go/amcl-go/PAIR.go - 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.
 */

 /* AMCL BN Curve Pairing functions */

 package amcl

 //import "fmt"

 /* Line function */
 func line(A *ECP2, B *ECP2, Qx *FP, Qy *FP) *FP12 {
 	P := NewECP2()

 	P.copy(A)
 	ZZ := NewFP2copy(P.getz())
 	ZZ.sqr()
 	var D int
 	if A == B {
 		D = A.dbl()
 	} else {
 		D = A.add(B)
 	}

 	if D < 0 {
 		return NewFP12int(1)
 	}

 	Z3 := NewFP2copy(A.getz())

 	var a *FP4
 	var b *FP4
 	c := NewFP4int(0)

 	if D == 0 { /* Addition */
 		X := NewFP2copy(B.getx())
 		Y := NewFP2copy(B.gety())
 		T := NewFP2copy(P.getz())
 		T.mul(Y)
 		ZZ.mul(T)

 		NY := NewFP2copy(P.gety())
 		NY.neg()
 		ZZ.add(NY)
 		Z3.pmul(Qy)
 		T.mul(P.getx())
 		X.mul(NY)
 		T.add(X)
 		a = NewFP4fp2s(Z3, T)
 		ZZ.neg()
 		ZZ.pmul(Qx)
 		b = NewFP4fp2(ZZ)
 	} else { /* Doubling */
 		X := NewFP2copy(P.getx())
 		Y := NewFP2copy(P.gety())
 		T := NewFP2copy(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 = NewFP4fp2s(Z3, X)
 		T.neg()
 		ZZ.mul(T)
 		ZZ.pmul(Qx)
 		b = NewFP4fp2(ZZ)
 	}
 	return NewFP12fp4s(a, b, c)
 }

 /* Optimal R-ate pairing */
 func ate(P *ECP2, Q *ECP) *FP12 {
 	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
 	x := NewBIGints(CURVE_Bnx)
 	n := NewBIGcopy(x)
 	K := NewECP2()
 	var lv *FP12
 	n.pmul(6)
 	n.dec(2)
 	n.norm()
 	P.affine()
 	Q.affine()
 	Qx := NewFPcopy(Q.getx())
 	Qy := NewFPcopy(Q.gety())

 	A := NewECP2()
 	r := NewFP12int(1)

 	A.copy(P)
 	nb := n.nbits()

 	for i := nb - 2; i >= 1; i-- {
 		lv = line(A, A, Qx, Qy)
 		r.smul(lv)

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

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

 	/* R-ate fixup */

 	r.conj()

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

 	return r
 }

 /* Optimal R-ate double pairing e(P,Q).e(R,S) */
 func ate2(P *ECP2, Q *ECP, R *ECP2, S *ECP) *FP12 {
 	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
 	x := NewBIGints(CURVE_Bnx)
 	n := NewBIGcopy(x)
 	K := NewECP2()
 	var lv *FP12
 	n.pmul(6)
 	n.dec(2)
 	n.norm()
 	P.affine()
 	Q.affine()
 	R.affine()
 	S.affine()

 	Qx := NewFPcopy(Q.getx())
 	Qy := NewFPcopy(Q.gety())
 	Sx := NewFPcopy(S.getx())
 	Sy := NewFPcopy(S.gety())

 	A := NewECP2()
 	B := NewECP2()
 	r := NewFP12int(1)

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

 	for i := nb - 2; i >= 1; i-- {
 		lv = line(A, A, Qx, Qy)
 		r.smul(lv)
 		lv = line(B, B, Sx, Sy)
 		r.smul(lv)

 		if n.bit(i) == 1 {
 			lv = line(A, P, Qx, Qy)
 			r.smul(lv)
 			lv = line(B, R, Sx, Sy)
 			r.smul(lv)
 		}
 		r.sqr()
 	}

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

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

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

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

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

 	return r
 }

 /* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
 func fexp(m *FP12) *FP12 {
 	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
 	x := NewBIGints(CURVE_Bnx)
 	r := NewFP12copy(m)

