BOOL fast_tate_pairing(ECn& P,ZZn6& Qx,ZZn6& Qy,Big& q,Big &cf,ZZn6& res)
{
int i,j,n,nb,nbw,nzs;
ECn A,P2,t[16];
ZZn6 w,hc,z2n,zn[16];
res=zn[0]=1;
t[0]=P2=A=P;
g(P2,P2,Qx,Qy,z2n,TRUE); // P2=P+P
//
// Build windowing table
//
for (i=1;i<16;i++)
{
g(A,P2,Qx,Qy,hc,TRUE);
t[i]=A;
zn[i]=z2n*zn[i-1]*hc;
}
A=P; // reset A
/* Left to right method */
nb=bits(q);
for (i=nb-2;i>=0;i-=(nbw+nzs))
{
n=window(q,i,&nbw,&nzs); // standard MIRACL windowing
for (j=0;j<nbw;j++)
{
res*=res;
g(A,A,Qx,Qy,res,FALSE);
}
if (n>0)
{
res*=zn[n/2];
g(A,t[n/2],Qx,Qy,res,FALSE);
}
for (j=0;j<nzs;j++)
{
res*=res;
g(A,A,Qx,Qy,res,FALSE);
}
if (res.iszero()) return FALSE;
}
if (!A.iszero() || res.iszero()) return FALSE;
res=pow(res,cf); // ^(p*p-p+1)/q
w=res;
w.powq();
res*=w; // ^(p+1)
w=res;
w.powq(); w.powq(); w.powq();
res=w/res; // ^(p^3-1)
if (res.isunity()) return FALSE;
return TRUE;
}
ZZn6 sqrt(const ZZn6& x)
{
// sqrt(a+xb) = sqrt((a+sqrt(a*a-n*b*b))/2)+x.b/(2*sqrt((a+sqrt(a*a-n*b*b))/2))
// sqrt(a) = x.sqrt(a/n)
// where x*x=n
ZZn6 w;
ZZn3 a,s,t;
if (x.iszero()) return w;
if (x.b.iszero())
{
w.unitary=x.unitary;
a=x.a;
if (qr(a))
{
s=sqrt(a);
w.a=s; w.b=0;
}
else
{
a=txd(a);
s=sqrt(a);
w.a=0; w.b=s;
}
return w;
}
s=x.b; s*=s;
a=x.a; a*=a; a-=tx(s);
s=sqrt(a);
if (s.iszero()) return w;
w.unitary=x.unitary;
if (qr((x.a+s)/2))
{
a=sqrt((x.a+s)/2);
}
else
{
a=sqrt((x.a-s)/2);
if (a.iszero()) return w;
}
w.a=a;
w.b=x.b/(2*a);
return w;
}
BOOL qr(const ZZn6& x)
{
ZZn3 a,s;
if (x.iszero()) return TRUE;
if (x.b.iszero()) return TRUE;
s=x.b; s*=s;
a=x.a; a*=a; a-=tx(s);
if (!qr(a)) return FALSE;
return TRUE;
/*
s=sqrt(a);
if (qr((x.a+s)/2) || qr((x.a-s)/2)) return TRUE;
exit(0);
return FALSE;
*/
}
BOOL ate(ECn3& Q,ECn& P,Big &x,ZZn2& X,ZZn6& res)
{
int i,j,n,nb,nbw,nzs;
ECn3 A;
ZZn Px,Py;
ZZn6 w;
Big q=x*x-x+1;
#ifdef MR_COUNT_OPS
fpc=fpa=fpx=0;
#endif
normalise(P);
#ifdef PROJECTIVE
Q.norm();
#endif
extract(P,Px,Py);
Px+=Px; // because x^6+2 is irreducible.. simplifies line function calculation
Py+=Py;
res=1;
A=Q; // reset A
nb=bits(x);
res.mark_as_miller();
for (i=nb-2;i>=0;i--)
{
res*=res;
res*=g(A,A,Px,Py);
if (bit(x,i)==1)
res*=g(A,Q,Px,Py);
if (res.iszero()) return FALSE;
}
#ifdef MR_COUNT_OPS
printf("After Miller fpc= %d fpa= %d fpx= %d\n",fpc,fpa,fpx);
#endif
// if (!A.iszero() || res.iszero()) return FALSE;
w=res;
w.powq(X);
res*=w; // ^(p+1)
w=res;
w.powq(X); w.powq(X); w.powq(X);
res=w/res; // ^(p^3-1)
// exploit the clever "trick" for a half-length exponentiation!
