js/PAIR.js - incubator-milagro-crypto - Git at Google

 /*
 Licensed to the Apache Software Foundation (ASF) under one
 or more contributor license agreements.  See the NOTICE file
 distributed with this work for additional information
 regarding copyright ownership.  The ASF licenses this file
 to you under the Apache License, Version 2.0 (the
 "License"); you may not use this file except in compliance
 with the License.  You may obtain a copy of the License at

   http://www.apache.org/licenses/LICENSE-2.0

 Unless required by applicable law or agreed to in writing,
 software distributed under the License is distributed on an
 "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
 KIND, either express or implied.  See the License for the
 specific language governing permissions and limitations
 under the License.
 */

 var PAIR = {
 /* Line function */
 	line: function(A,B,Qx,Qy)
 	{
 		var P=new ECP2();
 		var a,b,c;
 		var r=new FP12(1);
 		P.copy(A);

 		var ZZ=new FP2(P.getz()); //ZZ.copy(P.getz());
 		ZZ.sqr();
 		var D;
 		if (A==B) D=A.dbl();
 		else D=A.add(B);
 		if (D<0) return r;
 		var Z3=new FP2(A.getz()); //Z3.copy(A.getz());
 		c=new FP4(0);
 		var X,Y,T;
 		if (D===0)
 		{ /* Addition */
 			X=new FP2(B.getx()); //X.copy(B.getx());
 			Y=new FP2(B.gety()); //Y.copy(B.gety());
 			T=new FP2(P.getz()); //T.copy(P.getz());

 			T.mul(Y);
 			ZZ.mul(T);

 			var NY=new FP2(P.gety()); /*NY.copy(P.gety());*/ NY.neg();
 			ZZ.add(NY);
 			Z3.pmul(Qy);
 			T.mul(P.getx());
 			X.mul(NY);
 			T.add(X);
 			a=new FP4(Z3,T); //a.set(Z3,T);
 			ZZ.neg();
 			ZZ.pmul(Qx);
 			b=new FP4(ZZ); //b.seta(ZZ);
 		}
 		else
 		{ /* Doubling */
 			X=new FP2(P.getx()); //X.copy(P.getx());
 			Y=new FP2(P.gety()); //Y.copy(P.gety());
 			T=new FP2(P.getx()); //T.copy(P.getx());
 			T.sqr();
 			T.imul(3);

 			Y.sqr();
 			Y.add(Y);
 			Z3.mul(ZZ);
 			Z3.pmul(Qy);

 			X.mul(T);
 			X.sub(Y);
 			a=new FP4(Z3,X); //a.set(Z3,X);
 			T.neg();
 			ZZ.mul(T);

 			ZZ.pmul(Qx);

 			b=new FP4(ZZ); //b.seta(ZZ);
 		}
 		r.set(a,b,c);
 		return r;
 	},

 /* Optimal R-ate pairing */
 	ate: function(P,Q)
 	{
 		var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 		var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 		var f=new FP2(fa,fb); //f.bset(fa,fb);

 		var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);
 		var n=new BIG(x); //n.copy(x);
 		var K=new ECP2();
 		var lv;
 		n.pmul(6); n.dec(2); n.norm();
 		P.affine();
 		Q.affine();
 		var Qx=new FP(Q.getx()); //Qx.copy(Q.getx());
 		var Qy=new FP(Q.gety()); //Qy.copy(Q.gety());

 		var A=new ECP2();
 		var r=new FP12(1);

 		A.copy(P);
 		var nb=n.nbits();

 		for (var i=nb-2;i>=1;i--)
 		{
 			lv=PAIR.line(A,A,Qx,Qy);

 			r.smul(lv);

 			if (n.bit(i)==1)
 			{
 				lv=PAIR.line(A,P,Qx,Qy);
 				r.smul(lv);
 			}
 			r.sqr();
 		}
 		lv=PAIR.line(A,A,Qx,Qy);
 		r.smul(lv);