 	/* Easy part of final exp */
 	lv := NewFP12copy(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 := NewFP12copy(lv)
 	x0.frob(f)
 	lv.mul(r)
 	x0.mul(lv)
 	x0.frob(f)
 	x1 := NewFP12copy(r)
 	x1.conj()
 	x4 := r.pow(x)

 	x3 := NewFP12copy(x4)
 	x3.frob(f)

 	x2 := x4.pow(x)

 	x5 := NewFP12copy(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 */
 func glv(e *BIG) []*BIG {
 	t := NewBIGint(0)
 	q := NewBIGints(CURVE_Order)
 	var u []*BIG
 	var v []*BIG

 	for i := 0; i < 2; i++ {
 		t.copy(NewBIGints(CURVE_W[i])) // why not just t=new BIG(ROM.CURVE_W[i]);
 		d := mul(t, e)
 		v = append(v, NewBIGcopy(d.div(q)))
 		u = append(u, NewBIGint(0))
 	}
 	u[0].copy(e)
 	for i := 0; i < 2; i++ {
 		for j := 0; j < 2; j++ {
 			t.copy(NewBIGints(CURVE_SB[j][i]))
 			t.copy(modmul(v[j], t, q))
 			u[i].add(q)
 			u[i].sub(t)
 			u[i].mod(q)
 		}
 	}
 	return u
 }

 /* Galbraith & Scott Method */
 func gs(e *BIG) []*BIG {
 	t := NewBIGint(0)
 	q := NewBIGints(CURVE_Order)
 	var u []*BIG
 	var v []*BIG
 	for i := 0; i < 4; i++ {
 		t.copy(NewBIGints(CURVE_WB[i]))
 		d := mul(t, e)
 		v = append(v, NewBIGcopy(d.div(q)))
 		u = append(u, NewBIGint(0))
 	}
 	u[0].copy(e)
 	for i := 0; i < 4; i++ {
 		for j := 0; j < 4; j++ {
 			t.copy(NewBIGints(CURVE_BB[j][i]))
 			t.copy(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 */
 func G1mul(P *ECP, e *BIG) *ECP {
 	var R *ECP
 	if USE_GLV {
 		P.affine()
 		R = NewECP()
 		R.copy(P)
 		Q := NewECP()
 		Q.copy(P)
 		q := NewBIGints(CURVE_Order)
 		cru := NewFPbig(NewBIGints(CURVE_Cru))
 		t := NewBIGint(0)
 		u := glv(e)
 		Q.getx().mul(cru)

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

 		np = u[1].nbits()
 		t.copy(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 */
 func G2mul(P *ECP2, e *BIG) *ECP2 {
 	var R *ECP2
 	if USE_GS_G2 {
 		var Q []*ECP2
 		f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
 		q := NewBIGints(CURVE_Order)
 		u := gs(e)

 		t := NewBIGint(0)
 		P.affine()
 		Q = append(Q, NewECP2())
 		Q[0].copy(P)
 		for i := 1; i < 4; i++ {
 			Q = append(Q, NewECP2())
 			Q[i].copy(Q[i-1])
 			Q[i].frob(f)
 		}
 		for i := 0; i < 4; i++ {
 			np := u[i].nbits()
 			t.copy(modneg(u[i], q))
 			nn := t.nbits()
 			if nn < np {
 				u[i].copy(t)
 				Q[i].neg()
 			}
 		}
 		R = mul4(Q, u)

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

 /* f=f^e */
 /* Note that this method requires a lot of RAM! Better to use compressed XTR method, see FP4.java */
 func GTpow(d *FP12, e *BIG) *FP12 {
 	var r *FP12
 	if USE_GS_GT {
 		var g []*FP12
 		f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
 		q := NewBIGints(CURVE_Order)
 		t := NewBIGint(0)

 		u := gs(e)

 		g = append(g, NewFP12copy(d))
 		for i := 1; i < 4; i++ {
 			g = append(g, NewFP12int(0))
 			g[i].copy(g[i-1])
 			g[i].frob(f)
 		}
 		for i := 0; i < 4; i++ {
 			np := u[i].nbits()
 			t.copy(modneg(u[i], q))
 			nn := t.nbits()
 			if nn < np {
 				u[i].copy(t)
 				g[i].conj()
 			}
 		}
 		r = 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} */
 func GTmember(m *FP12) bool {
 	if m.isunity() {
 		return false
 	}
 	r := NewFP12copy(m)
 	r.conj()
 	r.mul(m)
 	if !r.isunity() {
 		return false
 	}

