crypto/bls12381: update kilic library

core/vm/contracts.go

@@ -679,11 +679,11 @@ func (c *bls12381G1Add) Run(input []byte) ([]byte, error) {
 	g := bls12381.NewG1()
 
 	// Decode G1 point p_0
-	if p0, err = g.DecodePoint(input[:128]); err != nil {
+	if p0, err = decodePointG1(g, input[:128]); err != nil {
 		return nil, err
 	}
 	// Decode G1 point p_1
-	if p1, err = g.DecodePoint(input[128:]); err != nil {
+	if p1, err = decodePointG1(g, input[128:]); err != nil {
 		return nil, err
 	}
 
@@ -692,7 +692,7 @@ func (c *bls12381G1Add) Run(input []byte) ([]byte, error) {
 	g.Add(r, p0, p1)
 
 	// Encode the G1 point result into 128 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG1(g, r), nil
 }
 
 // bls12381G1Mul implements EIP-2537 G1Mul precompile.
@@ -717,7 +717,7 @@ func (c *bls12381G1Mul) Run(input []byte) ([]byte, error) {
 	g := bls12381.NewG1()
 
 	// Decode G1 point
-	if p0, err = g.DecodePoint(input[:128]); err != nil {
+	if p0, err = decodePointG1(g, input[:128]); err != nil {
 		return nil, err
 	}
 	// Decode scalar value
@@ -725,10 +725,10 @@ func (c *bls12381G1Mul) Run(input []byte) ([]byte, error) {
 
 	// Compute r = e * p_0
 	r := g.New()
-	g.MulScalar(r, p0, e)
+	g.MulScalarBig(r, p0, e)
 
 	// Encode the G1 point into 128 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG1(g, r), nil
 }
 
 // bls12381G1MultiExp implements EIP-2537 G1MultiExp precompile.
@@ -773,7 +773,7 @@ func (c *bls12381G1MultiExp) Run(input []byte) ([]byte, error) {
 		off := 160 * i
 		t0, t1, t2 := off, off+128, off+160
 		// Decode G1 point
-		if points[i], err = g.DecodePoint(input[t0:t1]); err != nil {
+		if points[i], err = decodePointG1(g, input[t0:t1]); err != nil {
 			return nil, err
 		}
 		// Decode scalar value
@@ -782,10 +782,10 @@ func (c *bls12381G1MultiExp) Run(input []byte) ([]byte, error) {
 
 	// Compute r = e_0 * p_0 + e_1 * p_1 + ... + e_(k-1) * p_(k-1)
 	r := g.New()
-	g.MultiExp(r, points, scalars)
+	g.MultiExpBig(r, points, scalars)
 
 	// Encode the G1 point to 128 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG1(g, r), nil
 }
 
 // bls12381G2Add implements EIP-2537 G2Add precompile.
@@ -811,11 +811,11 @@ func (c *bls12381G2Add) Run(input []byte) ([]byte, error) {
 	r := g.New()
 
 	// Decode G2 point p_0
-	if p0, err = g.DecodePoint(input[:256]); err != nil {
+	if p0, err = decodePointG2(g, input[:256]); err != nil {
 		return nil, err
 	}
 	// Decode G2 point p_1
-	if p1, err = g.DecodePoint(input[256:]); err != nil {
+	if p1, err = decodePointG2(g, input[256:]); err != nil {
 		return nil, err
 	}
 
@@ -823,7 +823,7 @@ func (c *bls12381G2Add) Run(input []byte) ([]byte, error) {
 	g.Add(r, p0, p1)
 
 	// Encode the G2 point into 256 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG2(g, r), nil
 }
 
 // bls12381G2Mul implements EIP-2537 G2Mul precompile.
@@ -848,7 +848,7 @@ func (c *bls12381G2Mul) Run(input []byte) ([]byte, error) {
 	g := bls12381.NewG2()
 
 	// Decode G2 point
-	if p0, err = g.DecodePoint(input[:256]); err != nil {
+	if p0, err = decodePointG2(g, input[:256]); err != nil {
 		return nil, err
 	}
 	// Decode scalar value
@@ -856,10 +856,10 @@ func (c *bls12381G2Mul) Run(input []byte) ([]byte, error) {
 
 	// Compute r = e * p_0
 	r := g.New()
-	g.MulScalar(r, p0, e)
+	g.MulScalarBig(r, p0, e)
 
 	// Encode the G2 point into 256 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG2(g, r), nil
 }
 
 // bls12381G2MultiExp implements EIP-2537 G2MultiExp precompile.
@@ -903,8 +903,8 @@ func (c *bls12381G2MultiExp) Run(input []byte) ([]byte, error) {
 	for i := 0; i < k; i++ {
 		off := 288 * i
 		t0, t1, t2 := off, off+256, off+288
-		// Decode G1 point
+		// Decode G2 point
-		if points[i], err = g.DecodePoint(input[t0:t1]); err != nil {
+		if points[i], err = decodePointG2(g, input[t0:t1]); err != nil {
 			return nil, err
 		}
 		// Decode scalar value
@@ -913,10 +913,10 @@ func (c *bls12381G2MultiExp) Run(input []byte) ([]byte, error) {
 
 	// Compute r = e_0 * p_0 + e_1 * p_1 + ... + e_(k-1) * p_(k-1)
 	r := g.New()
-	g.MultiExp(r, points, scalars)
+	g.MultiExpBig(r, points, scalars)
 
 	// Encode the G2 point to 256 bytes.
-	return g.EncodePoint(r), nil
+	return encodePointG2(g, r), nil
 }
 
 // bls12381Pairing implements EIP-2537 Pairing precompile.
@@ -940,7 +940,7 @@ func (c *bls12381Pairing) Run(input []byte) ([]byte, error) {
 	}
 
 	// Initialize BLS12-381 pairing engine
-	e := bls12381.NewPairingEngine()
+	e := bls12381.NewEngine()
 	g1, g2 := e.G1, e.G2
 
 	// Decode pairs
@@ -949,12 +949,12 @@ func (c *bls12381Pairing) Run(input []byte) ([]byte, error) {
 		t0, t1, t2 := off, off+128, off+384
 
 		// Decode G1 point
-		p1, err := g1.DecodePoint(input[t0:t1])
+		p1, err := decodePointG1(g1, input[t0:t1])
 		if err != nil {
 			return nil, err
 		}
 		// Decode G2 point
-		p2, err := g2.DecodePoint(input[t1:t2])
+		p2, err := decodePointG2(g2, input[t1:t2])
 		if err != nil {
 			return nil, err
 		}
@@ -981,6 +981,55 @@ func (c *bls12381Pairing) Run(input []byte) ([]byte, error) {
 	return out, nil
 }
 
+func decodePointG1(g *bls12381.G1, in []byte) (*bls12381.PointG1, error) {
+	if len(in) != 128 {
+		return nil, errors.New("invalid g1 point length")
+	}
+	pointBytes := make([]byte, 96)
+	// decode x
+	xBytes, err := decodeBLS12381FieldElement(in[:64])
+	if err != nil {
+		return nil, err
+	}
+	// decode y
+	yBytes, err := decodeBLS12381FieldElement(in[64:])
+	if err != nil {
+		return nil, err
+	}
+	copy(pointBytes[:48], xBytes)
+	copy(pointBytes[48:], yBytes)
+	return g.FromBytes(pointBytes)
+}
+
+// decodePointG2 given encoded (x, y) coordinates in 256 bytes returns a valid G2 Point.
+func decodePointG2(g *bls12381.G2, in []byte) (*bls12381.PointG2, error) {
+	if len(in) != 256 {
+		return nil, errors.New("invalid g2 point length")
+	}
+	pointBytes := make([]byte, 192)
+	x0Bytes, err := decodeBLS12381FieldElement(in[:64])
+	if err != nil {
+		return nil, err
+	}
+	x1Bytes, err := decodeBLS12381FieldElement(in[64:128])
+	if err != nil {
+		return nil, err
+	}
+	y0Bytes, err := decodeBLS12381FieldElement(in[128:192])
+	if err != nil {
+		return nil, err
+	}
+	y1Bytes, err := decodeBLS12381FieldElement(in[192:])
+	if err != nil {
+		return nil, err
+	}
+	copy(pointBytes[:48], x1Bytes)
+	copy(pointBytes[48:96], x0Bytes)
+	copy(pointBytes[96:144], y1Bytes)
+	copy(pointBytes[144:192], y0Bytes)
+	return g.FromBytes(pointBytes)
+}
+
 // decodeBLS12381FieldElement decodes BLS12-381 elliptic curve field element.
 // Removes top 16 bytes of 64 byte input.
 func decodeBLS12381FieldElement(in []byte) ([]byte, error) {
@@ -998,6 +1047,31 @@ func decodeBLS12381FieldElement(in []byte) ([]byte, error) {
 	return out, nil
 }
 
+// encodePointG1 encodes a point into 128 bytes.
+func encodePointG1(g *bls12381.G1, p *bls12381.PointG1) []byte {
+	outRaw := g.ToBytes(p)
+	out := make([]byte, 128)
+	// encode x
+	copy(out[16:], outRaw[:48])
+	// encode y
+	copy(out[64+16:], outRaw[48:])
+	return out
+}
+
+// encodePointG2 encodes a point into 256 bytes.
+func encodePointG2(g *bls12381.G2, p *bls12381.PointG2) []byte {
+	// outRaw is 96 bytes
+	outRaw := g.ToBytes(p)
+	out := make([]byte, 256)
+	// encode x
+	copy(out[16:16+48], outRaw[48:96])
+	copy(out[80:80+48], outRaw[:48])
+	// encode y
+	copy(out[144:144+48], outRaw[144:])
+	copy(out[208:208+48], outRaw[96:144])
+	return out
+}
+
 // bls12381MapG1 implements EIP-2537 MapG1 precompile.
 type bls12381MapG1 struct{}
 
@@ -1030,7 +1104,7 @@ func (c *bls12381MapG1) Run(input []byte) ([]byte, error) {
 	}
 
 	// Encode the G1 point to 128 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG1(g, r), nil
 }
 
 // bls12381MapG2 implements EIP-2537 MapG2 precompile.
@@ -1072,7 +1146,7 @@ func (c *bls12381MapG2) Run(input []byte) ([]byte, error) {
 	}
 
 	// Encode the G2 point to 256 bytes
-	return g.EncodePoint(r), nil
+	return encodePointG2(g, r), nil
 }
 
 // kzgPointEvaluation implements the EIP-4844 point evaluation precompile.

core/vm/contracts_test.go

@@ -372,7 +372,7 @@ func BenchmarkPrecompiledBLS12381G1MultiExpWorstCase(b *testing.B) {
 		Name:        "WorstCaseG1",
 		NoBenchmark: false,
 	}
-	benchmarkPrecompiled("0c", testcase, b)
+	benchmarkPrecompiled("f0c", testcase, b)
 }
 
 // BenchmarkPrecompiledBLS12381G2MultiExpWorstCase benchmarks the worst case we could find that still fits a gaslimit of 10MGas.
@@ -393,5 +393,5 @@ func BenchmarkPrecompiledBLS12381G2MultiExpWorstCase(b *testing.B) {
 		Name:        "WorstCaseG2",
 		NoBenchmark: false,
 	}
-	benchmarkPrecompiled("0f", testcase, b)
+	benchmarkPrecompiled("f0f", testcase, b)
 }

crypto/bls12381/arithmetic_decl.go

@@ -1,21 +1,4 @@
-// Copyright 2020 The go-ethereum Authors
+// +build amd64,!generic
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
-//go:build (amd64 && blsasm) || (amd64 && blsadx)
-// +build amd64,blsasm amd64,blsadx
 
 package bls12381
 
@@ -24,13 +7,22 @@ import (
 )
 
 func init() {
-	if !enableADX || !cpu.X86.HasADX || !cpu.X86.HasBMI2 {
+	if !cpu.X86.HasADX || !cpu.X86.HasBMI2 {
 		mul = mulNoADX
+		wmul = wmulNoADX
+		fromWide = montRedNoADX
+		mulFR = mulNoADXFR
+		wmulFR = wmulNoADXFR
+		wfp2Mul = wfp2MulGeneric
+		wfp2Square = wfp2SquareGeneric
 	}
 }
 
-// Use ADX backend for default
 var mul func(c, a, b *fe) = mulADX
+var wmul func(c *wfe, a, b *fe) = wmulADX
+var fromWide func(c *fe, w *wfe) = montRedADX
+var wfp2Mul func(c *wfe2, a, b *fe2) = wfp2MulADX
+var wfp2Square func(c *wfe2, b *fe2) = wfp2SquareADX
 
 func square(c, a *fe) {
 	mul(c, a, a)
@@ -65,6 +57,9 @@ func doubleAssign(a *fe)
 //go:noescape
 func ldouble(c, a *fe)
 
+//go:noescape
+func ldoubleAssign(a *fe)
+
 //go:noescape
 func sub(c, a, b *fe)
 
@@ -82,3 +77,165 @@ func mulNoADX(c, a, b *fe)
 
 //go:noescape
 func mulADX(c, a, b *fe)
+
+//go:noescape
+func wmulNoADX(c *wfe, a, b *fe)
+
+//go:noescape
+func wmulADX(c *wfe, a, b *fe)
+
+//go:noescape
+func montRedNoADX(a *fe, w *wfe)
+
+//go:noescape
+func montRedADX(a *fe, w *wfe)
+
+//go:noescape
+func lwadd(c, a, b *wfe)
+
+//go:noescape
+func lwaddAssign(a, b *wfe)
+
+//go:noescape
+func wadd(c, a, b *wfe)
+
+//go:noescape
+func lwdouble(c, a *wfe)
+
+//go:noescape
+func wdouble(c, a *wfe)
+
+//go:noescape
+func lwsub(c, a, b *wfe)
+
+//go:noescape
+func lwsubAssign(a, b *wfe)
+
+//go:noescape
+func wsub(c, a, b *wfe)
+
+//go:noescape
+func fp2Add(c, a, b *fe2)
+
+//go:noescape
+func fp2AddAssign(a, b *fe2)
+
+//go:noescape
+func fp2Ladd(c, a, b *fe2)
+
+//go:noescape
+func fp2LaddAssign(a, b *fe2)
+
+//go:noescape
+func fp2DoubleAssign(a *fe2)
+
+//go:noescape
+func fp2Double(c, a *fe2)
+
+//go:noescape
+func fp2Sub(c, a, b *fe2)
+
+//go:noescape
+func fp2SubAssign(a, b *fe2)
+
+//go:noescape
+func mulByNonResidue(c, a *fe2)
+
+//go:noescape
+func mulByNonResidueAssign(a *fe2)
+
+//go:noescape
+func wfp2Add(c, a, b *wfe2)
+
+//go:noescape
+func wfp2AddAssign(a, b *wfe2)
+
+//go:noescape
+func wfp2Ladd(c, a, b *wfe2)
+
+//go:noescape
+func wfp2LaddAssign(a, b *wfe2)
+
+//go:noescape
+func wfp2AddMixed(c, a, b *wfe2)
+
+//go:noescape
+func wfp2AddMixedAssign(a, b *wfe2)
+
+//go:noescape
+func wfp2Sub(c, a, b *wfe2)
+
+//go:noescape
+func wfp2SubAssign(a, b *wfe2)
+
+//go:noescape
+func wfp2SubMixed(c, a, b *wfe2)
+
+//go:noescape
+func wfp2SubMixedAssign(a, b *wfe2)
+
+//go:noescape
+func wfp2Double(c, a *wfe2)
+
+//go:noescape
+func wfp2DoubleAssign(a *wfe2)
+
+//go:noescape
+func wfp2MulByNonResidue(c, a *wfe2)
+
+//go:noescape
+func wfp2MulByNonResidueAssign(a *wfe2)
+
+//go:noescape
+func wfp2SquareADX(c *wfe2, a *fe2)
+
+//go:noescape
+func wfp2MulADX(c *wfe2, a, b *fe2)
+
+var mulFR func(c, a, b *Fr) = mulADXFR
+var wmulFR func(c *wideFr, a, b *Fr) = wmulADXFR
+
+func squareFR(c, a *Fr) {
+	mulFR(c, a, a)
+}
+
+func negFR(c, a *Fr) {
+	if a.IsZero() {
+		c.Set(a)
+	} else {
+		_negFR(c, a)
+	}
+}
+
+//go:noescape
+func addFR(c, a, b *Fr)
+
+//go:noescape
+func laddAssignFR(a, b *Fr)
+
+//go:noescape
+func doubleFR(c, a *Fr)
+
+//go:noescape
+func subFR(c, a, b *Fr)
+
+//go:noescape
+func lsubAssignFR(a, b *Fr)
+
+//go:noescape
+func _negFR(c, a *Fr)
+
+//go:noescape
+func mulNoADXFR(c, a, b *Fr)
+
+//go:noescape
+func mulADXFR(c, a, b *Fr)
+
+//go:noescape
+func wmulADXFR(c *wideFr, a, b *Fr)
+
+//go:noescape
+func wmulNoADXFR(c *wideFr, a, b *Fr)
+
+//go:noescape
+func waddFR(a, b *wideFr)

crypto/bls12381/arithmetic_fallback.go

@@ -1,8 +1,6 @@
-// Native go field arithmetic code is generated with 'goff'
+// +build !amd64 generic
-// https://github.com/ConsenSys/goff
-// Many function signature of field operations are renamed.
 
-// Copyright 2020 ConsenSys AG
+// 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.
@@ -16,23 +14,7 @@
 // See the License for the specific language governing permissions and
 // limitations under the License.
 
-// field modulus q =
+// Code generated by goff (v0.3.5) DO NOT EDIT
-//
-// 4002409555221667393417789825735904156556882819939007885332058136124031650490837864442687629129015664037894272559787
-// Code generated by goff DO NOT EDIT
-// goff version: v0.1.0 - build: 790f1f56eac432441e043abff8819eacddd1d668
-// fe are assumed to be in Montgomery form in all methods
-
-// /!\ WARNING /!\
-// this code has not been audited and is provided as-is. In particular,
-// there is no security guarantees such as constant time implementation
-// or side-channel attack resistance
-// /!\ WARNING /!\
-
-// Package bls (generated by goff) contains field arithmetics operations
-
-//go:build !amd64 || (!blsasm && !blsadx)
-// +build !amd64 !blsasm,!blsadx
 
 package bls12381
 
@@ -40,440 +22,6 @@ import (
 	"math/bits"
 )
 
-func add(z, x, y *fe) {
-	var carry uint64
-
-	z[0], carry = bits.Add64(x[0], y[0], 0)
-	z[1], carry = bits.Add64(x[1], y[1], carry)
-	z[2], carry = bits.Add64(x[2], y[2], carry)
-	z[3], carry = bits.Add64(x[3], y[3], carry)
-	z[4], carry = bits.Add64(x[4], y[4], carry)
-	z[5], _ = bits.Add64(x[5], y[5], carry)
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(z[5] < 1873798617647539866 || (z[5] == 1873798617647539866 && (z[4] < 5412103778470702295 || (z[4] == 5412103778470702295 && (z[3] < 7239337960414712511 || (z[3] == 7239337960414712511 && (z[2] < 7435674573564081700 || (z[2] == 7435674573564081700 && (z[1] < 2210141511517208575 || (z[1] == 2210141511517208575 && (z[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 13402431016077863595, 0)
-		z[1], b = bits.Sub64(z[1], 2210141511517208575, b)
-		z[2], b = bits.Sub64(z[2], 7435674573564081700, b)
-		z[3], b = bits.Sub64(z[3], 7239337960414712511, b)
-		z[4], b = bits.Sub64(z[4], 5412103778470702295, b)
-		z[5], _ = bits.Sub64(z[5], 1873798617647539866, b)
-	}
-}
-
-func addAssign(x, y *fe) {
-	var carry uint64
-
-	x[0], carry = bits.Add64(x[0], y[0], 0)
-	x[1], carry = bits.Add64(x[1], y[1], carry)
-	x[2], carry = bits.Add64(x[2], y[2], carry)
-	x[3], carry = bits.Add64(x[3], y[3], carry)
-	x[4], carry = bits.Add64(x[4], y[4], carry)
-	x[5], _ = bits.Add64(x[5], y[5], carry)
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(x[5] < 1873798617647539866 || (x[5] == 1873798617647539866 && (x[4] < 5412103778470702295 || (x[4] == 5412103778470702295 && (x[3] < 7239337960414712511 || (x[3] == 7239337960414712511 && (x[2] < 7435674573564081700 || (x[2] == 7435674573564081700 && (x[1] < 2210141511517208575 || (x[1] == 2210141511517208575 && (x[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		x[0], b = bits.Sub64(x[0], 13402431016077863595, 0)
-		x[1], b = bits.Sub64(x[1], 2210141511517208575, b)
-		x[2], b = bits.Sub64(x[2], 7435674573564081700, b)
-		x[3], b = bits.Sub64(x[3], 7239337960414712511, b)
-		x[4], b = bits.Sub64(x[4], 5412103778470702295, b)
-		x[5], _ = bits.Sub64(x[5], 1873798617647539866, b)
-	}
-}
-
-func ladd(z, x, y *fe) {
-	var carry uint64
-	z[0], carry = bits.Add64(x[0], y[0], 0)
-	z[1], carry = bits.Add64(x[1], y[1], carry)
-	z[2], carry = bits.Add64(x[2], y[2], carry)
-	z[3], carry = bits.Add64(x[3], y[3], carry)
-	z[4], carry = bits.Add64(x[4], y[4], carry)
-	z[5], _ = bits.Add64(x[5], y[5], carry)
-}
-
-func laddAssign(x, y *fe) {
-	var carry uint64
-	x[0], carry = bits.Add64(x[0], y[0], 0)
-	x[1], carry = bits.Add64(x[1], y[1], carry)
-	x[2], carry = bits.Add64(x[2], y[2], carry)
-	x[3], carry = bits.Add64(x[3], y[3], carry)
-	x[4], carry = bits.Add64(x[4], y[4], carry)
-	x[5], _ = bits.Add64(x[5], y[5], carry)
-}
-
-func double(z, x *fe) {
-	var carry uint64
-
-	z[0], carry = bits.Add64(x[0], x[0], 0)
-	z[1], carry = bits.Add64(x[1], x[1], carry)
-	z[2], carry = bits.Add64(x[2], x[2], carry)
-	z[3], carry = bits.Add64(x[3], x[3], carry)
-	z[4], carry = bits.Add64(x[4], x[4], carry)
-	z[5], _ = bits.Add64(x[5], x[5], carry)
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(z[5] < 1873798617647539866 || (z[5] == 1873798617647539866 && (z[4] < 5412103778470702295 || (z[4] == 5412103778470702295 && (z[3] < 7239337960414712511 || (z[3] == 7239337960414712511 && (z[2] < 7435674573564081700 || (z[2] == 7435674573564081700 && (z[1] < 2210141511517208575 || (z[1] == 2210141511517208575 && (z[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 13402431016077863595, 0)
-		z[1], b = bits.Sub64(z[1], 2210141511517208575, b)
-		z[2], b = bits.Sub64(z[2], 7435674573564081700, b)
-		z[3], b = bits.Sub64(z[3], 7239337960414712511, b)
-		z[4], b = bits.Sub64(z[4], 5412103778470702295, b)
-		z[5], _ = bits.Sub64(z[5], 1873798617647539866, b)
-	}
-}
-
-func doubleAssign(z *fe) {
-	var carry uint64
-
-	z[0], carry = bits.Add64(z[0], z[0], 0)
-	z[1], carry = bits.Add64(z[1], z[1], carry)
-	z[2], carry = bits.Add64(z[2], z[2], carry)
-	z[3], carry = bits.Add64(z[3], z[3], carry)
-	z[4], carry = bits.Add64(z[4], z[4], carry)
-	z[5], _ = bits.Add64(z[5], z[5], carry)
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(z[5] < 1873798617647539866 || (z[5] == 1873798617647539866 && (z[4] < 5412103778470702295 || (z[4] == 5412103778470702295 && (z[3] < 7239337960414712511 || (z[3] == 7239337960414712511 && (z[2] < 7435674573564081700 || (z[2] == 7435674573564081700 && (z[1] < 2210141511517208575 || (z[1] == 2210141511517208575 && (z[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 13402431016077863595, 0)
-		z[1], b = bits.Sub64(z[1], 2210141511517208575, b)
-		z[2], b = bits.Sub64(z[2], 7435674573564081700, b)
-		z[3], b = bits.Sub64(z[3], 7239337960414712511, b)
-		z[4], b = bits.Sub64(z[4], 5412103778470702295, b)
-		z[5], _ = bits.Sub64(z[5], 1873798617647539866, b)
-	}
-}
-
-func ldouble(z, x *fe) {
-	var carry uint64
-
-	z[0], carry = bits.Add64(x[0], x[0], 0)
-	z[1], carry = bits.Add64(x[1], x[1], carry)
-	z[2], carry = bits.Add64(x[2], x[2], carry)
-	z[3], carry = bits.Add64(x[3], x[3], carry)
-	z[4], carry = bits.Add64(x[4], x[4], carry)
-	z[5], _ = bits.Add64(x[5], x[5], carry)
-}
-
-func sub(z, x, y *fe) {
-	var b uint64
-	z[0], b = bits.Sub64(x[0], y[0], 0)
-	z[1], b = bits.Sub64(x[1], y[1], b)
-	z[2], b = bits.Sub64(x[2], y[2], b)
-	z[3], b = bits.Sub64(x[3], y[3], b)
-	z[4], b = bits.Sub64(x[4], y[4], b)
-	z[5], b = bits.Sub64(x[5], y[5], b)
-	if b != 0 {
-		var c uint64
-		z[0], c = bits.Add64(z[0], 13402431016077863595, 0)
-		z[1], c = bits.Add64(z[1], 2210141511517208575, c)
-		z[2], c = bits.Add64(z[2], 7435674573564081700, c)
-		z[3], c = bits.Add64(z[3], 7239337960414712511, c)
-		z[4], c = bits.Add64(z[4], 5412103778470702295, c)
-		z[5], _ = bits.Add64(z[5], 1873798617647539866, c)
-	}
-}
-
-func subAssign(z, x *fe) {
-	var b uint64
-	z[0], b = bits.Sub64(z[0], x[0], 0)
-	z[1], b = bits.Sub64(z[1], x[1], b)
-	z[2], b = bits.Sub64(z[2], x[2], b)
-	z[3], b = bits.Sub64(z[3], x[3], b)
-	z[4], b = bits.Sub64(z[4], x[4], b)
-	z[5], b = bits.Sub64(z[5], x[5], b)
-	if b != 0 {
-		var c uint64
-		z[0], c = bits.Add64(z[0], 13402431016077863595, 0)
-		z[1], c = bits.Add64(z[1], 2210141511517208575, c)
-		z[2], c = bits.Add64(z[2], 7435674573564081700, c)
-		z[3], c = bits.Add64(z[3], 7239337960414712511, c)
-		z[4], c = bits.Add64(z[4], 5412103778470702295, c)
-		z[5], _ = bits.Add64(z[5], 1873798617647539866, c)
-	}
-}
-
-func lsubAssign(z, x *fe) {
-	var b uint64
-	z[0], b = bits.Sub64(z[0], x[0], 0)
-	z[1], b = bits.Sub64(z[1], x[1], b)
-	z[2], b = bits.Sub64(z[2], x[2], b)
-	z[3], b = bits.Sub64(z[3], x[3], b)
-	z[4], b = bits.Sub64(z[4], x[4], b)
-	z[5], _ = bits.Sub64(z[5], x[5], b)
-}
-
-func neg(z *fe, x *fe) {
-	if x.isZero() {
-		z.zero()
-		return
-	}
-	var borrow uint64
-	z[0], borrow = bits.Sub64(13402431016077863595, x[0], 0)
-	z[1], borrow = bits.Sub64(2210141511517208575, x[1], borrow)
-	z[2], borrow = bits.Sub64(7435674573564081700, x[2], borrow)
-	z[3], borrow = bits.Sub64(7239337960414712511, x[3], borrow)
-	z[4], borrow = bits.Sub64(5412103778470702295, x[4], borrow)
-	z[5], _ = bits.Sub64(1873798617647539866, x[5], borrow)
-}
-
-func mul(z, x, y *fe) {
-	var t [6]uint64
-	var c [3]uint64
-	{
-		// round 0
-		v := x[0]
-		c[1], c[0] = bits.Mul64(v, y[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd1(v, y[1], c[1])
-		c[2], t[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd1(v, y[2], c[1])
-		c[2], t[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd1(v, y[3], c[1])
-		c[2], t[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd1(v, y[4], c[1])
-		c[2], t[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd1(v, y[5], c[1])
-		t[5], t[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-	{
-		// round 1
-		v := x[1]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		c[2], t[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd2(v, y[4], c[1], t[4])
-		c[2], t[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd2(v, y[5], c[1], t[5])
-		t[5], t[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-	{
-		// round 2
-		v := x[2]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		c[2], t[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd2(v, y[4], c[1], t[4])
-		c[2], t[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd2(v, y[5], c[1], t[5])
-		t[5], t[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-	{
-		// round 3
-		v := x[3]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		c[2], t[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd2(v, y[4], c[1], t[4])
-		c[2], t[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd2(v, y[5], c[1], t[5])
-		t[5], t[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-	{
-		// round 4
-		v := x[4]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], t[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], t[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		c[2], t[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd2(v, y[4], c[1], t[4])
-		c[2], t[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd2(v, y[5], c[1], t[5])
-		t[5], t[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-	{
-		// round 5
-		v := x[5]
-		c[1], c[0] = madd1(v, y[0], t[0])
-		m := c[0] * 9940570264628428797
-		c[2] = madd0(m, 13402431016077863595, c[0])
-		c[1], c[0] = madd2(v, y[1], c[1], t[1])
-		c[2], z[0] = madd2(m, 2210141511517208575, c[2], c[0])
-		c[1], c[0] = madd2(v, y[2], c[1], t[2])
-		c[2], z[1] = madd2(m, 7435674573564081700, c[2], c[0])
-		c[1], c[0] = madd2(v, y[3], c[1], t[3])
-		c[2], z[2] = madd2(m, 7239337960414712511, c[2], c[0])
-		c[1], c[0] = madd2(v, y[4], c[1], t[4])
-		c[2], z[3] = madd2(m, 5412103778470702295, c[2], c[0])
-		c[1], c[0] = madd2(v, y[5], c[1], t[5])
-		z[5], z[4] = madd3(m, 1873798617647539866, c[0], c[2], c[1])
-	}
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(z[5] < 1873798617647539866 || (z[5] == 1873798617647539866 && (z[4] < 5412103778470702295 || (z[4] == 5412103778470702295 && (z[3] < 7239337960414712511 || (z[3] == 7239337960414712511 && (z[2] < 7435674573564081700 || (z[2] == 7435674573564081700 && (z[1] < 2210141511517208575 || (z[1] == 2210141511517208575 && (z[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 13402431016077863595, 0)
-		z[1], b = bits.Sub64(z[1], 2210141511517208575, b)
-		z[2], b = bits.Sub64(z[2], 7435674573564081700, b)
-		z[3], b = bits.Sub64(z[3], 7239337960414712511, b)
-		z[4], b = bits.Sub64(z[4], 5412103778470702295, b)
-		z[5], _ = bits.Sub64(z[5], 1873798617647539866, b)
-	}
-}
-
-func square(z, x *fe) {
-
-	var p [6]uint64
-
-	var u, v uint64
-	{
-		// round 0
-		u, p[0] = bits.Mul64(x[0], x[0])
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		var t uint64
-		t, u, v = madd1sb(x[0], x[1], u)
-		C, p[0] = madd2(m, 2210141511517208575, v, C)
-		t, u, v = madd1s(x[0], x[2], t, u)
-		C, p[1] = madd2(m, 7435674573564081700, v, C)
-		t, u, v = madd1s(x[0], x[3], t, u)
-		C, p[2] = madd2(m, 7239337960414712511, v, C)
-		t, u, v = madd1s(x[0], x[4], t, u)
-		C, p[3] = madd2(m, 5412103778470702295, v, C)
-		_, u, v = madd1s(x[0], x[5], t, u)
-		p[5], p[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-	{
-		// round 1
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		u, v = madd1(x[1], x[1], p[1])
-		C, p[0] = madd2(m, 2210141511517208575, v, C)
-		var t uint64
-		t, u, v = madd2sb(x[1], x[2], p[2], u)
-		C, p[1] = madd2(m, 7435674573564081700, v, C)
-		t, u, v = madd2s(x[1], x[3], p[3], t, u)
-		C, p[2] = madd2(m, 7239337960414712511, v, C)
-		t, u, v = madd2s(x[1], x[4], p[4], t, u)
-		C, p[3] = madd2(m, 5412103778470702295, v, C)
-		_, u, v = madd2s(x[1], x[5], p[5], t, u)
-		p[5], p[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-	{
-		// round 2
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		C, p[0] = madd2(m, 2210141511517208575, p[1], C)
-		u, v = madd1(x[2], x[2], p[2])
-		C, p[1] = madd2(m, 7435674573564081700, v, C)
-		var t uint64
-		t, u, v = madd2sb(x[2], x[3], p[3], u)
-		C, p[2] = madd2(m, 7239337960414712511, v, C)
-		t, u, v = madd2s(x[2], x[4], p[4], t, u)
-		C, p[3] = madd2(m, 5412103778470702295, v, C)
-		_, u, v = madd2s(x[2], x[5], p[5], t, u)
-		p[5], p[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-	{
-		// round 3
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		C, p[0] = madd2(m, 2210141511517208575, p[1], C)
-		C, p[1] = madd2(m, 7435674573564081700, p[2], C)
-		u, v = madd1(x[3], x[3], p[3])
-		C, p[2] = madd2(m, 7239337960414712511, v, C)
-		var t uint64
-		t, u, v = madd2sb(x[3], x[4], p[4], u)
-		C, p[3] = madd2(m, 5412103778470702295, v, C)
-		_, u, v = madd2s(x[3], x[5], p[5], t, u)
-		p[5], p[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-	{
-		// round 4
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		C, p[0] = madd2(m, 2210141511517208575, p[1], C)
-		C, p[1] = madd2(m, 7435674573564081700, p[2], C)
-		C, p[2] = madd2(m, 7239337960414712511, p[3], C)
-		u, v = madd1(x[4], x[4], p[4])
-		C, p[3] = madd2(m, 5412103778470702295, v, C)
-		_, u, v = madd2sb(x[4], x[5], p[5], u)
-		p[5], p[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-	{
-		// round 5
-		m := p[0] * 9940570264628428797
-		C := madd0(m, 13402431016077863595, p[0])
-		C, z[0] = madd2(m, 2210141511517208575, p[1], C)
-		C, z[1] = madd2(m, 7435674573564081700, p[2], C)
-		C, z[2] = madd2(m, 7239337960414712511, p[3], C)
-		C, z[3] = madd2(m, 5412103778470702295, p[4], C)
-		u, v = madd1(x[5], x[5], p[5])
-		z[5], z[4] = madd3(m, 1873798617647539866, v, C, u)
-	}
-
-	// if z > q --> z -= q
-	// note: this is NOT constant time
-	if !(z[5] < 1873798617647539866 || (z[5] == 1873798617647539866 && (z[4] < 5412103778470702295 || (z[4] == 5412103778470702295 && (z[3] < 7239337960414712511 || (z[3] == 7239337960414712511 && (z[2] < 7435674573564081700 || (z[2] == 7435674573564081700 && (z[1] < 2210141511517208575 || (z[1] == 2210141511517208575 && (z[0] < 13402431016077863595))))))))))) {
-		var b uint64
-		z[0], b = bits.Sub64(z[0], 13402431016077863595, 0)
-		z[1], b = bits.Sub64(z[1], 2210141511517208575, b)
-		z[2], b = bits.Sub64(z[2], 7435674573564081700, b)
-		z[3], b = bits.Sub64(z[3], 7239337960414712511, b)
-		z[4], b = bits.Sub64(z[4], 5412103778470702295, b)
-		z[5], _ = bits.Sub64(z[5], 1873798617647539866, b)
-	}
-}
-
-// arith.go
-// Copyright 2020 ConsenSys AG
-//
-// 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.
-
-// Code generated by goff DO NOT EDIT
-
-func madd(a, b, t, u, v uint64) (uint64, uint64, uint64) {
-	var carry uint64
-	hi, lo := bits.Mul64(a, b)
-	v, carry = bits.Add64(lo, v, 0)
-	u, carry = bits.Add64(hi, u, carry)
-	t, _ = bits.Add64(t, 0, carry)
-	return t, u, v
-}
-
 // madd0 hi = a*b + c (discards lo bits)
 func madd0(a, b, c uint64) (hi uint64) {
 	var carry, lo uint64
@@ -503,59 +51,6 @@ func madd2(a, b, c, d uint64) (hi uint64, lo uint64) {
 	return
 }
 
