ceremonyclient/ec/bls48581/ECP8.go

/*
 * Copyright (c) 2012-2020 MIRACL UK Ltd.
 *
 * This file is part of MIRACL Core
 * (see https://github.com/miracl/core).
 *
 * Licensed 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 Weierstrass elliptic curve functions over FP2 */

package bls48581

//import "fmt"

type ECP8 struct {
	x *FP8
	y *FP8
	z *FP8
}

func NewECP8() *ECP8 {
	E := new(ECP8)
	E.x = NewFP8()
	E.y = NewFP8int(1)
	E.z = NewFP8()
	return E
}

/* Test this=O? */
func (E *ECP8) Is_infinity() bool {
	return E.x.iszilch() && E.z.iszilch()
}

/* copy this=P */
func (E *ECP8) Copy(P *ECP8) {
	E.x.copy(P.x)
	E.y.copy(P.y)
	E.z.copy(P.z)
}

/* set this=O */
func (E *ECP8) inf() {
	E.x.zero()
	E.y.one()
	E.z.zero()
}

/* set this=-this */
func (E *ECP8) neg() {
	E.y.norm()
	E.y.neg()
	E.y.norm()
}

/* Conditional move of Q to P dependant on d */
func (E *ECP8) cmove(Q *ECP8, d int) {
	E.x.cmove(Q.x, d)
	E.y.cmove(Q.y, d)
	E.z.cmove(Q.z, d)
}

/* Constant time select from pre-computed table */
func (E *ECP8) selector(W []*ECP8, b int32) {
	MP := NewECP8()
	m := b >> 31
	babs := (b ^ m) - m

	babs = (babs - 1) / 2

	E.cmove(W[0], teq(babs, 0)) // conditional move
	E.cmove(W[1], teq(babs, 1))
	E.cmove(W[2], teq(babs, 2))
	E.cmove(W[3], teq(babs, 3))
	E.cmove(W[4], teq(babs, 4))
	E.cmove(W[5], teq(babs, 5))
	E.cmove(W[6], teq(babs, 6))
	E.cmove(W[7], teq(babs, 7))

	MP.Copy(E)
	MP.neg()
	E.cmove(MP, int(m&1))
}

/* Test if P == Q */
func (E *ECP8) Equals(Q *ECP8) bool {

	a := NewFP8copy(E.x)
	b := NewFP8copy(Q.x)
	a.mul(Q.z)
	b.mul(E.z)

	if !a.Equals(b) {
		return false
	}
	a.copy(E.y)
	b.copy(Q.y)
	a.mul(Q.z)
	b.mul(E.z)
	if !a.Equals(b) {
		return false
	}

	return true
}

/* set to Affine - (x,y,z) to (x,y) */
func (E *ECP8) Affine() {
	if E.Is_infinity() {
		return
	}
	one := NewFP8int(1)
	if E.z.Equals(one) {
		E.x.reduce()
		E.y.reduce()
		return
	}
	E.z.inverse(nil)

	E.x.mul(E.z)
	E.x.reduce()
	E.y.mul(E.z)
	E.y.reduce()
	E.z.copy(one)
}

/* extract affine x as FP2 */
func (E *ECP8) GetX() *FP8 {
	W := NewECP8()
	W.Copy(E)
	W.Affine()
	return W.x
}

/* extract affine y as FP2 */
func (E *ECP8) GetY() *FP8 {
	W := NewECP8()
	W.Copy(E)
	W.Affine()
	return W.y
}

/* extract projective x */
func (E *ECP8) getx() *FP8 {
	return E.x
}

/* extract projective y */
func (E *ECP8) gety() *FP8 {
	return E.y
}

/* extract projective z */
func (E *ECP8) getz() *FP8 {
	return E.z
}

/* convert to byte array */
func (E *ECP8) ToBytes(b []byte, compress bool) {
	var t [8 * int(MODBYTES)]byte
	MB := 8 * int(MODBYTES)
	alt := false
	W := NewECP8()
	W.Copy(E)
	W.Affine()
	W.x.ToBytes(t[:])

	if (MODBITS-1)%8 <= 4 && ALLOW_ALT_COMPRESS {
		alt = true
	}

	if alt {
		for i := 0; i < MB; i++ {
			b[i] = t[i]
		}
		if !compress {
			W.y.ToBytes(t[:])
			for i := 0; i < MB; i++ {
				b[i+MB] = t[i]
			}
		} else {
			b[0] |= 0x80
			if W.y.islarger() == 1 {
				b[0] |= 0x20
			}
		}

