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factor.c
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/*
* Program to factor Integers
*
* Current parameters assume a large 32-bit address space is available.
*
* This program is cobbled together from the implementations of various
* factoring algorithms better described in:-
*
* brute.c/cpp - brute force division by small primes
* brent.c/cpp - Pollard Rho method as improved by Brent
* pollard.c/cpp - Pollard (p-1) method
* williams.c/cpp - Williams (p+1) method
* lenstra.c/cpp - Lenstra's Elliptic Curve method
* qsieve.c/cpp - The Multiple polynomial quadratic sieve
*
* Note that the .cpp C++ implementations are easier to follow
*
* NOTE: The quadratic sieve program requires a lot of memory for
* bigger numbers. It may fail if you system cannot provide the memory
* requested.
*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <math.h>
#include "miracl.h"
#define LIMIT 15000
#define BTRIES 1000
#define MULT 2310 /* 2*3*5*7*11 */
#define NEXT 13 /* .. next prime */
#define mr_min(a,b) ((a) < (b)? (a) : (b))
static big *fu;
static BOOL *cp,*plus,*minus;
big n;
FILE *output;
static BOOL suppress=FALSE;
static int PADDING;
static miracl *mip;
void brute(void)
{ /* find factors by brute force division */
big x,y;
int m,p;
gprime(LIMIT);
x=mirvar(0);
y=mirvar(0);
m=0;
p=mip->PRIMES[0];
forever
{ /* try division by each prime in turn */
if (subdiv(n,p,y)==0)
{ /* factor found */
copy(y,n);
if (!suppress) printf("PRIME FACTOR ");
fprintf(output,"%d\n",p);
if (size(n)==1) exit(0);
continue;
}
if (size(y)<=p)
{ /* must be prime */
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
p=mip->PRIMES[++m];
if (p==0) break;
}
if (isprime(n))
{
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
mr_free(x);
mr_free(y);
gprime(0);
}
void brent(void)
{ /* factoring program using Brents method */
long k,r,i,m,iter;
big x,y,ys,z,q,c3,t;
x=mirvar(0);
y=mirvar(0);
ys=mirvar(0);
z=mirvar(0);
q=mirvar(0);
c3=mirvar(3);
t=mirvar(0);
m=10;
r=1;
iter=0;
do
{
if (!suppress) printf("iterations=%5ld",iter);
convert(1,q);
do
{
copy(y,x);
for (i=1;i<=r;i++)
mad(y,y,c3,n,n,y);
k=0;
do
{
iter++;
if (iter>BTRIES)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
fflush(stdout);
}
mr_free(t);
mr_free(x);
mr_free(y);
mr_free(ys);
mr_free(z);
mr_free(q);
mr_free(c3);
return;
}
if (iter%10==0) if (!suppress)
{
printf("\b\b\b\b\b%5ld",iter);
fflush(stdout);
}
copy(y,ys);
for (i=1;i<=mr_min(m,r-k);i++)
{
mad(y,y,c3,n,n,y);
subtract(y,x,z);
mad(z,q,q,n,n,q);
}
egcd(q,n,z);
k+=m;
} while (k<r && size(z)==1);
r*=2;
} while (size(z)==1);
if (mr_compare(z,n)==0) do
{ /* back-track */
mad(ys,ys,c3,n,n,ys);
subtract(ys,x,z);
} while (egcd(z,n,z)==1);
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
fflush(stdout);
}
forever
{
if (isprime(z))
{
if (!suppress) printf("PRIME FACTOR ");
cotnum(z,output);
}
else
{ /* if end of factorisation - pass it on... */
if (mr_compare(z,n)==0) return;
/* you will have to come back to it */
