version3/go/PAIR256.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.
 */

 /* MiotCL BN Curve Pairing functions */

 package XXX

 //import "fmt"

 /* Line function */
 func line(A *ECP8, B *ECP8, Qx *FP, Qy *FP) *FP48 {
 	var a *FP16
 	var b *FP16
 	var c *FP16

 	if A == B { /* Doubling */
 		XX := NewFP8copy(A.getx()) //X
 		YY := NewFP8copy(A.gety()) //Y
 		ZZ := NewFP8copy(A.getz()) //Z
 		YZ := NewFP8copy(YY)       //Y
 		YZ.mul(ZZ)                 //YZ
 		XX.sqr()                   //X^2
 		YY.sqr()                   //Y^2
 		ZZ.sqr()                   //Z^2

 		YZ.imul(4)
 		YZ.neg()
 		YZ.norm()   //-2YZ
 		YZ.tmul(Qy) //-2YZ.Ys

 		XX.imul(6)  //3X^2
 		XX.tmul(Qx) //3X^2.Xs

 		sb := 3 * CURVE_B_I
 		ZZ.imul(sb)
 		if SEXTIC_TWIST == D_TYPE {
 			ZZ.div_2i()
 		}
 		if SEXTIC_TWIST == M_TYPE {
 			ZZ.times_i()
 			ZZ.add(ZZ)
 			YZ.times_i()
 			YZ.norm()
 		}
 		ZZ.norm() // 3b.Z^2

 		YY.add(YY)
 		ZZ.sub(YY)
 		ZZ.norm() // 3b.Z^2-Y^2

 		a = NewFP16fp8s(YZ, ZZ) // -2YZ.Ys | 3b.Z^2-Y^2 | 3X^2.Xs
 		if SEXTIC_TWIST == D_TYPE {

 			b = NewFP16fp8(XX) // L(0,1) | L(0,0) | L(1,0)
 			c = NewFP16int(0)
 		}
 		if SEXTIC_TWIST == M_TYPE {
 			b = NewFP16int(0)
 			c = NewFP16fp8(XX)
 			c.times_i()
 		}
 		A.dbl()

 	} else { /* Addition */

 		X1 := NewFP8copy(A.getx()) // X1
 		Y1 := NewFP8copy(A.gety()) // Y1
 		T1 := NewFP8copy(A.getz()) // Z1
 		T2 := NewFP8copy(A.getz()) // Z1

 		T1.mul(B.gety()) // T1=Z1.Y2
 		T2.mul(B.getx()) // T2=Z1.X2

 		X1.sub(T2)
 		X1.norm() // X1=X1-Z1.X2
 		Y1.sub(T1)
 		Y1.norm() // Y1=Y1-Z1.Y2

 		T1.copy(X1) // T1=X1-Z1.X2
 		X1.tmul(Qy) // X1=(X1-Z1.X2).Ys

 		if SEXTIC_TWIST == M_TYPE {
 			X1.times_i()
 			X1.norm()
 		}

 		T1.mul(B.gety()) // T1=(X1-Z1.X2).Y2

 		T2.copy(Y1)      // T2=Y1-Z1.Y2
 		T2.mul(B.getx()) // T2=(Y1-Z1.Y2).X2
 		T2.sub(T1)
 		T2.norm() // T2=(Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2
 		Y1.tmul(Qx)
 		Y1.neg()
 		Y1.norm() // Y1=-(Y1-Z1.Y2).Xs

 		a = NewFP16fp8s(X1, T2) // (X1-Z1.X2).Ys  |  (Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2  | - (Y1-Z1.Y2).Xs
 		if SEXTIC_TWIST == D_TYPE {
 			b = NewFP16fp8(Y1)
 			c = NewFP16int(0)
 		}
 		if SEXTIC_TWIST == M_TYPE {
 			b = NewFP16int(0)
 			c = NewFP16fp8(Y1)
 			c.times_i()
 		}
 		A.Add(B)
 	}

 	return NewFP48fp16s(a, b, c)
 }

 /* Optimal R-ate pairing */
 func Ate(P1 *ECP8, Q1 *ECP) *FP48 {
 	x := NewBIGints(CURVE_Bnx)
 	n := NewBIGcopy(x)
 	var lv *FP48

 	n3 := NewBIGcopy(n)
 	n3.pmul(3)
 	n3.norm()

