void nres_powltr(_MIPD_ int x,big y,big w)
{ /* calculates w=x^y mod z using Left to Right Method */
/* uses only n^2 modular squarings, because x is small */
/* Note: x is NOT an nresidue */
int i,nb;
#ifndef MR_GENERIC_MT
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
copy(y,mr_mip->w1);
MR_IN(86)
zero(w);
if (x==0)
{
if (size(mr_mip->w1)==0)
{ /* 0^0 = 1 */
convert(_MIPP_ 1,w);
nres(_MIPP_ w,w);
}
MR_OUT
return;
}
convert(_MIPP_ 1,w);
nres(_MIPP_ w,w);
if (size(mr_mip->w1)==0)
{
MR_OUT
return;
}
BOOL epoint_set(_MIPD_ big x,big y,int cb,epoint *p)
{ /* initialise a point on active ecurve *
* if x or y == NULL, set to point at infinity *
* if x==y, a y co-ordinate is calculated - if *
* possible - and cb suggests LSB 0/1 of y *
* (which "decompresses" y). Otherwise, check *
* validity of given (x,y) point, ignoring cb. *
* Returns TRUE for valid point, otherwise FALSE. */
BOOL valid;
#ifndef MR_GENERIC_MT
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return FALSE;
MR_IN(97)
if (x==NULL || y==NULL)
{
convert(_MIPP_ 1,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,p->X);
nres(_MIPP_ mr_mip->w1,p->Y);
p->marker=MR_EPOINT_INFINITY;
MR_OUT
return TRUE;
}
void zzn2_out(_MIPD_ char *p,zzn2 *x)
{
printf(p);
printf("\n");
redc(_MIPP_ x->a,x->a);
redc(_MIPP_ x->b,x->b);
otnum(_MIPP_ x->a,stdout);
otnum(_MIPP_ x->b,stdout);
nres(_MIPP_ x->a,x->a);
nres(_MIPP_ x->b,x->b);
}
void zzn2_from_bigs(_MIPD_ big x,big y, zzn2 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(166)
nres(_MIPP_ x,w->a);
nres(_MIPP_ y,w->b);
MR_OUT
}
void zzn2_from_ints(_MIPD_ int i,int j,zzn2 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(168)
convert(_MIPP_ i,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,w->a);
convert(_MIPP_ j,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,w->b);
MR_OUT
}
void zzn4_powq(_MIPD_ big fr,zzn4 *w)
{
zzn2_conj(_MIPP_ &(w->x),&(w->x));
zzn2_conj(_MIPP_ &(w->y),&(w->y));
nres(_MIPP_ fr,mr_mip->w1);
zzn2_smul(_MIPP_ &(w->y),mr_mip->w1,&(w->y));
}
void zzn4_from_int(_MIPD_ int i,zzn4 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(FUNC_BASE+0)
if (i==1)
{
copy(mr_mip->one,w->a.a);
w->unitary=TRUE;
}
else
{
convert(_MIPP_ i,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,(w->a).a);
w->unitary=FALSE;
}
zero((w->a).b);
zero((w->b).a);
zero((w->b).b);
MR_OUT
}
void ecurve_init(_MIPD_ big a,big b,big p,int type)
{ /* Initialize the active ecurve *
* Asize indicate size of A *
* Bsize indicate size of B */
int as;
#ifndef MR_GENERIC_MT
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(93)
prepare_monty(_MIPP_ p);
mr_mip->Asize=size(a);
if (mr_abs(mr_mip->Asize)==MR_TOOBIG)
{
if (mr_mip->Asize>=0)
{ /* big positive number - check it isn't minus something small */
copy(a,mr_mip->w1);
divide(_MIPP_ mr_mip->w1,p,p);
subtract(_MIPP_ p,mr_mip->w1,mr_mip->w1);
as=size(mr_mip->w1);
if (as<MR_TOOBIG) mr_mip->Asize=-as;
else
{
if (mr_mip->A==NULL) mr_mip->A=mirvar(_MIPP_ 0);
nres(_MIPP_ a,mr_mip->A);
}
}
else
{
if (mr_mip->A==NULL) mr_mip->A=mirvar(_MIPP_ 0);
