FP12.go

/*
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 Fp^12 functions */
/* FP12 elements are of the form a+i.b+i^2.c */

package BLS381

//import "fmt"

type FP12 struct {
	a     *FP4
	b     *FP4
	c     *FP4
	stype int
}

/* Constructors */
func NewFP12fp4(d *FP4) *FP12 {
	F := new(FP12)
	F.a = NewFP4copy(d)
	F.b = NewFP4int(0)
	F.c = NewFP4int(0)
	F.stype = FP_SPARSER
	return F
}

func NewFP12int(d int) *FP12 {
	F := new(FP12)
	F.a = NewFP4int(d)
	F.b = NewFP4int(0)
	F.c = NewFP4int(0)
	//	if d==1 {
	//		F.stype=FP_ONE
	//	} else {
	//		F.stype=FP_SPARSER
	//	}
	return F
}

func NewFP12fp4s(d *FP4, e *FP4, f *FP4) *FP12 {
	F := new(FP12)
	F.a = NewFP4copy(d)
	F.b = NewFP4copy(e)
	F.c = NewFP4copy(f)
	F.stype = FP_DENSE
	return F
}

func NewFP12copy(x *FP12) *FP12 {
	F := new(FP12)
	F.a = NewFP4copy(x.a)
	F.b = NewFP4copy(x.b)
	F.c = NewFP4copy(x.c)
	F.stype = x.stype
	return F
}

/* reduce all components of this mod Modulus */
func (F *FP12) reduce() {
	F.a.reduce()
	F.b.reduce()
	F.c.reduce()
}

/* normalise all components of this */
func (F *FP12) norm() {
	F.a.norm()
	F.b.norm()
	F.c.norm()
}

/* test x==0 ? */
func (F *FP12) iszilch() bool {
	return (F.a.iszilch() && F.b.iszilch() && F.c.iszilch())
}

/* Conditional move */
func (F *FP12) cmove(g *FP12, d int) {
	F.a.cmove(g.a, d)
	F.b.cmove(g.b, d)
	F.c.cmove(g.c, d)
	d = ^(d - 1)
	F.stype ^= (F.stype ^ g.stype) & d
}

/* Constant time select from pre-computed table */
func (F *FP12) selector(g []*FP12, b int32) {

	m := b >> 31
	babs := (b ^ m) - m

	babs = (babs - 1) / 2

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

	invF := NewFP12copy(F)
	invF.conj()
	F.cmove(invF, int(m&1))
}

/* test x==1 ? */
func (F *FP12) Isunity() bool {
	one := NewFP4int(1)
	return (F.a.Equals(one) && F.b.iszilch() && F.c.iszilch())
}

/* return 1 if x==y, else 0 */
func (F *FP12) Equals(x *FP12) bool {
	return (F.a.Equals(x.a) && F.b.Equals(x.b) && F.c.Equals(x.c))
}

/* extract a from this */
func (F *FP12) geta() *FP4 {
	return F.a
}

/* extract b */
func (F *FP12) getb() *FP4 {
	return F.b
}

/* extract c */
func (F *FP12) getc() *FP4 {
	return F.c
}

/* copy this=x */
func (F *FP12) Copy(x *FP12) {
	F.a.copy(x.a)
	F.b.copy(x.b)
	F.c.copy(x.c)
	F.stype = x.stype
}

/* set this=1 */
func (F *FP12) one() {
	F.a.one()
	F.b.zero()
	F.c.zero()
	F.stype = FP_ONE
}

/* this=conj(this) */
func (F *FP12) conj() {
	F.a.conj()
	F.b.nconj()
	F.c.conj()
}

/* Granger-Scott Unitary Squaring */
func (F *FP12) usqr() {
	A := NewFP4copy(F.a)
	B := NewFP4copy(F.c)
	C := NewFP4copy(F.b)
	D := NewFP4int(0)

	F.a.sqr()
	D.copy(F.a)
	D.add(F.a)
	F.a.add(D)

	F.a.norm()
	A.nconj()

	A.add(A)
	F.a.add(A)
	B.sqr()
	B.times_i()

	D.copy(B)
	D.add(B)
	B.add(D)
	B.norm()

	C.sqr()
	D.copy(C)
	D.add(C)
	C.add(D)
	C.norm()