 	P := NewECP8()
 	P.Copy(P1)
 	P.Affine()
 	Q := NewECP()
 	Q.Copy(Q1)
 	Q.Affine()

 	Qx := NewFPcopy(Q.getx())
 	Qy := NewFPcopy(Q.gety())

 	A := NewECP8()
 	r := NewFP48int(1)

 	A.Copy(P)
 	NP := NewECP8()
 	NP.Copy(P)
 	NP.neg()

 	nb := n3.nbits()

 	for i := nb - 2; i >= 1; i-- {
 		r.sqr()
 		lv = line(A, A, Qx, Qy)
 		r.smul(lv, SEXTIC_TWIST)
 		bt := n3.bit(i) - n.bit(i)
 		if bt == 1 {
 			lv = line(A, P, Qx, Qy)
 			r.smul(lv, SEXTIC_TWIST)
 		}
 		if bt == -1 {
 			lv = line(A, NP, Qx, Qy)
 			r.smul(lv, SEXTIC_TWIST)
 		}
 	}

 	if SIGN_OF_X == NEGATIVEX {
 		r.conj()
 	}

 	return r
 }

 /* Optimal R-ate double pairing e(P,Q).e(R,S) */
 func Ate2(P1 *ECP8, Q1 *ECP, R1 *ECP8, S1 *ECP) *FP48 {
 	x := NewBIGints(CURVE_Bnx)
 	n := NewBIGcopy(x)
 	var lv *FP48

 	n3 := NewBIGcopy(n)
 	n3.pmul(3)
 	n3.norm()

 	P := NewECP8()
 	P.Copy(P1)
 	P.Affine()
 	Q := NewECP()
 	Q.Copy(Q1)
 	Q.Affine()
 	R := NewECP8()
 	R.Copy(R1)
 	R.Affine()
 	S := NewECP()
 	S.Copy(S1)
 	S.Affine()

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

 	A := NewECP8()
 	B := NewECP8()
 	r := NewFP48int(1)

 	A.Copy(P)
 	B.Copy(R)
 	NP := NewECP8()
 	NP.Copy(P)
 	NP.neg()
 	NR := NewECP8()
 	NR.Copy(R)
 	NR.neg()

 	nb := n3.nbits()

 	for i := nb - 2; i >= 1; i-- {
 		r.sqr()
 		lv = line(A, A, Qx, Qy)
 		r.smul(lv, SEXTIC_TWIST)
 		lv = line(B, B, Sx, Sy)
 		r.smul(lv, SEXTIC_TWIST)
 		bt := n3.bit(i) - n.bit(i)
 		if bt == 1 {
 			lv = line(A, P, Qx, Qy)
 			r.smul(lv, SEXTIC_TWIST)
 			lv = line(B, R, Sx, Sy)
 			r.smul(lv, SEXTIC_TWIST)
 		}
 		if bt == -1 {
 			lv = line(A, NP, Qx, Qy)
 			r.smul(lv, SEXTIC_TWIST)
 			lv = line(B, NR, Sx, Sy)
 			r.smul(lv, SEXTIC_TWIST)
 		}
 	}

 	if SIGN_OF_X == NEGATIVEX {
 		r.conj()
 	}

 	return r
 }

 /* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
 func Fexp(m *FP48) *FP48 {
 	f := NewFP2bigs(NewBIGints(Fra), NewBIGints(Frb))
 	x := NewBIGints(CURVE_Bnx)
 	r := NewFP48copy(m)

 	/* Easy part of final exp */
 	lv := NewFP48copy(r)

 	lv.Inverse()
 	r.conj()

 	r.Mul(lv)
 	lv.Copy(r)
 	r.frob(f, 8)
 	r.Mul(lv)