	} else {
		for i := 0; i < MB; i++ {
			b[i+1] = t[i]
		}
		if !compress {
			b[0] = 0x04
			W.y.ToBytes(t[:])
			for i := 0; i < MB; i++ {
				b[i+MB+1] = t[i]
			}
		} else {
			b[0] = 0x02
			if W.y.sign() == 1 {
				b[0] = 0x03
			}
		}
	}
}

/* convert from byte array to point */
func ECP8_fromBytes(b []byte) *ECP8 {
	var t [8 * int(MODBYTES)]byte
	MB := 8 * int(MODBYTES)
	typ := int(b[0])
	alt := false

	if (MODBITS-1)%8 <= 4 && ALLOW_ALT_COMPRESS {
		alt = true
	}

	if alt {
		for i := 0; i < MB; i++ {
			t[i] = b[i]
		}
		t[0] &= 0x1f
		rx := FP8_fromBytes(t[:])
		if (b[0] & 0x80) == 0 {
			for i := 0; i < MB; i++ {
				t[i] = b[i+MB]
			}
			ry := FP8_fromBytes(t[:])
			return NewECP8fp8s(rx, ry)
		} else {
			sgn := (b[0] & 0x20) >> 5
			P := NewECP8fp8(rx, 0)
			cmp := P.y.islarger()
			if (sgn == 1 && cmp != 1) || (sgn == 0 && cmp == 1) {
				P.neg()
			}
			return P
		}
	} else {
		for i := 0; i < MB; i++ {
			t[i] = b[i+1]
		}
		rx := FP8_fromBytes(t[:])
		if typ == 0x04 {
			for i := 0; i < MB; i++ {
				t[i] = b[i+MB+1]
			}
			ry := FP8_fromBytes(t[:])
			return NewECP8fp8s(rx, ry)
		} else {
			return NewECP8fp8(rx, typ&1)
		}
	}
}

/* convert this to hex string */
func (E *ECP8) ToString() string {
	W := NewECP8()
	W.Copy(E)
	W.Affine()
	if W.Is_infinity() {
		return "infinity"
	}
	return "(" + W.x.toString() + "," + W.y.toString() + ")"
}

/* Calculate RHS of twisted curve equation x^3+B/i */
func RHS8(x *FP8) *FP8 {
	r := NewFP8copy(x)
	r.sqr()
	b2 := NewFP2big(NewBIGints(CURVE_B))
	b4 := NewFP4fp2(b2)
	b := NewFP8fp4(b4)

	if SEXTIC_TWIST == D_TYPE {
		b.div_i()
	}
	if SEXTIC_TWIST == M_TYPE {
		b.times_i()
	}
	r.mul(x)
	r.add(b)

	r.reduce()
	return r
}

/* construct this from (x,y) - but set to O if not on curve */
func NewECP8fp8s(ix *FP8, iy *FP8) *ECP8 {
	E := new(ECP8)
	E.x = NewFP8copy(ix)
	E.y = NewFP8copy(iy)
	E.z = NewFP8int(1)
	E.x.norm()
	rhs := RHS8(E.x)
	y2 := NewFP8copy(E.y)
	y2.sqr()
	if !y2.Equals(rhs) {
		E.inf()
	}
	return E
}

/* construct this from x - but set to O if not on curve */
func NewECP8fp8(ix *FP8, s int) *ECP8 {
	E := new(ECP8)
	h := NewFP()
	E.x = NewFP8copy(ix)
	E.y = NewFP8int(1)
	E.z = NewFP8int(1)
	E.x.norm()
	rhs := RHS8(E.x)
	if rhs.qr(h) == 1 {
		rhs.sqrt(h)
		if rhs.sign() != s {
			rhs.neg()
		}
		rhs.reduce()
		E.y.copy(rhs)