if (!suppress) printf("COMPOSITE FACTOR ");
else printf("& ");
cotnum(z,output);
}
divide(n,z,n);
divide(y,n,n);
copy(n,t);
divide(t,z,z);
if (size(t)==0) continue;
break;
}
if (size(n)==1) exit(0);
} while (!isprime(n));
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
void marks(long start)
{ /* mark non-primes in this interval. Note *
* that those < NEXT are dealt with already */
int i,pr,j,k;
for (j=1;j<=MULT/2;j+=2) plus[j]=minus[j]=TRUE;
for (i=0;;i++)
{ /* mark in both directions */
pr=mip->PRIMES[i];
if (pr<NEXT) continue;
if ((long)pr*pr>start) break;
k=pr-start%pr;
for (j=k;j<=MULT/2;j+=pr)
plus[j]=FALSE;
k=start%pr;
for (j=k;j<=MULT/2;j+=pr)
minus[j]=FALSE;
}
}
void pollard(int lim1,long lim2)
{ /* factoring program using Pollards (p-1) method */
long i,p,pa,interval;
int phase,m,pos,btch,iv;
big t,b,bw,bvw,bd,bp,q;
t=mirvar(0);
b=mirvar(0);
q=mirvar(0);
bw=mirvar(0);
bvw=mirvar(0);
bd=mirvar(0);
bp=mirvar(0);
gprime(lim1);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1)
{
fu[m]=mirvar(0);
cp[m]=TRUE;
}
else cp[m]=FALSE;
phase=1;
p=0;
btch=50;
i=0;
convert(3,b);
if (!suppress) printf("phase 1 - trying all primes less than %d\n",lim1);
if (!suppress) printf("prime= %8ld",p);
forever
{
if (phase==1)
{ /* looking for all factors of (p-1) < lim1 */
p=mip->PRIMES[i];
if (mip->PRIMES[i+1]==0)
{
phase=2;
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf("phase 2 - trying last prime less than %ld\n"
,lim2);
printf("prime= %8ld",p);
}
power(b,8,n,bw);
convert(1,t);
copy(b,bp);
copy(b,fu[1]);
for (m=3;m<=MULT/2;m+=2)
{ /* store fu[m] = b^(m*m) */
mad(t,bw,bw,n,n,t);
mad(bp,t,t,n,n,bp);
if (cp[m]) copy(bp,fu[m]);
}
power(b,MULT,n,t);
power(t,MULT,n,t);
mad(t,t,t,n,n,bd); /* bd=b^(2*MULT*MULT) */
iv=p/MULT;
if (p%MULT>MULT/2) iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
power(t,2*iv-1,n,bw);
power(t,iv,n,bvw);
power(bvw,iv,n,bvw); /* bvw = b^(MULT*MULT*iv*iv) */
subtract(bvw,fu[p%MULT],q);
btch*=100;
i++;
continue;
}
pa=p;
while ((lim1/p) > pa) pa*=p;
power(b,(int)pa,n,b);
decr(b,1,q);
}
else
{ /* looking for last PRIME FACTOR of (p-1) < lim2 */
p+=2;
pos=p%MULT;
if (pos>MULT/2)
{ /* increment giant step */
iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
pos=1;
mad(bw,bd,bd,n,n,bw);
mad(bvw,bw,bw,n,n,bvw);
}
if (!cp[pos]) continue;
/* if neither interval+/-pos is prime, don't bother */
if (!plus[pos] && !minus[pos]) continue;
subtract(bvw,fu[pos],t);
mad(q,t,t,n,n,q); /* batching gcds */
}
if (i++%btch==0)
{ /* try for a solution */
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b%8ld",p);
fflush(stdout);
}
egcd(q,n,t);
if (size(t)==1)
{
if (p>lim2)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
fflush(stdout);
}
break;
}
else continue;
}
if (mr_compare(t,n)==0)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf("degenerate case\n");
}
break;
}
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
if (isprime(t)) printf("PRIME FACTOR ");
else printf("COMPOSITE FACTOR ");
}
else if (!isprime(t)) printf("& ");
cotnum(t,output);