 	/* Hard part of final exp */
 	// Ghamman & Fouotsa Method

 	t7 := NewFP48copy(r)
 	t7.usqr()
 	t1 := t7.Pow(x)

 	x.fshr(1)
 	t2 := t1.Pow(x)
 	x.fshl(1)

 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3 := NewFP48copy(t1)
 	t3.conj()
 	t2.Mul(t3)
 	t2.Mul(r)

 	r.Mul(t7)

 	t1 = t2.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}
 	t3.Copy(t1)
 	t3.frob(f, 14)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 13)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 12)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 11)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 10)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 9)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 8)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t2)
 	t3.conj()
 	t1.Mul(t3)
 	t3.Copy(t1)
 	t3.frob(f, 7)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 6)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 5)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 4)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 3)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 2)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	t3.Copy(t1)
 	t3.frob(f, 1)
 	r.Mul(t3)
 	t1 = t1.Pow(x)
 	if SIGN_OF_X == NEGATIVEX {
 		t1.conj()
 	}

 	r.Mul(t1)
 	t2.frob(f, 15)
 	r.Mul(t2)

 	r.reduce()

 	return r
 }

 /* GLV method */
 func glv(e *BIG) []*BIG {
 	var u []*BIG

 	q := NewBIGints(CURVE_Order)
 	x := NewBIGints(CURVE_Bnx)
 	x2 := smul(x, x)
 	x = smul(x2, x2)
 	x2 = smul(x, x)
 	u = append(u, NewBIGcopy(e))
 	u[0].Mod(x2)
 	u = append(u, NewBIGcopy(e))
 	u[1].div(x2)
 	u[1].rsub(q)
 	return u
 }

 /* Galbraith & Scott Method */
 func gs(e *BIG) []*BIG {
 	var u []*BIG

 	q := NewBIGints(CURVE_Order)
 	x := NewBIGints(CURVE_Bnx)
 	w := NewBIGcopy(e)
 	for i := 0; i < 15; i++ {
 		u = append(u, NewBIGcopy(w))
 		u[i].Mod(x)
 		w.div(x)
 	}
 	u = append(u, NewBIGcopy(w))
 	if SIGN_OF_X == NEGATIVEX {
 		u[1].copy(Modneg(u[1], q))
 		u[3].copy(Modneg(u[3], q))
 		u[5].copy(Modneg(u[5], q))
 		u[7].copy(Modneg(u[7], q))
 		u[9].copy(Modneg(u[9], q))
 		u[11].copy(Modneg(u[11], q))
 		u[13].copy(Modneg(u[13], q))
 		u[15].copy(Modneg(u[15], q))
 	}

 	return u
 }

 /* Multiply P by e in group G1 */
 func G1mul(P *ECP, e *BIG) *ECP {
 	var R *ECP
 	if USE_GLV {
 		R = NewECP()
 		R.Copy(P)
 		Q := NewECP()
 		Q.Copy(P)
 		Q.Affine()
 		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()
 		}
 		u[0].norm()
 		u[1].norm()
 		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 *ECP8, e *BIG) *ECP8 {
 	var R *ECP8
 	if USE_GS_G2 {
 		var Q []*ECP8

 		F := ECP8_frob_constants()

 		q := NewBIGints(CURVE_Order)
 		u := gs(e)

 		t := NewBIGint(0)

 		Q = append(Q, NewECP8())
 		Q[0].Copy(P)
 		for i := 1; i < 16; i++ {
 			Q = append(Q, NewECP8())
 			Q[i].Copy(Q[i-1])
 			Q[i].frob(F, 1)
 		}
 		for i := 0; i < 16; i++ {
 			np := u[i].nbits()
 			t.copy(Modneg(u[i], q))
 			nn := t.nbits()
 			if nn < np {
 				u[i].copy(t)
 				Q[i].neg()
 			}
 			u[i].norm()
 		}