	} else {
		E.inf()
	}
	return E
}

/* this+=this */
func (E *ECP8) dbl() int {
	iy := NewFP8copy(E.y)
	if SEXTIC_TWIST == D_TYPE {
		iy.times_i()
	}

	t0 := NewFP8copy(E.y)
	t0.sqr()
	if SEXTIC_TWIST == D_TYPE {
		t0.times_i()
	}
	t1 := NewFP8copy(iy)
	t1.mul(E.z)
	t2 := NewFP8copy(E.z)
	t2.sqr()

	E.z.copy(t0)
	E.z.add(t0)
	E.z.norm()
	E.z.add(E.z)
	E.z.add(E.z)
	E.z.norm()

	t2.imul(3 * CURVE_B_I)
	if SEXTIC_TWIST == M_TYPE {
		t2.times_i()
	}
	x3 := NewFP8copy(t2)
	x3.mul(E.z)

	y3 := NewFP8copy(t0)

	y3.add(t2)
	y3.norm()
	E.z.mul(t1)
	t1.copy(t2)
	t1.add(t2)
	t2.add(t1)
	t2.norm()
	t0.sub(t2)
	t0.norm() //y^2-9bz^2
	y3.mul(t0)
	y3.add(x3) //(y^2+3z*2)(y^2-9z^2)+3b.z^2.8y^2
	t1.copy(E.x)
	t1.mul(iy) //
	E.x.copy(t0)
	E.x.norm()
	E.x.mul(t1)
	E.x.add(E.x) //(y^2-9bz^2)xy2

	E.x.norm()
	E.y.copy(y3)
	E.y.norm()

	return 1
}

/* this+=Q - return 0 for add, 1 for double, -1 for O */
func (E *ECP8) Add(Q *ECP8) int {
	b := 3 * CURVE_B_I
	t0 := NewFP8copy(E.x)
	t0.mul(Q.x) // x.Q.x
	t1 := NewFP8copy(E.y)
	t1.mul(Q.y) // y.Q.y

	t2 := NewFP8copy(E.z)
	t2.mul(Q.z)
	t3 := NewFP8copy(E.x)
	t3.add(E.y)
	t3.norm() //t3=X1+Y1
	t4 := NewFP8copy(Q.x)
	t4.add(Q.y)
	t4.norm()  //t4=X2+Y2
	t3.mul(t4) //t3=(X1+Y1)(X2+Y2)
	t4.copy(t0)
	t4.add(t1) //t4=X1.X2+Y1.Y2

	t3.sub(t4)
	t3.norm()
	if SEXTIC_TWIST == D_TYPE {
		t3.times_i() //t3=(X1+Y1)(X2+Y2)-(X1.X2+Y1.Y2) = X1.Y2+X2.Y1
	}
	t4.copy(E.y)
	t4.add(E.z)
	t4.norm() //t4=Y1+Z1
	x3 := NewFP8copy(Q.y)
	x3.add(Q.z)
	x3.norm() //x3=Y2+Z2

	t4.mul(x3)  //t4=(Y1+Z1)(Y2+Z2)
	x3.copy(t1) //
	x3.add(t2)  //X3=Y1.Y2+Z1.Z2

	t4.sub(x3)
	t4.norm()
	if SEXTIC_TWIST == D_TYPE {
		t4.times_i() //t4=(Y1+Z1)(Y2+Z2) - (Y1.Y2+Z1.Z2) = Y1.Z2+Y2.Z1
	}
	x3.copy(E.x)
	x3.add(E.z)
	x3.norm() // x3=X1+Z1
	y3 := NewFP8copy(Q.x)
	y3.add(Q.z)
	y3.norm()  // y3=X2+Z2
	x3.mul(y3) // x3=(X1+Z1)(X2+Z2)
	y3.copy(t0)
	y3.add(t2) // y3=X1.X2+Z1+Z2
	y3.rsub(x3)
	y3.norm() // y3=(X1+Z1)(X2+Z2) - (X1.X2+Z1.Z2) = X1.Z2+X2.Z1