-// madd2s superhi, hi, lo = 2*a*b + c + d + e
-func madd2s(a, b, c, d, e uint64) (superhi, hi, lo uint64) {
-	var carry, sum uint64
-
-	hi, lo = bits.Mul64(a, b)
-	lo, carry = bits.Add64(lo, lo, 0)
-	hi, superhi = bits.Add64(hi, hi, carry)
-
-	sum, carry = bits.Add64(c, e, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	lo, carry = bits.Add64(lo, sum, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	hi, _ = bits.Add64(hi, 0, d)
-	return
-}
-
-func madd1s(a, b, d, e uint64) (superhi, hi, lo uint64) {
-	var carry uint64
-
-	hi, lo = bits.Mul64(a, b)
-	lo, carry = bits.Add64(lo, lo, 0)
-	hi, superhi = bits.Add64(hi, hi, carry)
-	lo, carry = bits.Add64(lo, e, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	hi, _ = bits.Add64(hi, 0, d)
-	return
-}
-
-func madd2sb(a, b, c, e uint64) (superhi, hi, lo uint64) {
-	var carry, sum uint64
-
-	hi, lo = bits.Mul64(a, b)
-	lo, carry = bits.Add64(lo, lo, 0)
-	hi, superhi = bits.Add64(hi, hi, carry)
-
-	sum, carry = bits.Add64(c, e, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	lo, carry = bits.Add64(lo, sum, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	return
-}
-
-func madd1sb(a, b, e uint64) (superhi, hi, lo uint64) {
-	var carry uint64
-
-	hi, lo = bits.Mul64(a, b)
-	lo, carry = bits.Add64(lo, lo, 0)
-	hi, superhi = bits.Add64(hi, hi, carry)
-	lo, carry = bits.Add64(lo, e, 0)
-	hi, _ = bits.Add64(hi, 0, carry)
-	return
-}
-
 func madd3(a, b, c, d, e uint64) (hi uint64, lo uint64) {
 	var carry uint64
 	hi, lo = bits.Mul64(a, b)

crypto/bls12381/bls12_381.go

@@ -1,79 +1,89 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
-/*
+const fpNumberOfLimbs = 6
-	Field Constants
+const fpByteSize = 48
-*/
+const fpBitSize = 381
+const sixWordBitSize = 384
 
-// Base field modulus
+// Base Field
 // p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab
-
-// Size of six words
 // r = 2 ^ 384
 
 // modulus = p
 var modulus = fe{0xb9feffffffffaaab, 0x1eabfffeb153ffff, 0x6730d2a0f6b0f624, 0x64774b84f38512bf, 0x4b1ba7b6434bacd7, 0x1a0111ea397fe69a}
 
-var (
+// -p^(-1) mod 2^64
-	// -p^(-1) mod 2^64
+var inp uint64 = 0x89f3fffcfffcfffd
-	inp uint64 = 0x89f3fffcfffcfffd
-	// This value is used in assembly code
-	_ = inp
-)
 
-// r mod p
+// r1 = r mod p
 var r1 = &fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493}
 
-// r^2 mod p
+// one =  mod p
+var one = r1
+
+// zero = 0
+var zero = &fe{}
+
+// r2 = r^2 mod p
 var r2 = &fe{
 	0xf4df1f341c341746, 0x0a76e6a609d104f1, 0x8de5476c4c95b6d5, 0x67eb88a9939d83c0, 0x9a793e85b519952d, 0x11988fe592cae3aa,
 }
 
-// -1 + 0 * u
+// negativeOne = -r mod p
+var negativeOne = &fe{
+	0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206,
+}
+
+// negativeOne2 = -1 + 0 * u
 var negativeOne2 = &fe2{
 	fe{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
 	fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 }
 
-// 2 ^ (-1)
+// twoInv = 2^(-1)
 var twoInv = &fe{0x1804000000015554, 0x855000053ab00001, 0x633cb57c253c276f, 0x6e22d1ec31ebb502, 0xd3916126f2d14ca2, 0x17fbb8571a006596}
 
-// (p - 3) / 4
+// pMinus3Over4 = (p - 3) / 4
 var pMinus3Over4 = bigFromHex("0x680447a8e5ff9a692c6e9ed90d2eb35d91dd2e13ce144afd9cc34a83dac3d8907aaffffac54ffffee7fbfffffffeaaa")
 
-// (p + 1) / 4
+// pPlus1Over4 = (p + 1) / 4
 var pPlus1Over4 = bigFromHex("0x680447a8e5ff9a692c6e9ed90d2eb35d91dd2e13ce144afd9cc34a83dac3d8907aaffffac54ffffee7fbfffffffeaab")
 
-// (p - 1) / 2
+// pMinus1Over2 = (p - 1) / 2
 var pMinus1Over2 = bigFromHex("0xd0088f51cbff34d258dd3db21a5d66bb23ba5c279c2895fb39869507b587b120f55ffff58a9ffffdcff7fffffffd555")
 
-// -1
+// nonResidue1 = -1
 var nonResidue1 = &fe{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206}
 
-// (1 + 1 * u)
+// nonResidue2 = (1 + 1 * u)
 var nonResidue2 = &fe2{
 	fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
 	fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
 }
 
-/*
+// Scalar Field
-	Curve Constants
+// q = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
-*/
+// Size of six words
+// qr = 2 ^ 256
+
+var qBig = bigFromHex("0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001")
+var q = Fr{0xffffffff00000001, 0x53bda402fffe5bfe, 0x3339d80809a1d805, 0x73eda753299d7d48}
+
+// var qmodulus = Fr{0xffffffff00000001, 0x53bda402fffe5bfe, 0x3339d80809a1d805, 0x73eda753299d7d48}
+
+// -q^(-1) mod 2^64
+var qinp uint64 = 0xfffffffeffffffff
+
+// supress warning: qinp is used in assembly code
+var _ = qinp
+
+// qr1 = qr mod q
+var qr1 = &Fr{0x00000001fffffffe, 0x5884b7fa00034802, 0x998c4fefecbc4ff5, 0x1824b159acc5056f}
+
+// qr2 = qr^2 mod q
+var qr2 = &Fr{0xc999e990f3f29c6d, 0x2b6cedcb87925c23, 0x05d314967254398f, 0x0748d9d99f59ff11}
+
+// Curve Constants
 
 // b coefficient for G1
 var b = &fe{0xaa270000000cfff3, 0x53cc0032fc34000a, 0x478fe97a6b0a807f, 0xb1d37ebee6ba24d7, 0x8ec9733bbf78ab2f, 0x09d645513d83de7e}
@@ -84,21 +94,28 @@ var b2 = &fe2{
 	fe{0xaa270000000cfff3, 0x53cc0032fc34000a, 0x478fe97a6b0a807f, 0xb1d37ebee6ba24d7, 0x8ec9733bbf78ab2f, 0x09d645513d83de7e},
 }
 
-// Curve order
+// G1 cofactor
-var q = bigFromHex("0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001")
+var cofactorG1 = bigFromHex("0x396c8c005555e1568c00aaab0000aaab")
 
-// Efficient cofactor of G1
+// G2 cofactor
+var cofactorG2 = bigFromHex("5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5")
+
+// Efficient G1 cofactor
 var cofactorEFFG1 = bigFromHex("0xd201000000010001")
 
-// Efficient cofactor of G2
+// Efficient G2 cofactor
 var cofactorEFFG2 = bigFromHex("0x0bc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff031508ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689f6a359894c0adebbf6b4e8020005aaa95551")
 
+// G1 generator
 var g1One = PointG1{
 	fe{0x5cb38790fd530c16, 0x7817fc679976fff5, 0x154f95c7143ba1c1, 0xf0ae6acdf3d0e747, 0xedce6ecc21dbf440, 0x120177419e0bfb75},
 	fe{0xbaac93d50ce72271, 0x8c22631a7918fd8e, 0xdd595f13570725ce, 0x51ac582950405194, 0x0e1c8c3fad0059c0, 0x0bbc3efc5008a26a},
 	fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
 }
 
+var G1One = g1One
+
+// G2 generator
 var g2One = PointG2{
 	fe2{
 		fe{0xf5f28fa202940a10, 0xb3f5fb2687b4961a, 0xa1a893b53e2ae580, 0x9894999d1a3caee9, 0x6f67b7631863366b, 0x058191924350bcd7},
@@ -114,117 +131,179 @@ var g2One = PointG2{
 	},
 }
 
-/*
+var G2One = g2One
-	Frobenious Coeffs
-*/
 
+// Psi values for faster cofactor clearing
+
+// psix = 1 / (nr ^ (p - 1)/3)
+var psix = fe2{
+	fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+	fe{0x890dc9e4867545c3, 0x2af322533285a5d5, 0x50880866309b7e2c, 0xa20d1b8c7e881024, 0x14e4f04fe2db9068, 0x14e56d3f1564853a},
+}
+
+// psiy = 1 / (nr ^ (p - 1)/2)
+var psiy = fe2{
+	fe{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
+	fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+}
+
+// Frobenius Coeffs
+
+// z = -1
+var frobeniusCoeffs2 = [2]fe{
+	// z ^ (( p ^ 0 - 1) / 2)
+	{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+	// z ^ (( p ^ 1 - 1) / 2)
+	{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
+}
+
+// z = u + 1
 var frobeniusCoeffs61 = [6]fe2{
+	// z ^ (( p ^ 0 - 1) / 3)
 	{
-		fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+		{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( p ^ 1 - 1) / 3)
 	{
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
-		fe{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
+		{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
 	},
+	// z ^ (( p ^ 2 - 1) / 3)
 	{
-		fe{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
+		{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( p ^ 3 - 1) / 3)
 	{
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
-		fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+		{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
 	},
+	// z ^ (( p ^ 4 - 1) / 3)
 	{
-		fe{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
+		{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( p ^ 5 - 1) / 3)
 	{
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
-		fe{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
+		{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
 	},
 }
 
+// z = u + 1
 var frobeniusCoeffs62 = [6]fe2{
+	// z ^ (( 2 * p ^ 0 - 2) / 3)
 	{
-		fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+		{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( 2 * p ^ 1 - 2) / 3)
 	{
-		fe{0x890dc9e4867545c3, 0x2af322533285a5d5, 0x50880866309b7e2c, 0xa20d1b8c7e881024, 0x14e4f04fe2db9068, 0x14e56d3f1564853a},
+		{0x890dc9e4867545c3, 0x2af322533285a5d5, 0x50880866309b7e2c, 0xa20d1b8c7e881024, 0x14e4f04fe2db9068, 0x14e56d3f1564853a},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( 2 * p ^ 2 - 2) / 3)
 	{
-		fe{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
+		{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( 2 * p ^ 3 - 2) / 3)
 	{
-		fe{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
+		{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( 2 * p ^ 4 - 2) / 3)
 	{
-		fe{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
+		{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ (( 2 * p ^ 5 - 2) / 3)
 	{
-		fe{0xecfb361b798dba3a, 0xc100ddb891865a2c, 0x0ec08ff1232bda8e, 0xd5c13cc6f1ca4721, 0x47222a47bf7b5c04, 0x0110f184e51c5f59},
+		{0xecfb361b798dba3a, 0xc100ddb891865a2c, 0x0ec08ff1232bda8e, 0xd5c13cc6f1ca4721, 0x47222a47bf7b5c04, 0x0110f184e51c5f59},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
 }
 
 var frobeniusCoeffs12 = [12]fe2{
+	// z = u + 1
+	// z ^ ((p ^ 0 - 1) / 6)
 	{
-		fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+		{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 1 - 1) / 6)
 	{
-		fe{0x07089552b319d465, 0xc6695f92b50a8313, 0x97e83cccd117228f, 0xa35baecab2dc29ee, 0x1ce393ea5daace4d, 0x08f2220fb0fb66eb},
+		{0x07089552b319d465, 0xc6695f92b50a8313, 0x97e83cccd117228f, 0xa35baecab2dc29ee, 0x1ce393ea5daace4d, 0x08f2220fb0fb66eb},
-		fe{0xb2f66aad4ce5d646, 0x5842a06bfc497cec, 0xcf4895d42599d394, 0xc11b9cba40a8e8d0, 0x2e3813cbe5a0de89, 0x110eefda88847faf},
+		{0xb2f66aad4ce5d646, 0x5842a06bfc497cec, 0xcf4895d42599d394, 0xc11b9cba40a8e8d0, 0x2e3813cbe5a0de89, 0x110eefda88847faf},
 	},
+	// z ^ ((p ^ 2 - 1) / 6)
 	{
-		fe{0xecfb361b798dba3a, 0xc100ddb891865a2c, 0x0ec08ff1232bda8e, 0xd5c13cc6f1ca4721, 0x47222a47bf7b5c04, 0x0110f184e51c5f59},
+		{0xecfb361b798dba3a, 0xc100ddb891865a2c, 0x0ec08ff1232bda8e, 0xd5c13cc6f1ca4721, 0x47222a47bf7b5c04, 0x0110f184e51c5f59},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 3 - 1) / 6)
 	{
-		fe{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
+		{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
-		fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+		{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
 	},
+	// z ^ ((p ^ 4 - 1) / 6)
 	{
-		fe{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
+		{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 5 - 1) / 6)
 	{
-		fe{0x3726c30af242c66c, 0x7c2ac1aad1b6fe70, 0xa04007fbba4b14a2, 0xef517c3266341429, 0x0095ba654ed2226b, 0x02e370eccc86f7dd},
+		{0x3726c30af242c66c, 0x7c2ac1aad1b6fe70, 0xa04007fbba4b14a2, 0xef517c3266341429, 0x0095ba654ed2226b, 0x02e370eccc86f7dd},
-		fe{0x82d83cf50dbce43f, 0xa2813e53df9d018f, 0xc6f0caa53c65e181, 0x7525cf528d50fe95, 0x4a85ed50f4798a6b, 0x171da0fd6cf8eebd},
+		{0x82d83cf50dbce43f, 0xa2813e53df9d018f, 0xc6f0caa53c65e181, 0x7525cf528d50fe95, 0x4a85ed50f4798a6b, 0x171da0fd6cf8eebd},
 	},
+	// z ^ ((p ^ 6 - 1) / 6)
 	{
-		fe{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
+		{0x43f5fffffffcaaae, 0x32b7fff2ed47fffd, 0x07e83a49a2e99d69, 0xeca8f3318332bb7a, 0xef148d1ea0f4c069, 0x040ab3263eff0206},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 7 - 1) / 6)
 	{
-		fe{0xb2f66aad4ce5d646, 0x5842a06bfc497cec, 0xcf4895d42599d394, 0xc11b9cba40a8e8d0, 0x2e3813cbe5a0de89, 0x110eefda88847faf},
+		{0xb2f66aad4ce5d646, 0x5842a06bfc497cec, 0xcf4895d42599d394, 0xc11b9cba40a8e8d0, 0x2e3813cbe5a0de89, 0x110eefda88847faf},
-		fe{0x07089552b319d465, 0xc6695f92b50a8313, 0x97e83cccd117228f, 0xa35baecab2dc29ee, 0x1ce393ea5daace4d, 0x08f2220fb0fb66eb},
+		{0x07089552b319d465, 0xc6695f92b50a8313, 0x97e83cccd117228f, 0xa35baecab2dc29ee, 0x1ce393ea5daace4d, 0x08f2220fb0fb66eb},
 	},
+	// z ^ ((p ^ 8 - 1) / 6)
 	{
-		fe{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
+		{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 9 - 1) / 6)
 	{
-		fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+		{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
-		fe{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
+		{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
 	},
+	// z ^ ((p ^ 10 - 1) / 6)
 	{
-		fe{0x890dc9e4867545c3, 0x2af322533285a5d5, 0x50880866309b7e2c, 0xa20d1b8c7e881024, 0x14e4f04fe2db9068, 0x14e56d3f1564853a},
+		{0x890dc9e4867545c3, 0x2af322533285a5d5, 0x50880866309b7e2c, 0xa20d1b8c7e881024, 0x14e4f04fe2db9068, 0x14e56d3f1564853a},
-		fe{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
+		{0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000, 0x0000000000000000},
 	},
+	// z ^ ((p ^ 11 - 1) / 6)
 	{
-		fe{0x82d83cf50dbce43f, 0xa2813e53df9d018f, 0xc6f0caa53c65e181, 0x7525cf528d50fe95, 0x4a85ed50f4798a6b, 0x171da0fd6cf8eebd},
+		{0x82d83cf50dbce43f, 0xa2813e53df9d018f, 0xc6f0caa53c65e181, 0x7525cf528d50fe95, 0x4a85ed50f4798a6b, 0x171da0fd6cf8eebd},
-		fe{0x3726c30af242c66c, 0x7c2ac1aad1b6fe70, 0xa04007fbba4b14a2, 0xef517c3266341429, 0x0095ba654ed2226b, 0x02e370eccc86f7dd},
+		{0x3726c30af242c66c, 0x7c2ac1aad1b6fe70, 0xa04007fbba4b14a2, 0xef517c3266341429, 0x0095ba654ed2226b, 0x02e370eccc86f7dd},
 	},
 }
 
-/*
+// x
-	x
-*/
 
-var x = bigFromHex("0xd201000000010000")
+// var x = bigFromHex("0xd201000000010000")
+var x uint64 = 0xd201000000010000
+
+// square root
+
+var sqrtMinus1 = &fe2{*new(fe).zero(), *new(fe).one()}
+
+var sqrtSqrtMinus1 = &fe2{
+	fe{0x3e2f585da55c9ad1, 0x4294213d86c18183, 0x382844c88b623732, 0x92ad2afd19103e18, 0x1d794e4fac7cf0b9, 0x0bd592fc7d825ec8},
+	fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+}
+
+var sqrtMinusSqrtMinus1 = &fe2{
+	fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+	fe{0x7bcfa7a25aa30fda, 0xdc17dec12a927e7c, 0x2f088dd86b4ebef1, 0xd1ca2087da74d4a7, 0x2da2596696cebc1d, 0x0e2b7eedbbfd87d2},
+}

crypto/bls12381/bls12_381_test.go

@@ -2,12 +2,66 @@ package bls12381
 
 import (
 	"crypto/rand"
+	"encoding/hex"
+	"errors"
+	"flag"
 	"math/big"
+	"os"
+	"testing"
 )
 
-var fuz = 10
+var fuz int
+
+func TestMain(m *testing.M) {
+	_fuz := flag.Int("fuzz", 10, "# of iterations")
+	flag.Parse()
+	fuz = *_fuz
+	os.Exit(m.Run())
+}
 
 func randScalar(max *big.Int) *big.Int {
-	a, _ := rand.Int(rand.Reader, max)
+	a, err := rand.Int(rand.Reader, max)
+	if err != nil {
+		panic(errors.New(""))
+	}
 	return a
 }
+
+func fromHex(size int, hexStrs ...string) []byte {
+	var out []byte
+	if size > 0 {
+		out = make([]byte, size*len(hexStrs))
+	}
+	for i := 0; i < len(hexStrs); i++ {
+		hexStr := hexStrs[i]
+		if hexStr[:2] == "0x" {
+			hexStr = hexStr[2:]
+		}
+		if len(hexStr)%2 == 1 {
+			hexStr = "0" + hexStr
+		}
+		bytes, err := hex.DecodeString(hexStr)
+		if err != nil {
+			return nil
+		}
+		if size <= 0 {
+			out = append(out, bytes...)
+		} else {
+			if len(bytes) > size {
+				return nil
+			}
+			offset := i*size + (size - len(bytes))
+			copy(out[offset:], bytes)
+		}
+	}
+	return out
+}
+
+func padBytes(in []byte, size int) []byte {
+	out := make([]byte, size)
+	if len(in) > size {
+		panic("bad input for padding")
+	}
+	copy(out[size-len(in):], in)
+	return out
+}

crypto/bls12381/field_element.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -25,31 +9,34 @@ import (
 )
 
 // fe is base field element representation
-type fe [6]uint64
+type fe /***			***/ [fpNumberOfLimbs]uint64
 
-// fe2 is element representation of 'fp2' which is quadratic extension of base field 'fp'
+// fe2 is element representation of 'fp2' which is quadratic extention of base field 'fp'
 // Representation follows c[0] + c[1] * u encoding order.
-type fe2 [2]fe
+type fe2 /**			***/ [2]fe
 
-// fe6 is element representation of 'fp6' field which is cubic extension of 'fp2'
+// fe6 is element representation of 'fp6' field which is cubic extention of 'fp2'
 // Representation follows c[0] + c[1] * v + c[2] * v^2 encoding order.
-type fe6 [3]fe2
+type fe6 /**			***/ [3]fe2
 
-// fe12 is element representation of 'fp12' field which is quadratic extension of 'fp6'
+// fe12 is element representation of 'fp12' field which is quadratic extention of 'fp6'
 // Representation follows c[0] + c[1] * w encoding order.
-type fe12 [2]fe6
+type fe12 /**			***/ [2]fe6
+
+type wfe /***			***/ [fpNumberOfLimbs * 2]uint64
+type wfe2 /**			***/ [2]wfe
+type wfe6 /**			***/ [3]wfe2
 
 func (fe *fe) setBytes(in []byte) *fe {
-	size := 48
 	l := len(in)
-	if l >= size {
+	if l >= fpByteSize {
-		l = size
+		l = fpByteSize
 	}
-	padded := make([]byte, size)
+	padded := make([]byte, fpByteSize)
-	copy(padded[size-l:], in[:])
+	copy(padded[fpByteSize-l:], in[:])
 	var a int
-	for i := 0; i < 6; i++ {
+	for i := 0; i < fpNumberOfLimbs; i++ {
-		a = size - i*8
+		a = fpByteSize - i*8
 		fe[i] = uint64(padded[a-1]) | uint64(padded[a-2])<<8 |
 			uint64(padded[a-3])<<16 | uint64(padded[a-4])<<24 |
 			uint64(padded[a-5])<<32 | uint64(padded[a-6])<<40 |
@@ -84,10 +71,10 @@ func (fe *fe) set(fe2 *fe) *fe {
 }
 
 func (fe *fe) bytes() []byte {
-	out := make([]byte, 48)
+	out := make([]byte, fpByteSize)
 	var a int
-	for i := 0; i < 6; i++ {
+	for i := 0; i < fpNumberOfLimbs; i++ {
-		a = 48 - i*8
+		a = fpByteSize - i*8
 		out[a-1] = byte(fe[i])
 		out[a-2] = byte(fe[i] >> 8)
 		out[a-3] = byte(fe[i] >> 16)
@@ -105,7 +92,7 @@ func (fe *fe) big() *big.Int {
 }
 
 func (fe *fe) string() (s string) {
-	for i := 5; i >= 0; i-- {
+	for i := fpNumberOfLimbs - 1; i >= 0; i-- {
 		s = fmt.Sprintf("%s%16.16x", s, fe[i])
 	}
 	return "0x" + s
@@ -134,7 +121,7 @@ func (fe *fe) rand(r io.Reader) (*fe, error) {
 }
 
 func (fe *fe) isValid() bool {
-	return fe.cmp(&modulus) < 0
+	return fe.cmp(&modulus) == -1
 }
 
 func (fe *fe) isOdd() bool {
@@ -156,7 +143,7 @@ func (fe *fe) isOne() bool {
 }
 
 func (fe *fe) cmp(fe2 *fe) int {
-	for i := 5; i >= 0; i-- {
+	for i := fpNumberOfLimbs - 1; i >= 0; i-- {
 		if fe[i] > fe2[i] {
 			return 1
 		} else if fe[i] < fe2[i] {
@@ -170,30 +157,37 @@ func (fe *fe) equal(fe2 *fe) bool {
 	return fe2[0] == fe[0] && fe2[1] == fe[1] && fe2[2] == fe[2] && fe2[3] == fe[3] && fe2[4] == fe[4] && fe2[5] == fe[5]
 }
 
+func (e *fe) signBE() bool {
+	negZ, z := new(fe), new(fe)
+	fromMont(z, e)
+	neg(negZ, z)
+	return negZ.cmp(z) > -1
+}
+
 func (e *fe) sign() bool {
 	r := new(fe)
 	fromMont(r, e)
 	return r[0]&1 == 0
 }
 
-func (fe *fe) div2(e uint64) {
+func (e *fe) div2(u uint64) {
-	fe[0] = fe[0]>>1 | fe[1]<<63
+	e[0] = e[0]>>1 | e[1]<<63
-	fe[1] = fe[1]>>1 | fe[2]<<63
+	e[1] = e[1]>>1 | e[2]<<63
-	fe[2] = fe[2]>>1 | fe[3]<<63
+	e[2] = e[2]>>1 | e[3]<<63
-	fe[3] = fe[3]>>1 | fe[4]<<63
+	e[3] = e[3]>>1 | e[4]<<63
-	fe[4] = fe[4]>>1 | fe[5]<<63
+	e[4] = e[4]>>1 | e[5]<<63
-	fe[5] = fe[5]>>1 | e<<63
+	e[5] = e[5]>>1 | u<<63
 }
 
-func (fe *fe) mul2() uint64 {
+func (e *fe) mul2() uint64 {
-	e := fe[5] >> 63
+	u := e[5] >> 63
-	fe[5] = fe[5]<<1 | fe[4]>>63
+	e[5] = e[5]<<1 | e[4]>>63
-	fe[4] = fe[4]<<1 | fe[3]>>63
+	e[4] = e[4]<<1 | e[3]>>63
-	fe[3] = fe[3]<<1 | fe[2]>>63
+	e[3] = e[3]<<1 | e[2]>>63
-	fe[2] = fe[2]<<1 | fe[1]>>63
+	e[2] = e[2]<<1 | e[1]>>63
-	fe[1] = fe[1]<<1 | fe[0]>>63
+	e[1] = e[1]<<1 | e[0]>>63
-	fe[0] = fe[0] << 1
+	e[0] = e[0] << 1
-	return e
+	return u
 }
 
 func (e *fe2) zero() *fe2 {
@@ -214,16 +208,28 @@ func (e *fe2) set(e2 *fe2) *fe2 {
 	return e
 }
 
+func (e *fe2) fromMont(a *fe2) {
+	fromMont(&e[0], &a[0])
+	fromMont(&e[1], &a[1])
+}
+
+func (e *fe2) fromWide(w *wfe2) {
+	fromWide(&e[0], &w[0])
+	fromWide(&e[1], &w[1])
+}
+
 func (e *fe2) rand(r io.Reader) (*fe2, error) {
 	a0, err := new(fe).rand(r)
 	if err != nil {
 		return nil, err
 	}
+	e[0].set(a0)
 	a1, err := new(fe).rand(r)
 	if err != nil {
 		return nil, err
 	}
-	return &fe2{*a0, *a1}, nil
+	e[1].set(a1)
+	return e, nil
 }
 
 func (e *fe2) isOne() bool {
@@ -238,6 +244,13 @@ func (e *fe2) equal(e2 *fe2) bool {
 	return e[0].equal(&e2[0]) && e[1].equal(&e2[1])
 }
 
+func (e *fe2) signBE() bool {
+	if !e[1].isZero() {
+		return e[1].signBE()
+	}
+	return e[0].signBE()
+}
+
 func (e *fe2) sign() bool {
 	r := new(fe)
 	if !e[0].isZero() {
@@ -269,20 +282,35 @@ func (e *fe6) set(e2 *fe6) *fe6 {
 	return e
 }
 
+func (e *fe6) fromMont(a *fe6) {
+	e[0].fromMont(&a[0])
+	e[1].fromMont(&a[1])
+	e[2].fromMont(&a[2])
+}
+
+func (e *fe6) fromWide(w *wfe6) {
+	e[0].fromWide(&w[0])
+	e[1].fromWide(&w[1])
+	e[2].fromWide(&w[2])
+}
+
 func (e *fe6) rand(r io.Reader) (*fe6, error) {
 	a0, err := new(fe2).rand(r)
 	if err != nil {
 		return nil, err
 	}
+	e[0].set(a0)
 	a1, err := new(fe2).rand(r)
 	if err != nil {
 		return nil, err
 	}
+	e[1].set(a1)
 	a2, err := new(fe2).rand(r)
 	if err != nil {
 		return nil, err
 	}
-	return &fe6{*a0, *a1, *a2}, nil
+	e[2].set(a2)
+	return e, nil
 }
 
 func (e *fe6) isOne() bool {
@@ -315,16 +343,23 @@ func (e *fe12) set(e2 *fe12) *fe12 {
 	return e
 }
 