nres(_MIPP_ a,mr_mip->A);
}
}
mr_mip->Bsize=size(b);
if (mr_abs(mr_mip->Bsize)==MR_TOOBIG)
{
if (mr_mip->B==NULL) mr_mip->B=mirvar(_MIPP_ 0);
nres(_MIPP_ b,mr_mip->B);
}
mr_mip->coord=type;
MR_OUT
return;
}
void zzn3_set(_MIPD_ int cnr,big sru)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
mr_mip->cnr=cnr;
nres(_MIPP_ sru,mr_mip->sru);
}
void zzn3_from_big(_MIPD_ big x, zzn3 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(177)
nres(_MIPP_ x,w->a);
zero(w->b); zero(w->c);
MR_OUT
}
void zzn3_from_int(_MIPD_ int i,zzn3 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(174)
convert(_MIPP_ i,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,w->a);
zero(w->b); zero(w->c);
MR_OUT
}
void zzn3_inv(_MIPD_ zzn3 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(187)
nres_modmult(_MIPP_ w->a,w->a,mr_mip->w1);
nres_modmult(_MIPP_ w->b,w->c,mr_mip->w2);
nres_premult(_MIPP_ mr_mip->w2,mr_mip->cnr,mr_mip->w2);
nres_modsub(_MIPP_ mr_mip->w1,mr_mip->w2,mr_mip->w3);
nres_modmult(_MIPP_ w->c,w->c,mr_mip->w1);
nres_modmult(_MIPP_ w->a,w->b,mr_mip->w2);
nres_premult(_MIPP_ mr_mip->w1,mr_mip->cnr,mr_mip->w1);
nres_modsub(_MIPP_ mr_mip->w2,mr_mip->w1,mr_mip->w4);
nres_negate(_MIPP_ mr_mip->w4,mr_mip->w4);
nres_modmult(_MIPP_ w->b,w->b,mr_mip->w1);
nres_modmult(_MIPP_ w->a,w->c,mr_mip->w2);
nres_modsub(_MIPP_ mr_mip->w1,mr_mip->w2,mr_mip->w5);
nres_modmult(_MIPP_ w->b,mr_mip->w5,mr_mip->w1);
nres_modmult(_MIPP_ w->c,mr_mip->w4,mr_mip->w2);
nres_modadd(_MIPP_ mr_mip->w2,mr_mip->w1,mr_mip->w2);
nres_premult(_MIPP_ mr_mip->w2,mr_mip->cnr,mr_mip->w2);
nres_modmult(_MIPP_ w->a,mr_mip->w3,mr_mip->w1);
nres_modadd(_MIPP_ mr_mip->w1,mr_mip->w2,mr_mip->w1);
copy(mr_mip->w3,w->a);
copy(mr_mip->w4,w->b);
copy(mr_mip->w5,w->c);
redc(_MIPP_ mr_mip->w1,mr_mip->w6);
invmodp(_MIPP_ mr_mip->w6,mr_mip->modulus,mr_mip->w6);
nres(_MIPP_ mr_mip->w6,mr_mip->w6);
nres_modmult(_MIPP_ w->a,mr_mip->w6,w->a);
nres_modmult(_MIPP_ w->b,mr_mip->w6,w->b);
nres_modmult(_MIPP_ w->c,mr_mip->w6,w->c);
MR_OUT
}
void nres_lucas(_MIPD_ big p,big r,big vp,big v)
{
int i,nb;
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(107)
if (size(r)==0)
{
zero(vp);
convert(_MIPP_ 2,v);
nres(_MIPP_ v,v);
MR_OUT
return;
}
void zzn2_from_int(_MIPD_ int i,zzn2 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(156)
if (i==1)
{
copy(mr_mip->one,w->a);
}
else
{
convert(_MIPP_ i,mr_mip->w1);
nres(_MIPP_ mr_mip->w1,w->a);
}
zero(w->b);
MR_OUT
}
BOOL nres_sqroot(_MIPD_ big x,big w)
{ /* w=sqrt(x) mod p. This depends on p being prime! */
int t,js;
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return FALSE;
copy(x,w);
if (size(w)==0) return TRUE;
MR_IN(100)
redc(_MIPP_ w,w); /* get it back into normal form */
if (size(w)==1) /* square root of 1 is 1 */
{
nres(_MIPP_ w,w);
MR_OUT
return TRUE;
}
/*
void zzn2_print(_MIPD_ char *label, zzn2 *x)
{
char s1[1024], s2[1024];
big a, b;
#ifdef MR_STATIC
char mem_big[MR_BIG_RESERVE(2)];
memset(mem_big, 0, MR_BIG_RESERVE(2));
a=mirvar_mem(_MIPP_ mem_big,0);
b=mirvar_mem(_MIPP_ mem_big,1);
#else
a = mirvar(_MIPP_ 0);