	if SEXTIC_TWIST == D_TYPE {
		t0.times_i() // x.Q.x
		t1.times_i() // y.Q.y
	}
	x3.copy(t0)
	x3.add(t0)
	t0.add(x3)
	t0.norm()
	t2.imul(b)
	if SEXTIC_TWIST == M_TYPE {
		t2.times_i()
	}
	z3 := NewFP8copy(t1)
	z3.add(t2)
	z3.norm()
	t1.sub(t2)
	t1.norm()
	y3.imul(b)
	if SEXTIC_TWIST == M_TYPE {
		y3.times_i()
	}
	x3.copy(y3)
	x3.mul(t4)
	t2.copy(t3)
	t2.mul(t1)
	x3.rsub(t2)
	y3.mul(t0)
	t1.mul(z3)
	y3.add(t1)
	t0.mul(t3)
	z3.mul(t4)
	z3.add(t0)

	E.x.copy(x3)
	E.x.norm()
	E.y.copy(y3)
	E.y.norm()
	E.z.copy(z3)
	E.z.norm()

	return 0
}

/* set this-=Q */
func (E *ECP8) Sub(Q *ECP8) int {
	NQ := NewECP8()
	NQ.Copy(Q)
	NQ.neg()
	D := E.Add(NQ)
	return D
}

func ECP8_frob_constants() [3]*FP2 {
	Fra := NewBIGints(Fra)
	Frb := NewBIGints(Frb)
	X := NewFP2bigs(Fra, Frb)

	F0 := NewFP2copy(X)
	F0.sqr()
	F2 := NewFP2copy(F0)
	F2.mul_ip()
	F2.norm()
	F1 := NewFP2copy(F2)
	F1.sqr()
	F2.mul(F1)

	F2.mul_ip()
	F2.norm()

	F1.copy(X)
	if SEXTIC_TWIST == M_TYPE {
		F1.mul_ip()
		F1.norm()
		F1.inverse(nil)
		F0.copy(F1)
		F0.sqr()
		F1.mul(F0)
	}
	if SEXTIC_TWIST == D_TYPE {
		F0.copy(F1)
		F0.sqr()
		F1.mul(F0)
		F0.mul_ip()
		F0.norm()
		F1.mul_ip()
		F1.norm()
		F1.mul_ip()
		F1.norm()
	}

	F := [3]*FP2{F0, F1, F2}
	return F
}

/* set this*=q, where q is Modulus, using Frobenius */
func (E *ECP8) frob(F [3]*FP2, n int) {
	for i := 0; i < n; i++ {
		E.x.frob(F[2])
		if SEXTIC_TWIST == M_TYPE {
			E.x.qmul(F[0])
			E.x.times_i2()
		}
		if SEXTIC_TWIST == D_TYPE {
			E.x.qmul(F[0])
			E.x.times_i2()
		}
		E.y.frob(F[2])
		if SEXTIC_TWIST == M_TYPE {
			E.y.qmul(F[1])
			E.y.times_i2()
			E.y.times_i()
		}
		if SEXTIC_TWIST == D_TYPE {
			E.y.qmul(F[1])
			E.y.times_i()
		}

		E.z.frob(F[2])
	}
}

/* P*=e */
func (E *ECP8) mul(e *BIG) *ECP8 {
	/* fixed size windows */
	mt := NewBIG()
	t := NewBIG()
	P := NewECP8()
	Q := NewECP8()
	C := NewECP8()

	if E.Is_infinity() {
		return NewECP8()
	}

	var W []*ECP8
	var w [1 + (NLEN*int(BASEBITS)+3)/4]int8

	/* precompute table */
	Q.Copy(E)
	Q.dbl()

	W = append(W, NewECP8())
	W[0].Copy(E)

	for i := 1; i < 8; i++ {
		W = append(W, NewECP8())
		W[i].Copy(W[i-1])
		W[i].Add(Q)
	}