+func (e *fe12) fromMont(a *fe12) {
+	e[0].fromMont(&a[0])
+	e[1].fromMont(&a[1])
+}
+
 func (e *fe12) rand(r io.Reader) (*fe12, error) {
 	a0, err := new(fe6).rand(r)
 	if err != nil {
 		return nil, err
 	}
+	e[0].set(a0)
 	a1, err := new(fe6).rand(r)
 	if err != nil {
 		return nil, err
 	}
-	return &fe12{*a0, *a1}, nil
+	e[1].set(a1)
+	return e, nil
 }
 
 func (e *fe12) isOne() bool {
@@ -338,3 +373,32 @@ func (e *fe12) isZero() bool {
 func (e *fe12) equal(e2 *fe12) bool {
 	return e[0].equal(&e2[0]) && e[1].equal(&e2[1])
 }
+
+func (fe *wfe) set(fe2 *wfe) *wfe {
+	fe[0] = fe2[0]
+	fe[1] = fe2[1]
+	fe[2] = fe2[2]
+	fe[3] = fe2[3]
+	fe[4] = fe2[4]
+	fe[5] = fe2[5]
+	fe[6] = fe2[6]
+	fe[7] = fe2[7]
+	fe[8] = fe2[8]
+	fe[9] = fe2[9]
+	fe[10] = fe2[10]
+	fe[11] = fe2[11]
+	return fe
+}
+
+func (fe *wfe2) set(fe2 *wfe2) *wfe2 {
+	fe[0].set(&fe2[0])
+	fe[1].set(&fe2[1])
+	return fe
+}
+
+func (fe *wfe6) set(fe2 *wfe6) *wfe6 {
+	fe[0].set(&fe2[0])
+	fe[1].set(&fe2[1])
+	fe[2].set(&fe2[2])
+	return fe
+}

crypto/bls12381/field_element_test.go

@@ -8,6 +8,7 @@ import (
 )
 
 func TestFieldElementValidation(t *testing.T) {
+	// fe
 	zero := new(fe).zero()
 	if !zero.isValid() {
 		t.Fatal("zero must be valid")
@@ -59,8 +60,7 @@ func TestFieldElementEquality(t *testing.T) {
 		t.Fatal("a == a")
 	}
 	b2 := new(fe2)
-	fp2 := newFp2()
+	fp2Add(b2, a2, one2)
-	fp2.add(b2, a2, one2)
 	if a2.equal(b2) {
 		t.Fatal("a != a + 1")
 	}
@@ -78,8 +78,7 @@ func TestFieldElementEquality(t *testing.T) {
 		t.Fatal("a == a")
 	}
 	b6 := new(fe6)
-	fp6 := newFp6(fp2)
+	fp6Add(b6, a6, one6)
-	fp6.add(b6, a6, one6)
 	if a6.equal(b6) {
 		t.Fatal("a != a + 1")
 	}
@@ -97,11 +96,11 @@ func TestFieldElementEquality(t *testing.T) {
 		t.Fatal("a == a")
 	}
 	b12 := new(fe12)
-	fp12 := newFp12(fp6)
+	fp12Add(b12, a12, one12)
-	fp12.add(b12, a12, one12)
 	if a12.equal(b12) {
 		t.Fatal("a != a + 1")
 	}
+
 }
 
 func TestFieldElementHelpers(t *testing.T) {
@@ -159,13 +158,13 @@ func TestFieldElementHelpers(t *testing.T) {
 
 func TestFieldElementSerialization(t *testing.T) {
 	t.Run("zero", func(t *testing.T) {
-		in := make([]byte, 48)
+		in := make([]byte, fpByteSize)
 		fe := new(fe).setBytes(in)
 		if !fe.isZero() {
-			t.Fatal("bad serialization")
+			t.Fatal("serialization failed")
 		}
 		if !bytes.Equal(in, fe.bytes()) {
-			t.Fatal("bad serialization")
+			t.Fatal("serialization failed")
 		}
 	})
 	t.Run("bytes", func(t *testing.T) {
@@ -173,7 +172,7 @@ func TestFieldElementSerialization(t *testing.T) {
 			a, _ := new(fe).rand(rand.Reader)
 			b := new(fe).setBytes(a.bytes())
 			if !a.equal(b) {
-				t.Fatal("bad serialization")
+				t.Fatal("serialization failed")
 			}
 		}
 	})
@@ -182,7 +181,7 @@ func TestFieldElementSerialization(t *testing.T) {
 			a, _ := new(fe).rand(rand.Reader)
 			b := new(fe).setBig(a.big())
 			if !a.equal(b) {
-				t.Fatal("bad encoding or decoding")
+				t.Fatal("encoding or decoding failed")
 			}
 		}
 	})
@@ -194,7 +193,7 @@ func TestFieldElementSerialization(t *testing.T) {
 				t.Fatal(err)
 			}
 			if !a.equal(b) {
-				t.Fatal("bad encoding or decoding")
+				t.Fatal("encoding or decoding failed")
 			}
 		}
 	})
@@ -205,24 +204,24 @@ func TestFieldElementByteInputs(t *testing.T) {
 	in := make([]byte, 0)
 	a := new(fe).setBytes(in)
 	if !a.equal(zero) {
-		t.Fatal("bad serialization")
+		t.Fatal("serialization failed")
 	}
-	in = make([]byte, 48)
+	in = make([]byte, fpByteSize)
 	a = new(fe).setBytes(in)
 	if !a.equal(zero) {
-		t.Fatal("bad serialization")
+		t.Fatal("serialization failed")
 	}
-	in = make([]byte, 64)
+	in = make([]byte, fpByteSize+200)
 	a = new(fe).setBytes(in)
 	if !a.equal(zero) {
-		t.Fatal("bad serialization")
+		t.Fatal("serialization failed")
 	}
-	in = make([]byte, 49)
+	in = make([]byte, fpByteSize+1)
-	in[47] = 1
+	in[fpByteSize-1] = 1
 	normalOne := &fe{1, 0, 0, 0, 0, 0}
 	a = new(fe).setBytes(in)
 	if !a.equal(normalOne) {
-		t.Fatal("bad serialization")
+		t.Fatal("serialization failed")
 	}
 }
 
@@ -230,21 +229,21 @@ func TestFieldElementCopy(t *testing.T) {
 	a, _ := new(fe).rand(rand.Reader)
 	b := new(fe).set(a)
 	if !a.equal(b) {
-		t.Fatal("bad copy, 1")
+		t.Fatal("copy failed")
 	}
 	a2, _ := new(fe2).rand(rand.Reader)
 	b2 := new(fe2).set(a2)
 	if !a2.equal(b2) {
-		t.Fatal("bad copy, 2")
+		t.Fatal("copy failed")
 	}
 	a6, _ := new(fe6).rand(rand.Reader)
 	b6 := new(fe6).set(a6)
 	if !a6.equal(b6) {
-		t.Fatal("bad copy, 6")
+		t.Fatal("copy failed")
 	}
 	a12, _ := new(fe12).rand(rand.Reader)
 	b12 := new(fe12).set(a12)
 	if !a12.equal(b12) {
-		t.Fatal("bad copy, 12")
+		t.Fatal("copy failed2")
 	}
 }

crypto/bls12381/fp.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -23,8 +7,8 @@ import (
 
 func fromBytes(in []byte) (*fe, error) {
 	fe := &fe{}
-	if len(in) != 48 {
+	if len(in) != fpByteSize {
-		return nil, errors.New("input string should be equal 48 bytes")
+		return nil, errors.New("input string must be equal 48 bytes")
 	}
 	fe.setBytes(in)
 	if !fe.isValid() {
@@ -34,6 +18,37 @@ func fromBytes(in []byte) (*fe, error) {
 	return fe, nil
 }
 
+func from64Bytes(in []byte) (*fe, error) {
+	if len(in) != 32*2 {
+		return nil, errors.New("input string must be equal 64 bytes")
+	}
+	a0 := make([]byte, fpByteSize)
+	copy(a0[fpByteSize-32:fpByteSize], in[:32])
+	a1 := make([]byte, fpByteSize)
+	copy(a1[fpByteSize-32:fpByteSize], in[32:])
+	e0, err := fromBytes(a0)
+	if err != nil {
+		return nil, err
+	}
+	e1, err := fromBytes(a1)
+	if err != nil {
+		return nil, err
+	}
+	// F = 2 ^ 256 * R
+	F := fe{
+		0x75b3cd7c5ce820f,
+		0x3ec6ba621c3edb0b,
+		0x168a13d82bff6bce,
+		0x87663c4bf8c449d2,
+		0x15f34c83ddc8d830,
+		0xf9628b49caa2e85,
+	}
+
+	mul(e0, e0, &F)
+	add(e1, e1, e0)
+	return e1, nil
+}
+
 func fromBig(in *big.Int) (*fe, error) {
 	fe := new(fe).setBig(in)
 	if !fe.isValid() {
@@ -81,6 +96,28 @@ func fromMont(c, a *fe) {
 	mul(c, a, &fe{1})
 }
 
+func wfp2MulGeneric(c *wfe2, a, b *fe2) {
+	wt0, wt1 := new(wfe), new(wfe)
+	t0, t1 := new(fe), new(fe)
+	wmul(wt0, &a[0], &b[0])
+	wmul(wt1, &a[1], &b[1])
+	wsub(&c[0], wt0, wt1)
+	lwaddAssign(wt0, wt1)
+	ladd(t0, &a[0], &a[1])
+	ladd(t1, &b[0], &b[1])
+	wmul(wt1, t0, t1)
+	lwsub(&c[1], wt1, wt0)
+}
+
+func wfp2SquareGeneric(c *wfe2, a *fe2) {
+	t0, t1, t2 := new(fe), new(fe), new(fe)
+	ladd(t0, &a[0], &a[1])
+	sub(t1, &a[0], &a[1])
+	ldouble(t2, &a[0])
+	wmul(&c[0], t1, t0)
+	wmul(&c[1], t2, &a[1])
+}
+
 func exp(c, a *fe, e *big.Int) {
 	z := new(fe).set(r1)
 	for i := e.BitLen(); i >= 0; i-- {
@@ -105,7 +142,7 @@ func inverse(inv, e *fe) {
 	var z uint64
 	var found = false
 	// Phase 1
-	for i := 0; i < 768; i++ {
+	for i := 0; i < sixWordBitSize*2; i++ {
 		if v.isZero() {
 			found = true
 			break
@@ -135,7 +172,7 @@ func inverse(inv, e *fe) {
 		return
 	}
 
-	if k < 381 || k > 381+384 {
+	if k < fpBitSize || k > fpBitSize+sixWordBitSize {
 		inv.zero()
 		return
 	}
@@ -147,21 +184,189 @@ func inverse(inv, e *fe) {
 	lsubAssign(u, r)
 
 	// Phase 2
-	for i := k; i < 384*2; i++ {
+	for i := k; i < 2*sixWordBitSize; i++ {
 		double(u, u)
 	}
 	inv.set(u)
 }
 
+func inverseBatch(in []fe) {
+
+	n, N, setFirst := 0, len(in), false
+
+	for i := 0; i < len(in); i++ {
+		if !in[i].isZero() {
+			n++
+		}
+	}
+	if n == 0 {
+		return
+	}
+
+	tA := make([]fe, n)
+	tB := make([]fe, n)
+
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].isZero() {
+			if !setFirst {
+				setFirst = true
+				tA[j].set(&in[i])
+			} else {
+				mul(&tA[j], &in[i], &tA[j-1])
+			}
+			j = j + 1
+		}
+	}
+
+	inverse(&tB[n-1], &tA[n-1])
+	for i, j := N-1, n-1; j != 0; i-- {
+		if !in[i].isZero() {
+			mul(&tB[j-1], &tB[j], &in[i])
+			j = j - 1
+		}
+	}
+
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].isZero() {
+			if setFirst {
+				setFirst = false
+				in[i].set(&tB[j])
+			} else {
+				mul(&in[i], &tA[j-1], &tB[j])
+			}
+			j = j + 1
+		}
+	}
+}
+
+func rsqrt(c, a *fe) bool {
+	t0, t1 := new(fe), new(fe)
+	sqrtAddchain(t0, a)
+	mul(t1, t0, a)
+	square(t1, t1)
+	ret := t1.equal(a)
+	c.set(t0)
+	return ret
+}
+
 func sqrt(c, a *fe) bool {
+	u, v := new(fe).set(a), new(fe)
+	// a ^ (p - 3) / 4
+	sqrtAddchain(c, a)
+	// a ^ (p + 1) / 4
+	mul(c, c, u)
+
+	square(v, c)
+	return u.equal(v)
+}
+
+func _sqrt(c, a *fe) bool {
 	u, v := new(fe).set(a), new(fe)
 	exp(c, a, pPlus1Over4)
 	square(v, c)
 	return u.equal(v)
 }
 
-func isQuadraticNonResidue(elem *fe) bool {
+func sqrtAddchain(c, a *fe) {
-	result := new(fe)
+	chain := func(c *fe, n int, a *fe) {
-	exp(result, elem, pMinus1Over2)
+		for i := 0; i < n; i++ {
-	return !result.isOne()
+			square(c, c)
+		}
+		mul(c, c, a)
+	}
+
+	t := make([]fe, 16)
+	t[13].set(a)
+	square(&t[0], &t[13])
+	mul(&t[8], &t[0], &t[13])
+	square(&t[4], &t[0])
+	mul(&t[1], &t[8], &t[0])
+	mul(&t[6], &t[4], &t[8])
+	mul(&t[9], &t[1], &t[4])
+	mul(&t[12], &t[6], &t[4])
+	mul(&t[3], &t[9], &t[4])
+	mul(&t[7], &t[12], &t[4])
+	mul(&t[15], &t[3], &t[4])
+	mul(&t[10], &t[7], &t[4])
+	mul(&t[2], &t[15], &t[4])
+	mul(&t[11], &t[10], &t[4])
+	square(&t[0], &t[3])
+	mul(&t[14], &t[11], &t[4])
+	mul(&t[5], &t[0], &t[8])
+	mul(&t[4], &t[0], &t[1])
+
+	chain(&t[0], 12, &t[15])
+	chain(&t[0], 7, &t[7])
+	chain(&t[0], 4, &t[1])
+	chain(&t[0], 6, &t[6])
+	chain(&t[0], 7, &t[11])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 2, &t[8])
+	chain(&t[0], 6, &t[3])
+	chain(&t[0], 6, &t[3])
+	chain(&t[0], 6, &t[9])
+	chain(&t[0], 3, &t[8])
+	chain(&t[0], 7, &t[3])
+	chain(&t[0], 4, &t[3])
+	chain(&t[0], 6, &t[7])
+	chain(&t[0], 6, &t[14])
+	chain(&t[0], 3, &t[13])
+	chain(&t[0], 8, &t[3])
+	chain(&t[0], 7, &t[11])
+	chain(&t[0], 5, &t[12])
+	chain(&t[0], 6, &t[3])
+	chain(&t[0], 6, &t[5])
+	chain(&t[0], 4, &t[9])
+	chain(&t[0], 8, &t[5])
+	chain(&t[0], 4, &t[3])
+	chain(&t[0], 7, &t[11])
+	chain(&t[0], 9, &t[10])
+	chain(&t[0], 2, &t[8])
+	chain(&t[0], 5, &t[6])
+	chain(&t[0], 7, &t[1])
+	chain(&t[0], 7, &t[9])
+	chain(&t[0], 6, &t[11])
+	chain(&t[0], 5, &t[5])
+	chain(&t[0], 5, &t[10])
+	chain(&t[0], 5, &t[10])
+	chain(&t[0], 8, &t[3])
+	chain(&t[0], 7, &t[2])
+	chain(&t[0], 9, &t[7])
+	chain(&t[0], 5, &t[3])
+	chain(&t[0], 3, &t[8])
+	chain(&t[0], 8, &t[7])
+	chain(&t[0], 3, &t[8])
+	chain(&t[0], 7, &t[9])
+	chain(&t[0], 9, &t[7])
+	chain(&t[0], 6, &t[2])
+	chain(&t[0], 6, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 4, &t[3])
+	chain(&t[0], 3, &t[8])
+	chain(&t[0], 8, &t[2])
+	chain(&t[0], 7, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 4, &t[7])
+	chain(&t[0], 4, &t[6])
+	chain(&t[0], 7, &t[4])
+	chain(&t[0], 5, &t[5])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 5, &t[4])
+	chain(&t[0], 4, &t[3])
+	chain(&t[0], 6, &t[2])
+	chain(&t[0], 4, &t[1])
+	square(c, &t[0])
+}
+
+func isQuadraticNonResidue(a *fe) bool {
+	if a.isZero() {
+		return true
+	}
+	return !sqrt(new(fe), a)
 }

crypto/bls12381/fp12.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -27,21 +11,30 @@ type fp12 struct {
 }
 
 type fp12temp struct {
-	t2  [9]*fe2
+	t2  [7]*fe2
-	t6  [5]*fe6
+	t6  [4]*fe6
-	t12 *fe12
+	wt2 [3]*wfe2
+	wt6 [3]*wfe6
 }
 
 func newFp12Temp() fp12temp {
-	t2 := [9]*fe2{}
+	t2 := [7]*fe2{}
-	t6 := [5]*fe6{}
+	t6 := [4]*fe6{}
 	for i := 0; i < len(t2); i++ {
 		t2[i] = &fe2{}
 	}
 	for i := 0; i < len(t6); i++ {
 		t6[i] = &fe6{}
 	}
-	return fp12temp{t2, t6, &fe12{}}
+	wt2 := [3]*wfe2{}
+	for i := 0; i < len(wt2); i++ {
+		wt2[i] = &wfe2{}
+	}
+	wt6 := [3]*wfe6{}
+	for i := 0; i < len(wt6); i++ {
+		wt6[i] = &wfe6{}
+	}
+	return fp12temp{t2, t6, wt2, wt6}
 }
 
 func newFp12(fp6 *fp6) *fp12 {
@@ -58,14 +51,14 @@ func (e *fp12) fp2() *fp2 {
 
 func (e *fp12) fromBytes(in []byte) (*fe12, error) {
 	if len(in) != 576 {
-		return nil, errors.New("input string should be larger than 96 bytes")
+		return nil, errors.New("input string length must be equal to 576 bytes")
 	}
 	fp6 := e.fp6
-	c1, err := fp6.fromBytes(in[:288])
+	c1, err := fp6.fromBytes(in[:6*fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	c0, err := fp6.fromBytes(in[288:])
+	c0, err := fp6.fromBytes(in[6*fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -74,9 +67,9 @@ func (e *fp12) fromBytes(in []byte) (*fe12, error) {
 
 func (e *fp12) toBytes(a *fe12) []byte {
 	fp6 := e.fp6
-	out := make([]byte, 576)
+	out := make([]byte, 12*fpByteSize)
-	copy(out[:288], fp6.toBytes(&a[1]))
+	copy(out[:6*fpByteSize], fp6.toBytes(&a[1]))
-	copy(out[288:], fp6.toBytes(&a[0]))
+	copy(out[6*fpByteSize:], fp6.toBytes(&a[0]))
 	return out
 }
 
@@ -92,138 +85,125 @@ func (e *fp12) one() *fe12 {
 	return new(fe12).one()
 }
 
-func (e *fp12) add(c, a, b *fe12) {
+func fp12Add(c, a, b *fe12) {
-	fp6 := e.fp6
+	fp6Add(&c[0], &a[0], &b[0])
-	fp6.add(&c[0], &a[0], &b[0])
+	fp6Add(&c[1], &a[1], &b[1])
-	fp6.add(&c[1], &a[1], &b[1])
 }
 
-func (e *fp12) double(c, a *fe12) {
+func fp12Double(c, a *fe12) {
-	fp6 := e.fp6
+	fp6Double(&c[0], &a[0])
-	fp6.double(&c[0], &a[0])
+	fp6Double(&c[1], &a[1])
-	fp6.double(&c[1], &a[1])
 }
 
-func (e *fp12) sub(c, a, b *fe12) {
+func fp12Sub(c, a, b *fe12) {
-	fp6 := e.fp6
+	fp6Sub(&c[0], &a[0], &b[0])
-	fp6.sub(&c[0], &a[0], &b[0])
+	fp6Sub(&c[1], &a[1], &b[1])
-	fp6.sub(&c[1], &a[1], &b[1])
+
 }
 
-func (e *fp12) neg(c, a *fe12) {
+func fp12Neg(c, a *fe12) {
-	fp6 := e.fp6
+	fp6Neg(&c[0], &a[0])
-	fp6.neg(&c[0], &a[0])
+	fp6Neg(&c[1], &a[1])
-	fp6.neg(&c[1], &a[1])
 }
 
-func (e *fp12) conjugate(c, a *fe12) {
+func fp12Conjugate(c, a *fe12) {
-	fp6 := e.fp6
 	c[0].set(&a[0])
-	fp6.neg(&c[1], &a[1])
+	fp6Neg(&c[1], &a[1])
-}
-
-func (e *fp12) square(c, a *fe12) {
-	fp6, t := e.fp6, e.t6
-	fp6.add(t[0], &a[0], &a[1])
-	fp6.mul(t[2], &a[0], &a[1])
-	fp6.mulByNonResidue(t[1], &a[1])
-	fp6.addAssign(t[1], &a[0])
-	fp6.mulByNonResidue(t[3], t[2])
-	fp6.mulAssign(t[0], t[1])
-	fp6.subAssign(t[0], t[2])
-	fp6.sub(&c[0], t[0], t[3])
-	fp6.double(&c[1], t[2])
-}
-
-func (e *fp12) cyclotomicSquare(c, a *fe12) {
-	t, fp2 := e.t2, e.fp2()
-	e.fp4Square(t[3], t[4], &a[0][0], &a[1][1])
-	fp2.sub(t[2], t[3], &a[0][0])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[0][0], t[2], t[3])
-	fp2.add(t[2], t[4], &a[1][1])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[1][1], t[2], t[4])
-	e.fp4Square(t[3], t[4], &a[1][0], &a[0][2])
-	e.fp4Square(t[5], t[6], &a[0][1], &a[1][2])
-	fp2.sub(t[2], t[3], &a[0][1])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[0][1], t[2], t[3])
-	fp2.add(t[2], t[4], &a[1][2])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[1][2], t[2], t[4])
-	fp2.mulByNonResidue(t[3], t[6])
-	fp2.add(t[2], t[3], &a[1][0])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[1][0], t[2], t[3])
-	fp2.sub(t[2], t[5], &a[0][2])
-	fp2.doubleAssign(t[2])
-	fp2.add(&c[0][2], t[2], t[5])
 }
 
 func (e *fp12) mul(c, a, b *fe12) {
-	t, fp6 := e.t6, e.fp6
+	wt, t := e.wt6, e.t6
-	fp6.mul(t[1], &a[0], &b[0])
+	e.fp6.wmul(wt[1], &a[0], &b[0])
-	fp6.mul(t[2], &a[1], &b[1])
+	e.fp6.wmul(wt[2], &a[1], &b[1])
-	fp6.add(t[0], t[1], t[2])
+	fp6Add(t[0], &a[0], &a[1])
-	fp6.mulByNonResidue(t[2], t[2])
+	fp6Add(t[3], &b[0], &b[1])
-	fp6.add(t[3], t[1], t[2])
+	e.fp6.wmul(wt[0], t[0], t[3])
-	fp6.add(t[1], &a[0], &a[1])
+	wfp6SubAssign(wt[0], wt[1])
-	fp6.add(t[2], &b[0], &b[1])
+	wfp6SubAssign(wt[0], wt[2])
-	fp6.mulAssign(t[1], t[2])
+	c[1].fromWide(wt[0])
-	c[0].set(t[3])
+	e.fp6.wmulByNonResidueAssign(wt[2])
-	fp6.sub(&c[1], t[1], t[0])
+	wfp6AddAssign(wt[1], wt[2])
+	c[0].fromWide(wt[1])
+
 }
 
 func (e *fp12) mulAssign(a, b *fe12) {
-	t, fp6 := e.t6, e.fp6
+	wt, t := e.wt6, e.t6
-	fp6.mul(t[1], &a[0], &b[0])
+	e.fp6.wmul(wt[1], &a[0], &b[0])
-	fp6.mul(t[2], &a[1], &b[1])
+	e.fp6.wmul(wt[2], &a[1], &b[1])
-	fp6.add(t[0], t[1], t[2])
+	fp6Add(t[0], &a[0], &a[1])
-	fp6.mulByNonResidue(t[2], t[2])
+	fp6Add(t[3], &b[0], &b[1])
-	fp6.add(t[3], t[1], t[2])
+	e.fp6.wmul(wt[0], t[0], t[3])
-	fp6.add(t[1], &a[0], &a[1])
+	wfp6SubAssign(wt[0], wt[1])
-	fp6.add(t[2], &b[0], &b[1])
+	wfp6SubAssign(wt[0], wt[2])
-	fp6.mulAssign(t[1], t[2])
+	a[1].fromWide(wt[0])
-	a[0].set(t[3])
+	e.fp6.wmulByNonResidueAssign(wt[2])
-	fp6.sub(&a[1], t[1], t[0])
+	wfp6AddAssign(wt[1], wt[2])
+	a[0].fromWide(wt[1])
 }
 
-func (e *fp12) fp4Square(c0, c1, a0, a1 *fe2) {
+func (e *fp12) mul014(a *fe12, b0, b1, b4 *fe2) {
-	t, fp2 := e.t2, e.fp2()
+	wt, t := e.wt6, e.t6
-	fp2.square(t[0], a0)
+	e.fp6.wmul01(wt[0], &a[0], b0, b1)
-	fp2.square(t[1], a1)
+	e.fp6.wmul1(wt[1], &a[1], b4)
-	fp2.mulByNonResidue(t[2], t[1])
+	fp2LaddAssign(b1, b4)
-	fp2.add(c0, t[2], t[0])
+	fp6Ladd(t[2], &a[1], &a[0])
-	fp2.add(t[2], a0, a1)
+	e.fp6.wmul01(wt[2], t[2], b0, b1)
-	fp2.squareAssign(t[2])
+	wfp6SubAssign(wt[2], wt[0])
-	fp2.subAssign(t[2], t[0])
+	wfp6SubAssign(wt[2], wt[1])
-	fp2.sub(c1, t[2], t[1])
+	a[1].fromWide(wt[2])
+	e.fp6.wmulByNonResidueAssign(wt[1])
+	wfp6AddAssign(wt[0], wt[1])
+	a[0].fromWide(wt[0])
+}
+
+func (e *fp12) square(c, a *fe12) {
+	t := e.t6
+	// Multiplication and Squaring on Pairing-Friendly Fields
+	// Complex squaring algorithm
+	// https://eprint.iacr.org/2006/471
+
+	fp6Add(t[0], &a[0], &a[1])
+	e.fp6.mul(t[2], &a[0], &a[1])
+	e.fp6.mulByNonResidue(t[1], &a[1])
+	fp6AddAssign(t[1], &a[0])
+	e.fp6.mulByNonResidue(t[3], t[2])
+	e.fp6.mul(t[0], t[0], t[1])
+	fp6SubAssign(t[0], t[2])
+	fp6Sub(&c[0], t[0], t[3])
+	fp6Double(&c[1], t[2])
+}
+
+func (e *fp12) squareAssign(a *fe12) {
+	t := e.t6
+	// Multiplication and Squaring on Pairing-Friendly Fields
+	// Complex squaring algorithm
+	// https://eprint.iacr.org/2006/471
+
+	fp6Add(t[0], &a[0], &a[1])
+	e.fp6.mul(t[2], &a[0], &a[1])
+	e.fp6.mulByNonResidue(t[1], &a[1])
+	fp6AddAssign(t[1], &a[0])
+	e.fp6.mulByNonResidue(t[3], t[2])
+	e.fp6.mul(t[0], t[0], t[1])
+	fp6SubAssign(t[0], t[2])
+	fp6Sub(&a[0], t[0], t[3])
+	fp6Double(&a[1], t[2])
 }
 
 func (e *fp12) inverse(c, a *fe12) {
-	fp6, t := e.fp6, e.t6
+	// Guide to Pairing Based Cryptography
-	fp6.square(t[0], &a[0])
+	// Algorithm 5.16
-	fp6.square(t[1], &a[1])
-	fp6.mulByNonResidue(t[1], t[1])
-	fp6.sub(t[1], t[0], t[1])
-	fp6.inverse(t[0], t[1])
-	fp6.mul(&c[0], &a[0], t[0])
-	fp6.mulAssign(t[0], &a[1])
-	fp6.neg(&c[1], t[0])
-}
 
-func (e *fp12) mulBy014Assign(a *fe12, c0, c1, c4 *fe2) {
+	t := e.t6
-	fp2, fp6, t, t2 := e.fp2(), e.fp6, e.t6, e.t2[0]
+	e.fp6.square(t[0], &a[0])         // a0^2
-	fp6.mulBy01(t[0], &a[0], c0, c1)
+	e.fp6.square(t[1], &a[1])         // a1^2
-	fp6.mulBy1(t[1], &a[1], c4)
+	e.fp6.mulByNonResidue(t[1], t[1]) // βa1^2
-	fp2.add(t2, c1, c4)
+	fp6SubAssign(t[0], t[1])          // v = (a0^2 - a1^2)
-	fp6.add(t[2], &a[1], &a[0])
+	e.fp6.inverse(t[1], t[0])         // v = v^-1
-	fp6.mulBy01Assign(t[2], c0, t2)
+	e.fp6.mul(&c[0], &a[0], t[1])     // c0 = a0v
-	fp6.subAssign(t[2], t[0])
+	e.fp6.mulAssign(t[1], &a[1])      //
-	fp6.sub(&a[1], t[2], t[1])
+	fp6Neg(&c[1], t[1])               // c1 = -a1v
-	fp6.mulByNonResidue(t[1], t[1])
-	fp6.add(&a[0], t[1], t[0])
 }
 
 func (e *fp12) exp(c, a *fe12, s *big.Int) {
@@ -240,7 +220,7 @@ func (e *fp12) exp(c, a *fe12, s *big.Int) {
 func (e *fp12) cyclotomicExp(c, a *fe12, s *big.Int) {
 	z := e.one()
 	for i := s.BitLen() - 1; i >= 0; i-- {
-		e.cyclotomicSquare(z, z)
+		e.cyclotomicSquare(z)
 		if s.Bit(i) == 1 {
 			e.mul(z, z, a)
 		}
@@ -248,30 +228,76 @@ func (e *fp12) cyclotomicExp(c, a *fe12, s *big.Int) {
 	c.set(z)
 }
 
-func (e *fp12) frobeniusMap(c, a *fe12, power uint) {
+func (e *fp12) cyclotomicSquare(a *fe12) {
-	fp6 := e.fp6
+	t := e.t2
-	fp6.frobeniusMap(&c[0], &a[0], power)
+	// Guide to Pairing Based Cryptography
-	fp6.frobeniusMap(&c[1], &a[1], power)
+	// 5.5.4 Airthmetic in Cyclotomic Groups
-	switch power {
+
-	case 0:
+	e.fp4Square(t[3], t[4], &a[0][0], &a[1][1])
-		return
+	fp2Sub(t[2], t[3], &a[0][0])
-	case 6:
+	fp2DoubleAssign(t[2])
-		fp6.neg(&c[1], &c[1])
+	fp2Add(&a[0][0], t[2], t[3])
-	default:
+	fp2Add(t[2], t[4], &a[1][1])
-		fp6.mulByBaseField(&c[1], &c[1], &frobeniusCoeffs12[power])
+	fp2DoubleAssign(t[2])
-	}
+	fp2Add(&a[1][1], t[2], t[4])
+	e.fp4Square(t[3], t[4], &a[1][0], &a[0][2])
+	e.fp4Square(t[5], t[6], &a[0][1], &a[1][2])
+	fp2Sub(t[2], t[3], &a[0][1])
+	fp2DoubleAssign(t[2])
+	fp2Add(&a[0][1], t[2], t[3])
+	fp2Add(t[2], t[4], &a[1][2])
+	fp2DoubleAssign(t[2])
+	fp2Add(&a[1][2], t[2], t[4])
+	mulByNonResidue(t[3], t[6])
+	fp2Add(t[2], t[3], &a[1][0])
+	fp2DoubleAssign(t[2])
+	fp2Add(&a[1][0], t[2], t[3])
+	fp2Sub(t[2], t[5], &a[0][2])
+	fp2DoubleAssign(t[2])
+	fp2Add(&a[0][2], t[2], t[5])
 }
 
-func (e *fp12) frobeniusMapAssign(a *fe12, power uint) {
+func (e *fp12) fp4Square(c0, c1, a0, a1 *fe2) {
-	fp6 := e.fp6
+	wt, t := e.wt2, e.t2
-	fp6.frobeniusMapAssign(&a[0], power)
+	// Multiplication and Squaring on Pairing-Friendly Fields
-	fp6.frobeniusMapAssign(&a[1], power)
+	// Karatsuba squaring algorithm
-	switch power {
+	// https://eprint.iacr.org/2006/471
-	case 0:
+
-		return
+	wfp2Square(wt[0], a0)
-	case 6:
+	wfp2Square(wt[1], a1)
-		fp6.neg(&a[1], &a[1])
+	wfp2MulByNonResidue(wt[2], wt[1])
-	default:
+	wfp2AddAssign(wt[2], wt[0])
-		fp6.mulByBaseField(&a[1], &a[1], &frobeniusCoeffs12[power])
+	c0.fromWide(wt[2])
-	}
+	fp2Add(t[0], a0, a1)
+	wfp2Square(wt[2], t[0])
+	wfp2SubAssign(wt[2], wt[0])
+	wfp2SubAssign(wt[2], wt[1])
+	c1.fromWide(wt[2])
+}
+
+func (e *fp12) frobeniusMap1(a *fe12) {
+	fp6, fp2 := e.fp6, e.fp6.fp2
+	fp6.frobeniusMap1(&a[0])
+	fp6.frobeniusMap1(&a[1])
+	fp2.mulAssign(&a[1][0], &frobeniusCoeffs12[1])
+	fp2.mulAssign(&a[1][1], &frobeniusCoeffs12[1])
+	fp2.mulAssign(&a[1][2], &frobeniusCoeffs12[1])
+}
+
+func (e *fp12) frobeniusMap2(a *fe12) {
+	fp6, fp2 := e.fp6, e.fp6.fp2
+	fp6.frobeniusMap2(&a[0])
+	fp6.frobeniusMap2(&a[1])
+	fp2.mulAssign(&a[1][0], &frobeniusCoeffs12[2])
+	fp2.mulAssign(&a[1][1], &frobeniusCoeffs12[2])
+	fp2.mulAssign(&a[1][2], &frobeniusCoeffs12[2])
+}
+
+func (e *fp12) frobeniusMap3(a *fe12) {
+	fp6, fp2 := e.fp6, e.fp6.fp2
+	fp6.frobeniusMap3(&a[0])
+	fp6.frobeniusMap3(&a[1])
+	fp2.mulAssign(&a[1][0], &frobeniusCoeffs12[3])
+	fp2.mulAssign(&a[1][1], &frobeniusCoeffs12[3])
+	fp2.mulAssign(&a[1][2], &frobeniusCoeffs12[3])
 }

crypto/bls12381/fp2.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -22,7 +6,8 @@ import (
 )
 
 type fp2Temp struct {
-	t [4]*fe
+	t [3]*fe
+	w *wfe2
 }
 
 type fp2 struct {
@@ -30,11 +15,11 @@ type fp2 struct {
 }
 
 func newFp2Temp() fp2Temp {
-	t := [4]*fe{}
+	t := [3]*fe{}
 	for i := 0; i < len(t); i++ {
 		t[i] = &fe{}
 	}
-	return fp2Temp{t}
+	return fp2Temp{t, &wfe2{}}
 }
 
 func newFp2() *fp2 {
@@ -43,14 +28,14 @@ func newFp2() *fp2 {
 }
 
 func (e *fp2) fromBytes(in []byte) (*fe2, error) {
-	if len(in) != 96 {
+	if len(in) != 2*fpByteSize {
-		return nil, errors.New("length of input string should be 96 bytes")
+		return nil, errors.New("input string must be equal to 96 bytes")
 	}
-	c1, err := fromBytes(in[:48])
+	c1, err := fromBytes(in[:fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	c0, err := fromBytes(in[48:])
+	c0, err := fromBytes(in[fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -58,9 +43,9 @@ func (e *fp2) fromBytes(in []byte) (*fe2, error) {
 }
 
 func (e *fp2) toBytes(a *fe2) []byte {
-	out := make([]byte, 96)
+	out := make([]byte, 2*fpByteSize)
-	copy(out[:48], toBytes(&a[1]))
+	copy(out[:fpByteSize], toBytes(&a[1]))
-	copy(out[48:], toBytes(&a[0]))
+	copy(out[fpByteSize:], toBytes(&a[0]))
 	return out
 }
 