res.mark_as_unitary();
w=res;
res.powq(X); // res*=res; // res=pow(res,CF);
if (x<0) res/=powu(w,-x);
else res*=powu(w,x);
#ifdef MR_COUNT_OPS
printf("After pairing fpc= %d fpa= %d fpx= %d\n",fpc,fpa,fpx);
fpa=fpc=fpx=0;
#endif
if (res==(ZZn6)1) return FALSE;
return TRUE;
}
GT PFC::multi_miller(int n,G2** QQ,G1** PP)
{
GT z;
ZZn *Px,*Py;
int i,j,*k,nb;
ECn3 *Q,*A;
ECn P;
ZZn6 res;
Big X=*x;
Px=new ZZn[n];
Py=new ZZn[n];
Q=new ECn3[n];
A=new ECn3[n];
k=new int[n];
nb=bits(X);
res=1;
for (j=0;j<n;j++)
{
k[j]=0;
P=PP[j]->g; normalise(P); Q[j]=QQ[j]->g;
extract(P,Px[j],Py[j]);
Px[j]+=Px[j];
Py[j]+=Py[j];
}
for (j=0;j<n;j++)
{
#ifdef MR_ECN3_PROJECTIVE
Q[j].norm();
#endif
A[j]=Q[j];
}
for (i=nb-2;i>=0;i--)
{
res*=res;
for (j=0;j<n;j++)
{
if (QQ[j]->ptable==NULL)
res*=g(A[j],A[j],Px[j],Py[j]);
else
res*=gp(QQ[j]->ptable,k[j],Px[j],Py[j]);
}
if (bit(X,i)==1)
for (j=0;j<n;j++)
{
if (QQ[j]->ptable==NULL)
res*=g(A[j],Q[j],Px[j],Py[j]);
else
res*=gp(QQ[j]->ptable,k[j],Px[j],Py[j]);
}
if (res.iszero()) return 0;
}
delete [] k;
delete [] A;
delete [] Q;
delete [] Py;
delete [] Px;
z.g=res;
return z;
}
BOOL fast_tate_pairing(ECn& P,ZZn3& Qx,ZZn3& Qy,Big &x,ZZn2& X,ZZn6& res)
{
int i,j,n,nb,nbw,nzs;
ECn A,P2,t[PRECOMP];
ZZn6 w,hc,z2n,zn[PRECOMP];
Big q=x*x-x+1;
res=zn[0]=1;
t[0]=P2=A=P;
z2n=g(P2,P2,Qx,Qy); // P2=P+P
normalise(P2);
//
// Build windowing table
//
for (i=1;i<PRECOMP;i++)
{
hc=g(A,P2,Qx,Qy);
t[i]=A;
zn[i]=z2n*zn[i-1]*hc;
}
multi_norm(PRECOMP,t); // make t points Affine
/*
A=P; // reset A
nb=bits(q);
for (i=nb-2;i>=0;i--)
{
res*=res;
res*=g(A,A,Qx,Qy);
if (bit(q,i)==1)
res*=g(A,P,Qx,Qy);
if (res.iszero()) return FALSE;
}
*/
A=P; // reset A
nb=bits(q);
for (i=nb-2;i>=0;i-=(nbw+nzs))
{ // windowing helps a little..
n=window(q,i,&nbw,&nzs,WINDOW_SIZE); // standard MIRACL windowing
for (j=0;j<nbw;j++)
{
res*=res;
res*=g(A,A,Qx,Qy);
}
if (n>0)
{
res*=zn[n/2];
res*=g(A,t[n/2],Qx,Qy);
}
for (j=0;j<nzs;j++)
{
res*=res;
res*=g(A,A,Qx,Qy);
}
if (res.iszero()) return FALSE;
}
#ifdef MR_COUNT_OPS
printf("After Miller fpc= %d fpa= %d fpx= %d\n",fpc,fpa,fpx);
#endif
// if (!A.iszero() || res.iszero()) return FALSE;
w=res;
w.powq(X);
res*=w; // ^(p+1)
w=res;
w.powq(X); w.powq(X); w.powq(X);
res=w/res; // ^(p^3-1)
// exploit the clever "trick" for a half-length exponentiation!
res.mark_as_unitary();
w=res;
res.powq(X); // res*=res; // res=pow(res,CF);
if (x<0) res/=powu(w,-x);
else res*=powu(w,x);
if (res==(ZZn6)1) return FALSE;
return TRUE;
}