 /* R-ate fixup */
 		r.conj();
 		K.copy(P);
 		K.frob(f);
 		A.neg();
 		lv=PAIR.line(A,K,Qx,Qy);
 		r.smul(lv);
 		K.frob(f);
 		K.neg();
 		lv=PAIR.line(A,K,Qx,Qy);
 		r.smul(lv);

 		return r;
 	},

 /* Optimal R-ate double pairing e(P,Q).e(R,S) */
 	ate2: function(P,Q,R,S)
 	{
 		var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 		var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 		var f=new FP2(fa,fb); //f.bset(fa,fb);
 		var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);

 		var n=new BIG(x); //n.copy(x);
 		var K=new ECP2();
 		var lv;
 		n.pmul(6); n.dec(2); n.norm();
 		P.affine();
 		Q.affine();
 		R.affine();
 		S.affine();

 		var Qx=new FP(Q.getx()); //Qx.copy(Q.getx());
 		var Qy=new FP(Q.gety()); //Qy.copy(Q.gety());

 		var Sx=new FP(S.getx()); //Sx.copy(S.getx());
 		var Sy=new FP(S.gety()); //Sy.copy(S.gety());

 		var A=new ECP2();
 		var B=new ECP2();
 		var r=new FP12(1);

 		A.copy(P);
 		B.copy(R);
 		var nb=n.nbits();

 		for (var i=nb-2;i>=1;i--)
 		{
 			lv=PAIR.line(A,A,Qx,Qy);
 			r.smul(lv);
 			lv=PAIR.line(B,B,Sx,Sy);
 			r.smul(lv);
 			if (n.bit(i)==1)
 			{
 				lv=PAIR.line(A,P,Qx,Qy);
 				r.smul(lv);
 				lv=PAIR.line(B,R,Sx,Sy);
 				r.smul(lv);
 			}
 			r.sqr();
 		}

 		lv=PAIR.line(A,A,Qx,Qy);
 		r.smul(lv);

 		lv=PAIR.line(B,B,Sx,Sy);
 		r.smul(lv);


 /* R-ate fixup */
 		r.conj();

 		K.copy(P);
 		K.frob(f);
 		A.neg();
 		lv=PAIR.line(A,K,Qx,Qy);
 		r.smul(lv);
 		K.frob(f);
 		K.neg();
 		lv=PAIR.line(A,K,Qx,Qy);
 		r.smul(lv);

 		K.copy(R);
 		K.frob(f);
 		B.neg();
 		lv=PAIR.line(B,K,Sx,Sy);
 		r.smul(lv);
 		K.frob(f);
 		K.neg();
 		lv=PAIR.line(B,K,Sx,Sy);
 		r.smul(lv);

 		return r;
 	},

 /* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
 	fexp: function(m)
 	{
 		var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 		var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 		var f=new FP2(fa,fb);
 		var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);

 		var r=new FP12(m); //r.copy(m);
 		var x0,x1,x2,x3,x4,x5;

 /* Easy part of final exp */
 		var lv=new FP12(r); //lv.copy(r);
 		lv.inverse();
 		r.conj();
 		r.mul(lv);
 		lv.copy(r);
 		r.frob(f);
 		r.frob(f);
 		r.mul(lv);

 /* Hard part of final exp */
 		lv.copy(r);
 		lv.frob(f);
 		x0=new FP12(lv); //x0.copy(lv);
 		x0.frob(f);
 		lv.mul(r);
 		x0.mul(lv);
 		x0.frob(f);
 		x1=new FP12(r); //x1.copy(r);
 		x1.conj();

 		x4=r.pow(x);

 		x3=new FP12(x4); //x3.copy(x4);
 		x3.frob(f);
 		x2=x4.pow(x);

 		x5=new FP12(x2); /*x5.copy(x2);*/  x5.conj();
 		lv=x2.pow(x);

 		x2.frob(f);
 		r.copy(x2); r.conj();

 		x4.mul(r);
 		x2.frob(f);

 		r.copy(lv);
 		r.frob(f);
 		lv.mul(r);

 		lv.usqr();
 		lv.mul(x4);
 		lv.mul(x5);
 		r.copy(x3);
 		r.mul(x5);
 		r.mul(lv);
 		lv.mul(x2);
 		r.usqr();
 		r.mul(lv);
 		r.usqr();
 		lv.copy(r);
 		lv.mul(x1);
 		r.mul(x0);
 		lv.usqr();
 		r.mul(lv);
 		r.reduce();
 		return r;
 	}
 };