	/* make exponent odd - add 2P if even, P if odd */
	t.copy(e)
	s := int(t.parity())
	t.inc(1)
	t.norm()
	ns := int(t.parity())
	mt.copy(t)
	mt.inc(1)
	mt.norm()
	t.cmove(mt, s)
	Q.cmove(E, ns)
	C.Copy(Q)

	nb := 1 + (t.nbits()+3)/4
	/* convert exponent to signed 4-bit window */
	for i := 0; i < nb; i++ {
		w[i] = int8(t.lastbits(5) - 16)
		t.dec(int(w[i]))
		t.norm()
		t.fshr(4)
	}
	w[nb] = int8(t.lastbits(5))

	//P.Copy(W[(w[nb]-1)/2])
	P.selector(W, int32(w[nb]))
	for i := nb - 1; i >= 0; i-- {
		Q.selector(W, int32(w[i]))
		P.dbl()
		P.dbl()
		P.dbl()
		P.dbl()
		P.Add(Q)
	}
	P.Sub(C)
	P.Affine()
	return P
}

/* Public version */
func (E *ECP8) Mul(e *BIG) *ECP8 {
	return E.mul(e)
}

/* needed for SOK */
func (E *ECP8) Cfp() {

	F := ECP8_frob_constants()
	x := NewBIGints(CURVE_Bnx)

	xQ := E.mul(x)
	x2Q := xQ.mul(x)
	x3Q := x2Q.mul(x)
	x4Q := x3Q.mul(x)
	x5Q := x4Q.mul(x)
	x6Q := x5Q.mul(x)
	x7Q := x6Q.mul(x)
	x8Q := x7Q.mul(x)

	if SIGN_OF_X == NEGATIVEX {
		xQ.neg()
		x3Q.neg()
		x5Q.neg()
		x7Q.neg()
	}
	x8Q.Sub(x7Q)
	x8Q.Sub(E)

	x7Q.Sub(x6Q)
	x7Q.frob(F, 1)

	x6Q.Sub(x5Q)
	x6Q.frob(F, 2)

	x5Q.Sub(x4Q)
	x5Q.frob(F, 3)

	x4Q.Sub(x3Q)
	x4Q.frob(F, 4)

	x3Q.Sub(x2Q)
	x3Q.frob(F, 5)

	x2Q.Sub(xQ)
	x2Q.frob(F, 6)

	xQ.Sub(E)
	xQ.frob(F, 7)

	E.dbl()
	E.frob(F, 8)

	E.Add(x8Q)
	E.Add(x7Q)
	E.Add(x6Q)
	E.Add(x5Q)

	E.Add(x4Q)
	E.Add(x3Q)
	E.Add(x2Q)
	E.Add(xQ)

	E.Affine()
}

func ECP8_generator() *ECP8 {
	var G *ECP8
	G = NewECP8fp8s(
		NewFP8fp4s(
			NewFP4fp2s(
				NewFP2bigs(NewBIGints(CURVE_Pxaaa), NewBIGints(CURVE_Pxaab)),
				NewFP2bigs(NewBIGints(CURVE_Pxaba), NewBIGints(CURVE_Pxabb))),
			NewFP4fp2s(
				NewFP2bigs(NewBIGints(CURVE_Pxbaa), NewBIGints(CURVE_Pxbab)),
				NewFP2bigs(NewBIGints(CURVE_Pxbba), NewBIGints(CURVE_Pxbbb)))),
		NewFP8fp4s(
			NewFP4fp2s(
				NewFP2bigs(NewBIGints(CURVE_Pyaaa), NewBIGints(CURVE_Pyaab)),
				NewFP2bigs(NewBIGints(CURVE_Pyaba), NewBIGints(CURVE_Pyabb))),
			NewFP4fp2s(
				NewFP2bigs(NewBIGints(CURVE_Pybaa), NewBIGints(CURVE_Pybab)),
				NewFP2bigs(NewBIGints(CURVE_Pybba), NewBIGints(CURVE_Pybbb)))))
	return G
}

func ECP8_hap2point(h *BIG) *ECP8 {
	one := NewBIGint(1)
	x := NewBIGcopy(h)
	var X2 *FP2
	var X4 *FP4
	var X8 *FP8
	var Q *ECP8
	for true {
		X2 = NewFP2bigs(one, x)
		X4 = NewFP4fp2(X2)
		X8 = NewFP8fp4(X4)
		Q = NewECP8fp8(X8, 0)
		if !Q.Is_infinity() {
			break
		}
		x.inc(1)
		x.norm()
	}
	return Q
}