	F.b.conj()
	F.b.add(F.b)
	F.c.nconj()

	F.c.add(F.c)
	F.b.add(B)
	F.c.add(C)
	F.reduce()
	F.stype = FP_DENSE

}

/* Chung-Hasan SQR2 method from http://cacr.uwaterloo.ca/techreports/2006/cacr2006-24.pdf */
func (F *FP12) sqr() {
	if F.stype == FP_ONE {
		return
	}

	A := NewFP4copy(F.a)
	B := NewFP4copy(F.b)
	C := NewFP4copy(F.c)
	D := NewFP4copy(F.a)

	A.sqr()
	B.mul(F.c)
	B.add(B)
	B.norm()
	C.sqr()
	D.mul(F.b)
	D.add(D)

	F.c.add(F.a)
	F.c.add(F.b)
	F.c.norm()
	F.c.sqr()

	F.a.copy(A)

	A.add(B)
	A.norm()
	A.add(C)
	A.add(D)
	A.norm()

	A.neg()
	B.times_i()
	C.times_i()

	F.a.add(B)

	F.b.copy(C)
	F.b.add(D)
	F.c.add(A)
	if F.stype == FP_SPARSER {
		F.stype = FP_SPARSE
	} else {
		F.stype = FP_DENSE
	}
	F.norm()
}

/* FP12 full multiplication this=this*y */
func (F *FP12) Mul(y *FP12) {
	z0 := NewFP4copy(F.a)
	z1 := NewFP4int(0)
	z2 := NewFP4copy(F.b)
	z3 := NewFP4int(0)
	t0 := NewFP4copy(F.a)
	t1 := NewFP4copy(y.a)

	z0.mul(y.a)
	z2.mul(y.b)

	t0.add(F.b)
	t0.norm()
	t1.add(y.b)
	t1.norm()

	z1.copy(t0)
	z1.mul(t1)
	t0.copy(F.b)
	t0.add(F.c)
	t0.norm()

	t1.copy(y.b)
	t1.add(y.c)
	t1.norm()
	z3.copy(t0)
	z3.mul(t1)

	t0.copy(z0)
	t0.neg()
	t1.copy(z2)
	t1.neg()

	z1.add(t0)
	F.b.copy(z1)
	F.b.add(t1)

	z3.add(t1)
	z2.add(t0)

	t0.copy(F.a)
	t0.add(F.c)
	t0.norm()
	t1.copy(y.a)
	t1.add(y.c)
	t1.norm()
	t0.mul(t1)
	z2.add(t0)

	t0.copy(F.c)
	t0.mul(y.c)
	t1.copy(t0)
	t1.neg()

	F.c.copy(z2)
	F.c.add(t1)
	z3.add(t1)
	t0.times_i()
	F.b.add(t0)
	z3.norm()
	z3.times_i()
	F.a.copy(z0)
	F.a.add(z3)
	F.stype = FP_DENSE
	F.norm()
}

/* FP12 full multiplication w=w*y */
/* Supports sparse multiplicands */
/* Usually w is denser than y */
func (F *FP12) ssmul(y *FP12) {
	if F.stype == FP_ONE {
		F.Copy(y)
		return
	}
	if y.stype == FP_ONE {
		return
	}
	if y.stype >= FP_SPARSE {
		z0 := NewFP4copy(F.a)
		z1 := NewFP4int(0)
		z2 := NewFP4int(0)
		z3 := NewFP4int(0)
		z0.mul(y.a)

		if SEXTIC_TWIST == M_TYPE {
			if y.stype == FP_SPARSE || F.stype == FP_SPARSE {
				z2.getb().copy(F.b.getb())
				z2.getb().mul(y.b.getb())
				z2.geta().zero()
				if y.stype != FP_SPARSE {
					z2.geta().copy(F.b.getb())
					z2.geta().mul(y.b.geta())
				}
				if F.stype != FP_SPARSE {
					z2.geta().copy(F.b.geta())
					z2.geta().mul(y.b.getb())
				}
				z2.times_i()
			} else {
				z2.copy(F.b)
				z2.mul(y.b)
			}
		} else {
			z2.copy(F.b)
			z2.mul(y.b)
		}
		t0 := NewFP4copy(F.a)
		t1 := NewFP4copy(y.a)
		t0.add(F.b)
		t0.norm()
		t1.add(y.b)
		t1.norm()

