[BLS24] Fp24 tower

ecc/bls24-315/bls24-315.go

@@ -79,10 +79,10 @@ type E2 = fptower.E2
 // E4 is a degree two finite field extension of fp2
 type E4 = fptower.E4
 
-// E8 is a degree two finite field extension of fp4
-type E8 = fptower.E8
+// E12 is a degree three finite field extension of fp4
+type E12 = fptower.E12
 
-// E24 is a degree three finite field extension of fp8
+// E24 is a degree two finite field extension of fp6
 type E24 = fptower.E24
 
 func init() {

ecc/bls24-315/internal/fptower/e12.go

@@ -0,0 +1,277 @@
+// Copyright 2020 ConsenSys Software Inc.
+//
+// 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.
+
+package fptower
+
+import (
+	"math/big"
+)
+
+// E12 is a degree three finite field extension of fp4
+type E12 struct {
+	C0, C1, C2 E4
+}
+
+// Equal returns true if z equals x, fasle otherwise
+func (z *E12) Equal(x *E12) bool {
+	return z.C0.Equal(&x.C0) && z.C1.Equal(&x.C1) && z.C2.Equal(&x.C2)
+}
+
+// String puts E12 elmt in string form
+func (z *E12) String() string {
+	return (z.C0.String() + "+(" + z.C1.String() + ")*w+(" + z.C2.String() + ")*w**2")
+}
+
+// SetString sets a E12 elmt from stringf
+func (z *E12) SetString(s0, s1, s2, s3, s4, s5, s6, s7, s8, s9, s10, s11 string) *E12 {
+	z.C0.SetString(s0, s1, s2, s3)
+	z.C1.SetString(s4, s5, s6, s7)
+	z.C2.SetString(s8, s9, s10, s11)
+	return z
+}
+
+// Set Sets a E12 elmt form another E12 elmt
+func (z *E12) Set(x *E12) *E12 {
+	z.C0 = x.C0
+	z.C1 = x.C1
+	z.C2 = x.C2
+	return z
+}
+
+// SetOne sets z to 1 in Montgomery form and returns z
+func (z *E12) SetOne() *E12 {
+	*z = E12{}
+	z.C0.B0.A0.SetOne()
+	return z
+}
+
+// SetRandom set z to a random elmt
+func (z *E12) SetRandom() (*E12, error) {
+	if _, err := z.C0.SetRandom(); err != nil {
+		return nil, err
+	}
+	if _, err := z.C1.SetRandom(); err != nil {
+		return nil, err
+	}
+	if _, err := z.C2.SetRandom(); err != nil {
+		return nil, err
+	}
+	return z, nil
+}
+
+// ToMont converts to Mont form
+func (z *E12) ToMont() *E12 {
+	z.C0.ToMont()
+	z.C1.ToMont()
+	z.C2.ToMont()
+	return z
+}
+
+// FromMont converts from Mont form
+func (z *E12) FromMont() *E12 {
+	z.C0.FromMont()
+	z.C1.FromMont()
+	z.C2.FromMont()
+	return z
+}
+
+// Add adds two elements of E12
+func (z *E12) Add(x, y *E12) *E12 {
+	z.C0.Add(&x.C0, &y.C0)
+	z.C1.Add(&x.C1, &y.C1)
+	z.C2.Add(&x.C2, &y.C2)
+	return z
+}
+
+// Neg negates the E12 number
+func (z *E12) Neg(x *E12) *E12 {
+	z.C0.Neg(&x.C0)
+	z.C1.Neg(&x.C1)
+	z.C2.Neg(&x.C2)
+	return z
+}
+
+// Sub two elements of E12
+func (z *E12) Sub(x, y *E12) *E12 {
+	z.C0.Sub(&x.C0, &y.C0)
+	z.C1.Sub(&x.C1, &y.C1)
+	z.C2.Sub(&x.C2, &y.C2)
+	return z
+}
+
+// Double doubles an element in E12
+func (z *E12) Double(x *E12) *E12 {
+	z.C0.Double(&x.C0)
+	z.C1.Double(&x.C1)
+	z.C2.Double(&x.C2)
+	return z
+}
+
+// MulByNonResidue mul x by (0,1,0)
+func (z *E12) MulByNonResidue(x *E12) *E12 {
+	z.C2, z.C1, z.C0 = x.C1, x.C0, x.C2
+	z.C0.MulByNonResidue(&z.C0)
+	return z
+}
+
+// Mul sets z to the E12 product of x,y, returns z
+func (z *E12) Mul(x, y *E12) *E12 {
+	// Algorithm 13 from https://eprint.iacr.org/2010/354.pdf
+	var t0, t1, t2, c0, c1, c2, tmp E4