 	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))

 	r.copy(m)
 	r.frob(f)
 	r.frob(f)
 	w := NewFP12copy(r)
 	w.frob(f)
 	w.frob(f)
 	w.mul(m)
 	if !GT_STRONG {
 		if !w.equals(r) {
 			return false
 		}
 		x := NewBIGints(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)
 }

 /*
 func main() {

 	Q:=NewECPbigs(NewBIGints(CURVE_Gx),NewBIGints(CURVE_Gy))
 	P:=NewECP2fp2s(NewFP2bigs(NewBIGints(CURVE_Pxa),NewBIGints(CURVE_Pxb)),NewFP2bigs(NewBIGints(CURVE_Pya),NewBIGints(CURVE_Pyb)))

 	r:=NewBIGints(CURVE_Order)
 	xa:=NewBIGints(CURVE_Pxa)

 	fmt.Printf("P= "+P.toString())
 	fmt.Printf("\n");
 	fmt.Printf("Q= "+Q.toString());
 	fmt.Printf("\n");

 	//m:=NewBIGint(17)

 	e:=ate(P,Q)
 	fmt.Printf("\ne= "+e.toString())
 	fmt.Printf("\n")

 	e=fexp(e)
 	//	e=GTpow(e,m);

 	fmt.Printf("\ne= "+e.toString())
 	fmt.Printf("\n");
 	GLV:=glv(r)

 	fmt.Printf("GLV[0]= "+GLV[0].toString())
 	fmt.Printf("\n")

 	fmt.Printf("GLV[0]= "+GLV[1].toString())
 	fmt.Printf("\n")

 	G:=NewECP(); G.copy(Q)
 	R:=NewECP2(); R.copy(P)


 	e=ate(R,Q)
 	e=fexp(e)

 	e=GTpow(e,xa)
 	fmt.Printf("\ne= "+e.toString());
 	fmt.Printf("\n")

 	R=G2mul(R,xa)
 	e=ate(R,G)
 	e=fexp(e)

 	fmt.Printf("\ne= "+e.toString())
 	fmt.Printf("\n")

 	G=G1mul(G,xa)
 	e=ate(P,G)
 	e=fexp(e)
 	fmt.Printf("\ne= "+e.toString())
 	fmt.Printf("\n")
 }
 */
	/*
	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.
	*/

	/* AMCL BN Curve Pairing functions */

	package amcl

	//import "fmt"

	/* Line function */
	func line(A ECP2, B ECP2, Qx FP, Qy FP) *FP12 {
	P := NewECP2()

	P.copy(A)
	ZZ := NewFP2copy(P.getz())
	ZZ.sqr()
	var D int
	if A == B {
	D = A.dbl()
	} else {
	D = A.add(B)
	}

	if D < 0 {
	return NewFP12int(1)
	}

	Z3 := NewFP2copy(A.getz())

	var a *FP4
	var b *FP4
	c := NewFP4int(0)

	if D == 0 { /* Addition */
	X := NewFP2copy(B.getx())
	Y := NewFP2copy(B.gety())
	T := NewFP2copy(P.getz())
	T.mul(Y)
	ZZ.mul(T)

	NY := NewFP2copy(P.gety())
	NY.neg()
	ZZ.add(NY)
	Z3.pmul(Qy)
	T.mul(P.getx())
	X.mul(NY)
	T.add(X)
	a = NewFP4fp2s(Z3, T)
	ZZ.neg()
	ZZ.pmul(Qx)
	b = NewFP4fp2(ZZ)
	} else { /* Doubling */
	X := NewFP2copy(P.getx())
	Y := NewFP2copy(P.gety())
	T := NewFP2copy(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 = NewFP4fp2s(Z3, X)
	T.neg()
	ZZ.mul(T)
	ZZ.pmul(Qx)
	b = NewFP4fp2(ZZ)
	}
	return NewFP12fp4s(a, b, c)
	}

