#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "ptypes.h"
#include "factor.h"
#include "util.h"
#include "sieve.h"
#include "mulmod.h"
/*
* You need to remember to use UV for unsigned and IV for signed types that
* are large enough to hold our data.
* If you use int, that's 32-bit on LP64 and LLP64 machines. You lose.
* If you use long, that's 32-bit on LLP64 machines. You lose.
* If you use long long, you may be too large which isn't so bad, but some
* compilers may not understand the type at all.
* perl.h already figured all this out, and provided us with these types which
* match the native integer type used inside our Perl, so just use those.
*/
int trial_factor(UV n, UV *factors, UV maxtrial)
{
UV f, m, limit;
int nfactors = 0;
if (maxtrial == 0) maxtrial = UV_MAX;
if ( (n < 2) || (maxtrial < 2) ) {
factors[0] = n;
return 1;
}
while ( (n & 1) == 0 ) { factors[nfactors++] = 2; n /= 2; }
if (3 <= maxtrial) while ( (n % 3) == 0 ) { factors[nfactors++] = 3; n /= 3; }
if (5 <= maxtrial) while ( (n % 5) == 0 ) { factors[nfactors++] = 5; n /= 5; }
if (7 <= maxtrial) while ( (n % 7) == 0 ) { factors[nfactors++] = 7; n /= 7; }
f = 11;
m = 11;
if ( (n < (f*f)) || (maxtrial < f) ) {
if (n != 1)
factors[nfactors++] = n;
return nfactors;
}
limit = (UV) (sqrt(n)+0.1);
if (limit > maxtrial)
limit = maxtrial;
/* wheel 30 */
while (f <= limit) {
if ( (n%f) == 0 ) {
UV newlimit;
do {
factors[nfactors++] = f;
n /= f;
} while ( (n%f) == 0 );
newlimit = (UV) (sqrt(n)+0.1);
if (newlimit < limit) limit = newlimit;
}
f += wheeladvance30[m];
m = nextwheel30[m];
}
if (n != 1)
factors[nfactors++] = n;
return nfactors;
}
/* Return 0 if n is not a perfect square. Set sqrtn to int(sqrt(n)) if so.
*
* Some simple solutions:
*
* return ( ((m&2)!= 0) || ((m&7)==5) || ((m&11) == 8) ) ? 0 : 1;
*
* or:
*
* m = n & 31;
* if ( m==0 || m==1 || m==4 || m==9 || m==16 || m==17 || m==25 )
* ...test for perfect square...
*
* or:
*
* if ( ((0x0202021202030213ULL >> (n & 63)) & 1) &&
* ((0x0402483012450293ULL >> (n % 63)) & 1) &&
* ((0x218a019866014613ULL >> ((n % 65) & 63)) & 1) &&
* ((0x23b >> (n % 11) & 1)) ) {
*
*
* The following Bloom filter cascade works very well indeed. Read all
* about it here: http://mersenneforum.org/showpost.php?p=110896
*/
static int is_perfect_square(UV n, UV* sqrtn)
{
UV m; /* lm */
m = n & 127;
if ((m*0x8bc40d7d) & (m*0xa1e2f5d1) & 0x14020a) return 0;
/* 82% of non-squares rejected here */
#if 0
/* The big deal with this technique is that you do two total operations,
* one cheap (the & 127 above), one expensive (the modulo below) on n.
* The rest of the operations are 32-bit operations. This is a huge win
* if n is multiprecision.
* However, in this file we're doing native precision sqrt, so it just
* isn't expensive enough to justify this second filter set.
*/
lm = n % UVCONST(63*25*11*17*19*23*31);
m = lm % 63;
if ((m*0x3d491df7) & (m*0xc824a9f9) & 0x10f14008) return 0;
m = lm % 25;
if ((m*0x1929fc1b) & (m*0x4c9ea3b2) & 0x51001005) return 0;
m = 0xd10d829a*(lm%31);
if (m & (m+0x672a5354) & 0x21025115) return 0;
m = lm % 23;
if ((m*0x7bd28629) & (m*0xe7180889) & 0xf8300) return 0;
m = lm % 19;
if ((m*0x1b8bead3) & (m*0x4d75a124) & 0x4280082b) return 0;
m = lm % 17;
if ((m*0x6736f323) & (m*0x9b1d499) & 0xc0000300) return 0;
m = lm % 11;
if ((m*0xabf1a3a7) & (m*0x2612bf93) & 0x45854000) return 0;
/* 99.92% of non-squares are rejected now */
#else
m = n % 63;
if ((m*0x3d491df7) & (m*0xc824a9f9) & 0x10f14008) return 0;
#endif
m = sqrt(n);
if (n != (m*m))
return 0;
if (sqrtn != 0) *sqrtn = m;
return 1;
}
/* Miller-Rabin probabilistic primality test
* Returns 1 if probably prime relative to the bases, 0 if composite.