@@ -76,82 +61,36 @@ func (e *fp2) one() *fe2 {
 	return new(fe2).one()
 }
 
-func (e *fp2) add(c, a, b *fe2) {
+func fp2Neg(c, a *fe2) {
-	add(&c[0], &a[0], &b[0])
-	add(&c[1], &a[1], &b[1])
-}
-
-func (e *fp2) addAssign(a, b *fe2) {
-	addAssign(&a[0], &b[0])
-	addAssign(&a[1], &b[1])
-}
-
-func (e *fp2) ladd(c, a, b *fe2) {
-	ladd(&c[0], &a[0], &b[0])
-	ladd(&c[1], &a[1], &b[1])
-}
-
-func (e *fp2) double(c, a *fe2) {
-	double(&c[0], &a[0])
-	double(&c[1], &a[1])
-}
-
-func (e *fp2) doubleAssign(a *fe2) {
-	doubleAssign(&a[0])
-	doubleAssign(&a[1])
-}
-
-func (e *fp2) ldouble(c, a *fe2) {
-	ldouble(&c[0], &a[0])
-	ldouble(&c[1], &a[1])
-}
-
-func (e *fp2) sub(c, a, b *fe2) {
-	sub(&c[0], &a[0], &b[0])
-	sub(&c[1], &a[1], &b[1])
-}
-
-func (e *fp2) subAssign(c, a *fe2) {
-	subAssign(&c[0], &a[0])
-	subAssign(&c[1], &a[1])
-}
-
-func (e *fp2) neg(c, a *fe2) {
 	neg(&c[0], &a[0])
 	neg(&c[1], &a[1])
 }
 
+func fp2Conjugate(c, a *fe2) {
+	c[0].set(&a[0])
+	neg(&c[1], &a[1])
+}
+
 func (e *fp2) mul(c, a, b *fe2) {
-	t := e.t
+	wfp2Mul(e.w, b, a)
-	mul(t[1], &a[0], &b[0])
+	c.fromWide(e.w)
-	mul(t[2], &a[1], &b[1])
-	add(t[0], &a[0], &a[1])
-	add(t[3], &b[0], &b[1])
-	sub(&c[0], t[1], t[2])
-	addAssign(t[1], t[2])
-	mul(t[0], t[0], t[3])
-	sub(&c[1], t[0], t[1])
 }
 
 func (e *fp2) mulAssign(a, b *fe2) {
-	t := e.t
+	wfp2Mul(e.w, b, a)
-	mul(t[1], &a[0], &b[0])
+	a.fromWide(e.w)
-	mul(t[2], &a[1], &b[1])
-	add(t[0], &a[0], &a[1])
-	add(t[3], &b[0], &b[1])
-	sub(&a[0], t[1], t[2])
-	addAssign(t[1], t[2])
-	mul(t[0], t[0], t[3])
-	sub(&a[1], t[0], t[1])
 }
 
 func (e *fp2) square(c, a *fe2) {
 	t := e.t
-	ladd(t[0], &a[0], &a[1])
+	// Guide to Pairing Based Cryptography
-	sub(t[1], &a[0], &a[1])
+	// Algorithm 5.16
-	ldouble(t[2], &a[0])
+
-	mul(&c[0], t[0], t[1])
+	ladd(t[0], &a[0], &a[1]) // (a0 + a1)
-	mul(&c[1], t[2], &a[1])
+	sub(t[1], &a[0], &a[1])  // (a0 - a1)
+	ldouble(t[2], &a[0])     // 2a0
+	mul(&c[0], t[0], t[1])   // c0 = (a0 + a1)(a0 - a1)
+	mul(&c[1], t[2], &a[1])  // c1 = 2a0a1
 }
 
 func (e *fp2) squareAssign(a *fe2) {
@@ -163,18 +102,23 @@ func (e *fp2) squareAssign(a *fe2) {
 	mul(&a[1], t[2], &a[1])
 }
 
-func (e *fp2) mulByNonResidue(c, a *fe2) {
+func (e *fp2) mul0(c, a *fe2, b *fe) {
-	t := e.t
+	mul(&c[0], &a[0], b)
-	sub(t[0], &a[0], &a[1])
+	mul(&c[1], &a[1], b)
-	add(&c[1], &a[0], &a[1])
+}
-	c[0].set(t[0])
+
+func (e *fp2) mul0Assign(a *fe2, b *fe) {
+	mul(&a[0], &a[0], b)
+	mul(&a[1], &a[1], b)
 }
 
 func (e *fp2) mulByB(c, a *fe2) {
 	t := e.t
+	// c0 = 4a0 - 4a1
+	// c1 = 4a0 + 4a1
 	double(t[0], &a[0])
-	double(t[1], &a[1])
 	doubleAssign(t[0])
+	double(t[1], &a[1])
 	doubleAssign(t[1])
 	sub(&c[0], t[0], t[1])
 	add(&c[1], t[0], t[1])
@@ -182,18 +126,70 @@ func (e *fp2) mulByB(c, a *fe2) {
 
 func (e *fp2) inverse(c, a *fe2) {
 	t := e.t
-	square(t[0], &a[0])
+	// Guide to Pairing Based Cryptography
-	square(t[1], &a[1])
+	// Algorithm 5.16
-	addAssign(t[0], t[1])
+
-	inverse(t[0], t[0])
+	square(t[0], &a[0])     // a0^2
-	mul(&c[0], &a[0], t[0])
+	square(t[1], &a[1])     // a1^2
-	mul(t[0], t[0], &a[1])
+	addAssign(t[0], t[1])   // a0^2 + a1^2
-	neg(&c[1], t[0])
+	inverse(t[0], t[0])     // (a0^2 + a1^2)^-1
+	mul(&c[0], &a[0], t[0]) // c0 = a0(a0^2 + a1^2)^-1
+	mul(t[0], t[0], &a[1])  // a1(a0^2 + a1^2)^-1
+	neg(&c[1], t[0])        // c1 = a1(a0^2 + a1^2)^-1
 }
 
-func (e *fp2) mulByFq(c, a *fe2, b *fe) {
+func (e *fp2) inverseBatch(in []fe2) {
-	mul(&c[0], &a[0], b)
+
-	mul(&c[1], &a[1], b)
+	n, N, setFirst := 0, len(in), false
+
+	for i := 0; i < len(in); i++ {
+		if !in[i].isZero() {
+			n++
+		}
+	}
+	if n == 0 {
+		return
+	}
+
+	tA := make([]fe2, n)
+	tB := make([]fe2, n)
+
+	// a, ab, abc, abcd, ...
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].isZero() {
+			if !setFirst {
+				setFirst = true
+				tA[j].set(&in[i])
+			} else {
+				e.mul(&tA[j], &in[i], &tA[j-1])
+			}
+			j = j + 1
+		}
+	}
+
+	// (abcd...)^-1
+	e.inverse(&tB[n-1], &tA[n-1])
+
+	// a^-1, ab^-1, abc^-1, abcd^-1, ...
+	for i, j := N-1, n-1; j != 0; i-- {
+		if !in[i].isZero() {
+			e.mul(&tB[j-1], &tB[j], &in[i])
+			j = j - 1
+		}
+	}
+
+	// a^-1, b^-1, c^-1, d^-1
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].isZero() {
+			if setFirst {
+				setFirst = false
+				in[i].set(&tB[j])
+			} else {
+				e.mul(&in[i], &tA[j-1], &tB[j])
+			}
+			j = j + 1
+		}
+	}
 }
 
 func (e *fp2) exp(c, a *fe2, s *big.Int) {
@@ -207,19 +203,13 @@ func (e *fp2) exp(c, a *fe2, s *big.Int) {
 	c.set(z)
 }
 
-func (e *fp2) frobeniusMap(c, a *fe2, power uint) {
+func (e *fp2) frobeniusMap1(a *fe2) {
-	c[0].set(&a[0])
+	fp2Conjugate(a, a)
-	if power%2 == 1 {
-		neg(&c[1], &a[1])
-		return
-	}
-	c[1].set(&a[1])
 }
 
-func (e *fp2) frobeniusMapAssign(a *fe2, power uint) {
+func (e *fp2) frobeniusMap(a *fe2, power int) {
-	if power%2 == 1 {
+	if power&1 == 1 {
-		neg(&a[1], &a[1])
+		fp2Conjugate(a, a)
-		return
 	}
 }
 
@@ -235,7 +225,7 @@ func (e *fp2) sqrt(c, a *fe2) bool {
 		c[1].set(&x0[0])
 		return true
 	}
-	e.add(alpha, alpha, e.one())
+	fp2Add(alpha, alpha, e.one())
 	e.exp(alpha, alpha, pMinus1Over2)
 	e.mul(c, alpha, x0)
 	e.square(alpha, c)
@@ -243,10 +233,74 @@ func (e *fp2) sqrt(c, a *fe2) bool {
 }
 
 func (e *fp2) isQuadraticNonResidue(a *fe2) bool {
-	// https://github.com/leovt/constructible/wiki/Taking-Square-Roots-in-quadratic-extension-Fields
 	c0, c1 := new(fe), new(fe)
 	square(c0, &a[0])
 	square(c1, &a[1])
 	add(c1, c1, c0)
 	return isQuadraticNonResidue(c1)
 }
+
+// faster square root algorith is adapted from blst library
+// https://github.com/supranational/blst/blob/master/src/sqrt.c
+
+func (e *fp2) sqrtBLST(out, inp *fe2) bool {
+	aa, bb := new(fe), new(fe)
+	ret := new(fe2)
+	square(aa, &inp[0])
+	square(bb, &inp[1])
+	add(aa, aa, bb)
+	sqrt(aa, aa)
+	sub(bb, &inp[0], aa)
+	add(aa, &inp[0], aa)
+	if aa.isZero() {
+		aa.set(bb)
+	}
+	mul(aa, aa, twoInv)
+	rsqrt(&ret[0], aa)
+	ret[1].set(&inp[1])
+	mul(&ret[1], &ret[1], twoInv)
+	mul(&ret[1], &ret[1], &ret[0])
+	mul(&ret[0], &ret[0], aa)
+	return e.sqrtAlignBLST(out, ret, ret, inp)
+}
+
+func (e *fp2) sqrtAlignBLST(out, ret, sqrt, inp *fe2) bool {
+
+	t0, t1 := new(fe2), new(fe2)
+	coeff := e.one()
+	e.square(t0, sqrt)
+
+	//
+	fp2Sub(t1, t0, inp)
+	isSqrt := t1.isZero()
+
+	//
+	fp2Add(t1, t0, inp)
+	flag := t1.isZero()
+	if flag {
+		coeff.set(sqrtMinus1)
+	}
+	isSqrt = flag || isSqrt
+
+	//
+	sub(&t1[0], &t0[0], &inp[1])
+	add(&t1[1], &t0[1], &inp[0])
+	flag = t1.isZero()
+	if flag {
+		coeff.set(sqrtSqrtMinus1)
+	}
+	isSqrt = flag || isSqrt
+
+	//
+	add(&t1[0], &t0[0], &inp[1])
+	sub(&t1[1], &t0[1], &inp[0])
+	flag = t1.isZero()
+	if flag {
+
+		coeff.set(sqrtMinusSqrtMinus1)
+	}
+	isSqrt = flag || isSqrt
+
+	e.mul(out, coeff, ret)
+	return isSqrt
+}

crypto/bls12381/fp6.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -22,7 +6,8 @@ import (
 )
 
 type fp6Temp struct {
-	t [6]*fe2
+	t  [5]*fe2
+	wt [6]*wfe2
 }
 
 type fp6 struct {
@@ -31,11 +16,15 @@ type fp6 struct {
 }
 
 func newFp6Temp() fp6Temp {
-	t := [6]*fe2{}
+	t := [5]*fe2{}
 	for i := 0; i < len(t); i++ {
 		t[i] = &fe2{}
 	}
-	return fp6Temp{t}
+	wt := [6]*wfe2{}
+	for i := 0; i < len(wt); i++ {
+		wt[i] = &wfe2{}
+	}
+	return fp6Temp{t, wt}
 }
 
 func newFp6(f *fp2) *fp6 {
@@ -47,19 +36,19 @@ func newFp6(f *fp2) *fp6 {
 }
 
 func (e *fp6) fromBytes(b []byte) (*fe6, error) {
-	if len(b) < 288 {
+	if len(b) != 288 {
-		return nil, errors.New("input string should be larger than 288 bytes")
+		return nil, errors.New("input string length must be equal to 288 bytes")
 	}
 	fp2 := e.fp2
-	u2, err := fp2.fromBytes(b[:96])
+	u2, err := fp2.fromBytes(b[:2*fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	u1, err := fp2.fromBytes(b[96:192])
+	u1, err := fp2.fromBytes(b[2*fpByteSize : 4*fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	u0, err := fp2.fromBytes(b[192:])
+	u0, err := fp2.fromBytes(b[4*fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -68,10 +57,10 @@ func (e *fp6) fromBytes(b []byte) (*fe6, error) {
 
 func (e *fp6) toBytes(a *fe6) []byte {
 	fp2 := e.fp2
-	out := make([]byte, 288)
+	out := make([]byte, 6*fpByteSize)
-	copy(out[:96], fp2.toBytes(&a[2]))
+	copy(out[:2*fpByteSize], fp2.toBytes(&a[2]))
-	copy(out[96:192], fp2.toBytes(&a[1]))
+	copy(out[2*fpByteSize:4*fpByteSize], fp2.toBytes(&a[1]))
-	copy(out[192:], fp2.toBytes(&a[0]))
+	copy(out[4*fpByteSize:], fp2.toBytes(&a[0]))
 	return out
 }
 
@@ -87,188 +76,368 @@ func (e *fp6) one() *fe6 {
 	return new(fe6).one()
 }
 
-func (e *fp6) add(c, a, b *fe6) {
+func fp6Ladd(c, a, b *fe6) {
-	fp2 := e.fp2
+	fp2Ladd(&c[0], &a[0], &b[0])
-	fp2.add(&c[0], &a[0], &b[0])
+	fp2Ladd(&c[1], &a[1], &b[1])
-	fp2.add(&c[1], &a[1], &b[1])
+	fp2Ladd(&c[2], &a[2], &b[2])
-	fp2.add(&c[2], &a[2], &b[2])
 }
 
-func (e *fp6) addAssign(a, b *fe6) {
+func wfp6SubAssign(a, b *wfe6) {
-	fp2 := e.fp2
+	wfp2SubAssign(&a[0], &b[0])
-	fp2.addAssign(&a[0], &b[0])
+	wfp2SubAssign(&a[1], &b[1])
-	fp2.addAssign(&a[1], &b[1])
+	wfp2SubAssign(&a[2], &b[2])
-	fp2.addAssign(&a[2], &b[2])
 }
 
-func (e *fp6) double(c, a *fe6) {
+func wfp6AddAssign(a, b *wfe6) {
-	fp2 := e.fp2
+	wfp2AddAssign(&a[0], &b[0])
-	fp2.double(&c[0], &a[0])
+	wfp2AddAssign(&a[1], &b[1])
-	fp2.double(&c[1], &a[1])
+	wfp2AddAssign(&a[2], &b[2])
-	fp2.double(&c[2], &a[2])
 }
 
-func (e *fp6) doubleAssign(a *fe6) {
+func fp6Add(c, a, b *fe6) {
-	fp2 := e.fp2
+	fp2Add(&c[0], &a[0], &b[0])
-	fp2.doubleAssign(&a[0])
+	fp2Add(&c[1], &a[1], &b[1])
-	fp2.doubleAssign(&a[1])
+	fp2Add(&c[2], &a[2], &b[2])
-	fp2.doubleAssign(&a[2])
 }
 
-func (e *fp6) sub(c, a, b *fe6) {
+func fp6AddAssign(a, b *fe6) {
-	fp2 := e.fp2
+	fp2AddAssign(&a[0], &b[0])
-	fp2.sub(&c[0], &a[0], &b[0])
+	fp2AddAssign(&a[1], &b[1])
-	fp2.sub(&c[1], &a[1], &b[1])
+	fp2AddAssign(&a[2], &b[2])
-	fp2.sub(&c[2], &a[2], &b[2])
 }
 
-func (e *fp6) subAssign(a, b *fe6) {
+func fp6Double(c, a *fe6) {
-	fp2 := e.fp2
+	fp2Double(&c[0], &a[0])
-	fp2.subAssign(&a[0], &b[0])
+	fp2Double(&c[1], &a[1])
-	fp2.subAssign(&a[1], &b[1])
+	fp2Double(&c[2], &a[2])
-	fp2.subAssign(&a[2], &b[2])
 }
 
-func (e *fp6) neg(c, a *fe6) {
+func fp6DoubleAssign(a *fe6) {
-	fp2 := e.fp2
+	fp2DoubleAssign(&a[0])
-	fp2.neg(&c[0], &a[0])
+	fp2DoubleAssign(&a[1])
-	fp2.neg(&c[1], &a[1])
+	fp2DoubleAssign(&a[2])
-	fp2.neg(&c[2], &a[2])
 }
 
-func (e *fp6) mul(c, a, b *fe6) {
+func fp6Sub(c, a, b *fe6) {
-	fp2, t := e.fp2, e.t
+	fp2Sub(&c[0], &a[0], &b[0])
-	fp2.mul(t[0], &a[0], &b[0])
+	fp2Sub(&c[1], &a[1], &b[1])
-	fp2.mul(t[1], &a[1], &b[1])
+	fp2Sub(&c[2], &a[2], &b[2])
-	fp2.mul(t[2], &a[2], &b[2])
+}
-	fp2.add(t[3], &a[1], &a[2])
+
-	fp2.add(t[4], &b[1], &b[2])
+func fp6SubAssign(a, b *fe6) {
-	fp2.mulAssign(t[3], t[4])
+	fp2SubAssign(&a[0], &b[0])
-	fp2.add(t[4], t[1], t[2])
+	fp2SubAssign(&a[1], &b[1])
-	fp2.subAssign(t[3], t[4])
+	fp2SubAssign(&a[2], &b[2])
-	fp2.mulByNonResidue(t[3], t[3])
+}
-	fp2.add(t[5], t[0], t[3])
+
-	fp2.add(t[3], &a[0], &a[1])
+func fp6Neg(c, a *fe6) {
-	fp2.add(t[4], &b[0], &b[1])
+	fp2Neg(&c[0], &a[0])
-	fp2.mulAssign(t[3], t[4])
+	fp2Neg(&c[1], &a[1])
-	fp2.add(t[4], t[0], t[1])
+	fp2Neg(&c[2], &a[2])
-	fp2.subAssign(t[3], t[4])
+}
-	fp2.mulByNonResidue(t[4], t[2])
+
-	fp2.add(&c[1], t[3], t[4])
+func (e *fp6) wmul01(c *wfe6, a *fe6, b0, b1 *fe2) {
-	fp2.add(t[3], &a[0], &a[2])
+	wt, t := e.wt, e.t
-	fp2.add(t[4], &b[0], &b[2])
+	wfp2Mul(wt[0], &a[0], b0)   // v0 = b0a0
-	fp2.mulAssign(t[3], t[4])
+	wfp2Mul(wt[1], &a[1], b1)   // v1 = a1b1
-	fp2.add(t[4], t[0], t[2])
+	fp2Ladd(t[2], &a[1], &a[2]) // a1 + a2
-	fp2.subAssign(t[3], t[4])
+	wfp2Mul(wt[2], t[2], b1)    // b1(a1 + a2)
-	fp2.add(&c[2], t[1], t[3])
+	wfp2SubAssign(wt[2], wt[1]) // b1(a1 + a2) - v1
-	c[0].set(t[5])
+	wfp2MulByNonResidueAssign(wt[2])
+	fp2Ladd(t[3], &a[0], &a[2]) // a0 + a2
+	wfp2Mul(wt[3], t[3], b0)    // b0(a0 + a2)
+	wfp2SubAssign(wt[3], wt[0])
+	wfp2Add(&c[2], wt[3], wt[1])
+	fp2Ladd(t[0], b0, b1)       // (b0 + b1)
+	fp2Ladd(t[1], &a[0], &a[1]) // (a0 + a1)
+	wfp2Mul(wt[4], t[0], t[1])  // (a0 + a1)(b0 + b1)
+	wfp2SubAssign(wt[4], wt[0])
+	wfp2Sub(&c[1], wt[4], wt[1])
+	wfp2Add(&c[0], wt[2], wt[0])
+}
+
+func (e *fp6) wmul1(c *wfe6, a *fe6, b1 *fe2) {
+	wt := e.wt
+	wfp2Mul(wt[0], &a[2], b1)
+	wfp2Mul(&c[2], &a[1], b1)
+	wfp2Mul(&c[1], &a[0], b1)
+	wfp2MulByNonResidue(&c[0], wt[0])
+}
+
+func (e *fp6) wmul(c *wfe6, a, b *fe6) {
+
+	wt, t := e.wt, e.t
+
+	// Faster Explicit Formulas for Computing Pairings over Ordinary Curves
+	// AKLGL
+	// https://eprint.iacr.org/2010/526.pdf
+	// Algorithm 3
+
+	// 1. T0 = a0b0,T1 = a1b1, T2 = a2b2
+	wfp2Mul(wt[0], &a[0], &b[0])
+	wfp2Mul(wt[1], &a[1], &b[1])
+	wfp2Mul(wt[2], &a[2], &b[2])
+	// 2. t0 = a1 + a2, t1 = b1 + b2
+	fp2Ladd(t[0], &a[1], &a[2])
+	fp2Ladd(t[1], &b[1], &b[2])
+	// 3. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+	// 4. T4 = T1 + T2
+	wfp2Add(wt[4], wt[1], wt[2])
+
+	// 5,6. T3 = T3 - T4
+	wfp2SubMixedAssign(wt[3], wt[4])
+
+	// 7. T4 = β * T3
+	wfp2MulByNonResidue(wt[4], wt[3])
+
+	// 8. T5 = T4 + T0
+	wfp2Add(wt[5], wt[4], wt[0])
+
+	// 9. t0 = a0 + a1, t1 = b0 + b1
+	fp2Ladd(t[0], &a[0], &a[1])
+	fp2Ladd(t[1], &b[0], &b[1])
+
+	// 10. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 11. T4 = T0 + T1
+	wfp2Add(wt[4], wt[0], wt[1])
+
+	// 12,13. T3 = T3 - T4
+	wfp2SubMixedAssign(wt[3], wt[4])
+
+	// 14,15. T4 = β * T2
+	wfp2MulByNonResidue(wt[4], wt[2])
+
+	// 17. t0 = a0 + a2, t1 = b0 + b2
+	fp2Ladd(t[0], &a[0], &a[2])
+	fp2Ladd(t[1], &b[0], &b[2])
+
+	// 16. T6 = T3 + T4
+	wfp2Add(&c[1], wt[3], wt[4])
+
+	// 18. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 19. T4 = T0 + T2
+	wfp2Add(wt[4], wt[0], wt[2])
+
+	// 20,21. T3 = T3 - T4
+	wfp2SubMixedAssign(wt[3], wt[4])
+
+	// 22,23. T7 = T3 + T1
+	wfp2AddMixed(&c[2], wt[3], wt[1])
+
+	// c = T5, T6, T7
+	c[0].set(wt[5])
+}
+
+func (e *fp6) mul(c *fe6, a, b *fe6) {
+	wt, t := e.wt, e.t
+
+	// 1. T0 = a0b0,T1 = a1b1, T2 = a2b2
+	wfp2Mul(wt[0], &a[0], &b[0])
+	wfp2Mul(wt[1], &a[1], &b[1])
+	wfp2Mul(wt[2], &a[2], &b[2])
+	// 2. t0 = a1 + a2, t1 = b1 + b2
+	fp2Ladd(t[0], &a[1], &a[2])
+	fp2Ladd(t[1], &b[1], &b[2])
+	// 3. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+	// 4. T4 = T1 + T2
+	wfp2Add(wt[4], wt[1], wt[2])
+
+	// 5,6. T3 = T3 - T4
+	wfp2SubMixedAssign(wt[3], wt[4])
+
+	// 7. T4 = β * T3
+	wfp2MulByNonResidue(wt[4], wt[3])
+
+	// 8. T5 = T4 + T0
+	wfp2Add(wt[5], wt[4], wt[0])
+
+	// 9. t0 = a0 + a1, t1 = b0 + b1
+	fp2Ladd(t[0], &a[0], &a[1])
+	fp2Ladd(t[1], &b[0], &b[1])
+
+	// 10. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 11. T4 = T0 + T1
+	wfp2Add(wt[4], wt[0], wt[1])
+
+	// 12,13. T3 = T3 - T4
+	wfp2SubMixed(wt[3], wt[3], wt[4])
+
+	// 14,15. T4 = β * T2
+	wfp2MulByNonResidue(wt[4], wt[2])
+
+	// 17. t0 = a0 + a2, t1 = b0 + b2
+	fp2Ladd(t[0], &a[0], &a[2])
+	fp2Ladd(t[1], &b[0], &b[2])
+
+	// 16. T6 = T3 + T4
+	wfp2Add(wt[3], wt[3], wt[4])
+	c[1].fromWide(wt[3])
+
+	// 18. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 19. T4 = T0 + T2
+	wfp2Add(wt[4], wt[0], wt[2])
+
+	// 20,21. T3 = T3 - T4
+	wfp2SubMixed(wt[3], wt[3], wt[4])
+
+	// 22,23. T7 = T3 + T1
+	wfp2AddMixed(wt[3], wt[3], wt[1])
+	c[2].fromWide(wt[3])
+
+	// c = T5, T6, T7
+	c[0].fromWide(wt[5])
 }
 
 func (e *fp6) mulAssign(a, b *fe6) {
-	fp2, t := e.fp2, e.t
+	wt, t := e.wt, e.t
-	fp2.mul(t[0], &a[0], &b[0])
+
-	fp2.mul(t[1], &a[1], &b[1])
+	// Faster Explicit Formulas for Computing Pairings over Ordinary Curves
-	fp2.mul(t[2], &a[2], &b[2])
+	// AKLGL
-	fp2.add(t[3], &a[1], &a[2])
+	// https://eprint.iacr.org/2010/526.pdf
-	fp2.add(t[4], &b[1], &b[2])
+	// Algorithm 3
-	fp2.mulAssign(t[3], t[4])
+
-	fp2.add(t[4], t[1], t[2])
+	// 1. T0 = a0b0,T1 = a1b1, T2 = a2b2
-	fp2.subAssign(t[3], t[4])
+	wfp2Mul(wt[0], &a[0], &b[0])
-	fp2.mulByNonResidue(t[3], t[3])
+	wfp2Mul(wt[1], &a[1], &b[1])
-	fp2.add(t[5], t[0], t[3])
+	wfp2Mul(wt[2], &a[2], &b[2])
-	fp2.add(t[3], &a[0], &a[1])
+	// 2. t0 = a1 + a2, t1 = b1 + b2
-	fp2.add(t[4], &b[0], &b[1])
+	fp2Ladd(t[0], &a[1], &a[2])
-	fp2.mulAssign(t[3], t[4])
+	fp2Ladd(t[1], &b[1], &b[2])
-	fp2.add(t[4], t[0], t[1])
+	// 3. T3 = t0 * t1
-	fp2.subAssign(t[3], t[4])
+	wfp2Mul(wt[3], t[0], t[1])
-	fp2.mulByNonResidue(t[4], t[2])
+	// 4. T4 = T1 + T2
-	fp2.add(&a[1], t[3], t[4])
+	wfp2Add(wt[4], wt[1], wt[2])
-	fp2.add(t[3], &a[0], &a[2])
+
-	fp2.add(t[4], &b[0], &b[2])
+	// 5,6. T3 = T3 - T4
-	fp2.mulAssign(t[3], t[4])
+	wfp2SubMixed(wt[3], wt[3], wt[4])
-	fp2.add(t[4], t[0], t[2])
+
-	fp2.subAssign(t[3], t[4])
+	// 7. T4 = β * T3
-	fp2.add(&a[2], t[1], t[3])
+	wfp2MulByNonResidue(wt[4], wt[3])
-	a[0].set(t[5])
+
+	// 8. T5 = T4 + T0
+	wfp2Add(wt[5], wt[4], wt[0])
+
+	// 9. t0 = a0 + a1, t1 = b0 + b1
+	fp2Ladd(t[0], &a[0], &a[1])
+	fp2Ladd(t[1], &b[0], &b[1])
+
+	// 10. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 11. T4 = T0 + T1
+	wfp2Add(wt[4], wt[0], wt[1])
+
+	// 12,13. T3 = T3 - T4
+	wfp2SubMixed(wt[3], wt[3], wt[4])
+
+	// 14,15. T4 = β * T2
+	wfp2MulByNonResidue(wt[4], wt[2])
+
+	// 17. t0 = a0 + a2, t1 = b0 + b2
+	fp2Ladd(t[0], &a[0], &a[2])
+	fp2Ladd(t[1], &b[0], &b[2])
+
+	// 16. T6 = T3 + T4
+	wfp2Add(wt[3], wt[3], wt[4])
+	a[1].fromWide(wt[3])
+
+	// 18. T3 = t0 * t1
+	wfp2Mul(wt[3], t[0], t[1])
+
+	// 19. T4 = T0 + T2
+	wfp2Add(wt[4], wt[0], wt[2])
+
+	// 20,21. T3 = T3 - T4
+	wfp2SubMixed(wt[3], wt[3], wt[4])
+
+	// 22,23. T7 = T3 + T1
+	wfp2AddMixed(wt[3], wt[3], wt[1])
+	a[2].fromWide(wt[3])
+
+	// a = T5, T6, T7
+	a[0].fromWide(wt[5])
 }
 
 func (e *fp6) square(c, a *fe6) {
-	fp2, t := e.fp2, e.t
+	wt, t := e.wt, e.t
-	fp2.square(t[0], &a[0])
+	wfp2Square(wt[0], &a[0])
-	fp2.mul(t[1], &a[0], &a[1])
+	wfp2Mul(wt[1], &a[0], &a[1])
-	fp2.doubleAssign(t[1])
+	wfp2DoubleAssign(wt[1])
-	fp2.sub(t[2], &a[0], &a[1])
+	fp2Sub(t[2], &a[0], &a[1])
-	fp2.addAssign(t[2], &a[2])
+	fp2AddAssign(t[2], &a[2])
-	fp2.squareAssign(t[2])
+	wfp2Square(wt[2], t[2])
-	fp2.mul(t[3], &a[1], &a[2])
+	wfp2Mul(wt[3], &a[1], &a[2])
-	fp2.doubleAssign(t[3])
+	wfp2DoubleAssign(wt[3])
-	fp2.square(t[4], &a[2])
+	wfp2Square(wt[4], &a[2])
-	fp2.mulByNonResidue(t[5], t[3])
+	wfp2MulByNonResidue(wt[5], wt[3])
-	fp2.add(&c[0], t[0], t[5])
+	wfp2AddAssign(wt[5], wt[0])
-	fp2.mulByNonResidue(t[5], t[4])
+	c[0].fromWide(wt[5])
-	fp2.add(&c[1], t[1], t[5])
+	wfp2MulByNonResidue(wt[5], wt[4])
-	fp2.addAssign(t[1], t[2])
+	wfp2AddAssign(wt[5], wt[1])
-	fp2.addAssign(t[1], t[3])
+	c[1].fromWide(wt[5])
-	fp2.addAssign(t[0], t[4])
+	wfp2AddAssign(wt[1], wt[2])
-	fp2.sub(&c[2], t[1], t[0])
+	wfp2AddAssign(wt[1], wt[3])
+	wfp2AddAssign(wt[0], wt[4])
+	wfp2SubAssign(wt[1], wt[0])
+	c[2].fromWide(wt[1])
+
 }
 