 		R = mul16(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 *FP48, e *BIG) *FP48 {
 	var r *FP48
 	if USE_GS_GT {
 		var g []*FP48
 		f := NewFP2bigs(NewBIGints(Fra), NewBIGints(Frb))
 		q := NewBIGints(CURVE_Order)
 		t := NewBIGint(0)

 		u := gs(e)

 		g = append(g, NewFP48copy(d))
 		for i := 1; i < 16; i++ {
 			g = append(g, NewFP48int(0))
 			g[i].Copy(g[i-1])
 			g[i].frob(f, 1)
 		}
 		for i := 0; i < 16; i++ {
 			np := u[i].nbits()
 			t.copy(Modneg(u[i], q))
 			nn := t.nbits()
 			if nn < np {
 				u[i].copy(t)
 				g[i].conj()
 			}
 			u[i].norm()
 		}
 		r = pow16(g, u)
 	} else {
 		r = d.Pow(e)
 	}
 	return 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.
	*/

	/* MiotCL BN Curve Pairing functions */

	package XXX

	//import "fmt"

	/* Line function */
	func line(A ECP8, B ECP8, Qx FP, Qy FP) *FP48 {
	var a *FP16
	var b *FP16
	var c *FP16

	if A == B { /* Doubling */
	XX := NewFP8copy(A.getx()) //X
	YY := NewFP8copy(A.gety()) //Y
	ZZ := NewFP8copy(A.getz()) //Z
	YZ := NewFP8copy(YY) //Y
	YZ.mul(ZZ) //YZ
	XX.sqr() //X^2
	YY.sqr() //Y^2
	ZZ.sqr() //Z^2

	YZ.imul(4)
	YZ.neg()
	YZ.norm() //-2YZ
	YZ.tmul(Qy) //-2YZ.Ys

	XX.imul(6) //3X^2
	XX.tmul(Qx) //3X^2.Xs

	sb := 3 * CURVE_B_I
	ZZ.imul(sb)
	if SEXTIC_TWIST == D_TYPE {
	ZZ.div_2i()
	}
	if SEXTIC_TWIST == M_TYPE {
	ZZ.times_i()
	ZZ.add(ZZ)
	YZ.times_i()
	YZ.norm()
	}
	ZZ.norm() // 3b.Z^2

	YY.add(YY)
	ZZ.sub(YY)
	ZZ.norm() // 3b.Z^2-Y^2

	a = NewFP16fp8s(YZ, ZZ) // -2YZ.Ys \| 3b.Z^2-Y^2 \| 3X^2.Xs
	if SEXTIC_TWIST == D_TYPE {

	b = NewFP16fp8(XX) // L(0,1) \| L(0,0) \| L(1,0)
	c = NewFP16int(0)
	}
	if SEXTIC_TWIST == M_TYPE {
	b = NewFP16int(0)
	c = NewFP16fp8(XX)
	c.times_i()
	}
	A.dbl()

	} else { /* Addition */

	X1 := NewFP8copy(A.getx()) // X1
	Y1 := NewFP8copy(A.gety()) // Y1
	T1 := NewFP8copy(A.getz()) // Z1
	T2 := NewFP8copy(A.getz()) // Z1

	T1.mul(B.gety()) // T1=Z1.Y2
	T2.mul(B.getx()) // T2=Z1.X2

	X1.sub(T2)
	X1.norm() // X1=X1-Z1.X2
	Y1.sub(T1)
	Y1.norm() // Y1=Y1-Z1.Y2

	T1.copy(X1) // T1=X1-Z1.X2
	X1.tmul(Qy) // X1=(X1-Z1.X2).Ys

	if SEXTIC_TWIST == M_TYPE {
	X1.times_i()
	X1.norm()
	}

	T1.mul(B.gety()) // T1=(X1-Z1.X2).Y2

	T2.copy(Y1) // T2=Y1-Z1.Y2
	T2.mul(B.getx()) // T2=(Y1-Z1.Y2).X2
	T2.sub(T1)
	T2.norm() // T2=(Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2
	Y1.tmul(Qx)
	Y1.neg()
	Y1.norm() // Y1=-(Y1-Z1.Y2).Xs

	a = NewFP16fp8s(X1, T2) // (X1-Z1.X2).Ys \| (Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2 \| - (Y1-Z1.Y2).Xs
	if SEXTIC_TWIST == D_TYPE {
	b = NewFP16fp8(Y1)
	c = NewFP16int(0)
	}
	if SEXTIC_TWIST == M_TYPE {
	b = NewFP16int(0)
	c = NewFP16fp8(Y1)
	c.times_i()
	}
	A.Add(B)
	}

	return NewFP48fp16s(a, b, c)
	}

	/* Optimal R-ate pairing */
	func Ate(P1 ECP8, Q1 ECP) *FP48 {
	x := NewBIGints(CURVE_Bnx)
	n := NewBIGcopy(x)
	var lv *FP48

	n3 := NewBIGcopy(n)
	n3.pmul(3)
	n3.norm()