		z1.copy(t0)
		z1.mul(t1)
		t0.copy(F.b)
		t0.add(F.c)
		t0.norm()
		t1.copy(y.b)
		t1.add(y.c)
		t1.norm()

		z3.copy(t0)
		z3.mul(t1)

		t0.copy(z0)
		t0.neg()
		t1.copy(z2)
		t1.neg()

		z1.add(t0)
		F.b.copy(z1)
		F.b.add(t1)

		z3.add(t1)
		z2.add(t0)

		t0.copy(F.a)
		t0.add(F.c)
		t0.norm()
		t1.copy(y.a)
		t1.add(y.c)
		t1.norm()

		t0.mul(t1)
		z2.add(t0)

		if SEXTIC_TWIST == D_TYPE {
			if y.stype == FP_SPARSE || F.stype == FP_SPARSE {
				t0.geta().copy(F.c.geta())
				t0.geta().mul(y.c.geta())
				t0.getb().zero()
				if y.stype != FP_SPARSE {
					t0.getb().copy(F.c.geta())
					t0.getb().mul(y.c.getb())
				}
				if F.stype != FP_SPARSE {
					t0.getb().copy(F.c.getb())
					t0.getb().mul(y.c.geta())
				}
			} else {
				t0.copy(F.c)
				t0.mul(y.c)
			}
		} else {
			t0.copy(F.c)
			t0.mul(y.c)
		}
		t1.copy(t0)
		t1.neg()

		F.c.copy(z2)
		F.c.add(t1)
		z3.add(t1)
		t0.times_i()
		F.b.add(t0)
		z3.norm()
		z3.times_i()
		F.a.copy(z0)
		F.a.add(z3)
	} else {
		if F.stype == FP_SPARSER {
			F.smul(y)
			return
		}
		if SEXTIC_TWIST == D_TYPE { // dense by sparser - 13m
			z0 := NewFP4copy(F.a)
			z2 := NewFP4copy(F.b)
			z3 := NewFP4copy(F.b)
			t0 := NewFP4int(0)
			t1 := NewFP4copy(y.a)
			z0.mul(y.a)
			z2.pmul(y.b.real())
			F.b.add(F.a)
			t1.real().add(y.b.real())

			t1.norm()
			F.b.norm()
			F.b.mul(t1)
			z3.add(F.c)
			z3.norm()
			z3.pmul(y.b.real())

			t0.copy(z0)
			t0.neg()
			t1.copy(z2)
			t1.neg()

			F.b.add(t0)

			F.b.add(t1)
			z3.add(t1)
			z2.add(t0)

			t0.copy(F.a)
			t0.add(F.c)
			t0.norm()
			z3.norm()
			t0.mul(y.a)
			F.c.copy(z2)
			F.c.add(t0)

			z3.times_i()
			F.a.copy(z0)
			F.a.add(z3)
		}
		if SEXTIC_TWIST == M_TYPE {
			z0 := NewFP4copy(F.a)
			z1 := NewFP4int(0)
			z2 := NewFP4int(0)
			z3 := NewFP4int(0)
			t0 := NewFP4copy(F.a)
			t1 := NewFP4int(0)

			z0.mul(y.a)
			t0.add(F.b)
			t0.norm()

			z1.copy(t0)
			z1.mul(y.a)
			t0.copy(F.b)
			t0.add(F.c)
			t0.norm()

			z3.copy(t0)
			z3.pmul(y.c.getb())
			z3.times_i()

			t0.copy(z0)
			t0.neg()
			z1.add(t0)
			F.b.copy(z1)
			z2.copy(t0)

			t0.copy(F.a)
			t0.add(F.c)
			t0.norm()
			t1.copy(y.a)
			t1.add(y.c)
			t1.norm()

			t0.mul(t1)
			z2.add(t0)
			t0.copy(F.c)

			t0.pmul(y.c.getb())
			t0.times_i()
			t1.copy(t0)
			t1.neg()

			F.c.copy(z2)
			F.c.add(t1)
			z3.add(t1)
			t0.times_i()
			F.b.add(t0)
			z3.norm()
			z3.times_i()
			F.a.copy(z0)
			F.a.add(z3)
		}
	}
	F.stype = FP_DENSE
	F.norm()
}