divide(n,t,n);
if (isprime(n))
{
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
break;
}
}
gprime(0);
mr_free(t);
mr_free(b);
mr_free(q);
mr_free(bw);
mr_free(bvw);
mr_free(bd);
mr_free(bp);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1) mr_free(fu[m]);
}
void williams(int lim1,long lim2,int ntrys)
{ /* factoring program using Williams (p+1) method */
int k,phase,m,nt,iv,pos,btch;
long i,p,pa,interval;
big b,q,fp,fvw,fd,fn,t;
b=mirvar(0);
q=mirvar(0);
t=mirvar(0);
fp=mirvar(0);
fvw=mirvar(0);
fd=mirvar(0);
fn=mirvar(0);
gprime(lim1);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1)
{
fu[m]=mirvar(0);
cp[m]=TRUE;
}
else cp[m]=FALSE;
for (nt=0,k=3;k<10;k++)
{ /* try more than once for p+1 condition (may be p-1) */
convert(k,b); /* try b=3,4,5.. */
convert((k*k-4),t);
if (egcd(t,n,t)!=1) continue; /* check (b*b-4,n)!=0 */
nt++;
phase=1;
p=0;
btch=50;
i=0;
if (!suppress) printf("phase 1 - trying all primes less than %d\n",lim1);
if (!suppress) printf("prime= %8ld",p);
forever
{ /* main loop */
if (phase==1)
{ /* looking for all factors of p+1 < lim1 */
p=mip->PRIMES[i];
if (mip->PRIMES[i+1]==0)
{ /* now change gear */
phase=2;
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf("phase 2 - trying last prime less than %ld\n"
,lim2);
printf("prime= %8ld",p);
}
copy(b,fu[1]);
copy(b,fp);
mad(b,b,b,n,n,fd);
decr(fd,2,fd);
negify(b,t);
mad(fd,b,t,n,n,fn);
for (m=5;m<=MULT/2;m+=2)
{ /* store fu[m] = Vm(b) */
negify(fp,t);
mad(fn,fd,t,n,n,t);
copy(fn,fp);
copy(t,fn);
if (!cp[m]) continue;
copy(t,fu[m]);
}
convert(MULT,t);
lucas(b,t,n,fp,fd);
iv=p/MULT;
if (p%MULT>MULT/2) iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
convert(iv,t);
lucas(fd,t,n,fp,fvw);
negify(fp,fp);
subtract(fvw,fu[p%MULT],q);
btch*=100;
i++;
continue;
}
pa=p;
while ((lim1/p) > pa) pa*=p;
convert((int)pa,t);
lucas(b,t,n,fp,q);
copy(q,b);
decr(q,2,q);
}
else
{ /* phase 2 - looking for last large PRIME FACTOR of (p+1) */
p+=2;
pos=p%MULT;
if (pos>MULT/2)
{ /* increment giant step */
iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
pos=1;
copy(fvw,t);
mad(fvw,fd,fp,n,n,fvw);
negify(t,fp);
}
if (!cp[pos]) continue;
/* if neither interval+/-pos is prime, don't bother */
if (!plus[pos] && !minus[pos]) continue;
subtract(fvw,fu[pos],t);
mad(q,t,t,n,n,q); /* batching gcds */
}
if (i++%btch==0)
{ /* try for a solution */
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b%8ld",p);
fflush(stdout);
}
egcd(q,n,t);
if (size(t)==1)
{
if (p>lim2)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
fflush(stdout);
}
break;
}
else continue;
}
if (mr_compare(t,n)==0)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf("degenerate case\n");
}
break;
}
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
if (isprime(t)) printf("PRIME FACTOR ");
else printf("COMPOSITE FACTOR ");
}
else if (!isprime(t)) printf("& ");
cotnum(t,output);
divide(n,t,n);
if (isprime(n))
{
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
nt=ntrys;
break;
}
}
if (nt>=ntrys) break;
}
gprime(0);