* Bases must be between 2 and n-2
*/
int _XS_miller_rabin(UV n, const UV *bases, int nbases)
{
UV const nm1 = n-1;
UV d = n-1;
int b, r, s = 0;
MPUassert(n > 3, "MR called with n <= 3");
while ( (d&1) == 0 ) {
s++;
d >>= 1;
}
for (b = 0; b < nbases; b++) {
UV x, a = bases[b];
if (a < 2)
croak("Base %"UVuf" is invalid", a);
#if 0
if (a > (n-2))
croak("Base %"UVuf" is invalid for input %"UVuf, a, n);
#else
if (a >= n)
a %= n;
if ( (a <= 1) || (a == nm1) )
continue;
#endif
/* n is a strong pseudoprime to this base if either
* - a^d = 1 mod n
* - a^(d2^r) = -1 mod n for some r: 0 <= r <= s-1
*/
x = powmod(a, d, n);
if ( (x == 1) || (x == nm1) ) continue;
/* cover r = 1 to s-1, r=0 was just done */
for (r = 1; r < s; r++) {
x = sqrmod(x, n);
if ( x == nm1 ) break;
if ( x == 1 ) return 0;
}
if (r >= s)
return 0;
}
return 1;
}
int _XS_is_prob_prime(UV n)
{
UV bases[12];
int nbases;
int prob_prime;
if ( (n == 2) || (n == 3) || (n == 5) || (n == 7) )
return 2;
if ( (n<2) || ((n% 2)==0) || ((n% 3)==0) || ((n% 5)==0) || ((n% 7)==0) )
return 0;
if (n < 121)
return 2;
#if BITS_PER_WORD == 32
if (n < UVCONST(9080191)) {
bases[0] = 31; bases[1] = 73; nbases = 2;
} else {
bases[0] = 2; bases[1] = 7; bases[2] = 61; nbases = 3;
}
#else
#if 1
/* Better bases from: http://miller-rabin.appspot.com/ */
if (n < UVCONST(212321)) {
bases[0] = UVCONST(1948244569546278);
nbases = 1;
} else if (n < UVCONST(360018361)) {
bases[0] = UVCONST( 1143370 );
bases[1] = UVCONST( 2350307676 );
nbases = 2;
} else if (n < UVCONST(105936894253)) {
bases[0] = 2;
bases[1] = UVCONST( 1005905886 );
bases[2] = UVCONST( 1340600841 );
nbases = 3;
} else if (n < UVCONST(31858317218647)) {
bases[0] = 2;
bases[1] = UVCONST( 642735 );
bases[2] = UVCONST( 553174392 );
bases[3] = UVCONST( 3046413974 );
nbases = 4;
} else if (n < UVCONST(3071837692357849)) {
bases[0] = 2;
bases[1] = UVCONST( 75088 );
bases[2] = UVCONST( 642735 );
bases[3] = UVCONST( 203659041 );
bases[4] = UVCONST( 3613982119 );
nbases = 5;
} else {
bases[0] = 2;
bases[1] = UVCONST( 325 );
bases[2] = UVCONST( 9375 );
bases[3] = UVCONST( 28178 );
bases[4] = UVCONST( 450775 );
bases[5] = UVCONST( 9780504 );
bases[6] = UVCONST( 1795265022 );
nbases = 7;
}
#else
/* Classic bases */
if (n < UVCONST(9080191)) {
bases[0] = 31; bases[1] = 73; nbases = 2;
} else if (n < UVCONST(4759123141)) {
bases[0] = 2; bases[1] = 7; bases[2] = 61; nbases = 3;
} else if (n < UVCONST(21652684502221)) {
bases[0] = 2; bases[1] = 1215; bases[2] = 34862; bases[3] = 574237825;
nbases = 4;
} else if (n < UVCONST(341550071728321)) {
bases[0] = 2; bases[1] = 3; bases[2] = 5; bases[3] = 7; bases[4] = 11;
bases[5] = 13; bases[6] = 17; nbases = 7;
} else if (n < UVCONST(3825123056546413051)) {
bases[0] = 2; bases[1] = 3; bases[2] = 5; bases[3] = 7; bases[4] = 11;
bases[5] = 13; bases[6] = 17; bases[7] = 19; bases[8] = 23; nbases = 9;
} else {
bases[0] = 2; bases[1] = 3; bases[2] = 5; bases[3] = 7; bases[4] = 11;
bases[5] = 13; bases[6] = 17; bases[7] = 19; bases[8] = 23; bases[9] = 29;
bases[10]= 31; bases[11]= 37;
nbases = 12;
}
#endif
#endif
prob_prime = _XS_miller_rabin(n, bases, nbases);
return 2*prob_prime;
}
/* Knuth volume 2, algorithm C.