/* Special case of multiplication arises from special form of ATE pairing line function */
func (F *FP12) smul(y *FP12) {
	if SEXTIC_TWIST == D_TYPE {
		w1 := NewFP2copy(F.a.geta())
		w2 := NewFP2copy(F.a.getb())
		w3 := NewFP2copy(F.b.geta())

		w1.mul(y.a.geta())
		w2.mul(y.a.getb())
		w3.mul(y.b.geta())

		ta := NewFP2copy(F.a.geta())
		tb := NewFP2copy(y.a.geta())
		ta.add(F.a.getb())
		ta.norm()
		tb.add(y.a.getb())
		tb.norm()
		tc := NewFP2copy(ta)
		tc.mul(tb)
		t := NewFP2copy(w1)
		t.add(w2)
		t.neg()
		tc.add(t)

		ta.copy(F.a.geta())
		ta.add(F.b.geta())
		ta.norm()
		tb.copy(y.a.geta())
		tb.add(y.b.geta())
		tb.norm()
		td := NewFP2copy(ta)
		td.mul(tb)
		t.copy(w1)
		t.add(w3)
		t.neg()
		td.add(t)

		ta.copy(F.a.getb())
		ta.add(F.b.geta())
		ta.norm()
		tb.copy(y.a.getb())
		tb.add(y.b.geta())
		tb.norm()
		te := NewFP2copy(ta)
		te.mul(tb)
		t.copy(w2)
		t.add(w3)
		t.neg()
		te.add(t)

		w2.mul_ip()
		w1.add(w2)

		F.a.geta().copy(w1)
		F.a.getb().copy(tc)
		F.b.geta().copy(td)
		F.b.getb().copy(te)
		F.c.geta().copy(w3)
		F.c.getb().zero()

		F.a.norm()
		F.b.norm()
	} else {
		w1 := NewFP2copy(F.a.geta())
		w2 := NewFP2copy(F.a.getb())
		w3 := NewFP2copy(F.c.getb())

		w1.mul(y.a.geta())
		w2.mul(y.a.getb())
		w3.mul(y.c.getb())

		ta := NewFP2copy(F.a.geta())
		tb := NewFP2copy(y.a.geta())
		ta.add(F.a.getb())
		ta.norm()
		tb.add(y.a.getb())
		tb.norm()
		tc := NewFP2copy(ta)
		tc.mul(tb)
		t := NewFP2copy(w1)
		t.add(w2)
		t.neg()
		tc.add(t)

		ta.copy(F.a.geta())
		ta.add(F.c.getb())
		ta.norm()
		tb.copy(y.a.geta())
		tb.add(y.c.getb())
		tb.norm()
		td := NewFP2copy(ta)
		td.mul(tb)
		t.copy(w1)
		t.add(w3)
		t.neg()
		td.add(t)

		ta.copy(F.a.getb())
		ta.add(F.c.getb())
		ta.norm()
		tb.copy(y.a.getb())
		tb.add(y.c.getb())
		tb.norm()
		te := NewFP2copy(ta)
		te.mul(tb)
		t.copy(w2)
		t.add(w3)
		t.neg()
		te.add(t)

		w2.mul_ip()
		w1.add(w2)
		F.a.geta().copy(w1)
		F.a.getb().copy(tc)

		w3.mul_ip()
		w3.norm()
		F.b.geta().zero()
		F.b.getb().copy(w3)

		te.norm()
		te.mul_ip()
		F.c.geta().copy(te)
		F.c.getb().copy(td)

		F.a.norm()
		F.c.norm()

	}
	F.stype = FP_SPARSE
}

/* this=1/this */
func (F *FP12) Inverse() {
	f0 := NewFP4copy(F.a)
	f1 := NewFP4copy(F.b)
	f2 := NewFP4copy(F.a)
	f3 := NewFP4int(0)

	F.norm()
	f0.sqr()
	f1.mul(F.c)
	f1.times_i()
	f0.sub(f1)
	f0.norm()

	f1.copy(F.c)
	f1.sqr()
	f1.times_i()
	f2.mul(F.b)
	f1.sub(f2)
	f1.norm()

	f2.copy(F.b)
	f2.sqr()
	f3.copy(F.a)
	f3.mul(F.c)
	f2.sub(f3)
	f2.norm()

	f3.copy(F.b)
	f3.mul(f2)
	f3.times_i()
	F.a.mul(f0)
	f3.add(F.a)
	F.c.mul(f1)
	F.c.times_i()

	f3.add(F.c)
	f3.norm()
	f3.inverse()
	F.a.copy(f0)
	F.a.mul(f3)
	F.b.copy(f1)
	F.b.mul(f3)
	F.c.copy(f2)
	F.c.mul(f3)
	F.stype = FP_DENSE
}