	P := NewECP8()
	P.Copy(P1)
	P.Affine()
	Q := NewECP()
	Q.Copy(Q1)
	Q.Affine()

	Qx := NewFPcopy(Q.getx())
	Qy := NewFPcopy(Q.gety())

	A := NewECP8()
	r := NewFP48int(1)

	A.Copy(P)
	NP := NewECP8()
	NP.Copy(P)
	NP.neg()

	nb := n3.nbits()

	for i := nb - 2; i >= 1; i-- {
	r.sqr()
	lv = line(A, A, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	bt := n3.bit(i) - n.bit(i)
	if bt == 1 {
	lv = line(A, P, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	}
	if bt == -1 {
	lv = line(A, NP, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	}
	}

	if SIGN_OF_X == NEGATIVEX {
	r.conj()
	}

	return r
	}

	/* Optimal R-ate double pairing e(P,Q).e(R,S) */
	func Ate2(P1 ECP8, Q1 ECP, R1 ECP8, S1 ECP) *FP48 {
	x := NewBIGints(CURVE_Bnx)
	n := NewBIGcopy(x)
	var lv *FP48

	n3 := NewBIGcopy(n)
	n3.pmul(3)
	n3.norm()

	P := NewECP8()
	P.Copy(P1)
	P.Affine()
	Q := NewECP()
	Q.Copy(Q1)
	Q.Affine()
	R := NewECP8()
	R.Copy(R1)
	R.Affine()
	S := NewECP()
	S.Copy(S1)
	S.Affine()

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

	A := NewECP8()
	B := NewECP8()
	r := NewFP48int(1)

	A.Copy(P)
	B.Copy(R)
	NP := NewECP8()
	NP.Copy(P)
	NP.neg()
	NR := NewECP8()
	NR.Copy(R)
	NR.neg()

	nb := n3.nbits()

	for i := nb - 2; i >= 1; i-- {
	r.sqr()
	lv = line(A, A, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	lv = line(B, B, Sx, Sy)
	r.smul(lv, SEXTIC_TWIST)
	bt := n3.bit(i) - n.bit(i)
	if bt == 1 {
	lv = line(A, P, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	lv = line(B, R, Sx, Sy)
	r.smul(lv, SEXTIC_TWIST)
	}
	if bt == -1 {
	lv = line(A, NP, Qx, Qy)
	r.smul(lv, SEXTIC_TWIST)
	lv = line(B, NR, Sx, Sy)
	r.smul(lv, SEXTIC_TWIST)
	}
	}

	if SIGN_OF_X == NEGATIVEX {
	r.conj()
	}

	return r
	}

	/* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */
	func Fexp(m FP48) FP48 {
	f := NewFP2bigs(NewBIGints(Fra), NewBIGints(Frb))
	x := NewBIGints(CURVE_Bnx)
	r := NewFP48copy(m)

	/* Easy part of final exp */
	lv := NewFP48copy(r)

	lv.Inverse()
	r.conj()

	r.Mul(lv)
	lv.Copy(r)
	r.frob(f, 8)
	r.Mul(lv)

	/* Hard part of final exp */
	// Ghamman & Fouotsa Method

	t7 := NewFP48copy(r)
	t7.usqr()
	t1 := t7.Pow(x)

	x.fshr(1)
	t2 := t1.Pow(x)
	x.fshl(1)