mr_free(b);
mr_free(q);
mr_free(t);
mr_free(fp);
mr_free(fvw);
mr_free(fd);
mr_free(fn);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1) mr_free(fu[m]);
}
static big ak,t,ww,s1,d1,s2,d2;
void duplication(big sum,big diff,big x,big z)
{ /* double a point on the curve P(x,z)=2.P(x1,z1) */
nres_modmult(sum,sum,t);
nres_modmult(diff,diff,z);
nres_modmult(t,z,x); /* x = sum^2.diff^2 */
nres_modsub(t,z,t); /* t = sum^2-diff^2 */
nres_modmult(ak,t,ww);
nres_modadd(z,ww,z); /* z = ak*t +diff^2 */
nres_modmult(z,t,z); /* z = z.t */
}
void addition(big xd,big zd,big sm1,big df1,big sm2,big df2,big x,big z)
{ /* add two points on the curve P(x,z)=P(x1,z1)+P(x2,z2) *
* given their difference P(xd,zd) */
nres_modmult(df2,sm1,x);
nres_modmult(df1,sm2,z);
nres_modadd(z,x,t);
nres_modsub(z,x,z);
nres_modmult(t,t,x);
nres_modmult(x,zd,x); /* x = zd.[df1.sm2+sm1.df2]^2 */
nres_modmult(z,z,z);
nres_modmult(z,xd,z); /* z = xd.[df1.sm2-sm1.df2]^2 */
}
void ellipse(big x,big z,int r,big x1,big z1,big x2,big z2)
{ /* calculate point r.P(x,z) on curve */
int k,rr;
k=1;
rr=r;
copy(x,x1);
copy(z,z1);
nres_modadd(x1,z1,s1);
nres_modsub(x1,z1,d1);
duplication(s1,d1,x2,z2); /* generate 2.P */
while ((rr/=2)>1) k*=2;
while (k>0)
{ /* use binary method */
nres_modadd(x1,z1,s1); /* form sums and differences */
nres_modsub(x1,z1,d1); /* x+z and x-z for P1 and P2 */
nres_modadd(x2,z2,s2);
nres_modsub(x2,z2,d2);
if ((r&k)==0)
{ /* double P(x1,z1) mP to 2mP */
addition(x,z,s1,d1,s2,d2,x2,z2);
duplication(s1,d1,x1,z1);
}
else
{ /* double P(x2,z2) (m+1)P to (2m+2)P */
addition(x,z,s1,d1,s2,d2,x1,z1);
duplication(s2,d2,x2,z2);
}
k/=2;
}
}
int lenstra(int lim1,long lim2,int nc,int kurve,int ncurves)
{ /* factoring program using Lenstras Elliptic Curve method */
int phase,m,iv,pos,btch,u,v,ncr;
long i,p,pa,interval;
big q,x,z,a,x1,z1,x2,z2,xt,zt,fvw;
q=mirvar(0);
x=mirvar(0);
z=mirvar(0);
a=mirvar(0);
x1=mirvar(0);
z1=mirvar(0);
x2=mirvar(0);
z2=mirvar(0);
s1=mirvar(0);
d1=mirvar(0);
s2=mirvar(0);
d2=mirvar(0);
ak=mirvar(0);
xt=mirvar(0);
zt=mirvar(0);
fvw=mirvar(0);
t=mirvar(0);
ww=mirvar(0);
gprime(lim1);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1)
{
fu[m]=mirvar(0);
cp[m]=TRUE;
}
else cp[m]=FALSE;
prepare_monty(n);
/* try a new curve */
/* generating an elliptic curve */
ncr=nc;
u=kurve*kurve-5;
v=4*kurve;
convert(u,x); nres(x,x);
convert(v,z); nres(z,z);
nres_modsub(z,x,a); /* a=v-u */
copy(x,t);
nres_modmult(x,x,x);
nres_modmult(x,t,x); /* x=u^3 */
copy(z,t);
nres_modmult(z,z,z);
nres_modmult(z,t,z); /* z=v^3 */
copy(a,t);
nres_modmult(t,t,t);
nres_modmult(t,a,t); /* t=(v-u)^3 */
convert(3*u,a); nres(a,a);
convert(v,ak); nres(ak,ak);
nres_modadd(a,ak,a);
nres_modmult(t,a,t); /* t=(v-u)^3.(3u+v) */
convert(u,a); nres(a,a);
copy(a,ak);
nres_modmult(a,a,a);
nres_modmult(a,ak,a); /* a=u^3 */
convert(v,ak); nres(ak,ak);
nres_modmult(a,ak,a); /* a=u^3.v */
nres_premult(a,16,a);
nres_moddiv(t,a,ak); /* ak=(v-u)^3.(3u+v)/16u^3v */
phase=1;
p=0;
i=0;
btch=50;