b = mirvar(_MIPP_ 0);
#endif
redc(_MIPP_ x->a, a); otstr(_MIPP_ a, s1);
redc(_MIPP_ x->b, b); otstr(_MIPP_ b, s2);
printf("%s: [%s,%s]\n", label, s1, s2);
#ifndef MR_STATIC
mr_free(a); mr_free(b);
#endif
}
static void nres_print(_MIPD_ char *label, big x)
{
char s[1024];
big a;
#ifdef MR_STATIC
char mem_big[MR_BIG_RESERVE(1)];
memset(mem_big, 0, MR_BIG_RESERVE(1));
a=mirvar_mem(_MIPP_ mem_big,0);
#else
a = mirvar(_MIPP_ 0);
#endif
redc(_MIPP_ x, a);
otstr(_MIPP_ a, s);
printf("%s: %s\n", label, s);
#ifndef MR_STATIC
mr_free(a);
#endif
}
*/
void zzn2_inv(_MIPD_ zzn2 *w)
{
#ifdef MR_OS_THREADS
miracl *mr_mip=get_mip();
#endif
if (mr_mip->ERNUM) return;
MR_IN(163)
nres_modmult(_MIPP_ w->a,w->a,mr_mip->w1);
nres_modmult(_MIPP_ w->b,w->b,mr_mip->w2);
nres_modadd(_MIPP_ mr_mip->w1,mr_mip->w2,mr_mip->w1);
if (mr_mip->qnr==-2)
nres_modadd(_MIPP_ mr_mip->w1,mr_mip->w2,mr_mip->w1);
redc(_MIPP_ mr_mip->w1,mr_mip->w6);
invmodp(_MIPP_ mr_mip->w6,mr_mip->modulus,mr_mip->w6);
nres(_MIPP_ mr_mip->w6,mr_mip->w6);
nres_modmult(_MIPP_ w->a,mr_mip->w6,w->a);
nres_negate(_MIPP_ mr_mip->w6,mr_mip->w6);
nres_modmult(_MIPP_ w->b,mr_mip->w6,w->b);
MR_OUT
}
int main()
{ /* factoring program using Lenstras Elliptic Curve method */
int phase,m,k,nc,iv,pos,btch,u,v;
long i,p,pa,interval;
big q,x,z,a,x1,z1,x2,z2,xt,zt,n,fvw;
static big fu[1+MULT/2];
static BOOL cp[1+MULT/2];
mip=mirsys(30,0);
q=mirvar(0);
x=mirvar(0);
z=mirvar(0);
a=mirvar(0);
x1=mirvar(0);
z1=mirvar(0);
x2=mirvar(0);
z2=mirvar(0);
n=mirvar(0);
t=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);
w=mirvar(0);
gprime(LIMIT1);
for (m=1;m<=MULT/2;m+=2)
if (igcd(MULT,m)==1)
{
fu[m]=mirvar(0);
cp[m]=TRUE;
}
else cp[m]=FALSE;
printf("input number to be factored\n");
cinnum(n,stdin);
if (isprime(n))
{
printf("this number is prime!\n");
return 0;
}
prepare_monty(n);
for (nc=1,k=6;k<100;k++)
{ /* try a new curve */
/* generating an elliptic curve */
u=k*k-5;
v=4*k;
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 */
nc++;
phase=1;
p=0;
i=0;
btch=50;
printf("phase 1 - trying all primes less than %d\n",LIMIT1);
printf("prime= %8ld",p);
forever
{ /* main loop */
if (phase==1)
{
p=mip->PRIMES[i];
if (mip->PRIMES[i+1]==0)
{ /* now change gear */
phase=2;
printf("\nphase 2 - trying last prime less than %ld\n",
LIMIT2);
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;
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);
marks(interval);
btch*=100;
i++;
continue;
}
pa=p;
while ((LIMIT1/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 */
printf("\b\b\b\b\b\b\b\b%8ld",p);
fflush(stdout);
egcd(q,n,t);
if (size(t)==1)
{
if (p>LIMIT2) break;
else continue;
}
if (compare(t,n)==0)
{
printf("\ndegenerate case");
break;
}
printf("\nfactors are\n");
if (isprime(t)) printf("prime factor ");
else printf("composite factor ");
cotnum(t,stdout);
divide(n,t,n);
if (isprime(n)) printf("prime factor ");
else printf("composite factor ");
cotnum(n,stdout);
return 0;
}
}
if (nc>NCURVES) break;
printf("\ntrying a different curve %d\n",nc);
}
printf("\nfailed to factor\n");
return 0;
}