* Very fast for small numbers, grows rapidly.
* SQUFOF is better for numbers nearing the 64-bit limit.
*/
int fermat_factor(UV n, UV *factors, UV rounds)
{
IV sqn, x, y, r;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in fermat_factor");
sqn = (UV) (sqrt(n)+0.1);
x = 2 * sqn + 1;
y = 1;
r = (sqn*sqn) - n;
while (r != 0) {
r += x;
x += 2;
do {
r -= y;
y += 2;
} while (r > 0);
}
r = (x-y)/2;
if ( (r != 1) && ((UV)r != n) ) {
factors[0] = r;
factors[1] = n/r;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
factors[0] = n;
return 1;
}
/* Hart's One Line Factorization.
* Missing premult (hard to do in native precision without overflow)
*/
int holf_factor(UV n, UV *factors, UV rounds)
{
UV i, s, m, f;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in holf_factor");
for (i = 1; i <= rounds; i++) {
s = sqrt( (double)n * (double)i );
/* Assume s^2 isn't a perfect square. We're rapidly losing precision
* so we won't be able to accurately detect it anyway. */
s++; /* s = ceil(sqrt(n*i)) */
m = sqrmod(s, n);
if (is_perfect_square(m, &f)) {
f = gcd_ui( (s>f) ? s-f : f-s, n);
/* This should always succeed, but with overflow concerns.... */
if ((f == 1) || (f == n))
break;
factors[0] = f;
factors[1] = n/f;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
}
factors[0] = n;
return 1;
}
/* Pollard / Brent. Brent's modifications to Pollard's Rho. Maybe faster. */
int pbrent_factor(UV n, UV *factors, UV rounds)
{
UV f, i, r;
UV Xi = 2;
UV Xm = 2;
UV a = 1;
const UV inner = 64;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pbrent_factor");
r = 1;
while (rounds > 0) {
UV rleft = (r > rounds) ? rounds : r;
UV saveXi;
/* Do rleft rounds, inner at a time */
while (rleft > 0) {
UV dorounds = (rleft > inner) ? inner : rleft;
UV m = 1;
saveXi = Xi;
for (i = 0; i < dorounds; i++) {
Xi = sqraddmod(Xi, a, n);
f = (Xi>Xm) ? Xi-Xm : Xm-Xi;
m = mulmod(m, f, n);
}
rleft -= dorounds;
rounds -= dorounds;
f = gcd_ui(m, n);
if (f != 1)
break;
}
/* If f == 1, then we didn't find a factor. Move on. */
if (f == 1) {
r *= 2;
Xm = Xi;
continue;
}
if (f == n) { /* back up, with safety */
Xi = saveXi;
do {
Xi = sqraddmod(Xi, a, n);
f = gcd_ui( (Xi>Xm) ? Xi-Xm : Xm-Xi, n);
} while (f == 1 && r-- != 0);
if ( (f == 1) || (f == n) ) break;
}
factors[0] = f;
factors[1] = n/f;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
factors[0] = n;
return 1;
}
/* Pollard's Rho. */
int prho_factor(UV n, UV *factors, UV rounds)
{
UV a, f, i, m, oldU, oldV;
const UV inner = 64;
UV U = 7;
UV V = 7;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in prho_factor");
/* We could just as well say a = 1 */
switch (n%8) {
case 1: a = 1; break;
case 3: a = 2; break;
case 5: a = 3; break;
case 7: a = 5; break;
default: a = 7; break;
}
rounds = (rounds + inner - 1) / inner;
while (rounds-- > 0) {
m = 1; oldU = U; oldV = V;
for (i = 0; i < inner; i++) {
U = sqraddmod(U, a, n);