/* Deterministic mapping of Fp to point on curve */
func ECP8_map2point(H *FP8) *ECP8 {
	// Shallue and van de Woestijne
	NY := NewFP8int(1)
	T := NewFP8copy(H)
	sgn := T.sign()

	Z := NewFPint(RIADZG2A)
	X1 := NewFP8fp(Z)
	X3 := NewFP8copy(X1)
	A := RHS8(X1)
	W := NewFP8copy(A)
	W.sqrt(nil)

	s := NewFPbig(NewBIGints(SQRTm3))
	Z.mul(s)

	T.sqr()
	Y := NewFP8copy(A)
	Y.mul(T)
	T.copy(NY)
	T.add(Y)
	T.norm()
	Y.rsub(NY)
	Y.norm()
	NY.copy(T)
	NY.mul(Y)

	NY.tmul(Z)
	NY.inverse(nil)

	W.tmul(Z)
	if W.sign() == 1 {
		W.neg()
		W.norm()
	}
	W.tmul(Z)
	W.mul(H)
	W.mul(Y)
	W.mul(NY)

	X1.neg()
	X1.norm()
	X1.div2()
	X2 := NewFP8copy(X1)
	X1.sub(W)
	X1.norm()
	X2.add(W)
	X2.norm()
	A.add(A)
	A.add(A)
	A.norm()
	T.sqr()
	T.mul(NY)
	T.sqr()
	A.mul(T)
	X3.add(A)
	X3.norm()

	Y.copy(RHS8(X2))
	X3.cmove(X2, Y.qr(nil))
	Y.copy(RHS8(X1))
	X3.cmove(X1, Y.qr(nil))
	Y.copy(RHS8(X3))
	Y.sqrt(nil)

	ne := Y.sign() ^ sgn
	W.copy(Y)
	W.neg()
	W.norm()
	Y.cmove(W, ne)

	return NewECP8fp8s(X3, Y)
}

/* Map octet string to curve point */
func ECP8_mapit(h []byte) *ECP8 {
	q := NewBIGints(Modulus)
	dx := DBIG_fromBytes(h)
	x := dx.Mod(q)

	Q := ECP8_hap2point(x)
	Q.Cfp()
	return Q
}

/* P=u0.Q0+u1*Q1+u2*Q2+u3*Q3.. */
// Bos & Costello https://eprint.iacr.org/2013/458.pdf
// Faz-Hernandez & Longa & Sanchez  https://eprint.iacr.org/2013/158.pdf
// Side channel attack secure
func mul16(Q []*ECP8, u []*BIG) *ECP8 {
	W := NewECP8()
	P := NewECP8()
	var T1 []*ECP8
	var T2 []*ECP8
	var T3 []*ECP8
	var T4 []*ECP8
	mt := NewBIG()
	var t []*BIG
	var bt int8
	var k int

	var w1 [NLEN*int(BASEBITS) + 1]int8
	var s1 [NLEN*int(BASEBITS) + 1]int8
	var w2 [NLEN*int(BASEBITS) + 1]int8
	var s2 [NLEN*int(BASEBITS) + 1]int8
	var w3 [NLEN*int(BASEBITS) + 1]int8
	var s3 [NLEN*int(BASEBITS) + 1]int8
	var w4 [NLEN*int(BASEBITS) + 1]int8
	var s4 [NLEN*int(BASEBITS) + 1]int8