+	t0.Mul(&x.C0, &y.C0)
+	t1.Mul(&x.C1, &y.C1)
+	t2.Mul(&x.C2, &y.C2)
+
+	c0.Add(&x.C1, &x.C2)
+	tmp.Add(&y.C1, &y.C2)
+	c0.Mul(&c0, &tmp).Sub(&c0, &t1).Sub(&c0, &t2).MulByNonResidue(&c0).Add(&c0, &t0)
+
+	c1.Add(&x.C0, &x.C1)
+	tmp.Add(&y.C0, &y.C1)
+	c1.Mul(&c1, &tmp).Sub(&c1, &t0).Sub(&c1, &t1)
+	tmp.MulByNonResidue(&t2)
+	c1.Add(&c1, &tmp)
+
+	tmp.Add(&x.C0, &x.C2)
+	c2.Add(&y.C0, &y.C2).Mul(&c2, &tmp).Sub(&c2, &t0).Sub(&c2, &t2).Add(&c2, &t1)
+
+	z.C0.Set(&c0)
+	z.C1.Set(&c1)
+	z.C2.Set(&c2)
+
+	return z
+}
+
+// Square sets z to the E12 product of x,x, returns z
+func (z *E12) Square(x *E12) *E12 {
+
+	// Algorithm 16 from https://eprint.iacr.org/2010/354.pdf
+	var c4, c5, c1, c2, c3, c0 E4
+	c4.Mul(&x.C0, &x.C1).Double(&c4)
+	c5.Square(&x.C2)
+	c1.MulByNonResidue(&c5).Add(&c1, &c4)
+	c2.Sub(&c4, &c5)
+	c3.Square(&x.C0)
+	c4.Sub(&x.C0, &x.C1).Add(&c4, &x.C2)
+	c5.Mul(&x.C1, &x.C2).Double(&c5)
+	c4.Square(&c4)
+	c0.MulByNonResidue(&c5).Add(&c0, &c3)
+	z.C2.Add(&c2, &c4).Add(&z.C2, &c5).Sub(&z.C2, &c3)
+	z.C0.Set(&c0)
+	z.C1.Set(&c1)
+
+	return z
+}
+
+// Inverse an element in E12
+func (z *E12) Inverse(x *E12) *E12 {
+	// Algorithm 17 from https://eprint.iacr.org/2010/354.pdf
+	// step 9 is wrong in the paper it's t1-t4
+	var t0, t1, t2, t3, t4, t5, t6, c0, c1, c2, d1, d2 E4
+	t0.Square(&x.C0)
+	t1.Square(&x.C1)
+	t2.Square(&x.C2)
+	t3.Mul(&x.C0, &x.C1)
+	t4.Mul(&x.C0, &x.C2)
+	t5.Mul(&x.C1, &x.C2)
+	c0.MulByNonResidue(&t5).Sub(&t0, &c0)
+	c1.MulByNonResidue(&t2).Sub(&c1, &t3)
+	c2.Sub(&t1, &t4)
+	t6.Mul(&x.C0, &c0)
+	d1.Mul(&x.C2, &c1)
+	d2.Mul(&x.C1, &c2)
+	d1.Add(&d1, &d2).MulByNonResidue(&d1)
+	t6.Add(&t6, &d1)
+	t6.Inverse(&t6)
+	z.C0.Mul(&c0, &t6)
+	z.C1.Mul(&c1, &t6)
+	z.C2.Mul(&c2, &t6)
+
+	return z
+}
+
+// Exp sets z=x**e and returns it
+func (z *E12) Exp(x *E12, e big.Int) *E12 {
+	var res E12
+	res.SetOne()
+	b := e.Bytes()
+	for i := range b {
+		w := b[i]
+		mask := byte(0x80)
+		for j := 7; j >= 0; j-- {
+			res.Square(&res)
+			if (w&mask)>>j != 0 {
+				res.Mul(&res, x)
+			}
+			mask = mask >> 1
+		}
+	}
+	z.Set(&res)
+	return z
+}
+
+// InverseUnitary inverse a unitary element
+func (z *E12) InverseUnitary(x *E12) *E12 {
+	return z.Conjugate(x)
+}
+
+// Conjugate set z to x conjugated and return z
+func (z *E12) Conjugate(x *E12) *E12 {
+	z.C0.Conjugate(&x.C0)
+	z.C1.Conjugate(&x.C1).Neg(&z.C1)
+	z.C2.Conjugate(&x.C2)
+	return z
+}
+
+// MulBy01 multiplication by sparse element (c0,c1,0)
+func (z *E12) MulBy01(c0, c1 *E4) *E12 {
+
+	var a, b, tmp, t0, t1, t2 E4
+
+	a.Mul(&z.C0, c0)
+	b.Mul(&z.C1, c1)
+
+	tmp.Add(&z.C1, &z.C2)
+	t0.Mul(c1, &tmp)
+	t0.Sub(&t0, &b)
+	t0.MulByNonResidue(&t0)
+	t0.Add(&t0, &a)
+
+	tmp.Add(&z.C0, &z.C2)
+	t2.Mul(c0, &tmp)
+	t2.Sub(&t2, &a)
+	t2.Add(&t2, &b)
+
+	t1.Add(c0, c1)
+	tmp.Add(&z.C0, &z.C1)
+	t1.Mul(&t1, &tmp)
+	t1.Sub(&t1, &a)
+	t1.Sub(&t1, &b)
+
+	z.C0.Set(&t0)
+	z.C1.Set(&t1)
+	z.C2.Set(&t2)
+
+	return z
+}
+
+// MulByE2 multiplies an element in E12 by an element in E2
+func (z *E12) MulByE2(x *E12, y *E4) *E12 {
+	var yCopy E4
+	yCopy.Set(y)
+	z.C0.Mul(&x.C0, &yCopy)
+	z.C1.Mul(&x.C1, &yCopy)
+	z.C2.Mul(&x.C2, &yCopy)
+	return z
+}