if (!suppress) printf("curve %3d phase 1 - trying all primes less than %d\n",nc,lim1);
if (!suppress) printf("prime= %8ld",p);
ncr++;
forever
{ /* main loop */
if (phase==1)
{
p=mip->PRIMES[i];
if (mip->PRIMES[i+1]==0)
{ /* now change gear */
phase=2;
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf(" phase 2 - trying last prime less than %ld\n",
lim2);
printf("prime= %8ld",p);
}
copy(x,xt);
copy(z,zt);
nres_modadd(x,z,s2);
nres_modsub(x,z,d2); /* P = (s2,d2) */
duplication(s2,d2,x,z);
nres_modadd(x,z,s1);
nres_modsub(x,z,d1); /* 2.P = (s1,d1) */
nres_moddiv(x1,z1,fu[1]); /* fu[1] = x1/z1 */
addition(x1,z1,s1,d1,s2,d2,x2,z2); /* 3.P = (x2,z2) */
for (m=5;m<=MULT/2;m+=2)
{ /* calculate m.P = (x,z) and store fu[m] = x/z */
nres_modadd(x2,z2,s2);
nres_modsub(x2,z2,d2);
addition(x1,z1,s2,d2,s1,d1,x,z);
copy(x2,x1);
copy(z2,z1);
copy(x,x2);
copy(z,z2);
if (!cp[m]) continue;
copy(z2,fu[m]);
nres_moddiv(x2,fu[m],fu[m]);
}
ellipse(xt,zt,MULT,x,z,x2,z2);
nres_modadd(x,z,xt);
nres_modsub(x,z,zt); /* MULT.P = (xt,zt) */
iv=(int)(p/MULT);
if (p%MULT>MULT/2) iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
ellipse(x,z,iv,x1,z1,x2,z2); /* (x1,z1) = iv.MULT.P */
nres_moddiv(x1,z1,fvw); /* fvw = x1/z1 */
nres_modsub(fvw,fu[p%MULT],q);
btch*=100;
i++;
continue;
}
pa=p;
while ((lim1/p) > pa) pa*=p;
ellipse(x,z,(int)pa,x1,z1,x2,z2);
copy(x1,x);
copy(z1,z);
copy(z,q);
}
else
{ /* phase 2 - looking for last large PRIME FACTOR of (p+1+d) */
p+=2;
pos=(int)(p%MULT);
if (pos>MULT/2)
{ /* increment giant step */
iv++;
interval=(long)iv*MULT;
p=interval+1;
marks(interval);
pos=1;
nres_moddiv(x2,z2,fvw);
nres_modadd(x2,z2,s2);
nres_modsub(x2,z2,d2);
addition(x1,z1,s2,d2,xt,zt,x,z);
copy(x2,x1);
copy(z2,z1);
copy(x,x2);
copy(z,z2);
}
if (!cp[pos]) continue;
/* if neither interval +/- pos is prime, don't bother */
if (!plus[pos] && !minus[pos]) continue;
nres_modsub(fvw,fu[pos],t);
nres_modmult(q,t,q);
}
if (i++%btch==0)
{ /* try for a solution */
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b%8ld",p);
fflush(stdout);
}
egcd(q,n,t);
if (size(t)==1)
{
if (p>lim2)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
fflush(stdout);
}
break;
}
else continue;
}
if (mr_compare(t,n)==0)
{
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
printf("degenerate case\n");
}
break;
}
if (!suppress)
{
printf("\b\b\b\b\b\b\b\b\b\b\b\b\b\b\b");
if (isprime(t)) printf("PRIME FACTOR ");
else printf("COMPOSITE FACTOR ");
}
else if (!isprime(t)) printf("& ");
cotnum(t,output);
divide(n,t,n);
if (isprime(n))
{
if (!suppress) printf("PRIME FACTOR ");
cotnum(n,output);
exit(0);
}
ncr=ncurves+1;
break;
}
}
gprime(0);
mr_free(ww);
mr_free(t);
mr_free(q);
mr_free(x);
mr_free(z);
mr_free(a);
mr_free(x1);
mr_free(z1);
mr_free(x2);
mr_free(z2);
mr_free(s1);
mr_free(d1);
mr_free(s2);
mr_free(d2);
mr_free(ak);
mr_free(xt);
mr_free(zt);
mr_free(fvw);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1) mr_free(fu[m]);
return ncr;
}
void do_lenstra(int lim1,long lim2,int ncurves)
{
int nc=1;
int kurve=5;