	/* Optimal R-ate pairing */
	func ate(P ECP2, Q ECP) *FP12 {
	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
	x := NewBIGints(CURVE_Bnx)
	n := NewBIGcopy(x)
	K := NewECP2()
	var lv *FP12
	n.pmul(6)
	n.dec(2)
	n.norm()
	P.affine()
	Q.affine()
	Qx := NewFPcopy(Q.getx())
	Qy := NewFPcopy(Q.gety())

	A := NewECP2()
	r := NewFP12int(1)

	A.copy(P)
	nb := n.nbits()

	for i := nb - 2; i >= 1; i-- {
	lv = line(A, A, Qx, Qy)
	r.smul(lv)

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

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

	/* R-ate fixup */

	r.conj()

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

	return r
	}

	/* Optimal R-ate double pairing e(P,Q).e(R,S) */
	func ate2(P ECP2, Q ECP, R ECP2, S ECP) *FP12 {
	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
	x := NewBIGints(CURVE_Bnx)
	n := NewBIGcopy(x)
	K := NewECP2()
	var lv *FP12
	n.pmul(6)
	n.dec(2)
	n.norm()
	P.affine()
	Q.affine()
	R.affine()
	S.affine()

	Qx := NewFPcopy(Q.getx())
	Qy := NewFPcopy(Q.gety())
	Sx := NewFPcopy(S.getx())
	Sy := NewFPcopy(S.gety())

	A := NewECP2()
	B := NewECP2()
	r := NewFP12int(1)

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

	for i := nb - 2; i >= 1; i-- {
	lv = line(A, A, Qx, Qy)
	r.smul(lv)
	lv = line(B, B, Sx, Sy)
	r.smul(lv)

	if n.bit(i) == 1 {
	lv = line(A, P, Qx, Qy)
	r.smul(lv)
	lv = line(B, R, Sx, Sy)
	r.smul(lv)
	}
	r.sqr()
	}

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

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

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

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

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

	return r
	}

	/* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
	func fexp(m FP12) FP12 {
	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
	x := NewBIGints(CURVE_Bnx)
	r := NewFP12copy(m)

	/* Easy part of final exp */
	lv := NewFP12copy(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 := NewFP12copy(lv)
	x0.frob(f)
	lv.mul(r)
	x0.mul(lv)
	x0.frob(f)
	x1 := NewFP12copy(r)
	x1.conj()
	x4 := r.pow(x)

	x3 := NewFP12copy(x4)
	x3.frob(f)

	x2 := x4.pow(x)

	x5 := NewFP12copy(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 */
	func glv(e BIG) []BIG {
	t := NewBIGint(0)
	q := NewBIGints(CURVE_Order)
	var u []*BIG
	var v []*BIG

	for i := 0; i < 2; i++ {
	t.copy(NewBIGints(CURVE_W[i])) // why not just t=new BIG(ROM.CURVE_W[i]);
	d := mul(t, e)
	v = append(v, NewBIGcopy(d.div(q)))
	u = append(u, NewBIGint(0))
	}
	u[0].copy(e)
	for i := 0; i < 2; i++ {
	for j := 0; j < 2; j++ {
	t.copy(NewBIGints(CURVE_SB[j][i]))
	t.copy(modmul(v[j], t, q))
	u[i].add(q)
	u[i].sub(t)
	u[i].mod(q)
	}
	}
	return u
	}

	/* Galbraith & Scott Method */
	func gs(e BIG) []BIG {
	t := NewBIGint(0)
	q := NewBIGints(CURVE_Order)
	var u []*BIG
	var v []*BIG
	for i := 0; i < 4; i++ {
	t.copy(NewBIGints(CURVE_WB[i]))
	d := mul(t, e)
	v = append(v, NewBIGcopy(d.div(q)))
	u = append(u, NewBIGint(0))
	}
	u[0].copy(e)
	for i := 0; i < 4; i++ {
	for j := 0; j < 4; j++ {
	t.copy(NewBIGints(CURVE_BB[j][i]))
	t.copy(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 */
	func G1mul(P ECP, e BIG) *ECP {
	var R *ECP
	if USE_GLV {
	P.affine()
	R = NewECP()
	R.copy(P)
	Q := NewECP()
	Q.copy(P)
	q := NewBIGints(CURVE_Order)
	cru := NewFPbig(NewBIGints(CURVE_Cru))
	t := NewBIGint(0)
	u := glv(e)
	Q.getx().mul(cru)