	for i := 0; i < 16; i++ {
		t = append(t, NewBIGcopy(u[i]))
	}

	T1 = append(T1, NewECP8())
	T1[0].Copy(Q[0]) // Q[0]
	T1 = append(T1, NewECP8())
	T1[1].Copy(T1[0])
	T1[1].Add(Q[1]) // Q[0]+Q[1]
	T1 = append(T1, NewECP8())
	T1[2].Copy(T1[0])
	T1[2].Add(Q[2]) // Q[0]+Q[2]
	T1 = append(T1, NewECP8())
	T1[3].Copy(T1[1])
	T1[3].Add(Q[2]) // Q[0]+Q[1]+Q[2]
	T1 = append(T1, NewECP8())
	T1[4].Copy(T1[0])
	T1[4].Add(Q[3]) // Q[0]+Q[3]
	T1 = append(T1, NewECP8())
	T1[5].Copy(T1[1])
	T1[5].Add(Q[3]) // Q[0]+Q[1]+Q[3]
	T1 = append(T1, NewECP8())
	T1[6].Copy(T1[2])
	T1[6].Add(Q[3]) // Q[0]+Q[2]+Q[3]
	T1 = append(T1, NewECP8())
	T1[7].Copy(T1[3])
	T1[7].Add(Q[3]) // Q[0]+Q[1]+Q[2]+Q[3]

	T2 = append(T2, NewECP8())
	T2[0].Copy(Q[4]) // Q[0]
	T2 = append(T2, NewECP8())
	T2[1].Copy(T2[0])
	T2[1].Add(Q[5]) // Q[0]+Q[1]
	T2 = append(T2, NewECP8())
	T2[2].Copy(T2[0])
	T2[2].Add(Q[6]) // Q[0]+Q[2]
	T2 = append(T2, NewECP8())
	T2[3].Copy(T2[1])
	T2[3].Add(Q[6]) // Q[0]+Q[1]+Q[2]
	T2 = append(T2, NewECP8())
	T2[4].Copy(T2[0])
	T2[4].Add(Q[7]) // Q[0]+Q[3]
	T2 = append(T2, NewECP8())
	T2[5].Copy(T2[1])
	T2[5].Add(Q[7]) // Q[0]+Q[1]+Q[3]
	T2 = append(T2, NewECP8())
	T2[6].Copy(T2[2])
	T2[6].Add(Q[7]) // Q[0]+Q[2]+Q[3]
	T2 = append(T2, NewECP8())
	T2[7].Copy(T2[3])
	T2[7].Add(Q[7]) // Q[0]+Q[1]+Q[2]+Q[3]

	T3 = append(T3, NewECP8())
	T3[0].Copy(Q[8]) // Q[0]
	T3 = append(T3, NewECP8())
	T3[1].Copy(T3[0])
	T3[1].Add(Q[9]) // Q[0]+Q[1]
	T3 = append(T3, NewECP8())
	T3[2].Copy(T3[0])
	T3[2].Add(Q[10]) // Q[0]+Q[2]
	T3 = append(T3, NewECP8())
	T3[3].Copy(T3[1])
	T3[3].Add(Q[10]) // Q[0]+Q[1]+Q[2]
	T3 = append(T3, NewECP8())
	T3[4].Copy(T3[0])
	T3[4].Add(Q[11]) // Q[0]+Q[3]
	T3 = append(T3, NewECP8())
	T3[5].Copy(T3[1])
	T3[5].Add(Q[11]) // Q[0]+Q[1]+Q[3]
	T3 = append(T3, NewECP8())
	T3[6].Copy(T3[2])
	T3[6].Add(Q[11]) // Q[0]+Q[2]+Q[3]
	T3 = append(T3, NewECP8())
	T3[7].Copy(T3[3])
	T3[7].Add(Q[11]) // Q[0]+Q[1]+Q[2]+Q[3]

	T4 = append(T4, NewECP8())
	T4[0].Copy(Q[12]) // Q[0]
	T4 = append(T4, NewECP8())
	T4[1].Copy(T4[0])
	T4[1].Add(Q[13]) // Q[0]+Q[1]
	T4 = append(T4, NewECP8())
	T4[2].Copy(T4[0])
	T4[2].Add(Q[14]) // Q[0]+Q[2]
	T4 = append(T4, NewECP8())
	T4[3].Copy(T4[1])
	T4[3].Add(Q[14]) // Q[0]+Q[1]+Q[2]
	T4 = append(T4, NewECP8())
	T4[4].Copy(T4[0])
	T4[4].Add(Q[15]) // Q[0]+Q[3]
	T4 = append(T4, NewECP8())
	T4[5].Copy(T4[1])
	T4[5].Add(Q[15]) // Q[0]+Q[1]+Q[3]
	T4 = append(T4, NewECP8())
	T4[6].Copy(T4[2])
	T4[6].Add(Q[15]) // Q[0]+Q[2]+Q[3]
	T4 = append(T4, NewECP8())
	T4[7].Copy(T4[3])
	T4[7].Add(Q[15]) // Q[0]+Q[1]+Q[2]+Q[3]