 /* GLV method */
 PAIR.glv= function(e)
 {
 	var i,j;
 	var t=new BIG(0);
 	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
 	var u=[];
 	var v=[];
 	for (i=0;i<2;i++)
 	{
 		t.rcopy(ROM.CURVE_W[i]);
 		var d=BIG.mul(t,e);
 		v[i]=new BIG(d.div(q));
 		u[i]=new BIG(0);
 	}
 	u[0].copy(e);
 	for (i=0;i<2;i++)
 		for (j=0;j<2;j++)
 		{
 			t.rcopy(ROM.CURVE_SB[j][i]);
 			t.copy(BIG.modmul(v[j],t,q));
 			u[i].add(q);
 			u[i].sub(t);
 			u[i].mod(q);
 		}
 	return u;
 };

 /* Galbraith & Scott Method */
 PAIR.gs= function(e)
 {
 	var i,j;
 	var t=new BIG(0);
 	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
 	var u=[];
 	var v=[];

 	for (i=0;i<4;i++)
 	{
 		t.rcopy(ROM.CURVE_WB[i]);
 		var d=BIG.mul(t,e);
 		v[i]=new BIG(d.div(q));
 		u[i]=new BIG(0);
 	}

 	u[0].copy(e);
 	for (i=0;i<4;i++)
 		for (j=0;j<4;j++)
 		{
 			t.rcopy(ROM.CURVE_BB[j][i]);
 			t.copy(BIG.modmul(v[j],t,q));
 			u[i].add(q);
 			u[i].sub(t);
 			u[i].mod(q);
 		}
 	return u;
 };

 /* Multiply P by e in group G1 */
 PAIR.G1mul= function(P,e)
 {
 	var R;
 	if (ROM.USE_GLV)
 	{
 		P.affine();
 		R=new ECP();
 		R.copy(P);
 		var np,nn;
 		var Q=new ECP();
 		Q.copy(P);
 		var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
 		var bcru=new BIG(0); bcru.rcopy(ROM.CURVE_Cru);
 		var cru=new FP(bcru);
 		var t=new BIG(0);
 		var u=PAIR.glv(e);
 		Q.getx().mul(cru);

 		np=u[0].nbits();
 		t.copy(BIG.modneg(u[0],q));
 		nn=t.nbits();
 		if (nn<np)
 		{
 			u[0].copy(t);
 			R.neg();
 		}

 		np=u[1].nbits();
 		t.copy(BIG.modneg(u[1],q));
 		nn=t.nbits();
 		if (nn<np)
 		{
 			u[1].copy(t);
 			Q.neg();
 		}

 		R=R.mul2(u[0],Q,u[1]);

 	}
 	else
 	{
 		R=P.mul(e);
 	}
 	return R;
 };

 /* Multiply P by e in group G2 */
 PAIR.G2mul= function(P,e)
 {
 	var R;
 	if (ROM.USE_GS_G2)
 	{
 		var Q=[];
 		var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 		var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 		var f=new FP2(fa,fb); //f.bset(fa,fb);
 		var q=new BIG(0); q.rcopy(ROM.CURVE_Order);

 		var u=PAIR.gs(e);
 		var t=new BIG(0);
 		var i,np,nn;
 		P.affine();
 		Q[0]=new ECP2(); Q[0].copy(P);
 		for (i=1;i<4;i++)
 		{
 			Q[i]=new ECP2(); Q[i].copy(Q[i-1]);
 			Q[i].frob(f);
 		}

 		for (i=0;i<4;i++)
 		{
 			np=u[i].nbits();
 			t.copy(BIG.modneg(u[i],q));
 			nn=t.nbits();
 			if (nn<np)
 			{
 				u[i].copy(t);
 				Q[i].neg();
 			}
 		}