-func (e *fp6) mulBy01Assign(a *fe6, b0, b1 *fe2) {
+func (e *fp6) wsquare(c *wfe6, a *fe6) {
-	fp2, t := e.fp2, e.t
+	wt, t := e.wt, e.t
-	fp2.mul(t[0], &a[0], b0)
+	wfp2Square(wt[0], &a[0])
-	fp2.mul(t[1], &a[1], b1)
+	wfp2Mul(wt[1], &a[0], &a[1])
-	fp2.add(t[5], &a[1], &a[2])
+	wfp2DoubleAssign(wt[1])
-	fp2.mul(t[2], b1, t[5])
+	fp2Sub(t[2], &a[0], &a[1])
-	fp2.subAssign(t[2], t[1])
+	fp2AddAssign(t[2], &a[2])
-	fp2.mulByNonResidue(t[2], t[2])
+	wfp2Square(wt[2], t[2])
-	fp2.add(t[5], &a[0], &a[2])
+	wfp2Mul(wt[3], &a[1], &a[2])
-	fp2.mul(t[3], b0, t[5])
+	wfp2DoubleAssign(wt[3])
-	fp2.subAssign(t[3], t[0])
+	wfp2Square(wt[4], &a[2])
-	fp2.add(&a[2], t[3], t[1])
+	wfp2MulByNonResidue(wt[5], wt[3])
-	fp2.add(t[4], b0, b1)
+	wfp2Add(&c[0], wt[5], wt[0])
-	fp2.add(t[5], &a[0], &a[1])
+	wfp2MulByNonResidue(wt[5], wt[4])
-	fp2.mulAssign(t[4], t[5])
+	wfp2Add(&c[1], wt[1], wt[5])
-	fp2.subAssign(t[4], t[0])
+	wfp2AddAssign(wt[1], wt[2])
-	fp2.sub(&a[1], t[4], t[1])
+	wfp2AddAssign(wt[1], wt[3])
-	fp2.add(&a[0], t[2], t[0])
+	wfp2AddAssign(wt[0], wt[4])
-}
+	wfp2Sub(&c[2], wt[1], wt[0])
-
-func (e *fp6) mulBy01(c, a *fe6, b0, b1 *fe2) {
-	fp2, t := e.fp2, e.t
-	fp2.mul(t[0], &a[0], b0)
-	fp2.mul(t[1], &a[1], b1)
-	fp2.add(t[2], &a[1], &a[2])
-	fp2.mulAssign(t[2], b1)
-	fp2.subAssign(t[2], t[1])
-	fp2.mulByNonResidue(t[2], t[2])
-	fp2.add(t[3], &a[0], &a[2])
-	fp2.mulAssign(t[3], b0)
-	fp2.subAssign(t[3], t[0])
-	fp2.add(&c[2], t[3], t[1])
-	fp2.add(t[4], b0, b1)
-	fp2.add(t[3], &a[0], &a[1])
-	fp2.mulAssign(t[4], t[3])
-	fp2.subAssign(t[4], t[0])
-	fp2.sub(&c[1], t[4], t[1])
-	fp2.add(&c[0], t[2], t[0])
-}
-
-func (e *fp6) mulBy1(c, a *fe6, b1 *fe2) {
-	fp2, t := e.fp2, e.t
-	fp2.mul(t[0], &a[2], b1)
-	fp2.mul(&c[2], &a[1], b1)
-	fp2.mul(&c[1], &a[0], b1)
-	fp2.mulByNonResidue(&c[0], t[0])
 }
 
 func (e *fp6) mulByNonResidue(c, a *fe6) {
-	fp2, t := e.fp2, e.t
+	t := e.t
 	t[0].set(&a[0])
-	fp2.mulByNonResidue(&c[0], &a[2])
+	mulByNonResidue(&c[0], &a[2])
 	c[2].set(&a[1])
 	c[1].set(t[0])
 }
 
+func (e *fp6) wmulByNonResidue(c, a *wfe6) {
+	t := e.wt
+	t[0].set(&a[0])
+	wfp2MulByNonResidue(&c[0], &a[2])
+	c[2].set(&a[1])
+	c[1].set(t[0])
+}
+
+func (e *fp6) wmulByNonResidueAssign(a *wfe6) {
+	t := e.wt
+	t[0].set(&a[0])
+	wfp2MulByNonResidue(&a[0], &a[2])
+	a[2].set(&a[1])
+	a[1].set(t[0])
+}
+
 func (e *fp6) mulByBaseField(c, a *fe6, b *fe2) {
 	fp2 := e.fp2
 	fp2.mul(&c[0], &a[0], b)
@@ -291,61 +460,57 @@ func (e *fp6) inverse(c, a *fe6) {
 	fp2, t := e.fp2, e.t
 	fp2.square(t[0], &a[0])
 	fp2.mul(t[1], &a[1], &a[2])
-	fp2.mulByNonResidue(t[1], t[1])
+	mulByNonResidueAssign(t[1])
-	fp2.subAssign(t[0], t[1])
+	fp2SubAssign(t[0], t[1])    // A = v0 - βv5
-	fp2.square(t[1], &a[1])
+	fp2.square(t[1], &a[1])     // v1 = a1^2
-	fp2.mul(t[2], &a[0], &a[2])
+	fp2.mul(t[2], &a[0], &a[2]) // v4 = a0a2
-	fp2.subAssign(t[1], t[2])
+	fp2SubAssign(t[1], t[2])    // C = v1 - v4
-	fp2.square(t[2], &a[2])
+	fp2.square(t[2], &a[2])     // v2 = a2^2
-	fp2.mulByNonResidue(t[2], t[2])
+	mulByNonResidueAssign(t[2]) // βv2
-	fp2.mul(t[3], &a[0], &a[1])
+	fp2.mul(t[3], &a[0], &a[1]) // v3 = a0a1
-	fp2.subAssign(t[2], t[3])
+	fp2SubAssign(t[2], t[3])    // B = βv2 - v3
-	fp2.mul(t[3], &a[2], t[2])
+	fp2.mul(t[3], &a[2], t[2])  // B * a2
-	fp2.mul(t[4], &a[1], t[1])
+	fp2.mul(t[4], &a[1], t[1])  // C * a1
-	fp2.addAssign(t[3], t[4])
+	fp2AddAssign(t[3], t[4])    // Ca1 + Ba2
-	fp2.mulByNonResidue(t[3], t[3])
+	mulByNonResidueAssign(t[3]) // β(Ca1 + Ba2)
-	fp2.mul(t[4], &a[0], t[0])
+	fp2.mul(t[4], &a[0], t[0])  // Aa0
-	fp2.addAssign(t[3], t[4])
+	fp2AddAssign(t[3], t[4])    // v6 = Aa0 + β(Ca1 + Ba2)
-	fp2.inverse(t[3], t[3])
+	fp2.inverse(t[3], t[3])     // F = v6^-1
-	fp2.mul(&c[0], t[0], t[3])
+	fp2.mul(&c[0], t[0], t[3])  // c0 = AF
-	fp2.mul(&c[1], t[2], t[3])
+	fp2.mul(&c[1], t[2], t[3])  // c1 = BF
-	fp2.mul(&c[2], t[1], t[3])
+	fp2.mul(&c[2], t[1], t[3])  // c2 = CF
 }
 
-func (e *fp6) frobeniusMap(c, a *fe6, power uint) {
+func (e *fp6) frobeniusMap(a *fe6, power int) {
 	fp2 := e.fp2
-	fp2.frobeniusMap(&c[0], &a[0], power)
+	fp2.frobeniusMap(&a[0], power)
-	fp2.frobeniusMap(&c[1], &a[1], power)
+	fp2.frobeniusMap(&a[1], power)
-	fp2.frobeniusMap(&c[2], &a[2], power)
+	fp2.frobeniusMap(&a[2], power)
-	switch power % 6 {
+	fp2.mulAssign(&a[1], &frobeniusCoeffs61[power%6])
-	case 0:
+	fp2.mulAssign(&a[2], &frobeniusCoeffs62[power%6])
-		return
-	case 3:
-		neg(&c[0][0], &a[1][1])
-		c[1][1].set(&a[1][0])
-		fp2.neg(&a[2], &a[2])
-	default:
-		fp2.mul(&c[1], &c[1], &frobeniusCoeffs61[power%6])
-		fp2.mul(&c[2], &c[2], &frobeniusCoeffs62[power%6])
-	}
 }
 
-func (e *fp6) frobeniusMapAssign(a *fe6, power uint) {
+func (e *fp6) frobeniusMap1(a *fe6) {
 	fp2 := e.fp2
-	fp2.frobeniusMapAssign(&a[0], power)
+	fp2.frobeniusMap1(&a[0])
-	fp2.frobeniusMapAssign(&a[1], power)
+	fp2.frobeniusMap1(&a[1])
-	fp2.frobeniusMapAssign(&a[2], power)
+	fp2.frobeniusMap1(&a[2])
+	fp2.mulAssign(&a[1], &frobeniusCoeffs61[1])
+	fp2.mulAssign(&a[2], &frobeniusCoeffs62[1])
+}
+
+func (e *fp6) frobeniusMap2(a *fe6) {
+	e.fp2.mulAssign(&a[1], &frobeniusCoeffs61[2])
+	e.fp2.mulAssign(&a[2], &frobeniusCoeffs62[2])
+}
+
+func (e *fp6) frobeniusMap3(a *fe6) {
 	t := e.t
-	switch power % 6 {
+	e.fp2.frobeniusMap1(&a[0])
-	case 0:
+	e.fp2.frobeniusMap1(&a[1])
-		return
+	e.fp2.frobeniusMap1(&a[2])
-	case 3:
 	neg(&t[0][0], &a[1][1])
 	a[1][1].set(&a[1][0])
 	a[1][0].set(&t[0][0])
-		fp2.neg(&a[2], &a[2])
+	fp2Neg(&a[2], &a[2])
-	default:
-		fp2.mulAssign(&a[1], &frobeniusCoeffs61[power%6])
-		fp2.mulAssign(&a[2], &frobeniusCoeffs62[power%6])
-	}
 }

crypto/bls12381/fr.go

@@ -0,0 +1,455 @@
+package bls12381
+
+import (
+	"crypto/rand"
+	"io"
+	"math/big"
+	"math/bits"
+)
+
+const frByteSize = 32
+const frBitSize = 255
+const frNumberOfLimbs = 4
+const fourWordBitSize = 256
+
+type Fr [4]uint64
+type wideFr [8]uint64
+
+func NewFr() *Fr {
+	return &Fr{}
+}
+
+func (e *Fr) Rand(r io.Reader) (*Fr, error) {
+	bi, err := rand.Int(r, qBig)
+	if err != nil {
+		return nil, err
+	}
+	_ = e.fromBig(bi)
+	return e, nil
+}
+
+func (e *Fr) Set(e2 *Fr) *Fr {
+	e[0] = e2[0]
+	e[1] = e2[1]
+	e[2] = e2[2]
+	e[3] = e2[3]
+	return e
+}
+
+func (e *Fr) Zero() *Fr {
+	e[0] = 0
+	e[1] = 0
+	e[2] = 0
+	e[3] = 0
+	return e
+}
+
+func (e *Fr) One() *Fr {
+	e.Set(&Fr{1})
+	return e
+}
+
+func (e *Fr) RedOne() *Fr {
+	e.Set(qr1)
+	return e
+}
+
+func (e *Fr) FromBytes(in []byte) *Fr {
+	e.fromBytes(in)
+	return e
+}
+
+func (e *Fr) RedFromBytes(in []byte) *Fr {
+	e.fromBytes(in)
+	e.toMont()
+	return e
+}
+
+func (e *Fr) fromBytes(in []byte) *Fr {
+	u := new(big.Int).SetBytes(in)
+	_ = e.fromBig(u)
+	return e
+}
+
+func (e *Fr) fromBig(in *big.Int) *Fr {
+	e.Zero()
+	_in := new(big.Int).Set(in)
+	zero := new(big.Int)
+	c0 := _in.Cmp(zero)
+	c1 := _in.Cmp(qBig)
+	if c0 == -1 || c1 == 1 {
+		_in.Mod(_in, qBig)
+	}
+
+	words := _in.Bits()      // a little-endian Word slice
+	if bits.UintSize == 64 { // in the 64-bit architecture
+		for i := 0; i < len(words); i++ {
+			e[i] = uint64(words[i])
+		}
+	} else { // in the 32-bit architecture
+		for i := 0; i < len(e); i++ {
+			j := i * 2
+			if j+1 < len(words) {
+				e[i] = uint64(words[j+1])<<32 | uint64(words[j])
+			} else if j < len(words) {
+				e[i] = uint64(words[j])
+			} else {
+				e[i] = uint64(0)
+			}
+		}
+	}
+
+	return e
+}
+
+func (e *Fr) setUint64(n uint64) *Fr {
+	e.Zero()
+	e[0] = n
+	return e
+}
+
+func (e *Fr) ToBytes() []byte {
+	return NewFr().Set(e).bytes()
+}
+
+func (e *Fr) RedToBytes() []byte {
+	out := NewFr().Set(e)
+	out.fromMont()
+	return out.bytes()
+}
+
+func (e *Fr) ToBig() *big.Int {
+	return new(big.Int).SetBytes(e.ToBytes())
+}
+
+func (e *Fr) RedToBig() *big.Int {
+	return new(big.Int).SetBytes(e.RedToBytes())
+}
+
+func (e *Fr) bytes() []byte {
+	out := make([]byte, frByteSize)
+	var a int
+	for i := 0; i < frNumberOfLimbs; i++ {
+		a = frByteSize - i*8
+		out[a-1] = byte(e[i])
+		out[a-2] = byte(e[i] >> 8)
+		out[a-3] = byte(e[i] >> 16)
+		out[a-4] = byte(e[i] >> 24)
+		out[a-5] = byte(e[i] >> 32)
+		out[a-6] = byte(e[i] >> 40)
+		out[a-7] = byte(e[i] >> 48)
+		out[a-8] = byte(e[i] >> 56)
+	}
+	return out
+}
+
+func (e *Fr) IsZero() bool {
+	return (e[3] | e[2] | e[1] | e[0]) == 0
+}
+
+func (e *Fr) IsOne() bool {
+	return e.Equal(&Fr{1})
+}
+
+func (e *Fr) IsRedOne() bool {
+	return e.Equal(qr1)
+}
+
+func (e *Fr) Equal(e2 *Fr) bool {
+	return e2[0] == e[0] && e2[1] == e[1] && e2[2] == e[2] && e2[3] == e[3]
+}
+
+func (e *Fr) Cmp(e1 *Fr) int {
+	for i := frNumberOfLimbs - 1; i >= 0; i-- {
+		if e[i] > e1[i] {
+			return 1
+		} else if e[i] < e1[i] {
+			return -1
+		}
+	}
+	return 0
+}
+
+func (e *Fr) sliceUint64(from int) uint64 {
+	if from < 64 {
+		return e[0]>>from | e[1]<<(64-from)
+	} else if from < 128 {
+		return e[1]>>(from-64) | e[2]<<(128-from)
+	} else if from < 192 {
+		return e[2]>>(from-128) | e[3]<<(192-from)
+	}
+	return e[3] >> (from - 192)
+}
+
+func (e *Fr) div2() {
+	e[0] = e[0]>>1 | e[1]<<63
+	e[1] = e[1]>>1 | e[2]<<63
+	e[2] = e[2]>>1 | e[3]<<63
+	e[3] = e[3] >> 1
+}
+
+func (e *Fr) mul2() uint64 {
+	c := e[3] >> 63
+	e[3] = e[3]<<1 | e[2]>>63
+	e[2] = e[2]<<1 | e[1]>>63
+	e[1] = e[1]<<1 | e[0]>>63
+	e[0] = e[0] << 1
+	return c
+}
+
+func (e *Fr) isEven() bool {
+	var mask uint64 = 1
+	return e[0]&mask == 0
+}
+
+func (e *Fr) Bit(at int) bool {
+	if at < 64 {
+		return (e[0]>>at)&1 == 1
+	} else if at < 128 {
+		return (e[1]>>(at-64))&1 == 1
+	} else if at < 192 {
+		return (e[2]>>(at-128))&1 == 1
+	} else if at < 256 {
+		return (e[3]>>(at-192))&1 == 1
+	}
+	return false
+}
+
+func (e *Fr) toMont() {
+	e.RedMul(e, qr2)
+}
+
+func (e *Fr) fromMont() {
+	e.RedMul(e, &Fr{1})
+}
+
+func (e *Fr) FromRed() {
+	e.fromMont()
+}
+
+func (e *Fr) ToRed() {
+	e.toMont()
+}
+
+func (e *Fr) Add(a, b *Fr) {
+	addFR(e, a, b)
+}
+
+func (e *Fr) Double(a *Fr) {
+	doubleFR(e, a)
+}
+
+func (e *Fr) Sub(a, b *Fr) {
+	subFR(e, a, b)
+}
+
+func (e *Fr) Neg(a *Fr) {
+	negFR(e, a)
+}
+
+func (e *Fr) Mul(a, b *Fr) {
+	e.RedMul(a, b)
+	e.toMont()
+}
+
+func (e *Fr) RedMul(a, b *Fr) {
+	mulFR(e, a, b)
+}
+
+func (e *Fr) Square(a *Fr) {
+	e.RedSquare(a)
+	e.toMont()
+}
+
+func (e *Fr) RedSquare(a *Fr) {
+	squareFR(e, a)
+}
+
+func (e *Fr) RedExp(a *Fr, ee *big.Int) {
+	z := new(Fr).RedOne()
+	for i := ee.BitLen(); i >= 0; i-- {
+		z.RedSquare(z)
+		if ee.Bit(i) == 1 {
+			z.RedMul(z, a)
+		}
+	}
+	e.Set(z)
+}
+
+func (e *Fr) Exp(a *Fr, ee *big.Int) {
+	e.Set(a).toMont()
+	e.RedExp(e, ee)
+	e.fromMont()
+
+}
+
+func RedInverseBatchFr(in []Fr) {
+	inverseBatchFr(in, func(a, b *Fr) { a.RedInverse(b) })
+}
+
+func InverseBatchFr(in []Fr) {
+	inverseBatchFr(in, func(a, b *Fr) { a.Inverse(b) })
+}
+
+func inverseBatchFr(in []Fr, invFn func(out *Fr, in *Fr)) {
+	n, N, setFirst := 0, len(in), false
+
+	for i := 0; i < len(in); i++ {
+		if !in[i].IsZero() {
+			n++
+		}
+	}
+	if n == 0 {
+		return
+	}
+
+	tA := make([]Fr, n)
+	tB := make([]Fr, n)
+
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].IsZero() {
+			if !setFirst {
+				setFirst = true
+				tA[j].Set(&in[i])
+			} else {
+				tA[j].Mul(&in[i], &tA[j-1])
+			}
+			j = j + 1
+		}
+	}
+
+	invFn(&tB[n-1], &tA[n-1])
+	for i, j := N-1, n-1; j != 0; i-- {
+		if !in[i].IsZero() {
+			tB[j-1].Mul(&tB[j], &in[i])
+			j = j - 1
+		}
+	}
+
+	for i, j := 0, 0; i < N; i++ {
+		if !in[i].IsZero() {
+			if setFirst {
+				setFirst = false
+				in[i].Set(&tB[j])
+			} else {
+				in[i].Mul(&tA[j-1], &tB[j])
+			}
+			j = j + 1
+		}
+	}
+}
+
+func (e *Fr) Inverse(a *Fr) {
+	e.Set(a).toMont()
+	e.RedInverse(e)
+	e.fromMont()
+}
+
+func (e *Fr) RedInverse(ei *Fr) {
+	if ei.IsZero() {
+		e.Zero()
+		return
+	}
+	u := new(Fr).Set(&q)
+	v := new(Fr).Set(ei)
+	s := &Fr{1}
+	r := &Fr{0}
+	var k int
+	var z uint64
+	var found = false
+	// Phase 1
+	for i := 0; i < fourWordBitSize*2; i++ {
+		if v.IsZero() {
+			found = true
+			break
+		}
+		if u.isEven() {
+			u.div2()
+			s.mul2()
+		} else if v.isEven() {
+			v.div2()
+			z += r.mul2()
+		} else if u.Cmp(v) == 1 {
+			lsubAssignFR(u, v)
+			u.div2()
+			laddAssignFR(r, s)
+			s.mul2()
+		} else {
+			lsubAssignFR(v, u)
+			v.div2()
+			laddAssignFR(s, r)
+			z += r.mul2()
+		}
+		k += 1
+	}
+
+	if !found {
+		e.Zero()
+		return
+	}
+
+	if k < frBitSize || k > frBitSize+fourWordBitSize {
+		e.Zero()
+		return
+	}
+
+	if r.Cmp(&q) != -1 || z > 0 {
+		lsubAssignFR(r, &q)
+	}
+	u.Set(&q)
+	lsubAssignFR(u, r)
+
+	// Phase 2
+	for i := k; i < 2*fourWordBitSize; i++ {
+		doubleFR(u, u)
+	}
+	e.Set(u)
+}
+
+func (ew *wideFr) mul(a, b *Fr) {
+	wmulFR(ew, a, b)
+}
+
+func (ew *wideFr) add(a *wideFr) {
+	waddFR(ew, a)
+}
+
+func (ew *wideFr) round() *Fr {
+	ew.add(halfR)
+	return ew.high()
+}
+
+func (ew *wideFr) high() *Fr {
+	e := new(Fr)
+	e[0] = ew[4]
+	e[1] = ew[5]
+	e[2] = ew[6]
+	e[3] = ew[7]
+	return e
+}
+
+func (ew *wideFr) low() *Fr {
+	e := new(Fr)
+	e[0] = ew[0]
+	e[1] = ew[1]
+	e[2] = ew[2]
+	e[3] = ew[3]
+	return e
+}
+
+func (e *wideFr) bytes() []byte {
+	out := make([]byte, frByteSize*2)
+	var a int
+	for i := 0; i < frNumberOfLimbs*2; i++ {
+		a = frByteSize*2 - i*8
+		out[a-1] = byte(e[i])
+		out[a-2] = byte(e[i] >> 8)
+		out[a-3] = byte(e[i] >> 16)
+		out[a-4] = byte(e[i] >> 24)
+		out[a-5] = byte(e[i] >> 32)
+		out[a-6] = byte(e[i] >> 40)
+		out[a-7] = byte(e[i] >> 48)
+		out[a-8] = byte(e[i] >> 56)
+	}
+	return out
+}

crypto/bls12381/fr_fallback.go

@@ -0,0 +1,383 @@
+// +build !amd64 generic
+
+// 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.
+
+// Code generated by goff (v0.3.5) DO NOT EDIT
+
+// /!\ WARNING /!\
+// this code has not been audited and is provided as-is. In particular,
+// there is no security guarantees such as constant time implementation
+// or side-channel attack resistance
+// /!\ WARNING /!\
+
+package bls12381
+
+import "math/bits"
+
+func addFR(z, x, y *Fr) {
+	var carry uint64
+
+	z[0], carry = bits.Add64(x[0], y[0], 0)
+	z[1], carry = bits.Add64(x[1], y[1], carry)
+	z[2], carry = bits.Add64(x[2], y[2], carry)
+	z[3], _ = bits.Add64(x[3], y[3], carry)
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 8353516859464449352 || (z[3] == 8353516859464449352 && (z[2] < 3691218898639771653 || (z[2] == 3691218898639771653 && (z[1] < 6034159408538082302 || (z[1] == 6034159408538082302 && (z[0] < 18446744069414584321))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 18446744069414584321, 0)
+		z[1], b = bits.Sub64(z[1], 6034159408538082302, b)
+		z[2], b = bits.Sub64(z[2], 3691218898639771653, b)
+		z[3], _ = bits.Sub64(z[3], 8353516859464449352, b)
+	}
+}
+
+func laddAssignFR(z, y *Fr) {
+	var carry uint64
+
+	z[0], carry = bits.Add64(z[0], y[0], 0)
+	z[1], carry = bits.Add64(z[1], y[1], carry)
+	z[2], carry = bits.Add64(z[2], y[2], carry)
+	z[3], _ = bits.Add64(z[3], y[3], carry)
+}
+
+func doubleFR(z, x *Fr) {
+	var carry uint64
+
+	z[0], carry = bits.Add64(x[0], x[0], 0)
+	z[1], carry = bits.Add64(x[1], x[1], carry)
+	z[2], carry = bits.Add64(x[2], x[2], carry)
+	z[3], _ = bits.Add64(x[3], x[3], carry)
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 8353516859464449352 || (z[3] == 8353516859464449352 && (z[2] < 3691218898639771653 || (z[2] == 3691218898639771653 && (z[1] < 6034159408538082302 || (z[1] == 6034159408538082302 && (z[0] < 18446744069414584321))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 18446744069414584321, 0)
+		z[1], b = bits.Sub64(z[1], 6034159408538082302, b)
+		z[2], b = bits.Sub64(z[2], 3691218898639771653, b)
+		z[3], _ = bits.Sub64(z[3], 8353516859464449352, b)
+	}
+}
+
+func subFR(z, x, y *Fr) {
+	var b uint64
+	z[0], b = bits.Sub64(x[0], y[0], 0)
+	z[1], b = bits.Sub64(x[1], y[1], b)
+	z[2], b = bits.Sub64(x[2], y[2], b)
+	z[3], b = bits.Sub64(x[3], y[3], b)
+	if b != 0 {
+		var c uint64
+		z[0], c = bits.Add64(z[0], 18446744069414584321, 0)
+		z[1], c = bits.Add64(z[1], 6034159408538082302, c)
+		z[2], c = bits.Add64(z[2], 3691218898639771653, c)
+		z[3], _ = bits.Add64(z[3], 8353516859464449352, c)
+	}
+}
+
+func lsubAssignFR(z, y *Fr) {
+	var b uint64
+	z[0], b = bits.Sub64(z[0], y[0], 0)
+	z[1], b = bits.Sub64(z[1], y[1], b)
+	z[2], b = bits.Sub64(z[2], y[2], b)
+	z[3], b = bits.Sub64(z[3], y[3], b)
+}
+
+func negFR(z, x *Fr) {
+	if x.IsZero() {
+		z.Zero()
+		return
+	}
+	var borrow uint64
+	z[0], borrow = bits.Sub64(18446744069414584321, x[0], 0)
+	z[1], borrow = bits.Sub64(6034159408538082302, x[1], borrow)
+	z[2], borrow = bits.Sub64(3691218898639771653, x[2], borrow)
+	z[3], _ = bits.Sub64(8353516859464449352, x[3], borrow)
+}
+
+func mulFR(z, x, y *Fr) {
+
+	var t [4]uint64
+	var c [3]uint64
+	{
+		// round 0
+		v := x[0]
+		c[1], c[0] = bits.Mul64(v, y[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd1(v, y[1], c[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd1(v, y[2], c[1])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd1(v, y[3], c[1])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 1
+		v := x[1]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 2
+		v := x[2]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 3
+		v := x[3]
+		c[1], c[0] = madd1(v, y[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, y[1], c[1], t[1])
+		c[2], z[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, y[2], c[1], t[2])
+		c[2], z[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, y[3], c[1], t[3])
+		z[3], z[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 8353516859464449352 || (z[3] == 8353516859464449352 && (z[2] < 3691218898639771653 || (z[2] == 3691218898639771653 && (z[1] < 6034159408538082302 || (z[1] == 6034159408538082302 && (z[0] < 18446744069414584321))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 18446744069414584321, 0)
+		z[1], b = bits.Sub64(z[1], 6034159408538082302, b)
+		z[2], b = bits.Sub64(z[2], 3691218898639771653, b)
+		z[3], _ = bits.Sub64(z[3], 8353516859464449352, b)
+	}
+}
+
+func squareFR(z, x *Fr) {
+
+	var t [4]uint64
+	var c [3]uint64
+	{
+		// round 0
+		v := x[0]
+		c[1], c[0] = bits.Mul64(v, x[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd1(v, x[1], c[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd1(v, x[2], c[1])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd1(v, x[3], c[1])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 1
+		v := x[1]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 2
+		v := x[2]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], t[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], t[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		t[3], t[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+	{
+		// round 3
+		v := x[3]
+		c[1], c[0] = madd1(v, x[0], t[0])
+		m := c[0] * 18446744069414584319
+		c[2] = madd0(m, 18446744069414584321, c[0])
+		c[1], c[0] = madd2(v, x[1], c[1], t[1])
+		c[2], z[0] = madd2(m, 6034159408538082302, c[2], c[0])
+		c[1], c[0] = madd2(v, x[2], c[1], t[2])
+		c[2], z[1] = madd2(m, 3691218898639771653, c[2], c[0])
+		c[1], c[0] = madd2(v, x[3], c[1], t[3])
+		z[3], z[2] = madd3(m, 8353516859464449352, c[0], c[2], c[1])
+	}
+
+	// if z > q --> z -= q
+	// note: this is NOT constant time
+	if !(z[3] < 8353516859464449352 || (z[3] == 8353516859464449352 && (z[2] < 3691218898639771653 || (z[2] == 3691218898639771653 && (z[1] < 6034159408538082302 || (z[1] == 6034159408538082302 && (z[0] < 18446744069414584321))))))) {
+		var b uint64
+		z[0], b = bits.Sub64(z[0], 18446744069414584321, 0)
+		z[1], b = bits.Sub64(z[1], 6034159408538082302, b)
+		z[2], b = bits.Sub64(z[2], 3691218898639771653, b)
+		z[3], _ = bits.Sub64(z[3], 8353516859464449352, b)
+	}
+}
+
+func waddFR(z, y *wideFr) {
+	var carry uint64
+	z[0], carry = bits.Add64(z[0], y[0], 0)
+	z[1], carry = bits.Add64(z[1], y[1], carry)
+	z[2], carry = bits.Add64(z[2], y[2], carry)
+	z[3], carry = bits.Add64(z[3], y[3], carry)
+	z[4], carry = bits.Add64(z[4], y[4], carry)
+	z[5], carry = bits.Add64(z[5], y[5], carry)
+	z[6], carry = bits.Add64(z[6], y[6], carry)
+	z[7], _ = bits.Add64(z[7], y[7], carry)
+}
+
+// We applied custom multiplication since goff does generate multiplication code nested with reduction
+func wmulFR(w *wideFr, a, b *Fr) {
+	// Handbook of Applied Cryptography
+	// Hankerson, Menezes, Vanstone
+	// 14.12 Algorithm Multiple-precision multiplication
+
+	var w0, w1, w2, w3, w4, w5, w6, w7 uint64
+	var a0 = a[0]
+	var a1 = a[1]
+	var a2 = a[2]
+	var a3 = a[3]
+	var b0 = b[0]
+	var b1 = b[1]
+	var b2 = b[2]
+	var b3 = b[3]
+	var u, v, c, t uint64
+
+	// i = 0, j = 0
+	c, w0 = bits.Mul64(a0, b0)
+
+	// i = 0, j = 1
+	u, v = bits.Mul64(a1, b0)
+	w1 = v + c
+	c = u + (v&c|(v|c)&^w1)>>63
+
+	// i = 0, j = 2
+	u, v = bits.Mul64(a2, b0)
+	w2 = v + c
+	c = u + (v&c|(v|c)&^w2)>>63
+
+	// i = 0, j = 3
+	u, v = bits.Mul64(a3, b0)
+	w3 = v + c
+	w4 = u + (v&c|(v|c)&^w3)>>63
+
+	// i = 1, j = 0
+	c, v = bits.Mul64(a0, b1)
+	t = v + w1
+	c += (v&w1 | (v|w1)&^t) >> 63
+	w1 = t
+
+	// i = 1, j = 1
+	u, v = bits.Mul64(a1, b1)
+	t = v + w2
+	u += (v&w2 | (v|w2)&^t) >> 63
+	w2 = t + c
+	c = u + (t&c|(t|c)&^w2)>>63
+
+	// i = 1, j = 2
+	u, v = bits.Mul64(a2, b1)
+	t = v + w3
+	u += (v&w3 | (v|w3)&^t) >> 63
+	w3 = t + c
+	c = u + (t&c|(t|c)&^w3)>>63
+
+	// i = 1, j = 3
+	u, v = bits.Mul64(a3, b1)
+	t = v + w4
+	u += (v&w4 | (v|w4)&^t) >> 63
+	w4 = t + c
+	w5 = u + (t&c|(t|c)&^w4)>>63
+
+	// i = 2, j = 0
+	c, v = bits.Mul64(a0, b2)
+	t = v + w2
+	c += (v&w2 | (v|w2)&^t) >> 63
+	w2 = t
+
+	// i = 2, j = 1
+	u, v = bits.Mul64(a1, b2)
+	t = v + w3
+	u += (v&w3 | (v|w3)&^t) >> 63
+	w3 = t + c
+	c = u + (t&c|(t|c)&^w3)>>63
+
+	// i = 2, j = 2
+	u, v = bits.Mul64(a2, b2)
+	t = v + w4
+	u += (v&w4 | (v|w4)&^t) >> 63
+	w4 = t + c
+	c = u + (t&c|(t|c)&^w4)>>63
+
+	// i = 2, j = 3
+	u, v = bits.Mul64(a3, b2)
+	t = v + w5
+	u += (v&w5 | (v|w5)&^t) >> 63
+	w5 = t + c
+	w6 = u + (t&c|(t|c)&^w5)>>63
+
+	// i = 3, j = 0
+	c, v = bits.Mul64(a0, b3)
+	t = v + w3
+	c += (v&w3 | (v|w3)&^t) >> 63
+	w3 = t
+
+	// i = 3, j = 1
+	u, v = bits.Mul64(a1, b3)
+	t = v + w4
+	u += (v&w4 | (v|w4)&^t) >> 63
+	w4 = t + c
+	c = u + (t&c|(t|c)&^w4)>>63
+
+	// i = 3, j = 2
+	u, v = bits.Mul64(a2, b3)
+	t = v + w5
+	u += (v&w5 | (v|w5)&^t) >> 63
+	w5 = t + c
+	c = u + (t&c|(t|c)&^w5)>>63
+
+	// i = 3, j = 3
+	u, v = bits.Mul64(a3, b3)
+	t = v + w6
+	u += (v&w6 | (v|w6)&^t) >> 63
+	w6 = t + c
+	w7 = u + (t&c|(t|c)&^w6)>>63
+
+	w[0] = w0
+	w[1] = w1
+	w[2] = w2
+	w[3] = w3
+	w[4] = w4
+	w[5] = w5
+	w[6] = w6
+	w[7] = w7
+}

