/* return this^e mod Modulus
* public FP pow(BIG e)
* {
* int bt;
* FP r=new FP(1);
* e.norm();
* x.norm();
* FP m=new FP(this);
* while (true)
* {
* bt=e.parity();
* e.fshr(1);
* if (bt==1) r.mul(m);
* if (e.iszilch()) break;
* m.sqr();
* }
* r.x.mod(p);
* return r;
* } */
/* return sqrt(this) mod Modulus */
public FP Sqrt()
{
Reduce();
BIG b = new BIG(ROM.Modulus);
if (MOD8 == 5)
{
b.Dec(5);
b.Norm();
b.Shr(3);
FP i = new FP(this);
i.x.Shl(1);
FP v = i.Pow(b);
i.Mul(v);
i.Mul(v);
i.x.Dec(1);
FP r = new FP(this);
r.Mul(v);
r.Mul(i);
r.Reduce();
return(r);
}
else
{
b.Inc(1);
b.Norm();
b.Shr(2);
return(Pow(b));
}
}
/* Optimal R-ate double pairing e(P,Q).e(R,S) */
public static FP12 Ate2(ECP2 P1, ECP Q1, ECP2 R1, ECP S1)
{
FP2 f;
BIG x = new BIG(ROM.CURVE_Bnx);
BIG n = new BIG(x);
ECP2 K = new ECP2();
FP12 lv;
int bt;
ECP2 P = new ECP2(P1);
ECP Q = new ECP(Q1);
P.Affine();
Q.Affine();
ECP2 R = new ECP2(R1);
ECP S = new ECP(S1);
R.Affine();
S.Affine();
if (ECP.CURVE_PAIRING_TYPE == ECP.BN)
{
f = new FP2(new BIG(ROM.Fra), new BIG(ROM.Frb));
if (ECP.SEXTIC_TWIST == ECP.M_TYPE)
{
f.Inverse();
f.Norm();
}
n.PMul(6);
if (ECP.SIGN_OF_X == ECP.POSITIVEX)
{
n.Inc(2);
}
else
{
n.Dec(2);
}
}
else
{
n.Copy(x);
}
n.Norm();
BIG n3 = new BIG(n);
n3.PMul(3);
n3.Norm();
FP Qx = new FP(Q.GetX());
FP Qy = new FP(Q.GetY());
FP Sx = new FP(S.GetX());
FP Sy = new FP(S.GetY());
ECP2 A = new ECP2();
ECP2 B = new ECP2();
FP12 r = new FP12(1);
A.Copy(P);
B.Copy(R);
ECP2 MP = new ECP2();
MP.Copy(P);
MP.Neg();
ECP2 MR = new ECP2();
MR.Copy(R);
MR.Neg();
int nb = n3.NBits();
for (int i = nb - 2; i >= 1; i--)
{
r.Sqr();
lv = Line(A, A, Qx, Qy);
r.SMul(lv, ECP.SEXTIC_TWIST);
lv = Line(B, B, Sx, Sy);
r.SMul(lv, ECP.SEXTIC_TWIST);
bt = n3.Bit(i) - n.Bit(i); // bt=n.bit(i);
if (bt == 1)
{
lv = Line(A, P, Qx, Qy);
r.SMul(lv, ECP.SEXTIC_TWIST);
lv = Line(B, R, Sx, Sy);
r.SMul(lv, ECP.SEXTIC_TWIST);
}
if (bt == -1)
{
//P.neg();
lv = Line(A, MP, Qx, Qy);
r.SMul(lv, ECP.SEXTIC_TWIST);
//P.neg();
//R.neg();
lv = Line(B, MR, Sx, Sy);
r.SMul(lv, ECP.SEXTIC_TWIST);
//R.neg();
}
}
if (ECP.SIGN_OF_X == ECP.NEGATIVEX)
{
r.Conj();
}
/* R-ate fixup required for BN curves */
if (ECP.CURVE_PAIRING_TYPE == ECP.BN)
{
if (ECP.SIGN_OF_X == ECP.NEGATIVEX)
{
// r.conj();
A.Neg();
B.Neg();
}
K.Copy(P);
K.Frob(f);
lv = Line(A, K, Qx, Qy);
r.SMul(lv, ECP.SEXTIC_TWIST);
K.Frob(f);
K.Neg();
lv = Line(A, K, Qx, Qy);
r.SMul(lv, ECP.SEXTIC_TWIST);
K.Copy(R);
K.Frob(f);
lv = Line(B, K, Sx, Sy);
r.SMul(lv, ECP.SEXTIC_TWIST);
K.Frob(f);
K.Neg();
lv = Line(B, K, Sx, Sy);
r.SMul(lv, ECP.SEXTIC_TWIST);
}
return(r);
}