 		R=ECP2.mul4(Q,u);
 	}
 	else
 	{
 		R=P.mul(e);
 	}
 	return R;
 };

 /* Note that this method requires a lot of RAM! Better to use compressed XTR method, see FP4.js */
 PAIR.GTpow= function(d,e)
 {
 	var r;
 	if (ROM.USE_GS_GT)
 	{
 		var g=[];
 		var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 		var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 		var f=new FP2(fa,fb);
 		var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
 		var t=new BIG(0);
 		var i,np,nn;
 		var u=PAIR.gs(e);

 		g[0]=new FP12(d);
 		for (i=1;i<4;i++)
 		{
 			g[i]=new FP12(0); g[i].copy(g[i-1]);
 			g[i].frob(f);
 		}
 		for (i=0;i<4;i++)
 		{
 			np=u[i].nbits();
 			t.copy(BIG.modneg(u[i],q));
 			nn=t.nbits();
 			if (nn<np)
 			{
 				u[i].copy(t);
 				g[i].conj();
 			}
 		}
 		r=FP12.pow4(g,u);
 	}
 	else
 	{
 		r=d.pow(e);
 	}
 	return r;
 };

 /* test group membership */
 /* with GT-Strong curve, now only check that m!=1, conj(m)*m==1, and m.m^{p^4}=m^{p^2} */
 PAIR.GTmember= function(m)
 {
 	if (m.isunity()) return false;
 	var r=new FP12(m);
 	r.conj();
 	r.mul(m);
 	if (!r.isunity()) return false;

 	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
 	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
 	var f=new FP2(fa,fb); //f.bset(fa,fb);

 	r.copy(m); r.frob(f); r.frob(f);
 	var w=new FP12(r); w.frob(f); w.frob(f);
 	w.mul(m);
 	if (!ROM.GT_STRONG)
 	{
 		if (!w.equals(r)) return false;
 		var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);
 		r.copy(m); w=r.pow(x); w=w.pow(x);
 		r.copy(w); r.sqr(); r.mul(w); r.sqr();
 		w.copy(m); w.frob(f);
 	}
 	return w.equals(r);
 };
	/*
	Licensed to the Apache Software Foundation (ASF) under one
	or more contributor license agreements. See the NOTICE file
	distributed with this work for additional information
	regarding copyright ownership. The ASF licenses this file
	to you under the Apache License, Version 2.0 (the
	"License"); you may not use this file except in compliance
	with the License. You may obtain a copy of the License at

	http://www.apache.org/licenses/LICENSE-2.0

	Unless required by applicable law or agreed to in writing,
	software distributed under the License is distributed on an
	"AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
	KIND, either express or implied. See the License for the
	specific language governing permissions and limitations
	under the License.
	*/

	var PAIR = {
	/* Line function */
	line: function(A,B,Qx,Qy)
	{
	var P=new ECP2();
	var a,b,c;
	var r=new FP12(1);
	P.copy(A);

	var ZZ=new FP2(P.getz()); //ZZ.copy(P.getz());
	ZZ.sqr();
	var D;
	if (A==B) D=A.dbl();
	else D=A.add(B);
	if (D<0) return r;
	var Z3=new FP2(A.getz()); //Z3.copy(A.getz());
	c=new FP4(0);
	var X,Y,T;
	if (D===0)
	{ /* Addition */
	X=new FP2(B.getx()); //X.copy(B.getx());
	Y=new FP2(B.gety()); //Y.copy(B.gety());
	T=new FP2(P.getz()); //T.copy(P.getz());