crypto/bls12381/fr_test.go

@@ -0,0 +1,417 @@
+package bls12381
+
+import (
+	"bytes"
+	"crypto/rand"
+	"math/big"
+	"testing"
+)
+
+func TestScalarField(t *testing.T) {
+	r := new(Fr).Set(qr1)
+	r.fromMont()
+	if r[0] != 1 && r[1] != 0 && r[2] != 0 && r[3] != 0 {
+		t.Fatal("bad r value")
+	}
+	r.Set(qr2)
+	r.fromMont()
+	r.fromMont()
+	if r[0] != 1 && r[1] != 0 && r[2] != 0 && r[3] != 0 {
+		t.Fatal("bad r2 value")
+	}
+	r = &Fr{1}
+	r.toMont()
+	if !r.Equal(qr1) {
+		t.Fatal("mont transformaition failed")
+	}
+}
+
+func TestFrSerialization(t *testing.T) {
+	in := make([]byte, frByteSize)
+
+	e := new(Fr).FromBytes(in)
+	if !e.IsZero() {
+		t.Fatal("serialization failed, from bytes zero")
+	}
+	if !bytes.Equal(in, e.ToBytes()) {
+		t.Fatal("serialization failed, to bytes zero")
+	}
+
+	e = new(Fr).RedFromBytes(in)
+	if !e.IsZero() {
+		t.Fatal("serialization failed, from bytes zero, reduced")
+	}
+	if !bytes.Equal(in, e.RedToBytes()) {
+		t.Fatal("serialization failed, to bytes zero, reduced")
+	}
+
+	a, err := new(Fr).Rand(rand.Reader)
+	if err != nil {
+		t.Fatal(err)
+	}
+	b := new(Fr)
+	b.fromBytes(a.bytes())
+	if !a.Equal(b) {
+		t.Fatal("serialization failed, set bytes")
+	}
+
+	b = new(Fr).FromBytes(a.ToBytes())
+	if !a.Equal(b) {
+		t.Fatal("serialization failed, from/to bytes")
+	}
+
+	b = new(Fr).RedFromBytes(a.RedToBytes())
+	if !a.Equal(b) {
+		t.Fatal("serialization failed, from/to bytes, reduced")
+	}
+}
+
+func TestFrSliceUint(t *testing.T) {
+	s, err := new(Fr).Rand(rand.Reader)
+	if err != nil {
+		t.Fatal(err)
+	}
+	sBig := s.ToBig()
+	for offset := 0; offset < 260; offset++ {
+		a0 := new(big.Int).Rsh(sBig, uint(offset)).Uint64()
+		a1 := s.sliceUint64(offset)
+		if a0 != a1 {
+			t.Fatal("uint slice failed", offset)
+		}
+	}
+}
+
+func TestFrBitTest(t *testing.T) {
+	s, err := new(Fr).Rand(rand.Reader)
+	if err != nil {
+		t.Fatal(err)
+	}
+	sBig := s.ToBig()
+	for i := 0; i < 260; i++ {
+		a0 := sBig.Bit(i) == 1
+		a1 := s.Bit(i)
+		if a0 != a1 {
+			t.Fatal("bit test failed", i)
+		}
+	}
+}
+
+func TestFrBitShift(t *testing.T) {
+	a, _ := new(Fr).Rand(rand.Reader)
+	b := new(Fr).Set(a)
+	b.mul2()
+	b.div2()
+	if !b.Equal(a) {
+		t.Fatal("mul2 div2 failed")
+	}
+	a, _ = new(Fr).Rand(rand.Reader)
+	a[0] = a[0] & 0xfffffffffffffffe
+	b.Set(a)
+	b.div2()
+	b.mul2()
+	if !b.Equal(a) {
+		t.Fatal("mul2 div2 failed")
+	}
+}
+
+func TestFrAdditionCrossAgainstBigInt(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		c := new(Fr)
+		bigA := a.ToBig()
+		bigB := b.ToBig()
+		bigC := new(big.Int)
+		c.Add(a, b)
+		out1 := c.ToBytes()
+		out2 := padBytes(bigC.Add(bigA, bigB).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed, add")
+		}
+		c.Double(a)
+		out1 = c.ToBytes()
+		out2 = padBytes(bigC.Add(bigA, bigA).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed, double")
+		}
+		c.Sub(a, b)
+		out1 = c.ToBytes()
+		out2 = padBytes(bigC.Sub(bigA, bigB).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed, sub")
+		}
+		c.Neg(a)
+		out1 = c.ToBytes()
+		out2 = padBytes(bigC.Neg(bigA).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed, neg")
+		}
+	}
+}
+
+func TestFrAdditionProperties(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		zero := new(Fr)
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		c1, c2 := new(Fr), new(Fr)
+		c1.Add(a, zero)
+		if !c1.Equal(a) {
+			t.Fatal("a + 0 == a")
+		}
+		c1.Sub(a, zero)
+		if !c1.Equal(a) {
+			t.Fatal("a - 0 == a")
+		}
+		c1.Double(zero)
+		if !c1.Equal(zero) {
+			t.Fatal("2 * 0 == 0")
+		}
+		c1.Neg(zero)
+		if !c1.Equal(zero) {
+			t.Fatal("-0 == 0")
+		}
+		c1.Sub(zero, a)
+		c2.Neg(a)
+		if !c1.Equal(c2) {
+			t.Fatal("0-a == -a")
+		}
+		c1.Double(a)
+		c2.Add(a, a)
+		if !c1.Equal(c2) {
+			t.Fatal("2 * a == a + a")
+		}
+		c1.Add(a, b)
+		c2.Add(b, a)
+		if !c1.Equal(c2) {
+			t.Fatal("a + b = b + a")
+		}
+		c1.Sub(a, b)
+		c2.Sub(b, a)
+		c2.Neg(c2)
+		if !c1.Equal(c2) {
+			t.Fatal("a - b = - ( b - a )")
+		}
+		c0, _ := new(Fr).Rand(rand.Reader)
+		c1.Add(a, b)
+		c1.Add(c1, c0)
+		c2.Add(a, c0)
+		c2.Add(c2, b)
+		if !c1.Equal(c2) {
+			t.Fatal("(a + b) + c == (a + c ) + b")
+		}
+		c1.Sub(a, b)
+		c1.Sub(c1, c0)
+		c2.Sub(a, c0)
+		c2.Sub(c2, b)
+		if !c1.Equal(c2) {
+			t.Fatal("(a - b) - c == (a - c ) -b")
+		}
+	}
+}
+
+func TestFrMultiplicationCrossAgainstBigInt(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		c := new(Fr)
+		bigA := a.ToBig()
+		bigB := b.ToBig()
+		bigC := new(big.Int)
+		c.Mul(a, b)
+		out1 := c.ToBytes()
+		out2 := padBytes(bigC.Mul(bigA, bigB).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed")
+		}
+	}
+}
+
+func TestFrMultiplicationCrossAgainstBigIntReduced(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		c := new(Fr)
+		bigA := a.RedToBig()
+		bigB := b.RedToBig()
+		bigC := new(big.Int)
+		c.RedMul(a, b)
+		out1 := c.RedToBytes()
+		out2 := padBytes(bigC.Mul(bigA, bigB).Mod(bigC, qBig).Bytes(), frByteSize)
+		if !bytes.Equal(out1, out2) {
+			t.Fatal("cross test against big.Int is failed, reduced")
+		}
+	}
+}
+
+func TestFrMultiplicationProperties(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		zero, one := new(Fr).Zero(), new(Fr).One()
+		c1, c2 := new(Fr), new(Fr)
+		c1.Mul(a, zero)
+		if !c1.Equal(zero) {
+			t.Fatal("a * 0 == 0")
+		}
+		c1.Mul(a, one)
+		if !c1.Equal(a) {
+			t.Fatal("a * 1 == a")
+		}
+		c1.Mul(a, b)
+		c2.Mul(b, a)
+		if !c1.Equal(c2) {
+			t.Fatal("a * b == b * a")
+		}
+		c0, _ := new(Fr).Rand(rand.Reader)
+		c1.Mul(a, b)
+		c1.Mul(c1, c0)
+		c2.Mul(c0, b)
+		c2.Mul(c2, a)
+		if !c1.Equal(c2) {
+			t.Fatal("(a * b) * c == (a * c) * b")
+		}
+		a.Square(zero)
+		if !a.Equal(zero) {
+			t.Fatal("0^2 == 0")
+		}
+		a.Square(one)
+		if !a.Equal(one) {
+			t.Fatal("1^2 == 1")
+		}
+		_, _ = a.Rand(rand.Reader)
+		c1.Square(a)
+		c2.Mul(a, a)
+		if !c1.Equal(c1) {
+			t.Fatal("a^2 == a*a")
+		}
+	}
+}
+
+func TestFrMultiplicationPropertiesReduced(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		b, _ := new(Fr).Rand(rand.Reader)
+		zero, one := new(Fr).Zero(), new(Fr).RedOne()
+		c1, c2 := new(Fr), new(Fr)
+		c1.RedMul(a, zero)
+		if !c1.Equal(zero) {
+			t.Fatal("a * 0 == 0")
+		}
+		c1.RedMul(a, one)
+		if !c1.Equal(a) {
+			t.Fatal("a * 1 == a")
+		}
+		c1.RedMul(a, b)
+		c2.RedMul(b, a)
+		if !c1.Equal(c2) {
+			t.Fatal("a * b == b * a")
+		}
+		c0, _ := new(Fr).Rand(rand.Reader)
+		c1.RedMul(a, b)
+		c1.RedMul(c1, c0)
+		c2.RedMul(c0, b)
+		c2.RedMul(c2, a)
+		if !c1.Equal(c2) {
+			t.Fatal("(a * b) * c == (a * c) * b")
+		}
+		a.RedSquare(zero)
+		if !a.Equal(zero) {
+			t.Fatal("0^2 == 0")
+		}
+		a.RedSquare(one)
+		if !a.Equal(one) {
+			t.Fatal("1^2 == 1")
+		}
+		_, _ = a.Rand(rand.Reader)
+		c1.RedSquare(a)
+		c2.RedMul(a, a)
+		if !c1.Equal(c1) {
+			t.Fatal("a^2 == a*a")
+		}
+	}
+}
+
+func TestFrExponentiation(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		a, _ := new(Fr).Rand(rand.Reader)
+		u := new(Fr)
+		u.Exp(a, big.NewInt(0))
+		if !u.IsOne() {
+			t.Fatal("a^0 == 1")
+		}
+		u.Exp(a, big.NewInt(1))
+		if !u.Equal(a) {
+			t.Fatal("a^1 == a")
+		}
+		v := new(Fr)
+		u.Mul(a, a)
+		u.Mul(u, u)
+		u.Mul(u, u)
+		v.Exp(a, big.NewInt(8))
+		if !u.Equal(v) {
+			t.Fatal("((a^2)^2)^2 == a^8")
+		}
+		u.Exp(a, qBig)
+		if !u.Equal(a) {
+			t.Fatal("a^p == a")
+		}
+		qMinus1 := new(big.Int).Sub(qBig, big.NewInt(1))
+		u.Exp(a, qMinus1)
+		if !u.IsOne() {
+			t.Fatal("a^(p-1) == 1")
+		}
+	}
+}
+
+func TestFrInversion(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		u := new(Fr)
+		zero, one := new(Fr).Zero(), new(Fr).One()
+		u.Inverse(zero)
+		if !u.Equal(zero) {
+			t.Fatal("(0^-1) == 0)")
+		}
+		u.Inverse(one)
+		if !u.IsOne() {
+			t.Fatal("(1^-1) == 1)")
+		}
+		a, _ := new(Fr).Rand(rand.Reader)
+		u.Inverse(a)
+		u.Mul(u, a)
+		if !u.IsOne() {
+			t.Fatal("a * a^-1 == 1")
+		}
+		v := new(Fr)
+		z := new(big.Int)
+		u.Exp(a, z.Sub(qBig, big.NewInt(2)))
+		v.Inverse(a)
+		if !v.Equal(u) {
+			t.Fatal("a^(p-2) == a^-1")
+		}
+	}
+}
+
+func TestFnBatchInversion(t *testing.T) {
+	for i := 0; i < fuz; i++ {
+		zero, one := new(Fr).Zero(), new(Fr).One()
+
+		a, _ := new(Fr).Rand(rand.Reader)
+		u := new(Fr)
+		z := new(big.Int)
+		u.Exp(a, z.Sub(qBig, big.NewInt(2)))
+
+		var arr []Fr
+		arr = append(arr, *zero, *one, *a)
+		InverseBatchFr(arr)
+		if !arr[0].Equal(zero) {
+			t.Fatal("(0^-1) == 0)")
+		}
+		if !arr[1].IsOne() {
+			t.Fatal("(1^-1) == 1)")
+		}
+		if !arr[2].Equal(u) {
+			t.Fatal("a^(p-2) == a^-1")
+		}
+	}
+}

crypto/bls12381/g1.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -22,11 +6,12 @@ import (
 	"math/big"
 )
 
-// PointG1 is type for point in G1.
+// PointG1 is type for point in G1 and used for both Affine and Jacobian point representation.
-// PointG1 is both used for Affine and Jacobian point representation.
+// A point is accounted as in affine form if z is equal to one.
-// If z is equal to one the point is considered as in affine form.
 type PointG1 [3]fe
 
+var wnafMulWindowG1 uint = 5
+
 func (p *PointG1) Set(p2 *PointG1) *PointG1 {
 	p[0].set(&p2[0])
 	p[1].set(&p2[1])
@@ -34,7 +19,6 @@ func (p *PointG1) Set(p2 *PointG1) *PointG1 {
 	return p
 }
 
-// Zero returns G1 point in point at infinity representation
 func (p *PointG1) Zero() *PointG1 {
 	p[0].zero()
 	p[1].one()
@@ -42,6 +26,11 @@ func (p *PointG1) Zero() *PointG1 {
 	return p
 }
 
+// IsAffine checks a G1 point whether it is in affine form.
+func (p *PointG1) IsAffine() bool {
+	return p[2].isOne()
+}
+
 type tempG1 struct {
 	t [9]*fe
 }
@@ -67,15 +56,141 @@ func newTempG1() tempG1 {
 
 // Q returns group order in big.Int.
 func (g *G1) Q() *big.Int {
-	return new(big.Int).Set(q)
+	return new(big.Int).Set(qBig)
 }
 
-func (g *G1) fromBytesUnchecked(in []byte) (*PointG1, error) {
+// FromUncompressed expects byte slice at least 96 bytes and given bytes returns a new point in G1.
-	p0, err := fromBytes(in[:48])
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G1) FromUncompressed(uncompressed []byte) (*PointG1, error) {
+	if len(uncompressed) != 2*fpByteSize {
+		return nil, errors.New("input string length must be equal to 96 bytes")
+	}
+	var in [2 * fpByteSize]byte
+	copy(in[:], uncompressed[:2*fpByteSize])
+	if in[0]&(1<<7) != 0 {
+		return nil, errors.New("compression flag must be zero")
+	}
+	if in[0]&(1<<5) != 0 {
+		return nil, errors.New("sort flag must be zero")
+	}
+	if in[0]&(1<<6) != 0 {
+		for i, v := range in {
+			if (i == 0 && v != 0x40) || (i != 0 && v != 0x00) {
+				return nil, errors.New("input string must be zero when infinity flag is set")
+			}
+		}
+		return g.Zero(), nil
+	}
+	in[0] &= 0x1f
+	x, err := fromBytes(in[:fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	p1, err := fromBytes(in[48:])
+	y, err := fromBytes(in[fpByteSize:])
+	if err != nil {
+		return nil, err
+	}
+	z := new(fe).one()
+	p := &PointG1{*x, *y, *z}
+	if !g.IsOnCurve(p) {
+		return nil, errors.New("point is not on curve")
+	}
+	if !g.InCorrectSubgroup(p) {
+		return nil, errors.New("point is not on correct subgroup")
+	}
+	return p, nil
+}
+
+// ToUncompressed given a G1 point returns bytes in uncompressed (x, y) form of the point.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G1) ToUncompressed(p *PointG1) []byte {
+	out := make([]byte, 2*fpByteSize)
+	if g.IsZero(p) {
+		out[0] |= 1 << 6
+		return out
+	}
+	g.Affine(p)
+	copy(out[:fpByteSize], toBytes(&p[0]))
+	copy(out[fpByteSize:], toBytes(&p[1]))
+	return out
+}
+
+// FromCompressed expects byte slice at least 48 bytes and given bytes returns a new point in G1.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G1) FromCompressed(compressed []byte) (*PointG1, error) {
+	if len(compressed) != fpByteSize {
+		return nil, errors.New("input string length must be equal to 48 bytes")
+	}
+	var in [fpByteSize]byte
+	copy(in[:], compressed[:])
+	if in[0]&(1<<7) == 0 {
+		return nil, errors.New("compression flag must be set")
+	}
+	if in[0]&(1<<6) != 0 {
+		// in[0] == (1 << 6) + (1 << 7)
+		for i, v := range in {
+			if (i == 0 && v != 0xc0) || (i != 0 && v != 0x00) {
+				return nil, errors.New("input string must be zero when infinity flag is set")
+			}
+		}
+		return g.Zero(), nil
+	}
+	a := in[0]&(1<<5) != 0
+	in[0] &= 0x1f
+	x, err := fromBytes(in[:])
+	if err != nil {
+		return nil, err
+	}
+	// solve curve equation
+	y := &fe{}
+	square(y, x)
+	mul(y, y, x)
+	add(y, y, b)
+	if ok := sqrt(y, y); !ok {
+		return nil, errors.New("point is not on curve")
+	}
+	if y.signBE() == a {
+		neg(y, y)
+	}
+	z := new(fe).one()
+	p := &PointG1{*x, *y, *z}
+	if !g.InCorrectSubgroup(p) {
+		return nil, errors.New("point is not on correct subgroup")
+	}
+	return p, nil
+}
+
+// ToCompressed given a G1 point returns bytes in compressed form of the point.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G1) ToCompressed(p *PointG1) []byte {
+	out := make([]byte, fpByteSize)
+	g.Affine(p)
+	if g.IsZero(p) {
+		out[0] |= 1 << 6
+	} else {
+		copy(out[:], toBytes(&p[0]))
+		if !p[1].signBE() {
+			out[0] |= 1 << 5
+		}
+	}
+	out[0] |= 1 << 7
+	return out
+}
+
+func (g *G1) fromBytesUnchecked(in []byte) (*PointG1, error) {
+	p0, err := fromBytes(in[:fpByteSize])
+	if err != nil {
+		return nil, err
+	}
+	p1, err := fromBytes(in[fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -84,19 +199,17 @@ func (g *G1) fromBytesUnchecked(in []byte) (*PointG1, error) {
 }
 
 // FromBytes constructs a new point given uncompressed byte input.
-// FromBytes does not take zcash flags into account.
+// Input string is expected to be equal to 96 bytes and concatenation of x and y cooridanates.
-// Byte input expected to be larger than 96 bytes.
+// (0, 0) is considered as infinity.
-// First 96 bytes should be concatenation of x and y values.
-// Point (0, 0) is considered as infinity.
 func (g *G1) FromBytes(in []byte) (*PointG1, error) {
-	if len(in) != 96 {
+	if len(in) != 2*fpByteSize {
-		return nil, errors.New("input string should be equal or larger than 96")
+		return nil, errors.New("input string length must be equal to 96 bytes")
 	}
-	p0, err := fromBytes(in[:48])
+	p0, err := fromBytes(in[:fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	p1, err := fromBytes(in[48:])
+	p1, err := fromBytes(in[fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -112,49 +225,16 @@ func (g *G1) FromBytes(in []byte) (*PointG1, error) {
 	return p, nil
 }
 
-// DecodePoint given encoded (x, y) coordinates in 128 bytes returns a valid G1 Point.
-func (g *G1) DecodePoint(in []byte) (*PointG1, error) {
-	if len(in) != 128 {
-		return nil, errors.New("invalid g1 point length")
-	}
-	pointBytes := make([]byte, 96)
-	// decode x
-	xBytes, err := decodeFieldElement(in[:64])
-	if err != nil {
-		return nil, err
-	}
-	// decode y
-	yBytes, err := decodeFieldElement(in[64:])
-	if err != nil {
-		return nil, err
-	}
-	copy(pointBytes[:48], xBytes)
-	copy(pointBytes[48:], yBytes)
-	return g.FromBytes(pointBytes)
-}
-
 // ToBytes serializes a point into bytes in uncompressed form.
-// ToBytes does not take zcash flags into account.
 // ToBytes returns (0, 0) if point is infinity.
 func (g *G1) ToBytes(p *PointG1) []byte {
-	out := make([]byte, 96)
+	out := make([]byte, 2*fpByteSize)
 	if g.IsZero(p) {
 		return out
 	}
 	g.Affine(p)
-	copy(out[:48], toBytes(&p[0]))
+	copy(out[:fpByteSize], toBytes(&p[0]))
-	copy(out[48:], toBytes(&p[1]))
+	copy(out[fpByteSize:], toBytes(&p[1]))
-	return out
-}
-
-// EncodePoint encodes a point into 128 bytes.
-func (g *G1) EncodePoint(p *PointG1) []byte {
-	outRaw := g.ToBytes(p)
-	out := make([]byte, 128)
-	// encode x
-	copy(out[16:], outRaw[:48])
-	// encode y
-	copy(out[64+16:], outRaw[48:])
 	return out
 }
 
@@ -201,9 +281,29 @@ func (g *G1) Equal(p1, p2 *PointG1) bool {
 
 // InCorrectSubgroup checks whether given point is in correct subgroup.
 func (g *G1) InCorrectSubgroup(p *PointG1) bool {
-	tmp := &PointG1{}
+
-	g.MulScalar(tmp, p, q)
+	// Faster Subgroup Checks for BLS12-381
-	return g.IsZero(tmp)
+	// S. Bowe
+	// https://eprint.iacr.org/2019/814.pdf
+
+	mulZ := func(p *PointG1) {
+		// z = [(x^2 − 1)/3]
+		z := &Fr{0x0000000055555555, 0x396c8c005555e156}
+		e := z.toWNAF(wnafMulWindowG1)
+		g.wnafMul(p, p, e)
+	}
+
+	// [(x^2 − 1)/3](2σ(P) − P − σ^2(P)) − σ^2(P) ?= O
+	t0 := g.New().Set(p)
+	g.glvEndomorphism(t0, t0)
+	t1 := g.New().Set(t0)     // σ(P)
+	g.glvEndomorphism(t0, t0) // σ^2(P)
+	g.Double(t1, t1)          // 2σ(P)
+	g.Sub(t1, t1, p)          // 2σ(P) − P
+	g.Sub(t1, t1, t0)         // 2σ(P) − P − σ^2(P)
+	mulZ(t1)                  // [(x^2 − 1)/3](2σ(P) − P − σ^2(P))
+	g.Sub(t1, t1, t0)         // [(x^2 − 1)/3](2σ(P) − P − σ^2(P)) − σ^2(P)
+	return g.IsZero(t1)
 }
 
 // IsOnCurve checks a G1 point is on curve.
@@ -212,15 +312,19 @@ func (g *G1) IsOnCurve(p *PointG1) bool {
 		return true
 	}
 	t := g.t
-	square(t[0], &p[1])
+	square(t[0], &p[1])    // y^2
-	square(t[1], &p[0])
+	square(t[1], &p[0])    // x^2
-	mul(t[1], t[1], &p[0])
+	mul(t[1], t[1], &p[0]) // x^3
-	square(t[2], &p[2])
+	if p.IsAffine() {
-	square(t[3], t[2])
+		addAssign(t[1], b)      // x^2 + b
-	mul(t[2], t[2], t[3])
+		return t[0].equal(t[1]) // y^2 ?= x^3 + b
-	mul(t[2], b, t[2])
+	}
-	add(t[1], t[1], t[2])
+	square(t[2], &p[2])     // z^2
-	return t[0].equal(t[1])
+	square(t[3], t[2])      // z^4
+	mul(t[2], t[2], t[3])   // z^6
+	mul(t[2], b, t[2])      // b * z^6
+	add(t[1], t[1], t[2])   // x^3 + b * z^6
+	return t[0].equal(t[1]) // y^2 ?= x^3 + b * z^6
 }
 
 // IsAffine checks a G1 point whether it is in affine form.
@@ -228,26 +332,105 @@ func (g *G1) IsAffine(p *PointG1) bool {
 	return p[2].isOne()
 }
 
-// Affine calculates affine form of given G1 point.
+// Affine returns the affine representation of the given point
 func (g *G1) Affine(p *PointG1) *PointG1 {
+	return g.affine(p, p)
+}
+
+func (g *G1) affine(r, p *PointG1) *PointG1 {
 	if g.IsZero(p) {
-		return p
+		return r.Zero()
 	}
 	if !g.IsAffine(p) {
 		t := g.t
-		inverse(t[0], &p[2])
+		inverse(t[0], &p[2])    // z^-1
-		square(t[1], t[0])
+		square(t[1], t[0])      // z^-2
-		mul(&p[0], &p[0], t[1])
+		mul(&r[0], &p[0], t[1]) // x = x * z^-2
-		mul(t[0], t[0], t[1])
+		mul(t[0], t[0], t[1])   // z^-3
-		mul(&p[1], &p[1], t[0])
+		mul(&r[1], &p[1], t[0]) // y = y * z^-3
-		p[2].one()
+		r[2].one()              // z = 1
+	} else {
+		r.Set(p)
+	}
+	return r
+}
+
+// AffineBatch given multiple of points returns affine representations
+func (g *G1) AffineBatch(p []*PointG1) {
+	inverses := make([]fe, len(p))
+	for i := 0; i < len(p); i++ {
+		inverses[i].set(&p[i][2])
+	}
+	inverseBatch(inverses)
+	t := g.t
+	for i := 0; i < len(p); i++ {
+		if !g.IsAffine(p[i]) && !g.IsZero(p[i]) {
+			square(t[1], &inverses[i])
+			mul(&p[i][0], &p[i][0], t[1])
+			mul(t[0], &inverses[i], t[1])
+			mul(&p[i][1], &p[i][1], t[0])
+			p[i][2].one()
+		}
 	}
-	return p
 }
 
 // Add adds two G1 points p1, p2 and assigns the result to point at first argument.
 func (g *G1) Add(r, p1, p2 *PointG1) *PointG1 {
-	// www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+
+	// http://www.hyperelliptic.org/EFD/gp/auto-shortw-jacobian-0.html#addition-add-2007-bl
+	if g.IsZero(p1) {
+		return r.Set(p2)
+	}
+	if g.IsZero(p2) {
+		return r.Set(p1)
+	}
+	if g.IsAffine(p2) {
+		return g.AddMixed(r, p1, p2)
+	}
+	t := g.t
+	square(t[7], &p1[2])    // z1z1
+	mul(t[1], &p2[0], t[7]) // u2 = x2 * z1z1
+	mul(t[2], &p1[2], t[7]) // z1z1 * z1
+	mul(t[0], &p2[1], t[2]) // s2 = y2 * z1z1 * z1
+	square(t[8], &p2[2])    // z2z2
+	mul(t[3], &p1[0], t[8]) // u1 = x1 * z2z2
+	mul(t[4], &p2[2], t[8]) // z2z2 * z2
+	mul(t[2], &p1[1], t[4]) // s1 = y1 * z2z2 * z2
+	if t[1].equal(t[3]) {
+		if t[0].equal(t[2]) {
+			return g.Double(r, p1)
+		} else {
+			return r.Zero()
+		}
+	}
+	subAssign(t[1], t[3])     // h = u2 - u1
+	double(t[4], t[1])        // 2h
+	square(t[4], t[4])        // i = 2h^2
+	mul(t[5], t[1], t[4])     // j = h*i
+	subAssign(t[0], t[2])     // s2 - s1
+	doubleAssign(t[0])        // r = 2*(s2 - s1)
+	square(t[6], t[0])        // r^2
+	subAssign(t[6], t[5])     // r^2 - j
+	mul(t[3], t[3], t[4])     // v = u1 * i
+	double(t[4], t[3])        // 2*v
+	sub(&r[0], t[6], t[4])    // x3 = r^2 - j - 2*v
+	sub(t[4], t[3], &r[0])    // v - x3
+	mul(t[6], t[2], t[5])     // s1 * j
+	doubleAssign(t[6])        // 2 * s1 * j
+	mul(t[0], t[0], t[4])     // r * (v - x3)
+	sub(&r[1], t[0], t[6])    // y3 = r * (v - x3) - (2 * s1 * j)
+	add(t[0], &p1[2], &p2[2]) // z1 + z2
+	square(t[0], t[0])        // (z1 + z2)^2
+	subAssign(t[0], t[7])     // (z1 + z2)^2 - z1z1
+	subAssign(t[0], t[8])     // (z1 + z2)^2 - z1z1 - z2z2
+	mul(&r[2], t[0], t[1])    // z3 = ((z1 + z2)^2 - z1z1 - z2z2) * h
+	return r
+}
+
+// Add adds two G1 points p1, p2 and assigns the result to point at first argument.
+// Expects the second point p2 in affine form.
+func (g *G1) AddMixed(r, p1, p2 *PointG1) *PointG1 {
+	// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
 	if g.IsZero(p1) {
 		return r.Set(p2)
 	}
@@ -255,73 +438,68 @@ func (g *G1) Add(r, p1, p2 *PointG1) *PointG1 {
 		return r.Set(p1)
 	}
 	t := g.t
-	square(t[7], &p1[2])
+	square(t[7], &p1[2])    // z1z1
-	mul(t[1], &p2[0], t[7])
+	mul(t[1], &p2[0], t[7]) // u2 = x2 * z1z1
-	mul(t[2], &p1[2], t[7])
+	mul(t[2], &p1[2], t[7]) // z1z1 * z1
-	mul(t[0], &p2[1], t[2])
+	mul(t[0], &p2[1], t[2]) // s2 = y2 * z1z1 * z1
-	square(t[8], &p2[2])
+
-	mul(t[3], &p1[0], t[8])
+	if p1[0].equal(t[1]) && p1[1].equal(t[0]) {
-	mul(t[4], &p2[2], t[8])
-	mul(t[2], &p1[1], t[4])
-	if t[1].equal(t[3]) {
-		if t[0].equal(t[2]) {
 		return g.Double(r, p1)
 	}
-		return r.Zero()
+
-	}
+	sub(t[1], t[1], &p1[0]) // h = u2 - x1
-	sub(t[1], t[1], t[3])
+	square(t[2], t[1])      // hh
-	double(t[4], t[1])
+	double(t[4], t[2])
-	square(t[4], t[4])
+	doubleAssign(t[4])      // 4hh
-	mul(t[5], t[1], t[4])
+	mul(t[5], t[1], t[4])   // j = h*i
-	sub(t[0], t[0], t[2])
+	subAssign(t[0], &p1[1]) // s2 - y1
-	double(t[0], t[0])
+	doubleAssign(t[0])      // r = 2*(s2 - y1)
-	square(t[6], t[0])
+	square(t[6], t[0])      // r^2
-	sub(t[6], t[6], t[5])
+	subAssign(t[6], t[5])   // r^2 - j
-	mul(t[3], t[3], t[4])
+	mul(t[3], &p1[0], t[4]) // v = x1 * i
-	double(t[4], t[3])
+	double(t[4], t[3])      // 2*v
-	sub(&r[0], t[6], t[4])
+	sub(&r[0], t[6], t[4])  // x3 = r^2 - j - 2*v
-	sub(t[4], t[3], &r[0])
+	sub(t[4], t[3], &r[0])  // v - x3
-	mul(t[6], t[2], t[5])
+	mul(t[6], &p1[1], t[5]) // y1 * j
-	double(t[6], t[6])
+	doubleAssign(t[6])      // 2 * y1 * j
-	mul(t[0], t[0], t[4])
+	mul(t[0], t[0], t[4])   // r * (v - x3)
-	sub(&r[1], t[0], t[6])
+	sub(&r[1], t[0], t[6])  // y3 = r * (v - x3) - (2 * y1 * j)
-	add(t[0], &p1[2], &p2[2])
+	add(t[0], &p1[2], t[1]) // z1 + h
-	square(t[0], t[0])
+	square(t[0], t[0])      // (z1 + h)^2
-	sub(t[0], t[0], t[7])
+	subAssign(t[0], t[7])   // (z1 + h)^2 - z1z1
-	sub(t[0], t[0], t[8])
+	sub(&r[2], t[0], t[2])  // z3 = (z1 + z2)^2 - z1z1 - hh
-	mul(&r[2], t[0], t[1])
 	return r
 }
 