/* this=this^p using Frobenius */
func (F *FP12) frob(f *FP2) {
	f2 := NewFP2copy(f)
	f3 := NewFP2copy(f)

	f2.sqr()
	f3.mul(f2)

	F.a.frob(f3)
	F.b.frob(f3)
	F.c.frob(f3)

	F.b.pmul(f)
	F.c.pmul(f2)
	F.stype = FP_DENSE
}

/* trace function */
func (F *FP12) trace() *FP4 {
	t := NewFP4int(0)
	t.copy(F.a)
	t.imul(3)
	t.reduce()
	return t
}

/* convert from byte array to FP12 */
func FP12_fromBytes(w []byte) *FP12 {
	var t [int(MODBYTES)]byte
	MB := int(MODBYTES)

	for i := 0; i < MB; i++ {
		t[i] = w[i]
	}
	a := FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+MB]
	}
	b := FromBytes(t[:])
	c := NewFP2bigs(a, b)

	for i := 0; i < MB; i++ {
		t[i] = w[i+2*MB]
	}
	a = FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+3*MB]
	}
	b = FromBytes(t[:])
	d := NewFP2bigs(a, b)

	e := NewFP4fp2s(c, d)

	for i := 0; i < MB; i++ {
		t[i] = w[i+4*MB]
	}
	a = FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+5*MB]
	}
	b = FromBytes(t[:])
	c = NewFP2bigs(a, b)

	for i := 0; i < MB; i++ {
		t[i] = w[i+6*MB]
	}
	a = FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+7*MB]
	}
	b = FromBytes(t[:])
	d = NewFP2bigs(a, b)

	f := NewFP4fp2s(c, d)

	for i := 0; i < MB; i++ {
		t[i] = w[i+8*MB]
	}
	a = FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+9*MB]
	}
	b = FromBytes(t[:])

	c = NewFP2bigs(a, b)

	for i := 0; i < MB; i++ {
		t[i] = w[i+10*MB]
	}
	a = FromBytes(t[:])
	for i := 0; i < MB; i++ {
		t[i] = w[i+11*MB]
	}
	b = FromBytes(t[:])
	d = NewFP2bigs(a, b)

	g := NewFP4fp2s(c, d)

	return NewFP12fp4s(e, f, g)
}

/* convert this to byte array */
func (F *FP12) ToBytes(w []byte) {
	var t [int(MODBYTES)]byte
	MB := int(MODBYTES)
	F.a.geta().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i] = t[i]
	}
	F.a.geta().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+MB] = t[i]
	}
	F.a.getb().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+2*MB] = t[i]
	}
	F.a.getb().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+3*MB] = t[i]
	}

	F.b.geta().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+4*MB] = t[i]
	}
	F.b.geta().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+5*MB] = t[i]
	}
	F.b.getb().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+6*MB] = t[i]
	}
	F.b.getb().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+7*MB] = t[i]
	}

	F.c.geta().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+8*MB] = t[i]
	}
	F.c.geta().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+9*MB] = t[i]
	}
	F.c.getb().GetA().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+10*MB] = t[i]
	}
	F.c.getb().GetB().ToBytes(t[:])
	for i := 0; i < MB; i++ {
		w[i+11*MB] = t[i]
	}
}

/* convert to hex string */
func (F *FP12) ToString() string {
	return ("[" + F.a.toString() + "," + F.b.toString() + "," + F.c.toString() + "]")
}

/* this=this^e */
func (F *FP12) Pow(e *BIG) *FP12 {
	//F.norm()
	e1 := NewBIGcopy(e)
	e1.norm()
	e3 := NewBIGcopy(e1)
	e3.pmul(3)
	e3.norm()
	sf := NewFP12copy(F)
	sf.norm()
	w := NewFP12copy(sf)

	nb := e3.nbits()
	for i := nb - 2; i >= 1; i-- {
		w.usqr()
		bt := e3.bit(i) - e1.bit(i)
		if bt == 1 {
			w.Mul(sf)
		}
		if bt == -1 {
			sf.conj()
			w.Mul(sf)
			sf.conj()
		}
	}
	w.reduce()
	return w
}