	T.mul(Y);
	ZZ.mul(T);

	var NY=new FP2(P.gety()); /NY.copy(P.gety());/ NY.neg();
	ZZ.add(NY);
	Z3.pmul(Qy);
	T.mul(P.getx());
	X.mul(NY);
	T.add(X);
	a=new FP4(Z3,T); //a.set(Z3,T);
	ZZ.neg();
	ZZ.pmul(Qx);
	b=new FP4(ZZ); //b.seta(ZZ);
	}
	else
	{ /* Doubling */
	X=new FP2(P.getx()); //X.copy(P.getx());
	Y=new FP2(P.gety()); //Y.copy(P.gety());
	T=new FP2(P.getx()); //T.copy(P.getx());
	T.sqr();
	T.imul(3);

	Y.sqr();
	Y.add(Y);
	Z3.mul(ZZ);
	Z3.pmul(Qy);

	X.mul(T);
	X.sub(Y);
	a=new FP4(Z3,X); //a.set(Z3,X);
	T.neg();
	ZZ.mul(T);

	ZZ.pmul(Qx);

	b=new FP4(ZZ); //b.seta(ZZ);
	}
	r.set(a,b,c);
	return r;
	},

	/* Optimal R-ate pairing */
	ate: function(P,Q)
	{
	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb); //f.bset(fa,fb);

	var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);
	var n=new BIG(x); //n.copy(x);
	var K=new ECP2();
	var lv;
	n.pmul(6); n.dec(2); n.norm();
	P.affine();
	Q.affine();
	var Qx=new FP(Q.getx()); //Qx.copy(Q.getx());
	var Qy=new FP(Q.gety()); //Qy.copy(Q.gety());

	var A=new ECP2();
	var r=new FP12(1);

	A.copy(P);
	var nb=n.nbits();

	for (var i=nb-2;i>=1;i--)
	{
	lv=PAIR.line(A,A,Qx,Qy);

	r.smul(lv);

	if (n.bit(i)==1)
	{
	lv=PAIR.line(A,P,Qx,Qy);
	r.smul(lv);
	}
	r.sqr();
	}
	lv=PAIR.line(A,A,Qx,Qy);
	r.smul(lv);

	/* R-ate fixup */
	r.conj();
	K.copy(P);
	K.frob(f);
	A.neg();
	lv=PAIR.line(A,K,Qx,Qy);
	r.smul(lv);
	K.frob(f);
	K.neg();
	lv=PAIR.line(A,K,Qx,Qy);
	r.smul(lv);

	return r;
	},

	/* Optimal R-ate double pairing e(P,Q).e(R,S) */
	ate2: function(P,Q,R,S)
	{
	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb); //f.bset(fa,fb);
	var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);

	var n=new BIG(x); //n.copy(x);
	var K=new ECP2();
	var lv;
	n.pmul(6); n.dec(2); n.norm();
	P.affine();
	Q.affine();
	R.affine();
	S.affine();

	var Qx=new FP(Q.getx()); //Qx.copy(Q.getx());
	var Qy=new FP(Q.gety()); //Qy.copy(Q.gety());

	var Sx=new FP(S.getx()); //Sx.copy(S.getx());
	var Sy=new FP(S.gety()); //Sy.copy(S.gety());

	var A=new ECP2();
	var B=new ECP2();
	var r=new FP12(1);

	A.copy(P);
	B.copy(R);
	var nb=n.nbits();

	for (var i=nb-2;i>=1;i--)
	{
	lv=PAIR.line(A,A,Qx,Qy);
	r.smul(lv);
	lv=PAIR.line(B,B,Sx,Sy);
	r.smul(lv);
	if (n.bit(i)==1)
	{
	lv=PAIR.line(A,P,Qx,Qy);
	r.smul(lv);
	lv=PAIR.line(B,R,Sx,Sy);
	r.smul(lv);
	}
	r.sqr();
	}

	lv=PAIR.line(A,A,Qx,Qy);
	r.smul(lv);

	lv=PAIR.line(B,B,Sx,Sy);
	r.smul(lv);