V = sqraddmod(V, a, n);
V = sqraddmod(V, a, n);
f = (U > V) ? U-V : V-U;
m = mulmod(m, f, n);
}
f = gcd_ui(m, n);
if (f == 1)
continue;
if (f == n) { /* back up to find a factor*/
U = oldU; V = oldV;
i = inner;
do {
U = sqraddmod(U, a, n);
V = sqraddmod(V, a, n);
V = sqraddmod(V, a, n);
f = gcd_ui( (U > V) ? U-V : V-U, n);
} while (f == 1 && i-- != 0);
if ( (f == 1) || (f == n) )
break;
}
factors[0] = f;
factors[1] = n/f;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
factors[0] = n;
return 1;
}
/* Pollard's P-1 */
int pminus1_factor(UV n, UV *factors, UV B1, UV B2)
{
UV q, f;
UV a = 2;
UV savea = 2;
UV saveq = 2;
UV j = 1;
UV sqrtB1 = sqrt(B1);
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in pminus1_factor");
for (q = 2; q <= sqrtB1; q = _XS_next_prime(q)) {
UV k = q*q;
UV kmin = B1/q;
while (k <= kmin)
k *= q;
a = powmod(a, k, n);
}
if (a == 0) { factors[0] = n; return 1; }
f = gcd_ui(a-1, n);
if (f == 1) {
savea = a;
saveq = q;
for (; q <= B1; q = _XS_next_prime(q)) {
a = powmod(a, q, n);
if ( (j++ % 32) == 0) {
if (a == 0 || gcd_ui(a-1, n) != 1)
break;
savea = a;
saveq = q;
}
}
if (a == 0) { factors[0] = n; return 1; }
f = gcd_ui(a-1, n);
}
/* If we found more than one factor in stage 1, backup and single step */
if (f == n) {
a = savea;
for (q = saveq; q <= B1; q = _XS_next_prime(q)) {
UV k = q;
UV kmin = B1/q;
while (k <= kmin)
k *= q;
a = powmod(a, k, n);
f = gcd_ui(a-1, n);
if (f != 1)
break;
}
/* If f == n again, we could do:
* for (savea = 3; f == n && savea < 100; savea = _XS_next_prime(savea)) {
* a = savea;
* for (q = 2; q <= B1; q = _XS_next_prime(q)) {
* ...
* }
* }
* but this could be a huge time sink if B1 is large, so just fail.
*/
}
/* STAGE 2 */
if (f == 1 && B2 > B1) {
UV bm = a;
UV b = 1;
UV bmdiff;
UV precomp_bm[111] = {0}; /* Enough for B2 = 189M */
/* calculate (a^q)^2, (a^q)^4, etc. */
bmdiff = sqrmod(bm, n);
precomp_bm[0] = bmdiff;
for (j = 1; j < 20; j++) {
bmdiff = mulmod(bmdiff,bm,n);
bmdiff = mulmod(bmdiff,bm,n);
precomp_bm[j] = bmdiff;
}
a = powmod(a, q, n);
j = 1;
while (q <= B2) {
UV lastq = q;
UV qdiff;
q = _XS_next_prime(q);
/* compute a^q = a^lastq * a^(q-lastq) */
qdiff = (q - lastq) / 2 - 1;
if (qdiff >= 111) {
bmdiff = powmod(bm, q-lastq, n); /* Big gap */
} else {
bmdiff = precomp_bm[qdiff];
if (bmdiff == 0) {
if (precomp_bm[qdiff-1] != 0)
bmdiff = mulmod(mulmod(precomp_bm[qdiff-1],bm,n),bm,n);
else
bmdiff = powmod(bm, q-lastq, n);
precomp_bm[qdiff] = bmdiff;
}
}
a = mulmod(a, bmdiff, n);
if (a == 0) break;
b = mulmod(b, a-1, n); /* if b == 0, we found multiple factors */
if ( (j++ % 64) == 0 ) {
f = gcd_ui(b, n);
if (f != 1)
break;
}
}
f = gcd_ui(b, n);
}
if ( (f != 1) && (f != n) ) {
factors[0] = f;
factors[1] = n/f;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
factors[0] = n;
return 1;
}
/* My modification of Ben Buhrow's modification of Bob Silverman's SQUFOF code.