/* constant time powering by small integer of max length bts */
func (F *FP12) pinpow(e int, bts int) {
	var R []*FP12
	R = append(R, NewFP12int(1))
	R = append(R, NewFP12copy(F))

	for i := bts - 1; i >= 0; i-- {
		b := (e >> uint(i)) & 1
		R[1-b].Mul(R[b])
		R[b].usqr()
	}
	F.Copy(R[0])
}

/* Fast compressed FP4 power of unitary FP12 */
func (F *FP12) Compow(e *BIG, r *BIG) *FP4 {
	q := NewBIGints(Modulus)
	f := NewFP2bigs(NewBIGints(Fra), NewBIGints(Frb))

	m := NewBIGcopy(q)
	m.Mod(r)

	a := NewBIGcopy(e)
	a.Mod(m)

	b := NewBIGcopy(e)
	b.div(m)

	g1 := NewFP12copy(F)
	c := g1.trace()

	if b.iszilch() {
		c = c.xtr_pow(e)
		return c
	}

	g2 := NewFP12copy(F)
	g2.frob(f)
	cp := g2.trace()

	g1.conj()
	g2.Mul(g1)
	cpm1 := g2.trace()
	g2.Mul(g1)
	cpm2 := g2.trace()

	c = c.xtr_pow2(cp, cpm1, cpm2, a, b)
	return c
}

/* p=q0^u0.q1^u1.q2^u2.q3^u3 */
// 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 pow4(q []*FP12, u []*BIG) *FP12 {
	var g []*FP12
	var w [NLEN*int(BASEBITS) + 1]int8
	var s [NLEN*int(BASEBITS) + 1]int8
	var t []*BIG
	r := NewFP12int(0)
	p := NewFP12int(0)
	mt := NewBIGint(0)

	for i := 0; i < 4; i++ {
		t = append(t, NewBIGcopy(u[i]))
	}

	g = append(g, NewFP12copy(q[0])) // q[0]
	g = append(g, NewFP12copy(g[0]))
	g[1].Mul(q[1]) // q[0].q[1]
	g = append(g, NewFP12copy(g[0]))
	g[2].Mul(q[2]) // q[0].q[2]
	g = append(g, NewFP12copy(g[1]))
	g[3].Mul(q[2]) // q[0].q[1].q[2]
	g = append(g, NewFP12copy(g[0]))
	g[4].Mul(q[3]) // q[0].q[3]
	g = append(g, NewFP12copy(g[1]))
	g[5].Mul(q[3]) // q[0].q[1].q[3]
	g = append(g, NewFP12copy(g[2]))
	g[6].Mul(q[3]) // q[0].q[2].q[3]
	g = append(g, NewFP12copy(g[3]))
	g[7].Mul(q[3]) // q[0].q[1].q[2].q[3]

	// Make it odd
	pb := 1 - t[0].parity()
	t[0].inc(pb)
	//	t[0].norm();

	// Number of bits
	mt.zero()
	for i := 0; i < 4; i++ {
		t[i].norm()
		mt.or(t[i])
	}

	nb := 1 + mt.nbits()

	// Sign pivot
	s[nb-1] = 1
	for i := 0; i < nb-1; i++ {
		t[0].fshr(1)
		s[i] = 2*int8(t[0].parity()) - 1
	}

	// Recoded exponent
	for i := 0; i < nb; i++ {
		w[i] = 0
		k := 1
		for j := 1; j < 4; j++ {
			bt := s[i] * int8(t[j].parity())
			t[j].fshr(1)
			t[j].dec(int(bt) >> 1)
			t[j].norm()
			w[i] += bt * int8(k)
			k *= 2
		}
	}

	// Main loop
	p.selector(g, int32(2*w[nb-1]+1))
	for i := nb - 2; i >= 0; i-- {
		p.usqr()
		r.selector(g, int32(2*w[i]+s[i]))
		p.Mul(r)
	}

	// apply correction
	r.Copy(q[0])
	r.conj()
	r.Mul(p)
	p.cmove(r, pb)

	p.reduce()
	return p
}