 // Double doubles a G1 point p and assigns the result to the point at first argument.
 func (g *G1) Double(r, p *PointG1) *PointG1 {
-	// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
+	// http://www.hyperelliptic.org/EFD/gp/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
 	if g.IsZero(p) {
-		return r.Set(p)
+		return r.Zero()
 	}
 	t := g.t
-	square(t[0], &p[0])
+	square(t[0], &p[0])     // a = x^2
-	square(t[1], &p[1])
+	square(t[1], &p[1])     // b = y^2
-	square(t[2], t[1])
+	square(t[2], t[1])      // c = b^2
-	add(t[1], &p[0], t[1])
+	add(t[1], &p[0], t[1])  // b + x1
-	square(t[1], t[1])
+	square(t[1], t[1])      // (b + x1)^2
-	sub(t[1], t[1], t[0])
+	subAssign(t[1], t[0])   // (b + x1)^2 - a
-	sub(t[1], t[1], t[2])
+	subAssign(t[1], t[2])   // (b + x1)^2 - a - c
-	double(t[1], t[1])
+	doubleAssign(t[1])      // d = 2((b+x1)^2 - a - c)
-	double(t[3], t[0])
+	double(t[3], t[0])      // 2a
-	add(t[0], t[3], t[0])
+	addAssign(t[0], t[3])   // e = 3a
-	square(t[4], t[0])
+	square(t[4], t[0])      // f = e^2
-	double(t[3], t[1])
+	double(t[3], t[1])      // 2d
-	sub(&r[0], t[4], t[3])
+	sub(&r[0], t[4], t[3])  // x3 = f - 2d
-	sub(t[1], t[1], &r[0])
+	subAssign(t[1], &r[0])  // d-x3
-	double(t[2], t[2])
+	doubleAssign(t[2])      //
-	double(t[2], t[2])
+	doubleAssign(t[2])      //
-	double(t[2], t[2])
+	doubleAssign(t[2])      // 8c
-	mul(t[0], t[0], t[1])
+	mul(t[0], t[0], t[1])   // e * (d - x3)
-	sub(t[1], t[0], t[2])
+	sub(t[1], t[0], t[2])   // x3 = e * (d - x3) - 8c
-	mul(t[0], &p[1], &p[2])
+	mul(t[0], &p[1], &p[2]) // y1 * z1
-	r[1].set(t[1])
+	r[1].set(t[1])          //
-	double(&r[2], t[0])
+	double(&r[2], t[0])     // z3 = 2(y1 * z1)
 	return r
 }
 
@@ -341,12 +519,32 @@ func (g *G1) Sub(c, a, b *PointG1) *PointG1 {
 	return c
 }
 
+// MulScalar multiplies a point by given scalar value and assigns the result to point at first argument.
+func (g *G1) MulScalar(r, p *PointG1, e *Fr) *PointG1 {
+	return g.glvMulFr(r, p, e)
+}
+
 // MulScalar multiplies a point by given scalar value in big.Int and assigns the result to point at first argument.
-func (g *G1) MulScalar(c, p *PointG1, e *big.Int) *PointG1 {
+func (g *G1) MulScalarBig(r, p *PointG1, e *big.Int) *PointG1 {
+	return g.glvMulBig(r, p, e)
+}
+
+func (g *G1) mulScalar(c, p *PointG1, e *Fr) *PointG1 {
 	q, n := &PointG1{}, &PointG1{}
 	n.Set(p)
-	l := e.BitLen()
+	for i := 0; i < frBitSize; i++ {
-	for i := 0; i < l; i++ {
+		if e.Bit(i) {
+			g.Add(q, q, n)
+		}
+		g.Double(n, n)
+	}
+	return c.Set(q)
+}
+
+func (g *G1) mulScalarBig(c, p *PointG1, e *big.Int) *PointG1 {
+	q, n := &PointG1{}, &PointG1{}
+	n.Set(p)
+	for i := 0; i < frBitSize; i++ {
 		if e.Bit(i) == 1 {
 			g.Add(q, q, n)
 		}
@@ -355,67 +553,242 @@ func (g *G1) MulScalar(c, p *PointG1, e *big.Int) *PointG1 {
 	return c.Set(q)
 }
 
-// ClearCofactor maps given a G1 point to correct subgroup
+func (g *G1) wnafMulFr(r, p *PointG1, e *Fr) *PointG1 {
-func (g *G1) ClearCofactor(p *PointG1) {
+	wnaf := e.toWNAF(wnafMulWindowG1)
-	g.MulScalar(p, p, cofactorEFFG1)
+	return g.wnafMul(r, p, wnaf)
 }
 
-// MultiExp calculates multi exponentiation. Given pairs of G1 point and scalar values
+func (g *G1) wnafMulBig(r, p *PointG1, e *big.Int) *PointG1 {
-// (P_0, e_0), (P_1, e_1), ... (P_n, e_n) calculates r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n
+	wnaf := bigToWNAF(e, wnafMulWindowG1)
-// Length of points and scalars are expected to be equal, otherwise an error is returned.
+	return g.wnafMul(r, p, wnaf)
-// Result is assigned to point at first argument.
+}
-func (g *G1) MultiExp(r *PointG1, points []*PointG1, powers []*big.Int) (*PointG1, error) {
+
-	if len(points) != len(powers) {
+func (g *G1) wnafMul(c, p *PointG1, wnaf nafNumber) *PointG1 {
-		return nil, errors.New("point and scalar vectors should be in same length")
+
+	l := (1 << (wnafMulWindowG1 - 1))
+
+	twoP, acc := g.New(), new(PointG1).Set(p)
+	g.Double(twoP, p)
+	g.Affine(twoP)
+
+	// table = {p, 3p, 5p, ..., -p, -3p, -5p}
+	table := make([]*PointG1, l*2)
+	table[0], table[l] = g.New(), g.New()
+	table[0].Set(p)
+	g.Neg(table[l], table[0])
+
+	for i := 1; i < l; i++ {
+		g.AddMixed(acc, acc, twoP)
+		table[i], table[i+l] = g.New(), g.New()
+		table[i].Set(acc)
+		g.Neg(table[i+l], table[i])
 	}
-	var c uint32 = 3
+
-	if len(powers) >= 32 {
+	q := g.Zero()
-		c = uint32(math.Ceil(math.Log10(float64(len(powers)))))
+	for i := len(wnaf) - 1; i >= 0; i-- {
+		if wnaf[i] > 0 {
+			g.Add(q, q, table[wnaf[i]>>1])
+		} else if wnaf[i] < 0 {
+			g.Add(q, q, table[((-wnaf[i])>>1)+l])
 		}
-	bucketSize, numBits := (1<<c)-1, uint32(g.Q().BitLen())
+		if i != 0 {
-	windows := make([]*PointG1, numBits/c+1)
+			g.Double(q, q)
-	bucket := make([]*PointG1, bucketSize)
-	acc, sum := g.New(), g.New()
-	for i := 0; i < bucketSize; i++ {
-		bucket[i] = g.New()
 		}
-	mask := (uint64(1) << c) - 1
-	j := 0
-	var cur uint32
-	for cur <= numBits {
-		acc.Zero()
-		bucket = make([]*PointG1, (1<<c)-1)
-		for i := 0; i < len(bucket); i++ {
-			bucket[i] = g.New()
 	}
-		for i := 0; i < len(powers); i++ {
+	return c.Set(q)
-			s0 := powers[i].Uint64()
+}
-			index := uint(s0 & mask)
+
-			if index != 0 {
+func (g *G1) glvMulFr(r, p *PointG1, e *Fr) *PointG1 {
-				g.Add(bucket[index-1], bucket[index-1], points[i])
+	return g.glvMul(r, p, new(glvVectorFr).new(e))
+}
+
+func (g *G1) glvMulBig(r, p *PointG1, e *big.Int) *PointG1 {
+	return g.glvMul(r, p, new(glvVectorBig).new(e))
+}
+
+func (g *G1) glvMul(r, p0 *PointG1, v glvVector) *PointG1 {
+
+	w := glvMulWindowG1
+	l := 1 << (w - 1)
+
+	// prepare tables
+	// tableK1 = {P, 3P, 5P, ...}
+	// tableK2 = {λP, 3λP, 5λP, ...}
+	tableK1, tableK2 := make([]*PointG1, l), make([]*PointG1, l)
+	double := g.New()
+	g.Double(double, p0)
+	g.affine(double, double)
+	tableK1[0] = new(PointG1)
+	tableK1[0].Set(p0)
+	for i := 1; i < l; i++ {
+		tableK1[i] = new(PointG1)
+		g.AddMixed(tableK1[i], tableK1[i-1], double)
 	}
-			powers[i] = new(big.Int).Rsh(powers[i], uint(c))
+	g.AffineBatch(tableK1)
+	for i := 0; i < l; i++ {
+		tableK2[i] = new(PointG1)
+		g.glvEndomorphism(tableK2[i], tableK1[i])
 	}
-		sum.Zero()
+
-		for i := len(bucket) - 1; i >= 0; i-- {
+	// recode small scalars
-			g.Add(sum, sum, bucket[i])
+	naf1, naf2 := v.wnaf(w)
-			g.Add(acc, acc, sum)
+	lenNAF1, lenNAF2 := len(naf1), len(naf2)
+	lenNAF := lenNAF1
+	if lenNAF2 > lenNAF {
+		lenNAF = lenNAF2
 	}
-		windows[j] = g.New()
+
-		windows[j].Set(acc)
+	acc, p1 := g.New(), g.New()
-		j++
+
-		cur += c
+	// function for naf addition
+	add := func(table []*PointG1, naf int) {
+		if naf != 0 {
+			nafAbs := naf
+			if nafAbs < 0 {
+				nafAbs = -nafAbs
 			}
-	acc.Zero()
+			p1.Set(table[nafAbs>>1])
-	for i := len(windows) - 1; i >= 0; i-- {
+			if naf < 0 {
-		for j := uint32(0); j < c; j++ {
+				g.Neg(p1, p1)
+			}
+			g.AddMixed(acc, acc, p1)
+		}
+	}
+
+	// sliding
+	for i := lenNAF - 1; i >= 0; i-- {
+		if i < lenNAF1 {
+			add(tableK1, naf1[i])
+		}
+		if i < lenNAF2 {
+			add(tableK2, naf2[i])
+		}
+		if i != 0 {
 			g.Double(acc, acc)
 		}
-		g.Add(acc, acc, windows[i])
+	}
+	return r.Set(acc)
+}
+
+// MultiExpBig calculates multi exponentiation. Scalar values are received as big.Int type.
+// Given pairs of G1 point and scalar values `(P_0, e_0), (P_1, e_1), ... (P_n, e_n)`,
+// calculates `r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n`.
+// Length of points and scalars are expected to be equal, otherwise an error is returned.
+// Result is assigned to point at first argument.
+func (g *G1) MultiExpBig(r *PointG1, points []*PointG1, scalars []*big.Int) (*PointG1, error) {
+	if len(points) != len(scalars) {
+		return nil, errors.New("point and scalar vectors should be in same length")
+	}
+
+	c := 3
+	if len(scalars) >= 32 {
+		c = int(math.Ceil(math.Log(float64(len(scalars)))))
+	}
+
+	bucketSize := (1 << c) - 1
+	windows := make([]PointG1, 255/c+1)
+	bucket := make([]PointG1, bucketSize)
+
+	for j := 0; j < len(windows); j++ {
+
+		for i := 0; i < bucketSize; i++ {
+			bucket[i].Zero()
+		}
+
+		for i := 0; i < len(scalars); i++ {
+			index := bucketSize & int(new(big.Int).Rsh(scalars[i], uint(c*j)).Int64())
+			if index != 0 {
+				g.Add(&bucket[index-1], &bucket[index-1], points[i])
+			}
+		}
+
+		acc, sum := g.New(), g.New()
+		for i := bucketSize - 1; i >= 0; i-- {
+			g.Add(sum, sum, &bucket[i])
+			g.Add(acc, acc, sum)
+		}
+		windows[j].Set(acc)
+	}
+
+	acc := g.New()
+	for i := len(windows) - 1; i >= 0; i-- {
+		for j := 0; j < c; j++ {
+			g.Double(acc, acc)
+		}
+		g.Add(acc, acc, &windows[i])
 	}
 	return r.Set(acc), nil
 }
 
+// MultiExp calculates multi exponentiation. Given pairs of G1 point and scalar values `(P_0, e_0), (P_1, e_1), ... (P_n, e_n)`,
+// calculates `r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n`. Length of points and scalars are expected to be equal,
+// otherwise an error is returned. Result is assigned to point at first argument.
+func (g *G1) MultiExp(r *PointG1, points []*PointG1, scalars []*Fr) (*PointG1, error) {
+	if len(points) != len(scalars) {
+		return nil, errors.New("point and scalar vectors should be in same length")
+	}
+
+	g.AffineBatch(points)
+
+	c := 3
+	if len(scalars) >= 32 {
+		c = int(math.Ceil(math.Log(float64(len(scalars)))))
+	}
+
+	bucketSize := (1 << c) - 1
+	windows := make([]*PointG1, 255/c+1)
+	bucket := make([]PointG1, bucketSize)
+
+	for j := 0; j < len(windows); j++ {
+
+		for i := 0; i < bucketSize; i++ {
+			bucket[i].Zero()
+		}
+
+		for i := 0; i < len(scalars); i++ {
+			index := bucketSize & int(scalars[i].sliceUint64(c*j))
+			if index != 0 {
+				g.AddMixed(&bucket[index-1], &bucket[index-1], points[i])
+			}
+		}
+
+		acc, sum := g.New(), g.New()
+		for i := bucketSize - 1; i >= 0; i-- {
+			g.Add(sum, sum, &bucket[i])
+			g.Add(acc, acc, sum)
+		}
+		windows[j] = g.New().Set(acc)
+	}
+
+	g.AffineBatch(windows)
+
+	acc := g.New()
+	for i := len(windows) - 1; i >= 0; i-- {
+		for j := 0; j < c; j++ {
+			g.Double(acc, acc)
+		}
+		g.AddMixed(acc, acc, windows[i])
+	}
+	return r.Set(acc), nil
+}
+
+func (g *G1) ClearCofactor(p *PointG1) *PointG1 {
+	chain := func(p0 *PointG1, n int, p1 *PointG1) {
+		for i := 0; i < n; i++ {
+			g.Double(p0, p0)
+		}
+		g.Add(p0, p0, p1)
+	}
+	t := g.New().Set(p)
+	chain(p, 1, t)
+	chain(p, 2, t)
+	chain(p, 3, t)
+	chain(p, 9, t)
+	chain(p, 32, t)
+	chain(p, 16, t)
+	return p
+}
+
 // MapToCurve given a byte slice returns a valid G1 point.
 // This mapping function implements the Simplified Shallue-van de Woestijne-Ulas method.
 // https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06
@@ -432,3 +805,42 @@ func (g *G1) MapToCurve(in []byte) (*PointG1, error) {
 	g.ClearCofactor(p)
 	return g.Affine(p), nil
 }
+
+// EncodeToCurve given a message and domain seperator tag returns the hash result
+// which is a valid curve point.
+// Implementation follows BLS12381G1_XMD:SHA-256_SSWU_NU_ suite at
+// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06
+func (g *G1) EncodeToCurve(msg, domain []byte) (*PointG1, error) {
+	hashRes, err := hashToFpXMDSHA256(msg, domain, 1)
+	if err != nil {
+		return nil, err
+	}
+	u := hashRes[0]
+	x, y := swuMapG1(u)
+	isogenyMapG1(x, y)
+	one := new(fe).one()
+	p := &PointG1{*x, *y, *one}
+	g.ClearCofactor(p)
+	return g.Affine(p), nil
+}
+
+// HashToCurve given a message and domain seperator tag returns the hash result
+// which is a valid curve point.
+// Implementation follows BLS12381G1_XMD:SHA-256_SSWU_RO_ suite at
+// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06
+func (g *G1) HashToCurve(msg, domain []byte) (*PointG1, error) {
+	hashRes, err := hashToFpXMDSHA256(msg, domain, 2)
+	if err != nil {
+		return nil, err
+	}
+	u0, u1 := hashRes[0], hashRes[1]
+	x0, y0 := swuMapG1(u0)
+	x1, y1 := swuMapG1(u1)
+	one := new(fe).one()
+	p0, p1 := &PointG1{*x0, *y0, *one}, &PointG1{*x1, *y1, *one}
+	g.Add(p0, p0, p1)
+	g.Affine(p0)
+	isogenyMapG1(&p0[0], &p0[1])
+	g.ClearCofactor(p0)
+	return g.Affine(p0), nil
+}

crypto/bls12381/g1_test.go

@@ -3,52 +3,117 @@ package bls12381
 import (
 	"bytes"
 	"crypto/rand"
+	"fmt"
+	"io/ioutil"
 	"math/big"
 	"testing"
-
-	"github.com/ethereum/go-ethereum/common"
 )
 
 func (g *G1) one() *PointG1 {
-	one, _ := g.fromBytesUnchecked(
+	return g.New().Set(&g1One)
-		common.FromHex("" +
-			"17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb" +
-			"08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1",
-		),
-	)
-	return one
 }
 
 func (g *G1) rand() *PointG1 {
-	k, err := rand.Int(rand.Reader, q)
+	p := &PointG1{}
-	if err != nil {
+	z, _ := new(fe).rand(rand.Reader)
-		panic(err)
+	z6, bz6 := new(fe), new(fe)
+	square(z6, z)
+	square(z6, z6)
+	mul(z6, z6, z)
+	mul(z6, z6, z)
+	mul(bz6, z6, b)
+	for {
+		x, _ := new(fe).rand(rand.Reader)
+		y := new(fe)
+		square(y, x)
+		mul(y, y, x)
+		add(y, y, bz6)
+		if sqrt(y, y) {
+			p.Set(&PointG1{*x, *y, *z})
+			break
 		}
-	return g.MulScalar(&PointG1{}, g.one(), k)
+	}
+	if !g.IsOnCurve(p) {
+		panic("rand point must be on curve")
+	}
+	if g.InCorrectSubgroup(p) {
+		panic("rand point must be out of correct subgroup")
+	}
+	return p
+}
+
+func (g *G1) randCorrect() *PointG1 {
+	p := g.ClearCofactor(g.rand())
+	if !g.InCorrectSubgroup(p) {
+		panic("must be in correct subgroup")
+	}
+	return p
+}
+
+func (g *G1) randAffine() *PointG1 {
+	return g.Affine(g.randCorrect())
+}
+
+func (g *G1) new() *PointG1 {
+	return g.Zero()
 }
 
 func TestG1Serialization(t *testing.T) {
-	g1 := NewG1()
+	var err error
-	for i := 0; i < fuz; i++ {
+	g := NewG1()
-		a := g1.rand()
+	zero := g.Zero()
-		buf := g1.ToBytes(a)
+	b0 := g.ToUncompressed(zero)
-		b, err := g1.FromBytes(buf)
+	p0, err := g.FromUncompressed(b0)
 	if err != nil {
 		t.Fatal(err)
 	}
-		if !g1.Equal(a, b) {
+	if !g.IsZero(p0) {
-			t.Fatal("bad serialization from/to")
+		t.Fatal("infinity serialization failed")
 	}
-	}
+	b0 = g.ToCompressed(zero)
-	for i := 0; i < fuz; i++ {
+	p0, err = g.FromCompressed(b0)
-		a := g1.rand()
-		encoded := g1.EncodePoint(a)
-		b, err := g1.DecodePoint(encoded)
 	if err != nil {
 		t.Fatal(err)
 	}
-		if !g1.Equal(a, b) {
+	if !g.IsZero(p0) {
-			t.Fatal("bad serialization encode/decode")
+		t.Fatal("infinity serialization failed")
+	}
+	b0 = g.ToBytes(zero)
+	p0, err = g.FromBytes(b0)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !g.IsZero(p0) {
+		t.Fatal("infinity serialization failed")
+	}
+	for i := 0; i < fuz; i++ {
+		a := g.randAffine()
+		uncompressed := g.ToUncompressed(a)
+		b, err := g.FromUncompressed(uncompressed)
+		if err != nil {
+			t.Fatal(err)
+		}
+		if !g.Equal(a, b) {
+			t.Fatal("serialization failed")
+		}
+		compressed := g.ToCompressed(b)
+		a, err = g.FromCompressed(compressed)
+		if err != nil {
+			t.Fatal(err)
+		}
+		if !g.Equal(a, b) {
+			t.Fatal("serialization failed")
+		}
+	}
+	for i := 0; i < fuz; i++ {
+		a := g.randAffine()
+		uncompressed := g.ToBytes(a)
+		b, err := g.FromBytes(uncompressed)
+		if err != nil {
+			t.Fatal(err)
+		}
+		if !g.Equal(a, b) {
+			t.Fatal("serialization failed")
 		}
 	}
 }
@@ -66,6 +131,26 @@ func TestG1IsOnCurve(t *testing.T) {
 	}
 }
 
+func TestG1BatchAffine(t *testing.T) {
+	n := 20
+	g := NewG1()
+	points0 := make([]*PointG1, n)
+	points1 := make([]*PointG1, n)
+	for i := 0; i < n; i++ {
+		points0[i] = g.rand()
+		points1[i] = g.New().Set(points0[i])
+		if g.IsAffine(points0[i]) {
+			t.Fatal("expect non affine point")
+		}
+	}
+	g.AffineBatch(points0)
+	for i := 0; i < n; i++ {
+		if !g.Equal(points0[i], points1[i]) {
+			t.Fatal("batch affine failed")
+		}
+	}
+}
+
 func TestG1AdditiveProperties(t *testing.T) {
 	g := NewG1()
 	t0, t1 := g.New(), g.New()
@@ -135,14 +220,71 @@ func TestG1AdditiveProperties(t *testing.T) {
 	}
 }
 
+func TestG1MixedAdd(t *testing.T) {
+	g := NewG1()
+	for i := 0; i < fuz; i++ {
+		a, b := g.rand(), g.rand()
+		if g.IsAffine(a) || g.IsAffine(b) {
+			t.Fatal("expect non affine points")
+		}
+		bAffine := g.New().Set(b)
+		g.Affine(bAffine)
+		r0, r1 := g.New(), g.New()
+		g.Add(r0, a, b)
+		g.AddMixed(r1, a, bAffine)
+		if !g.Equal(r0, r1) {
+			t.Fatal("mixed addition failed")
+		}
+		aAffine := g.New().Set(a)
+		g.Affine(aAffine)
+		g.AddMixed(r0, a, aAffine)
+		g.Double(r1, a)
+		if !g.Equal(r0, r1) {
+			t.Fatal("mixed addition must double where points are equal")
+		}
+	}
+}
+
+func TestG1MultiplicationCross(t *testing.T) {
+	g := NewG1()
+	for i := 0; i < fuz; i++ {
+
+		a := g.randCorrect()
+		s, _ := new(Fr).Rand(rand.Reader)
+		sBig := s.ToBig()
+		res0, res1, res2, res3, res4 := g.New(), g.New(), g.New(), g.New(), g.New()
+
+		g.mulScalar(res0, a, s)
+		g.glvMulFr(res1, a, s)
+		g.glvMulBig(res2, a, sBig)
+		g.wnafMulFr(res3, a, s)
+		g.wnafMulBig(res4, a, sBig)
+
+		if !g.Equal(res0, res1) {
+			t.Fatal("cross multiplication failed (glv, fr)", i)
+		}
+		if !g.Equal(res0, res2) {
+			t.Fatal("cross multiplication failed (glv, big)", i)
+		}
+		if !g.Equal(res0, res3) {
+			t.Fatal("cross multiplication failed (wnaf, fr)", i)
+		}
+		if !g.Equal(res0, res4) {
+			t.Fatal("cross multiplication failed (wnaf, big)", i)
+		}
+	}
+}
+
 func TestG1MultiplicativeProperties(t *testing.T) {
 	g := NewG1()
 	t0, t1 := g.New(), g.New()
 	zero := g.Zero()
 	for i := 0; i < fuz; i++ {
-		a := g.rand()
+		a := g.randCorrect()
-		s1, s2, s3 := randScalar(q), randScalar(q), randScalar(q)
+		s1, _ := new(Fr).Rand(rand.Reader)
-		sone := big.NewInt(1)
+		s2, _ := new(Fr).Rand(rand.Reader)
+		s3, _ := new(Fr).Rand(rand.Reader)
+		sone := &Fr{1}
 		g.MulScalar(t0, zero, s1)
 		if !g.Equal(t0, zero) {
 			t.Fatal(" 0 ^ s == 0")
@@ -160,7 +302,7 @@ func TestG1MultiplicativeProperties(t *testing.T) {
 		s3.Mul(s1, s2)
 		g.MulScalar(t1, a, s3)
 		if !g.Equal(t0, t1) {
-			t.Errorf(" (a ^ s1) ^ s2 == a ^ (s1 * s2)")
+			t.Fatal(" (a ^ s1) ^ s2 == a ^ (s1 * s2)")
 		}
 		g.MulScalar(t0, a, s1)
 		g.MulScalar(t1, a, s2)
@@ -168,12 +310,71 @@ func TestG1MultiplicativeProperties(t *testing.T) {
 		s3.Add(s1, s2)
 		g.MulScalar(t1, a, s3)
 		if !g.Equal(t0, t1) {
-			t.Errorf(" (a ^ s1) + (a ^ s2) == a ^ (s1 + s2)")
+			t.Fatal(" (a ^ s1) + (a ^ s2) == a ^ (s1 + s2)")
 		}
 	}
 }
 
+func TestZKCryptoVectorsG1UncompressedValid(t *testing.T) {
+	data, err := ioutil.ReadFile("tests/g1_uncompressed_valid_test_vectors.dat")
+	if err != nil {
+		panic(err)
+	}
+	g := NewG1()
+	p1 := g.Zero()
+	for i := 0; i < 1000; i++ {
+		vector := data[i*2*fpByteSize : (i+1)*2*fpByteSize]
+		p2, err := g.FromUncompressed(vector)
+		if err != nil {
+			t.Fatal("decoing fails", err, i)
+		}
+		uncompressed := g.ToUncompressed(p2)
+		if !bytes.Equal(vector, uncompressed) || !g.Equal(p1, p2) {
+			t.Fatal("serialization failed")
+		}
+
+		g.Add(p1, p1, &g1One)
+	}
+}
+
+func TestZKCryptoVectorsG1CompressedValid(t *testing.T) {
+	data, err := ioutil.ReadFile("tests/g1_compressed_valid_test_vectors.dat")
+	if err != nil {
+		panic(err)
+	}
+	g := NewG1()
+	p1 := g.Zero()
+	for i := 0; i < 1000; i++ {
+		vector := data[i*fpByteSize : (i+1)*fpByteSize]
+		p2, err := g.FromCompressed(vector)
+		if err != nil {
+			t.Fatal("decoing fails", err, i)
+		}
+		compressed := g.ToCompressed(p2)
+		if !bytes.Equal(vector, compressed) || !g.Equal(p1, p2) {
+			t.Fatal("serialization failed")
+		}
+		g.Add(p1, p1, &g1One)
+	}
+}
+
 func TestG1MultiExpExpected(t *testing.T) {
+	g := NewG1()
+	one := g.one()
+	var scalars [2]*Fr
+	var bases [2]*PointG1
+	scalars[0] = &Fr{2}
+	scalars[1] = &Fr{3}
+	bases[0], bases[1] = new(PointG1).Set(one), new(PointG1).Set(one)
+	expected, result := g.New(), g.New()
+	g.mulScalar(expected, one, &Fr{5})
+	_, _ = g.MultiExp(result, bases[:], scalars[:])
+	if !g.Equal(expected, result) {
+		t.Fatal("multi-exponentiation failed")
+	}
+}
+
+func TestG1MultiExpBigExpected(t *testing.T) {
 	g := NewG1()
 	one := g.one()
 	var scalars [2]*big.Int
@@ -182,36 +383,76 @@ func TestG1MultiExpExpected(t *testing.T) {
 	scalars[1] = big.NewInt(3)
 	bases[0], bases[1] = new(PointG1).Set(one), new(PointG1).Set(one)
 	expected, result := g.New(), g.New()
-	g.MulScalar(expected, one, big.NewInt(5))
+	g.mulScalarBig(expected, one, big.NewInt(5))
-	_, _ = g.MultiExp(result, bases[:], scalars[:])
+	_, _ = g.MultiExpBig(result, bases[:], scalars[:])
 	if !g.Equal(expected, result) {
-		t.Fatal("bad multi-exponentiation")
+		t.Fatal("multi-exponentiation failed")
 	}
 }
 
-func TestG1MultiExpBatch(t *testing.T) {
+func TestG1MultiExpBig(t *testing.T) {
 	g := NewG1()
-	one := g.one()
+	for n := 1; n < 1024+1; n = n * 2 {
-	n := 1000
 		bases := make([]*PointG1, n)
 		scalars := make([]*big.Int, n)
-	// scalars: [s0,s1 ... s(n-1)]
+		var err error
-	// bases: [P0,P1,..P(n-1)] = [s(n-1)*G, s(n-2)*G ... s0*G]
+		for i := 0; i < n; i++ {
-	for i, j := 0, n-1; i < n; i, j = i+1, j-1 {
+			scalars[i], err = rand.Int(rand.Reader, qBig)
-		scalars[j], _ = rand.Int(rand.Reader, big.NewInt(100000))
+			if err != nil {
-		bases[i] = g.New()
+				t.Fatal(err)
-		g.MulScalar(bases[i], one, scalars[j])
+			}
+			bases[i] = g.randAffine()
 		}
-	// expected: s(n-1)*P0 + s(n-2)*P1 + s0*P(n-1)
 		expected, tmp := g.New(), g.New()
 		for i := 0; i < n; i++ {
-		g.MulScalar(tmp, bases[i], scalars[i])
+			g.mulScalarBig(tmp, bases[i], scalars[i])
+			g.Add(expected, expected, tmp)
+		}
+		result := g.New()
+		_, _ = g.MultiExpBig(result, bases, scalars)
+		if !g.Equal(expected, result) {
+			t.Fatal("multi-exponentiation failed")
+		}
+	}
+}
+
+func TestG1MultiExp(t *testing.T) {
+	g := NewG1()
+	for n := 1; n < 1024+1; n = n * 2 {
+		bases := make([]*PointG1, n)
+		scalars := make([]*Fr, n)
+		var err error
+		for i := 0; i < n; i++ {
+			scalars[i], err = new(Fr).Rand(rand.Reader)
+			if err != nil {
+				t.Fatal(err)
+			}
+			bases[i] = g.randAffine()
+		}
+		expected, tmp := g.New(), g.New()
+		for i := 0; i < n; i++ {
+			g.mulScalar(tmp, bases[i], scalars[i])
 			g.Add(expected, expected, tmp)
 		}
 		result := g.New()
 		_, _ = g.MultiExp(result, bases, scalars)
 		if !g.Equal(expected, result) {
-		t.Fatal("bad multi-exponentiation")
+			t.Fatal("multi-exponentiation failed")
+		}
+	}
+}
+
+func TestG1ClearCofactor(t *testing.T) {
+	g := NewG1()
+	for i := 0; i < fuz; i++ {
+		p0 := g.rand()
+		if g.InCorrectSubgroup(p0) {
+			t.Fatal("rand point should be out of correct subgroup")
+		}
+		g.ClearCofactor(p0)
+		if !g.InCorrectSubgroup(p0) {
+			t.Fatal("cofactor clearing is failed")
+		}
 	}
 }
 