*/
static IV qqueue[100+1];
static IV qpoint;
static void enqu(IV q, IV *iter) {
qqueue[qpoint] = q;
if (++qpoint >= 100) *iter = -1;
}
int squfof_factor(UV n, UV *factors, UV rounds)
{
IV rounds2 = (IV) (rounds/16);
UV temp;
IV iq,ll,l2,p,pnext,q,qlast,r,s,t,i;
IV jter, iter;
int reloop;
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in squfof_factor");
/* TODO: What value of n leads to overflow? */
qlast = 1;
s = sqrt(n);
p = s;
temp = n - (s*s); /* temp = n - floor(sqrt(n))^2 */
if (temp == 0) {
factors[0] = s;
factors[1] = s;
return 2;
}
q = temp; /* q = excess of n over next smaller square */
ll = 1 + 2*(IV)sqrt((double)(p+p));
l2 = ll/2;
qpoint = 0;
/* In the loop below, we need to check if q is a square right before */
/* the end of the loop. Is there a faster way? The current way is */
/* EXPENSIVE! (many branches and double prec sqrt) */
for (jter=0; (UV)jter < rounds; jter++) {
iq = (s + p)/q;
pnext = iq*q - p;
if (q <= ll) {
if ((q & 1) == 0)
enqu(q/2,&jter);
else if (q <= l2)
enqu(q,&jter);
if (jter < 0) {
factors[0] = n; return 1;
}
}
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext; /* check for square; even iter */
if (jter & 1) continue; /* jter is odd:omit square test */
r = (int)sqrt((double)q); /* r = floor(sqrt(q)) */
if (q != r*r) continue;
if (qpoint == 0) break;
qqueue[qpoint] = 0;
reloop = 0;
for (i=0; i<qpoint-1; i+=2) { /* treat queue as list for simplicity*/
if (r == qqueue[i]) { reloop = 1; break; }
if (r == qqueue[i+1]) { reloop = 1; break; }
}
if (reloop || (r == qqueue[qpoint-1])) continue;
break;
} /* end of main loop */
if ((UV)jter >= rounds) {
factors[0] = n; return 1;
}
qlast = r;
p = p + r*((s - p)/r);
q = (n - (p*p)) / qlast; /* q = (n - p*p)/qlast (div is exact)*/
for (iter=0; iter<rounds2; iter++) { /* unrolled second main loop */
iq = (s + p)/q;
pnext = iq*q - p;
if (p == pnext) break;
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext;
iq = (s + p)/q;
pnext = iq*q - p;
if (p == pnext) break;
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext;
iq = (s + p)/q;
pnext = iq*q - p;
if (p == pnext) break;
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext;
iq = (s + p)/q;
pnext = iq*q - p;
if (p == pnext) break;
t = qlast + iq*(p - pnext);
qlast = q;
q = t;
p = pnext;
}
if (iter >= rounds2) {
factors[0] = n; return 1;
}
if ((q & 1) == 0) q/=2; /* q was factor or 2*factor */
if ( (q == 1) || ((UV)q == n) ) {
factors[0] = n; return 1;
}
p = n/q;
/* printf(" squfof found %lu = %lu * %lu in %ld/%ld rounds\n", n, p, q, jter, iter); */
factors[0] = p;
factors[1] = q;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
/* Another version, based on Ben Buhrow's racing SQUFOF. */
typedef struct
{
UV mult;
UV valid;
UV P;
UV bn;
UV Qn;
UV Q0;
UV b0;
UV it;
UV imax;
} mult_t;
// N < 2^63 (or 2^31). *f == 1 if no factor found
static void squfof_unit(UV n, mult_t* mult_save, UV* f)
{
UV imax,i,Q0,b0,Qn,bn,P,bbn,Ro,S,So,t1,t2;
int j;
*f = 0;
P = mult_save->P;
bn = mult_save->bn;
Qn = mult_save->Qn;
Q0 = mult_save->Q0;
b0 = mult_save->b0;
i = mult_save->it;
imax = i + mult_save->imax;
#define SQUARE_SEARCH_ITERATION \
t1 = P; \
P = bn*Qn - P; \
t2 = Qn; \
Qn = Q0 + bn*(t1-P); \
Q0 = t2; \
bn = (b0 + P) / Qn; \
i++;
while (1) {
j = 0;
if (i & 0x1) {
SQUARE_SEARCH_ITERATION;
}
// i is now even
while (1) {
// We need to know P, bn, Qn, Q0, iteration count, i from prev
if (i >= imax) {
// save state and try another multiplier.