	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3 := NewFP48copy(t1)
	t3.conj()
	t2.Mul(t3)
	t2.Mul(r)

	r.Mul(t7)

	t1 = t2.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}
	t3.Copy(t1)
	t3.frob(f, 14)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 13)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 12)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 11)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 10)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 9)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 8)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t2)
	t3.conj()
	t1.Mul(t3)
	t3.Copy(t1)
	t3.frob(f, 7)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 6)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 5)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 4)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 3)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 2)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	t3.Copy(t1)
	t3.frob(f, 1)
	r.Mul(t3)
	t1 = t1.Pow(x)
	if SIGN_OF_X == NEGATIVEX {
	t1.conj()
	}

	r.Mul(t1)
	t2.frob(f, 15)
	r.Mul(t2)

	r.reduce()

	return r
	}

	/* GLV method */
	func glv(e BIG) []BIG {
	var u []*BIG

	q := NewBIGints(CURVE_Order)
	x := NewBIGints(CURVE_Bnx)
	x2 := smul(x, x)
	x = smul(x2, x2)
	x2 = smul(x, x)
	u = append(u, NewBIGcopy(e))
	u[0].Mod(x2)
	u = append(u, NewBIGcopy(e))
	u[1].div(x2)
	u[1].rsub(q)
	return u
	}

	/* Galbraith & Scott Method */
	func gs(e BIG) []BIG {
	var u []*BIG

	q := NewBIGints(CURVE_Order)
	x := NewBIGints(CURVE_Bnx)
	w := NewBIGcopy(e)
	for i := 0; i < 15; i++ {
	u = append(u, NewBIGcopy(w))
	u[i].Mod(x)
	w.div(x)
	}
	u = append(u, NewBIGcopy(w))
	if SIGN_OF_X == NEGATIVEX {
	u[1].copy(Modneg(u[1], q))
	u[3].copy(Modneg(u[3], q))
	u[5].copy(Modneg(u[5], q))
	u[7].copy(Modneg(u[7], q))
	u[9].copy(Modneg(u[9], q))
	u[11].copy(Modneg(u[11], q))
	u[13].copy(Modneg(u[13], q))
	u[15].copy(Modneg(u[15], q))
	}

	return u
	}

	/* Multiply P by e in group G1 */
	func G1mul(P ECP, e BIG) *ECP {
	var R *ECP
	if USE_GLV {
	R = NewECP()
	R.Copy(P)
	Q := NewECP()
	Q.Copy(P)
	Q.Affine()
	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()
	}
	u[0].norm()
	u[1].norm()
	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 ECP8, e BIG) *ECP8 {
	var R *ECP8
	if USE_GS_G2 {
	var Q []*ECP8

	F := ECP8_frob_constants()

	q := NewBIGints(CURVE_Order)
	u := gs(e)

	t := NewBIGint(0)

	Q = append(Q, NewECP8())
	Q[0].Copy(P)
	for i := 1; i < 16; i++ {
	Q = append(Q, NewECP8())
	Q[i].Copy(Q[i-1])
	Q[i].frob(F, 1)
	}
	for i := 0; i < 16; i++ {
	np := u[i].nbits()
	t.copy(Modneg(u[i], q))
	nn := t.nbits()
	if nn < np {
	u[i].copy(t)
	Q[i].neg()
	}
	u[i].norm()
	}

	R = mul16(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 FP48, e BIG) *FP48 {
	var r *FP48
	if USE_GS_GT {
	var g []*FP48
	f := NewFP2bigs(NewBIGints(Fra), NewBIGints(Frb))
	q := NewBIGints(CURVE_Order)
	t := NewBIGint(0)

	u := gs(e)

	g = append(g, NewFP48copy(d))
	for i := 1; i < 16; i++ {
	g = append(g, NewFP48int(0))
	g[i].Copy(g[i-1])
	g[i].frob(f, 1)
	}
	for i := 0; i < 16; i++ {
	np := u[i].nbits()
	t.copy(Modneg(u[i], q))
	nn := t.nbits()
	if nn < np {
	u[i].copy(t)
	g[i].conj()
	}
	u[i].norm()
	}
	r = pow16(g, u)
	} else {
	r = d.Pow(e)
	}
	return r
	}