	/* R-ate fixup */
	r.conj();

	K.copy(P);
	K.frob(f);
	A.neg();
	lv=PAIR.line(A,K,Qx,Qy);
	r.smul(lv);
	K.frob(f);
	K.neg();
	lv=PAIR.line(A,K,Qx,Qy);
	r.smul(lv);

	K.copy(R);
	K.frob(f);
	B.neg();
	lv=PAIR.line(B,K,Sx,Sy);
	r.smul(lv);
	K.frob(f);
	K.neg();
	lv=PAIR.line(B,K,Sx,Sy);
	r.smul(lv);

	return r;
	},

	/* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
	fexp: function(m)
	{
	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb);
	var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);

	var r=new FP12(m); //r.copy(m);
	var x0,x1,x2,x3,x4,x5;

	/* Easy part of final exp */
	var lv=new FP12(r); //lv.copy(r);
	lv.inverse();
	r.conj();
	r.mul(lv);
	lv.copy(r);
	r.frob(f);
	r.frob(f);
	r.mul(lv);

	/* Hard part of final exp */
	lv.copy(r);
	lv.frob(f);
	x0=new FP12(lv); //x0.copy(lv);
	x0.frob(f);
	lv.mul(r);
	x0.mul(lv);
	x0.frob(f);
	x1=new FP12(r); //x1.copy(r);
	x1.conj();

	x4=r.pow(x);

	x3=new FP12(x4); //x3.copy(x4);
	x3.frob(f);
	x2=x4.pow(x);

	x5=new FP12(x2); /x5.copy(x2);/ x5.conj();
	lv=x2.pow(x);

	x2.frob(f);
	r.copy(x2); r.conj();

	x4.mul(r);
	x2.frob(f);

	r.copy(lv);
	r.frob(f);
	lv.mul(r);

	lv.usqr();
	lv.mul(x4);
	lv.mul(x5);
	r.copy(x3);
	r.mul(x5);
	r.mul(lv);
	lv.mul(x2);
	r.usqr();
	r.mul(lv);
	r.usqr();
	lv.copy(r);
	lv.mul(x1);
	r.mul(x0);
	lv.usqr();
	r.mul(lv);
	r.reduce();
	return r;
	}
	};

	/* GLV method */
	PAIR.glv= function(e)
	{
	var i,j;
	var t=new BIG(0);
	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
	var u=[];
	var v=[];
	for (i=0;i<2;i++)
	{
	t.rcopy(ROM.CURVE_W[i]);
	var d=BIG.mul(t,e);
	v[i]=new BIG(d.div(q));
	u[i]=new BIG(0);
	}
	u[0].copy(e);
	for (i=0;i<2;i++)
	for (j=0;j<2;j++)
	{
	t.rcopy(ROM.CURVE_SB[j][i]);
	t.copy(BIG.modmul(v[j],t,q));
	u[i].add(q);
	u[i].sub(t);
	u[i].mod(q);
	}
	return u;
	};

	/* Galbraith & Scott Method */
	PAIR.gs= function(e)
	{
	var i,j;
	var t=new BIG(0);
	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
	var u=[];
	var v=[];

	for (i=0;i<4;i++)
	{
	t.rcopy(ROM.CURVE_WB[i]);
	var d=BIG.mul(t,e);
	v[i]=new BIG(d.div(q));
	u[i]=new BIG(0);
	}

	u[0].copy(e);
	for (i=0;i<4;i++)
	for (j=0;j<4;j++)
	{
	t.rcopy(ROM.CURVE_BB[j][i]);
	t.copy(BIG.modmul(v[j],t,q));
	u[i].add(q);
	u[i].sub(t);
	u[i].mod(q);
	}
	return u;
	};