mult_save->P = P;
mult_save->bn = bn;
mult_save->Qn = Qn;
mult_save->Q0 = Q0;
mult_save->it = i;
*f = 0;
return;
}
SQUARE_SEARCH_ITERATION;
// Even iteration. Check for square: Qn = S*S
if (is_perfect_square( Qn, &S ))
break;
// Odd iteration.
SQUARE_SEARCH_ITERATION;
}
/* printf("found square %lu after %lu iterations with mult %d\n", Qn, i, mult_save->mult); */
// Reduce to G0
Ro = P + S*((b0 - P)/S);
t1 = Ro;
So = (n - t1*t1)/S;
bbn = (b0+Ro)/So;
// Search for symmetry point
#define SYMMETRY_POINT_ITERATION \
t1 = Ro; \
Ro = bbn*So - Ro; \
t2 = So; \
So = S + bbn*(t1-Ro); \
S = t2; \
bbn = (b0+Ro)/So; \
if (Ro == t1) break;
j = 0;
while (1) {
SYMMETRY_POINT_ITERATION;
SYMMETRY_POINT_ITERATION;
SYMMETRY_POINT_ITERATION;
SYMMETRY_POINT_ITERATION;
if (j++ > 2000000) {
mult_save->valid = 0;
*f = 0;
return;
}
}
*f = gcd_ui(Ro, n);
if (*f > 1)
return;
}
}
#define NSQUFOF_MULT (sizeof(multipliers)/sizeof(multipliers[0]))
int racing_squfof_factor(UV n, UV *factors, 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 UV big2 = UV_MAX >> 2;
mult_t mult_save[NSQUFOF_MULT];
int i, still_racing;
UV nn64, mult, f64;
UV rounds_done = 0;
/* Caller should have handled these trivial cases */
MPUassert( (n >= 3) && ((n%2) != 0) , "bad n in racing_squfof_factor");
/* Too big */
if (n > big2) {
factors[0] = n; return 1;
}
for (i = 0; i < NSQUFOF_MULT; i++) {
mult = multipliers[i];
nn64 = n * mult;
mult_save[i].mult = mult;
if ((big2 / mult) < n) {
mult_save[i].valid = 0; /* This multiplier would overflow 64-bit */
continue;
}
mult_save[i].valid = 1;
mult_save[i].b0 = sqrt( (double)nn64 );
mult_save[i].imax = sqrt( (double)mult_save[i].b0 ) / 3;
if (mult_save[i].imax < 20) mult_save[i].imax = 20;
if (mult_save[i].imax > rounds) mult_save[i].imax = rounds;
mult_save[i].Q0 = 1;
mult_save[i].P = mult_save[i].b0;
mult_save[i].Qn = nn64 - (mult_save[i].b0 * mult_save[i].b0);
if (mult_save[i].Qn == 0) {
factors[0] = mult_save[i].b0;
factors[1] = n / mult_save[i].b0;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
mult_save[i].bn = (mult_save[i].b0 + mult_save[i].P) / mult_save[i].Qn;
mult_save[i].it = 0;
}
/* Process the multipliers a little at a time: 0.33*(n*mult)^1/4: 20-20k */
do {
still_racing = 0;
for (i = 0; i < NSQUFOF_MULT; i++) {
if (!mult_save[i].valid)
continue;
nn64 = n * (UV)multipliers[i];
squfof_unit(nn64, &mult_save[i], &f64);
if (f64 > 1) {
if (f64 != multipliers[i]) {
f64 /= gcd_ui(f64, multipliers[i]);
if (f64 != 1) {
factors[0] = f64;
factors[1] = n / f64;
MPUassert( factors[0] * factors[1] == n , "incorrect factoring");
return 2;
}
}
/* Found trivial factor. Quit working with this multiplier. */
mult_save[i].valid = 0;
}
if (mult_save[i].valid == 1)
still_racing = 1;
rounds_done += mult_save[i].imax;
if (rounds_done >= rounds)
break;
}
} while (still_racing && rounds_done < rounds);
/* No factors found */
factors[0] = n;
return 1;
}