	// Make them odd
	pb1 := 1 - t[0].parity()
	t[0].inc(pb1)

	pb2 := 1 - t[4].parity()
	t[4].inc(pb2)

	pb3 := 1 - t[8].parity()
	t[8].inc(pb3)

	pb4 := 1 - t[12].parity()
	t[12].inc(pb4)

	// Number of bits
	mt.zero()
	for i := 0; i < 16; i++ {
		t[i].norm()
		mt.or(t[i])
	}

	nb := 1 + mt.nbits()

	// Sign pivot
	s1[nb-1] = 1
	s2[nb-1] = 1
	s3[nb-1] = 1
	s4[nb-1] = 1
	for i := 0; i < nb-1; i++ {
		t[0].fshr(1)
		s1[i] = 2*int8(t[0].parity()) - 1
		t[4].fshr(1)
		s2[i] = 2*int8(t[4].parity()) - 1
		t[8].fshr(1)
		s3[i] = 2*int8(t[8].parity()) - 1
		t[12].fshr(1)
		s4[i] = 2*int8(t[12].parity()) - 1

	}

	// Recoded exponents
	for i := 0; i < nb; i++ {
		w1[i] = 0
		k = 1
		for j := 1; j < 4; j++ {
			bt = s1[i] * int8(t[j].parity())
			t[j].fshr(1)
			t[j].dec(int(bt) >> 1)
			t[j].norm()
			w1[i] += bt * int8(k)
			k *= 2
		}
		w2[i] = 0
		k = 1
		for j := 5; j < 8; j++ {
			bt = s2[i] * int8(t[j].parity())
			t[j].fshr(1)
			t[j].dec(int(bt) >> 1)
			t[j].norm()
			w2[i] += bt * int8(k)
			k *= 2
		}
		w3[i] = 0
		k = 1
		for j := 9; j < 12; j++ {
			bt = s3[i] * int8(t[j].parity())
			t[j].fshr(1)
			t[j].dec(int(bt) >> 1)
			t[j].norm()
			w3[i] += bt * int8(k)
			k *= 2
		}
		w4[i] = 0
		k = 1
		for j := 13; j < 16; j++ {
			bt = s4[i] * int8(t[j].parity())
			t[j].fshr(1)
			t[j].dec(int(bt) >> 1)
			t[j].norm()
			w4[i] += bt * int8(k)
			k *= 2
		}
	}

	// Main loop
	P.selector(T1, int32(2*w1[nb-1]+1))
	W.selector(T2, int32(2*w2[nb-1]+1))
	P.Add(W)
	W.selector(T3, int32(2*w3[nb-1]+1))
	P.Add(W)
	W.selector(T4, int32(2*w4[nb-1]+1))
	P.Add(W)
	for i := nb - 2; i >= 0; i-- {
		P.dbl()
		W.selector(T1, int32(2*w1[i]+s1[i]))
		P.Add(W)
		W.selector(T2, int32(2*w2[i]+s2[i]))
		P.Add(W)
		W.selector(T3, int32(2*w3[i]+s3[i]))
		P.Add(W)
		W.selector(T4, int32(2*w4[i]+s4[i]))
		P.Add(W)

	}

	// apply correction
	W.Copy(P)
	W.Sub(Q[0])
	P.cmove(W, pb1)
	W.Copy(P)
	W.Sub(Q[4])
	P.cmove(W, pb2)
	W.Copy(P)
	W.Sub(Q[8])
	P.cmove(W, pb3)
	W.Copy(P)
	W.Sub(Q[12])
	P.cmove(W, pb4)

	P.Affine()
	return P
}