for (nc=1;nc<=ncurves;)
{
kurve++;
nc=lenstra(lim1,lim2,nc,kurve,ncurves);
}
}
#define SSIZE 100000 /* Maximum sieve size */
static big NN,TT,DD,RR,VV,PP,XX,YY,DG,IG,AA,BB;
static big *x,*y,*z,*w;
static unsigned int **EE,**G;
static int *epr,*r1,*r2,*rp,*b,*pr,*e,*hash;
static unsigned char *logp,*sieve;
static int mm,mlf,jj,nbts,nlp,lp,hmod,hmod2;
static BOOL partial;
int knuth(int mm,int *epr,big N,big D)
{ /* Input number to be factored N and find best multiplier k *
* for use over a factor base epr[] of size mm. Set D=k.N. */
double dp,fks,top;
BOOL found;
int i,j,bk,nk,kk,r,p;
static int K[]={0,1,2,3,5,6,7,10,11,13,14,15,17,0};
top=(-10.0e0);
found=FALSE;
nk=0;
bk=0;
epr[0]=1;
epr[1]=2;
do
{ /* search for best Knuth-Schroepel multiplier */
kk=K[++nk];
if (kk==0)
{ /* finished */
kk=K[bk];
found=TRUE;
}
premult(N,kk,D);
fks=log(2.0e0)/(2.0e0);
r=remain(D,8);
if (r==1) fks*=(4.0e0);
if (r==5) fks*=(2.0e0);
fks-=log((double)kk)/(2.0e0);
i=0;
j=1;
while (j<mm)
{ /* select small primes */
p=mip->PRIMES[++i];
r=remain(D,p);
if (spmd(r,(p-1)/2,p)<=1)
{ /* use only if Jacobi symbol = 0 or 1 */
epr[++j]=p;
dp=(double)p;
if (kk%p==0) fks+=log(dp)/dp;
else fks+=2*log(dp)/(dp-1.0e0);
}
}
if (fks>top)
{ /* find biggest fks */
top=fks;
bk=nk;
}
} while (!found);
return kk;
}
BOOL factored(long lptr,big T)
{ /* factor quadratic residue */
BOOL facted;
int i,j,r,st;
partial=FALSE;
facted=FALSE;
for (j=1;j<=mm;j++)
{ /* now attempt complete factorisation of T */
r=(int)(lptr%epr[j]);
if (r<0) r+=epr[j];
if (r!=r1[j] && r!=r2[j]) continue;
while (subdiv(T,epr[j],XX)==0)
{ /* cast out epr[j] */
e[j]++;
copy(XX,T);
}
st=size(T);
if (st==1)
{
facted=TRUE;
break;
}
if (size(XX)<=epr[j])
{ /* st is prime < epr[mm]^2 */
if (st>=MR_TOOBIG || (st/epr[mm])>(1+mlf/50)) break;
if (st<=epr[mm])
for (i=j;i<=mm;i++)
if (st==epr[i])
{
e[i]++;
facted=TRUE;
break;
}
if (facted) break;
lp=st; /* factored with large prime */
partial=TRUE;
facted=TRUE;
break;
}
}
return facted;
}
BOOL gotcha(void)
{ /* use new factorisation */
int r,j,i,k,n,rb,had,hp;
unsigned int t;
BOOL found;
found=TRUE;
if (partial)
{ /* check partial factorisation for usefulness */
had=lp%hmod;
forever
{ /* hash search for matching large prime */
hp=hash[had];
if (hp<0)
{ /* failed to find match */
found=FALSE;
break;
}
if (pr[hp]==lp) break; /* hash hit! */
had=(had+(hmod2-lp%hmod2))%hmod;
}
if (!found && nlp>=mlf) return FALSE;
}
copy(PP,XX);
convert(1,YY);
for (k=1;k<=mm;k++)
{ /* build up square part in YY *
* reducing e[k] to 0s and 1s */
if (e[k]<2) continue;
r=e[k]/2;
e[k]%=2;
expint(epr[k],r,TT);
multiply(TT,YY,YY);
}
/* debug only
cotnum(XX,stdout);
cotnum(YY,stdout);
if (e[0]==1) printf("-1");
else printf("1");
for (k=1;k<=mm;k++)
{
if (e[k]==0) continue;
printf(".%d",epr[k]);
}
if (partial) printf(".%d\n",lp);
else printf("\n");
*/
if (partial)
{ /* factored with large prime */
if (!found)
{ /* store new partial factorization */
hash[had]=nlp;
pr[nlp]=lp;
copy(XX,z[nlp]);
copy(YY,w[nlp]);
for (n=0,rb=0,j=0;j<=mm;j++)
{