	/* Multiply P by e in group G1 */
	PAIR.G1mul= function(P,e)
	{
	var R;
	if (ROM.USE_GLV)
	{
	P.affine();
	R=new ECP();
	R.copy(P);
	var np,nn;
	var Q=new ECP();
	Q.copy(P);
	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
	var bcru=new BIG(0); bcru.rcopy(ROM.CURVE_Cru);
	var cru=new FP(bcru);
	var t=new BIG(0);
	var u=PAIR.glv(e);
	Q.getx().mul(cru);

	np=u[0].nbits();
	t.copy(BIG.modneg(u[0],q));
	nn=t.nbits();
	if (nn<np)
	{
	u[0].copy(t);
	R.neg();
	}

	np=u[1].nbits();
	t.copy(BIG.modneg(u[1],q));
	nn=t.nbits();
	if (nn<np)
	{
	u[1].copy(t);
	Q.neg();
	}

	R=R.mul2(u[0],Q,u[1]);

	}
	else
	{
	R=P.mul(e);
	}
	return R;
	};

	/* Multiply P by e in group G2 */
	PAIR.G2mul= function(P,e)
	{
	var R;
	if (ROM.USE_GS_G2)
	{
	var Q=[];
	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb); //f.bset(fa,fb);
	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);

	var u=PAIR.gs(e);
	var t=new BIG(0);
	var i,np,nn;
	P.affine();
	Q[0]=new ECP2(); Q[0].copy(P);
	for (i=1;i<4;i++)
	{
	Q[i]=new ECP2(); Q[i].copy(Q[i-1]);
	Q[i].frob(f);
	}

	for (i=0;i<4;i++)
	{
	np=u[i].nbits();
	t.copy(BIG.modneg(u[i],q));
	nn=t.nbits();
	if (nn<np)
	{
	u[i].copy(t);
	Q[i].neg();
	}
	}

	R=ECP2.mul4(Q,u);
	}
	else
	{
	R=P.mul(e);
	}
	return R;
	};

	/* Note that this method requires a lot of RAM! Better to use compressed XTR method, see FP4.js */
	PAIR.GTpow= function(d,e)
	{
	var r;
	if (ROM.USE_GS_GT)
	{
	var g=[];
	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb);
	var q=new BIG(0); q.rcopy(ROM.CURVE_Order);
	var t=new BIG(0);
	var i,np,nn;
	var u=PAIR.gs(e);

	g[0]=new FP12(d);
	for (i=1;i<4;i++)
	{
	g[i]=new FP12(0); g[i].copy(g[i-1]);
	g[i].frob(f);
	}
	for (i=0;i<4;i++)
	{
	np=u[i].nbits();
	t.copy(BIG.modneg(u[i],q));
	nn=t.nbits();
	if (nn<np)
	{
	u[i].copy(t);
	g[i].conj();
	}
	}
	r=FP12.pow4(g,u);
	}
	else
	{
	r=d.pow(e);
	}
	return r;
	};

	/* test group membership */
	/* with GT-Strong curve, now only check that m!=1, conj(m)m==1, and m.m^{p^4}=m^{p^2} /
	PAIR.GTmember= function(m)
	{
	if (m.isunity()) return false;
	var r=new FP12(m);
	r.conj();
	r.mul(m);
	if (!r.isunity()) return false;

	var fa=new BIG(0); fa.rcopy(ROM.CURVE_Fra);
	var fb=new BIG(0); fb.rcopy(ROM.CURVE_Frb);
	var f=new FP2(fa,fb); //f.bset(fa,fb);

	r.copy(m); r.frob(f); r.frob(f);
	var w=new FP12(r); w.frob(f); w.frob(f);
	w.mul(m);
	if (!ROM.GT_STRONG)
	{
	if (!w.equals(r)) return false;
	var x=new BIG(0); x.rcopy(ROM.CURVE_Bnx);
	r.copy(m); w=r.pow(x); w=w.pow(x);
	r.copy(w); r.sqr(); r.mul(w); r.sqr();
	w.copy(m); w.frob(f);
	}
	return w.equals(r);
	};