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

	np = u[1].nbits()
	t.copy(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 */
	func G2mul(P ECP2, e BIG) *ECP2 {
	var R *ECP2
	if USE_GS_G2 {
	var Q []*ECP2
	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
	q := NewBIGints(CURVE_Order)
	u := gs(e)

	t := NewBIGint(0)
	P.affine()
	Q = append(Q, NewECP2())
	Q[0].copy(P)
	for i := 1; i < 4; i++ {
	Q = append(Q, NewECP2())
	Q[i].copy(Q[i-1])
	Q[i].frob(f)
	}
	for i := 0; i < 4; i++ {
	np := u[i].nbits()
	t.copy(modneg(u[i], q))
	nn := t.nbits()
	if nn < np {
	u[i].copy(t)
	Q[i].neg()
	}
	}
	R = mul4(Q, u)

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

	/* f=f^e */
	/* Note that this method requires a lot of RAM! Better to use compressed XTR method, see FP4.java */
	func GTpow(d FP12, e BIG) *FP12 {
	var r *FP12
	if USE_GS_GT {
	var g []*FP12
	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))
	q := NewBIGints(CURVE_Order)
	t := NewBIGint(0)

	u := gs(e)

	g = append(g, NewFP12copy(d))
	for i := 1; i < 4; i++ {
	g = append(g, NewFP12int(0))
	g[i].copy(g[i-1])
	g[i].frob(f)
	}
	for i := 0; i < 4; i++ {
	np := u[i].nbits()
	t.copy(modneg(u[i], q))
	nn := t.nbits()
	if nn < np {
	u[i].copy(t)
	g[i].conj()
	}
	}
	r = 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} /
	func GTmember(m *FP12) bool {
	if m.isunity() {
	return false
	}
	r := NewFP12copy(m)
	r.conj()
	r.mul(m)
	if !r.isunity() {
	return false
	}

	f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb))

	r.copy(m)
	r.frob(f)
	r.frob(f)
	w := NewFP12copy(r)
	w.frob(f)
	w.frob(f)
	w.mul(m)
	if !GT_STRONG {
	if !w.equals(r) {
	return false
	}
	x := NewBIGints(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)
	}

	/*
	func main() {

	Q:=NewECPbigs(NewBIGints(CURVE_Gx),NewBIGints(CURVE_Gy))
	P:=NewECP2fp2s(NewFP2bigs(NewBIGints(CURVE_Pxa),NewBIGints(CURVE_Pxb)),NewFP2bigs(NewBIGints(CURVE_Pya),NewBIGints(CURVE_Pyb)))

	r:=NewBIGints(CURVE_Order)
	xa:=NewBIGints(CURVE_Pxa)

	fmt.Printf("P= "+P.toString())
	fmt.Printf("\n");
	fmt.Printf("Q= "+Q.toString());
	fmt.Printf("\n");

	//m:=NewBIGint(17)

	e:=ate(P,Q)
	fmt.Printf("\ne= "+e.toString())
	fmt.Printf("\n")

	e=fexp(e)
	// e=GTpow(e,m);

	fmt.Printf("\ne= "+e.toString())
	fmt.Printf("\n");
	GLV:=glv(r)

	fmt.Printf("GLV[0]= "+GLV[0].toString())
	fmt.Printf("\n")

	fmt.Printf("GLV[0]= "+GLV[1].toString())
	fmt.Printf("\n")

	G:=NewECP(); G.copy(Q)
	R:=NewECP2(); R.copy(P)


	e=ate(R,Q)
	e=fexp(e)

	e=GTpow(e,xa)
	fmt.Printf("\ne= "+e.toString());
	fmt.Printf("\n")

	R=G2mul(R,xa)
	e=ate(R,G)
	e=fexp(e)

	fmt.Printf("\ne= "+e.toString())
	fmt.Printf("\n")

	G=G1mul(G,xa)
	e=ate(P,G)
	e=fexp(e)
	fmt.Printf("\ne= "+e.toString())
	fmt.Printf("\n")
	}
	*/