */
mpz_ui_pow_ui(pows, 10, n+20);
#if 0
/* Machin 1706 */
mpz_arctan(term1, 5, pows, t1, t2); mpz_mul_ui(term1, term1, 4);
mpz_arctan(term2, 239, pows, t1, t2);
mpz_sub(term1, term1, term2);
#elif 0
/* Störmer 1896 */
mpz_arctan(term1, 57, pows, t1, t2); mpz_mul_ui(term1, term1, 44);
mpz_arctan(term2, 239, pows, t1, t2); mpz_mul_ui(term2, term2, 7);
mpz_add(term1, term1, term2);
mpz_arctan(term2, 682, pows, t1, t2); mpz_mul_ui(term2, term2, 12);
mpz_sub(term1, term1, term2);
mpz_arctan(term2, 12943, pows, t1, t2); mpz_mul_ui(term2, term2, 24);
mpz_add(term1, term1, term2);
#else
/* Chien-Lih 1997 */
mpz_arctan(term1, 239, pows, t1, t2); mpz_mul_ui(term1, term1, 183);
mpz_arctan(term2, 1023, pows, t1, t2); mpz_mul_ui(term2, term2, 32);
mpz_add(term1, term1, term2);
mpz_arctan(term2, 5832, pows, t1, t2); mpz_mul_ui(term2, term2, 68);
mpz_sub(term1, term1, term2);
mpz_arctan(term2, 110443, pows, t1, t2); mpz_mul_ui(term2, term2, 12);
mpz_add(term1, term1, term2);
mpz_arctan(term2, 4841182, pows, t1, t2); mpz_mul_ui(term2, term2, 12);
mpz_sub(term1, term1, term2);
mpz_arctan(term2, 6826318, pows, t1, t2); mpz_mul_ui(term2, term2, 100);
mpz_sub(term1, term1, term2);
#endif
mpz_mul_ui(term1, term1, 4);
mpz_ui_pow_ui(pows, 10, 20);
mpz_tdiv_q(term1, term1, pows);
mpz_clear(t1); mpz_clear(t2); mpz_clear(term2); mpz_clear(pows);
k = mpz_sizeinbase(term1, 10); /* Copy result to out string */
while (k > n+1) { /* making sure we don't overflow */
mpz_tdiv_q_ui(term1, term1, 10);
k = mpz_sizeinbase(term1, 10);
}
(void) mpz_get_str(out+1, 10, term1);
mpz_clear(term1);
if (out[n] >= '5') out[n-1]++; /* Round */
for (i = n-1; out[i] == '9'+1; i--) /* Keep rounding */
{ out[i] = '0'; out[i-1]++; }
n--; /* Undo the extra digit we used for rounding */
out[1] = '.';
out[n+1] = '\0';
}
#else
/* AGM using GMP's floating point. Fast and very good growth. */
{
mpf_t t, an, bn, tn, prev_an;
UV k = 0;
unsigned long oldprec = mpf_get_default_prec();
mpf_set_default_prec(10 + n * 3.322);
mpf_init(t); mpf_init(prev_an);
mpf_init_set_d(an, 1); mpf_init_set_d(bn, 0.5); mpf_init_set_d(tn, 0.25);
mpf_sqrt(bn, bn);
while ((n >> k) > 0) {
mpf_set(prev_an, an);
mpf_add(t, an, bn);
mpf_div_ui(an, t, 2);
mpf_mul(t, bn, prev_an);
mpf_sqrt(bn, t);
mpf_sub(prev_an, prev_an, an);
mpf_mul(t, prev_an, prev_an);
mpf_mul_2exp(t, t, k);
mpf_sub(tn, tn, t);
k++;
}
mpf_add(t, an, bn);
mpf_mul(an, t, t);
mpf_mul_2exp(t, tn, 2);
mpf_div(bn, an, t);
gmp_sprintf(out, "%.*Ff", (int)(n-1), bn);
mpf_clear(tn); mpf_clear(bn); mpf_clear(an);
mpf_clear(prev_an); mpf_clear(t);
mpf_set_default_prec(oldprec);
}
#endif
return out;
}