@@ -221,24 +462,39 @@ func TestG1MapToCurve(t *testing.T) {
 		expected []byte
 	}{
 		{
-			u:        make([]byte, 48),
+			u: make([]byte, fpByteSize),
-			expected: common.FromHex("11a9a0372b8f332d5c30de9ad14e50372a73fa4c45d5f2fa5097f2d6fb93bcac592f2e1711ac43db0519870c7d0ea415" + "092c0f994164a0719f51c24ba3788de240ff926b55f58c445116e8bc6a47cd63392fd4e8e22bdf9feaa96ee773222133"),
+			expected: fromHex(-1,
+				"11a9a0372b8f332d5c30de9ad14e50372a73fa4c45d5f2fa5097f2d6fb93bcac592f2e1711ac43db0519870c7d0ea415",
+				"092c0f994164a0719f51c24ba3788de240ff926b55f58c445116e8bc6a47cd63392fd4e8e22bdf9feaa96ee773222133",
+			),
 		},
 		{
-			u:        common.FromHex("07fdf49ea58e96015d61f6b5c9d1c8f277146a533ae7fbca2a8ef4c41055cd961fbc6e26979b5554e4b4f22330c0e16d"),
+			u: fromHex(-1, "07fdf49ea58e96015d61f6b5c9d1c8f277146a533ae7fbca2a8ef4c41055cd961fbc6e26979b5554e4b4f22330c0e16d"),
-			expected: common.FromHex("1223effdbb2d38152495a864d78eee14cb0992d89a241707abb03819a91a6d2fd65854ab9a69e9aacb0cbebfd490732c" + "0f925d61e0b235ecd945cbf0309291878df0d06e5d80d6b84aa4ff3e00633b26f9a7cb3523ef737d90e6d71e8b98b2d5"),
+			expected: fromHex(-1,
+				"1223effdbb2d38152495a864d78eee14cb0992d89a241707abb03819a91a6d2fd65854ab9a69e9aacb0cbebfd490732c",
+				"0f925d61e0b235ecd945cbf0309291878df0d06e5d80d6b84aa4ff3e00633b26f9a7cb3523ef737d90e6d71e8b98b2d5",
+			),
 		},
 		{
-			u:        common.FromHex("1275ab3adbf824a169ed4b1fd669b49cf406d822f7fe90d6b2f8c601b5348436f89761bb1ad89a6fb1137cd91810e5d2"),
+			u: fromHex(-1, "1275ab3adbf824a169ed4b1fd669b49cf406d822f7fe90d6b2f8c601b5348436f89761bb1ad89a6fb1137cd91810e5d2"),
-			expected: common.FromHex("179d3fd0b4fb1da43aad06cea1fb3f828806ddb1b1fa9424b1e3944dfdbab6e763c42636404017da03099af0dcca0fd6" + "0d037cb1c6d495c0f5f22b061d23f1be3d7fe64d3c6820cfcd99b6b36fa69f7b4c1f4addba2ae7aa46fb25901ab483e4"),
+			expected: fromHex(-1,
+				"179d3fd0b4fb1da43aad06cea1fb3f828806ddb1b1fa9424b1e3944dfdbab6e763c42636404017da03099af0dcca0fd6",
+				"0d037cb1c6d495c0f5f22b061d23f1be3d7fe64d3c6820cfcd99b6b36fa69f7b4c1f4addba2ae7aa46fb25901ab483e4",
+			),
 		},
 		{
-			u:        common.FromHex("0e93d11d30de6d84b8578827856f5c05feef36083eef0b7b263e35ecb9b56e86299614a042e57d467fa20948e8564909"),
+			u: fromHex(-1, "0e93d11d30de6d84b8578827856f5c05feef36083eef0b7b263e35ecb9b56e86299614a042e57d467fa20948e8564909"),
-			expected: common.FromHex("15aa66c77eded1209db694e8b1ba49daf8b686733afaa7b68c683d0b01788dfb0617a2e2d04c0856db4981921d3004af" + "0952bb2f61739dd1d201dd0a79d74cda3285403d47655ee886afe860593a8a4e51c5b77a22d2133e3a4280eaaaa8b788"),
+			expected: fromHex(-1,
+				"15aa66c77eded1209db694e8b1ba49daf8b686733afaa7b68c683d0b01788dfb0617a2e2d04c0856db4981921d3004af",
+				"0952bb2f61739dd1d201dd0a79d74cda3285403d47655ee886afe860593a8a4e51c5b77a22d2133e3a4280eaaaa8b788",
+			),
 		},
 		{
-			u:        common.FromHex("015a41481155d17074d20be6d8ec4d46632a51521cd9c916e265bd9b47343b3689979b50708c8546cbc2916b86cb1a3a"),
+			u: fromHex(-1, "015a41481155d17074d20be6d8ec4d46632a51521cd9c916e265bd9b47343b3689979b50708c8546cbc2916b86cb1a3a"),
-			expected: common.FromHex("06328ce5106e837935e8da84bd9af473422e62492930aa5f460369baad9545defa468d9399854c23a75495d2a80487ee" + "094bfdfe3e552447433b5a00967498a3f1314b86ce7a7164c8a8f4131f99333b30a574607e301d5f774172c627fd0bca"),
+			expected: fromHex(-1,
+				"06328ce5106e837935e8da84bd9af473422e62492930aa5f460369baad9545defa468d9399854c23a75495d2a80487ee",
+				"094bfdfe3e552447433b5a00967498a3f1314b86ce7a7164c8a8f4131f99333b30a574607e301d5f774172c627fd0bca",
+			),
 		},
 	} {
 		g := NewG1()
@@ -252,31 +508,217 @@ func TestG1MapToCurve(t *testing.T) {
 	}
 }
 
-func BenchmarkG1Add(t *testing.B) {
+func TestG1EncodeToCurve(t *testing.T) {
-	g1 := NewG1()
+	domain := []byte("BLS12381G1_XMD:SHA-256_SSWU_NU_TESTGEN")
-	a, b, c := g1.rand(), g1.rand(), PointG1{}
+	for i, v := range []struct {
-	t.ResetTimer()
+		msg      []byte
-	for i := 0; i < t.N; i++ {
+		expected []byte
-		g1.Add(&c, a, b)
+	}{
+		{
+			msg: []byte(""),
+			expected: fromHex(-1,
+				"1223effdbb2d38152495a864d78eee14cb0992d89a241707abb03819a91a6d2fd65854ab9a69e9aacb0cbebfd490732c",
+				"0f925d61e0b235ecd945cbf0309291878df0d06e5d80d6b84aa4ff3e00633b26f9a7cb3523ef737d90e6d71e8b98b2d5",
+			),
+		},
+		{
+			msg: []byte("abc"),
+			expected: fromHex(-1,
+				"179d3fd0b4fb1da43aad06cea1fb3f828806ddb1b1fa9424b1e3944dfdbab6e763c42636404017da03099af0dcca0fd6",
+				"0d037cb1c6d495c0f5f22b061d23f1be3d7fe64d3c6820cfcd99b6b36fa69f7b4c1f4addba2ae7aa46fb25901ab483e4",
+			),
+		},
+		{
+			msg: []byte("abcdef0123456789"),
+			expected: fromHex(-1,
+				"15aa66c77eded1209db694e8b1ba49daf8b686733afaa7b68c683d0b01788dfb0617a2e2d04c0856db4981921d3004af",
+				"0952bb2f61739dd1d201dd0a79d74cda3285403d47655ee886afe860593a8a4e51c5b77a22d2133e3a4280eaaaa8b788",
+			),
+		},
+		{
+			msg: []byte("a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"),
+			expected: fromHex(-1,
+				"06328ce5106e837935e8da84bd9af473422e62492930aa5f460369baad9545defa468d9399854c23a75495d2a80487ee",
+				"094bfdfe3e552447433b5a00967498a3f1314b86ce7a7164c8a8f4131f99333b30a574607e301d5f774172c627fd0bca",
+			),
+		},
+	} {
+		g := NewG1()
+		p0, err := g.EncodeToCurve(v.msg, domain)
+		if err != nil {
+			t.Fatal("encode to point fails", i, err)
+		}
+		if !bytes.Equal(g.ToBytes(p0), v.expected) {
+			t.Fatal("encode to point fails", i)
+		}
 	}
 }
 
-func BenchmarkG1Mul(t *testing.B) {
+func TestG1HashToCurve(t *testing.T) {
-	worstCaseScalar, _ := new(big.Int).SetString("ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", 16)
+	domain := []byte("BLS12381G1_XMD:SHA-256_SSWU_RO_TESTGEN")
-	g1 := NewG1()
+	for i, v := range []struct {
-	a, e, c := g1.rand(), worstCaseScalar, PointG1{}
+		msg      []byte
+		expected []byte
+	}{
+		{
+			msg: []byte(""),
+			expected: fromHex(-1,
+				"0576730ab036cbac1d95b38dca905586f28d0a59048db4e8778782d89bff856ddef89277ead5a21e2975c4a6e3d8c79e",
+				"1273e568bebf1864393c517f999b87c1eaa1b8432f95aea8160cd981b5b05d8cd4a7cf00103b6ef87f728e4b547dd7ae",
+			),
+		},
+		{
+			msg: []byte("abc"),
+			expected: fromHex(-1,
+				"061daf0cc00d8912dac1d4cf5a7c32fca97f8b3bf3f805121888e5eb89f77f9a9f406569027ac6d0e61b1229f42c43d6",
+				"0de1601e5ba02cb637c1d35266f5700acee9850796dc88e860d022d7b9e7e3dce5950952e97861e5bb16d215c87f030d",
+			),
+		},
+		{
+			msg: []byte("abcdef0123456789"),
+			expected: fromHex(-1,
+				"0fb3455436843e76079c7cf3dfef75e5a104dfe257a29a850c145568d500ad31ccfe79be9ae0ea31a722548070cf98cd",
+				"177989f7e2c751658df1b26943ee829d3ebcf131d8f805571712f3a7527ee5334ecff8a97fc2a50cea86f5e6212e9a57",
+			),
+		},
+		{
+			msg: []byte("a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"),
+			expected: fromHex(-1,
+				"0514af2137c1ae1d78d5cb97ee606ea142824c199f0f25ac463a0c78200de57640d34686521d3e9cf6b3721834f8a038",
+				"047a85d6898416a0899e26219bca7c4f0fa682717199de196b02b95eaf9fb55456ac3b810e78571a1b7f5692b7c58ab6",
+			),
+		},
+	} {
+		g := NewG1()
+		p0, err := g.HashToCurve(v.msg, domain)
+		if err != nil {
+			t.Fatal("hash to point fails", i, err)
+		}
+		if !bytes.Equal(g.ToBytes(p0), v.expected) {
+			t.Fatal("hash to point fails", i)
+		}
+	}
+}
+
+func BenchmarkG1Add(t *testing.B) {
+	g := NewG1()
+	a, b, c := g.rand(), g.rand(), PointG1{}
 	t.ResetTimer()
 	for i := 0; i < t.N; i++ {
-		g1.MulScalar(&c, a, e)
+		g.Add(&c, a, b)
+	}
+}
+
+func BenchmarkG1MulWNAF(t *testing.B) {
+	g := NewG1()
+	p := new(PointG1).Set(&g1One)
+	s, _ := new(Fr).Rand(rand.Reader)
+	sBig := s.ToBig()
+	res := new(PointG1)
+	t.Run("Naive", func(t *testing.B) {
+		t.ResetTimer()
+		for i := 0; i < t.N; i++ {
+			g.mulScalar(res, p, s)
+		}
+	})
+	for i := 1; i < 8; i++ {
+		wnafMulWindowG1 = uint(i)
+		t.Run(fmt.Sprintf("Fr, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.wnafMulFr(res, p, s)
+			}
+		})
+		t.Run(fmt.Sprintf("Big, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.wnafMulBig(res, p, sBig)
+			}
+		})
+	}
+}
+
+func BenchmarkG1MulGLV(t *testing.B) {
+
+	g := NewG1()
+	p := new(PointG1).Set(&g1One)
+	s, _ := new(Fr).Rand(rand.Reader)
+	sBig := s.ToBig()
+	res := new(PointG1)
+	t.Run("Naive", func(t *testing.B) {
+		t.ResetTimer()
+		for i := 0; i < t.N; i++ {
+			g.mulScalar(res, p, s)
+		}
+	})
+	for i := 1; i < 8; i++ {
+		glvMulWindowG1 = uint(i)
+		t.Run(fmt.Sprintf("Fr, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.glvMulFr(res, p, s)
+			}
+		})
+		t.Run(fmt.Sprintf("Big, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.glvMulBig(res, p, sBig)
+			}
+		})
+	}
+}
+
+func BenchmarkG1MultiExp(t *testing.B) {
+	g := NewG1()
+	v := func(n int) ([]*PointG1, []*Fr) {
+		bases := make([]*PointG1, n)
+		scalars := make([]*Fr, n)
+		var err error
+		for i := 0; i < n; i++ {
+			scalars[i], err = new(Fr).Rand(rand.Reader)
+			if err != nil {
+				t.Fatal(err)
+			}
+			bases[i] = g.randAffine()
+		}
+		return bases, scalars
+	}
+	for _, i := range []int{2, 10, 100, 1000} {
+		t.Run(fmt.Sprint(i), func(t *testing.B) {
+			bases, scalars := v(i)
+			result := g.New()
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				_, _ = g.MultiExp(result, bases, scalars)
+			}
+		})
+	}
+}
+
+func BenchmarkG1ClearCofactor(t *testing.B) {
+	g := NewG1()
+	a := g.rand()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		g.ClearCofactor(a)
+	}
+}
+
+func BenchmarkG1SubgroupCheck(t *testing.B) {
+	g := NewG1()
+	a := g.rand()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		g.InCorrectSubgroup(a)
 	}
 }
 
 func BenchmarkG1MapToCurve(t *testing.B) {
-	a := make([]byte, 48)
+	a := fromHex(fpByteSize, "0x1234")
-	g1 := NewG1()
+	g := NewG1()
 	t.ResetTimer()
 	for i := 0; i < t.N; i++ {
-		_, err := g1.MapToCurve(a)
+		_, err := g.MapToCurve(a)
 		if err != nil {
 			t.Fatal(err)
 		}

crypto/bls12381/g2.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -22,9 +6,8 @@ import (
 	"math/big"
 )
 
-// PointG2 is type for point in G2.
+// PointG2 is type for point in G2 and used for both affine and Jacobian representation.
-// PointG2 is both used for Affine and Jacobian point representation.
+// A point is accounted as in affine form if z is equal to one.
-// If z is equal to one the point is considered as in affine form.
 type PointG2 [3]fe2
 
 // Set copies values of one point to another.
@@ -35,7 +18,6 @@ func (p *PointG2) Set(p2 *PointG2) *PointG2 {
 	return p
 }
 
-// Zero returns G2 point in point at infinity representation
 func (p *PointG2) Zero() *PointG2 {
 	p[0].zero()
 	p[1].one()
@@ -43,6 +25,11 @@ func (p *PointG2) Zero() *PointG2 {
 	return p
 }
 
+// IsAffine checks a G1 point whether it is in affine form.
+func (p *PointG2) IsAffine() bool {
+	return p[2].isOne()
+}
+
 type tempG2 struct {
 	t [9]*fe2
 }
@@ -76,15 +63,141 @@ func newTempG2() tempG2 {
 
 // Q returns group order in big.Int.
 func (g *G2) Q() *big.Int {
-	return new(big.Int).Set(q)
+	return new(big.Int).Set(qBig)
 }
 
-func (g *G2) fromBytesUnchecked(in []byte) (*PointG2, error) {
+// FromUncompressed expects byte slice at least 192 bytes and given bytes returns a new point in G2.
-	p0, err := g.f.fromBytes(in[:96])
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G2) FromUncompressed(uncompressed []byte) (*PointG2, error) {
+	if len(uncompressed) != 4*fpByteSize {
+		return nil, errors.New("input string length must be equal to 192 bytes")
+	}
+	var in [4 * fpByteSize]byte
+	copy(in[:], uncompressed[:4*fpByteSize])
+	if in[0]&(1<<7) != 0 {
+		return nil, errors.New("compression flag must be zero")
+	}
+	if in[0]&(1<<5) != 0 {
+		return nil, errors.New("sort flag must be zero")
+	}
+	if in[0]&(1<<6) != 0 {
+		for i, v := range in {
+			if (i == 0 && v != 0x40) || (i != 0 && v != 0x00) {
+				return nil, errors.New("input string must be zero when infinity flag is set")
+			}
+		}
+		return g.Zero(), nil
+	}
+	in[0] &= 0x1f
+	x, err := g.f.fromBytes(in[:2*fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	p1, err := g.f.fromBytes(in[96:])
+	y, err := g.f.fromBytes(in[2*fpByteSize:])
+	if err != nil {
+		return nil, err
+	}
+	z := new(fe2).one()
+	p := &PointG2{*x, *y, *z}
+	if !g.IsOnCurve(p) {
+		return nil, errors.New("point is not on curve")
+	}
+	if !g.InCorrectSubgroup(p) {
+		return nil, errors.New("point is not on correct subgroup")
+	}
+	return p, nil
+}
+
+// ToUncompressed given a G2 point returns bytes in uncompressed (x, y) form of the point.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G2) ToUncompressed(p *PointG2) []byte {
+	out := make([]byte, 4*fpByteSize)
+	g.Affine(p)
+	if g.IsZero(p) {
+		out[0] |= 1 << 6
+		return out
+	}
+	copy(out[:2*fpByteSize], g.f.toBytes(&p[0]))
+	copy(out[2*fpByteSize:], g.f.toBytes(&p[1]))
+	return out
+}
+
+// FromCompressed expects byte slice at least 96 bytes and given bytes returns a new point in G2.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G2) FromCompressed(compressed []byte) (*PointG2, error) {
+	if len(compressed) != 2*fpByteSize {
+		return nil, errors.New("input string length must be equal to 96 bytes")
+	}
+	var in [2 * fpByteSize]byte
+	copy(in[:], compressed[:])
+	if in[0]&(1<<7) == 0 {
+		return nil, errors.New("compression flag must be set")
+	}
+	if in[0]&(1<<6) != 0 {
+		// in[0] == (1 << 6) + (1 << 7)
+		for i, v := range in {
+			if (i == 0 && v != 0xc0) || (i != 0 && v != 0x00) {
+				return nil, errors.New("input string must be zero when infinity flag is set")
+			}
+		}
+		return g.Zero(), nil
+	}
+	a := in[0]&(1<<5) != 0
+	in[0] &= 0x1f
+	x, err := g.f.fromBytes(in[:])
+	if err != nil {
+		return nil, err
+	}
+	// solve curve equation
+	y := &fe2{}
+	g.f.square(y, x)
+	g.f.mul(y, y, x)
+	fp2Add(y, y, b2)
+	if ok := g.f.sqrt(y, y); !ok {
+		return nil, errors.New("point is not on curve")
+	}
+	if y.signBE() == a {
+		fp2Neg(y, y)
+	}
+	z := new(fe2).one()
+	p := &PointG2{*x, *y, *z}
+	if !g.InCorrectSubgroup(p) {
+		return nil, errors.New("point is not on correct subgroup")
+	}
+	return p, nil
+}
+
+// ToCompressed given a G2 point returns bytes in compressed form of the point.
+// Serialization rules are in line with zcash library. See below for details.
+// https://github.com/zcash/librustzcash/blob/master/pairing/src/bls12_381/README.md#serialization
+// https://docs.rs/bls12_381/0.1.1/bls12_381/notes/serialization/index.html
+func (g *G2) ToCompressed(p *PointG2) []byte {
+	out := make([]byte, 2*fpByteSize)
+	g.Affine(p)
+	if g.IsZero(p) {
+		out[0] |= 1 << 6
+	} else {
+		copy(out[:], g.f.toBytes(&p[0]))
+		if !p[1].signBE() {
+			out[0] |= 1 << 5
+		}
+	}
+	out[0] |= 1 << 7
+	return out
+}
+
+func (g *G2) fromBytesUnchecked(in []byte) (*PointG2, error) {
+	p0, err := g.f.fromBytes(in[:2*fpByteSize])
+	if err != nil {
+		return nil, err
+	}
+	p1, err := g.f.fromBytes(in[2*fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -93,19 +206,17 @@ func (g *G2) fromBytesUnchecked(in []byte) (*PointG2, error) {
 }
 
 // FromBytes constructs a new point given uncompressed byte input.
-// FromBytes does not take zcash flags into account.
+// Input string expected to be 192 bytes and concatenation of x and y values
-// Byte input expected to be larger than 96 bytes.
-// First 192 bytes should be concatenation of x and y values
 // Point (0, 0) is considered as infinity.
 func (g *G2) FromBytes(in []byte) (*PointG2, error) {
-	if len(in) != 192 {
+	if len(in) != 4*fpByteSize {
-		return nil, errors.New("input string should be equal or larger than 192")
+		return nil, errors.New("input string length must be equal to 192 bytes")
 	}
-	p0, err := g.f.fromBytes(in[:96])
+	p0, err := g.f.fromBytes(in[:2*fpByteSize])
 	if err != nil {
 		return nil, err
 	}
-	p1, err := g.f.fromBytes(in[96:])
+	p1, err := g.f.fromBytes(in[2*fpByteSize:])
 	if err != nil {
 		return nil, err
 	}
@@ -121,60 +232,16 @@ func (g *G2) FromBytes(in []byte) (*PointG2, error) {
 	return p, nil
 }
 
-// DecodePoint given encoded (x, y) coordinates in 256 bytes returns a valid G2 Point.
-func (g *G2) DecodePoint(in []byte) (*PointG2, error) {
-	if len(in) != 256 {
-		return nil, errors.New("invalid g2 point length")
-	}
-	pointBytes := make([]byte, 192)
-	x0Bytes, err := decodeFieldElement(in[:64])
-	if err != nil {
-		return nil, err
-	}
-	x1Bytes, err := decodeFieldElement(in[64:128])
-	if err != nil {
-		return nil, err
-	}
-	y0Bytes, err := decodeFieldElement(in[128:192])
-	if err != nil {
-		return nil, err
-	}
-	y1Bytes, err := decodeFieldElement(in[192:])
-	if err != nil {
-		return nil, err
-	}
-	copy(pointBytes[:48], x1Bytes)
-	copy(pointBytes[48:96], x0Bytes)
-	copy(pointBytes[96:144], y1Bytes)
-	copy(pointBytes[144:192], y0Bytes)
-	return g.FromBytes(pointBytes)
-}
-
 // ToBytes serializes a point into bytes in uncompressed form,
-// does not take zcash flags into account,
 // returns (0, 0) if point is infinity.
 func (g *G2) ToBytes(p *PointG2) []byte {
-	out := make([]byte, 192)
+	out := make([]byte, 4*fpByteSize)
 	if g.IsZero(p) {
 		return out
 	}
 	g.Affine(p)
-	copy(out[:96], g.f.toBytes(&p[0]))
+	copy(out[:2*fpByteSize], g.f.toBytes(&p[0]))
-	copy(out[96:], g.f.toBytes(&p[1]))
+	copy(out[2*fpByteSize:], g.f.toBytes(&p[1]))
-	return out
-}
-
-// EncodePoint encodes a point into 256 bytes.
-func (g *G2) EncodePoint(p *PointG2) []byte {
-	// outRaw is 96 bytes
-	outRaw := g.ToBytes(p)
-	out := make([]byte, 256)
-	// encode x
-	copy(out[16:16+48], outRaw[48:96])
-	copy(out[80:80+48], outRaw[:48])
-	// encode y
-	copy(out[144:144+48], outRaw[144:])
-	copy(out[208:208+48], outRaw[96:144])
 	return out
 }
 
@@ -212,35 +279,32 @@ func (g *G2) Equal(p1, p2 *PointG2) bool {
 	g.f.square(t[1], &p2[2])
 	g.f.mul(t[2], t[0], &p2[0])
 	g.f.mul(t[3], t[1], &p1[0])
-	g.f.mul(t[0], t[0], &p1[2])
+	g.f.mulAssign(t[0], &p1[2])
-	g.f.mul(t[1], t[1], &p2[2])
+	g.f.mulAssign(t[1], &p2[2])
-	g.f.mul(t[1], t[1], &p1[1])
+	g.f.mulAssign(t[1], &p1[1])
-	g.f.mul(t[0], t[0], &p2[1])
+	g.f.mulAssign(t[0], &p2[1])
 	return t[0].equal(t[1]) && t[2].equal(t[3])
 }
 
-// InCorrectSubgroup checks whether given point is in correct subgroup.
-func (g *G2) InCorrectSubgroup(p *PointG2) bool {
-	tmp := &PointG2{}
-	g.MulScalar(tmp, p, q)
-	return g.IsZero(tmp)
-}
-
 // IsOnCurve checks a G2 point is on curve.
 func (g *G2) IsOnCurve(p *PointG2) bool {
 	if g.IsZero(p) {
 		return true
 	}
 	t := g.t
-	g.f.square(t[0], &p[1])
+	g.f.square(t[0], &p[1])    // y^2
-	g.f.square(t[1], &p[0])
+	g.f.square(t[1], &p[0])    // x^2
-	g.f.mul(t[1], t[1], &p[0])
+	g.f.mul(t[1], t[1], &p[0]) // x^3
-	g.f.square(t[2], &p[2])
+	if p.IsAffine() {
-	g.f.square(t[3], t[2])
+		fp2Add(t[1], t[1], b2)  // x^2 + b
-	g.f.mul(t[2], t[2], t[3])
+		return t[0].equal(t[1]) // y^2 ?= x^3 + b
-	g.f.mul(t[2], b2, t[2])
+	}
-	g.f.add(t[1], t[1], t[2])
+	g.f.square(t[2], &p[2])   // z^2
-	return t[0].equal(t[1])
+	g.f.square(t[3], t[2])    // z^4
+	g.f.mulAssign(t[2], t[3]) // z^6
+	g.f.mulAssign(t[2], b2)   // b*z^6
+	fp2AddAssign(t[1], t[2])  // x^3 + b * z^6
+	return t[0].equal(t[1])   // y^2 ?= x^3 + b * z^6
 }
 
 // IsAffine checks a G2 point whether it is in affine form.
@@ -250,24 +314,102 @@ func (g *G2) IsAffine(p *PointG2) bool {
 
 // Affine calculates affine form of given G2 point.
 func (g *G2) Affine(p *PointG2) *PointG2 {
+	return g.affine(p, p)
+}
+
+func (g *G2) affine(r, p *PointG2) *PointG2 {
 	if g.IsZero(p) {
-		return p
+		return r.Zero()
 	}
 	if !g.IsAffine(p) {
 		t := g.t
-		g.f.inverse(t[0], &p[2])
+		g.f.inverse(t[0], &p[2])   // z^-1
-		g.f.square(t[1], t[0])
+		g.f.square(t[1], t[0])     // z^-2
-		g.f.mul(&p[0], &p[0], t[1])
+		g.f.mulAssign(&r[0], t[1]) // x = x * z^-2
-		g.f.mul(t[0], t[0], t[1])
+		g.f.mulAssign(t[0], t[1])  // z^-3
-		g.f.mul(&p[1], &p[1], t[0])
+		g.f.mulAssign(&r[1], t[0]) // y = y * z^-3
-		p[2].one()
+		r[2].one()                 // z = 1
+	} else {
+		r.Set(p)
+	}
+	return r
+}
+
+// AffineBatch given multiple of points returns affine representations
+func (g *G2) AffineBatch(p []*PointG2) {
+	inverses := make([]fe2, len(p))
+	for i := 0; i < len(p); i++ {
+		inverses[i].set(&p[i][2])
+	}
+	g.f.inverseBatch(inverses)
+	t := g.t
+	for i := 0; i < len(p); i++ {
+		if !g.IsAffine(p[i]) && !g.IsZero(p[i]) {
+			g.f.square(t[1], &inverses[i])
+			g.f.mulAssign(&p[i][0], t[1])
+			g.f.mul(t[0], &inverses[i], t[1])
+			g.f.mulAssign(&p[i][1], t[0])
+			p[i][2].one()
+		}
 	}
-	return p
 }
 
 // Add adds two G2 points p1, p2 and assigns the result to point at first argument.
 func (g *G2) Add(r, p1, p2 *PointG2) *PointG2 {
-	// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+	// http://www.hyperelliptic.org/EFD/gp/auto-shortw-jacobian-0.html#addition-add-2007-bl
+	if g.IsZero(p1) {
+		return r.Set(p2)
+	}
+	if g.IsZero(p2) {
+		return r.Set(p1)
+	}
+	if g.IsAffine(p2) {
+		return g.AddMixed(r, p1, p2)
+	}
+	t := g.t
+	g.f.square(t[7], &p1[2])    // z1z1
+	g.f.mul(t[1], &p2[0], t[7]) // u2 = x2 * z1z1
+	g.f.mul(t[2], &p1[2], t[7]) // z1z1 * z1
+	g.f.mul(t[0], &p2[1], t[2]) // s2 = y2 * z1z1 * z1
+	g.f.square(t[8], &p2[2])    // z2z2
+	g.f.mul(t[3], &p1[0], t[8]) // u1 = x1 * z2z2
+	g.f.mul(t[4], &p2[2], t[8]) // z2z2 * z2
+	g.f.mul(t[2], &p1[1], t[4]) // s1 = y1 * z2z2 * z2
+	if t[1].equal(t[3]) {
+		if t[0].equal(t[2]) {
+			return g.Double(r, p1)
+		} else {
+			return r.Zero()
+		}
+	}
+	fp2SubAssign(t[1], t[3])     // h = u2 - u1
+	fp2Double(t[4], t[1])        // 2h
+	g.f.squareAssign(t[4])       // i = 2h^2
+	g.f.mul(t[5], t[1], t[4])    // j = h*i
+	fp2SubAssign(t[0], t[2])     // s2 - s1
+	fp2DoubleAssign(t[0])        // r = 2*(s2 - s1)
+	g.f.square(t[6], t[0])       // r^2
+	fp2SubAssign(t[6], t[5])     // r^2 - j
+	g.f.mulAssign(t[3], t[4])    // v = u1 * i
+	fp2Double(t[4], t[3])        // 2*v
+	fp2Sub(&r[0], t[6], t[4])    // x3 = r^2 - j - 2*v
+	fp2Sub(t[4], t[3], &r[0])    // v - x3
+	g.f.mul(t[6], t[2], t[5])    // s1 * j
+	fp2DoubleAssign(t[6])        // 2 * s1 * j
+	g.f.mulAssign(t[0], t[4])    // r * (v - x3)
+	fp2Sub(&r[1], t[0], t[6])    // y3 = r * (v - x3) - (2 * s1 * j)
+	fp2Add(t[0], &p1[2], &p2[2]) // z1 + z2
+	g.f.squareAssign(t[0])       // (z1 + z2)^2
+	fp2SubAssign(t[0], t[7])     // (z1 + z2)^2 - z1z1
+	fp2SubAssign(t[0], t[8])     // (z1 + z2)^2 - z1z1 - z2z2
+	g.f.mul(&r[2], t[0], t[1])   // z3 = ((z1 + z2)^2 - z1z1 - z2z2) * h
+	return r
+}
+
+// Add adds two G1 points p1, p2 and assigns the result to point at first argument.
+// Expects the second point p2 in affine form.
+func (g *G2) AddMixed(r, p1, p2 *PointG2) *PointG2 {
+	// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
 	if g.IsZero(p1) {
 		return r.Set(p2)
 	}
@@ -275,80 +417,75 @@ func (g *G2) Add(r, p1, p2 *PointG2) *PointG2 {
 		return r.Set(p1)
 	}
 	t := g.t
-	g.f.square(t[7], &p1[2])
+	g.f.square(t[7], &p1[2])    // z1z1
-	g.f.mul(t[1], &p2[0], t[7])
+	g.f.mul(t[1], &p2[0], t[7]) // u2 = x2 * z1z1
-	g.f.mul(t[2], &p1[2], t[7])
+	g.f.mul(t[2], &p1[2], t[7]) // z1z1 * z1
-	g.f.mul(t[0], &p2[1], t[2])
+	g.f.mul(t[0], &p2[1], t[2]) // s2 = y2 * z1z1 * z1
-	g.f.square(t[8], &p2[2])
+
-	g.f.mul(t[3], &p1[0], t[8])
+	if p1[0].equal(t[1]) && p1[1].equal(t[0]) {
-	g.f.mul(t[4], &p2[2], t[8])
-	g.f.mul(t[2], &p1[1], t[4])
-	if t[1].equal(t[3]) {
-		if t[0].equal(t[2]) {
 		return g.Double(r, p1)
 	}
-		return r.Zero()
+
-	}
+	fp2SubAssign(t[1], &p1[0]) // h = u2 - x1
-	g.f.sub(t[1], t[1], t[3])
+	g.f.square(t[2], t[1])     // hh
-	g.f.double(t[4], t[1])
+	fp2Double(t[4], t[2])
-	g.f.square(t[4], t[4])
+	fp2DoubleAssign(t[4])       // 4hh
-	g.f.mul(t[5], t[1], t[4])
+	g.f.mul(t[5], t[1], t[4])   // j = h*i
-	g.f.sub(t[0], t[0], t[2])
+	fp2SubAssign(t[0], &p1[1])  // s2 - y1
-	g.f.double(t[0], t[0])
+	fp2DoubleAssign(t[0])       // r = 2*(s2 - y1)
-	g.f.square(t[6], t[0])
+	g.f.square(t[6], t[0])      // r^2
-	g.f.sub(t[6], t[6], t[5])
+	fp2SubAssign(t[6], t[5])    // r^2 - j
-	g.f.mul(t[3], t[3], t[4])
+	g.f.mul(t[3], &p1[0], t[4]) // v = x1 * i
-	g.f.double(t[4], t[3])
+	fp2Double(t[4], t[3])       // 2*v
-	g.f.sub(&r[0], t[6], t[4])
+	fp2Sub(&r[0], t[6], t[4])   // x3 = r^2 - j - 2*v
-	g.f.sub(t[4], t[3], &r[0])
+	fp2Sub(t[4], t[3], &r[0])   // v - x3
-	g.f.mul(t[6], t[2], t[5])
+	g.f.mul(t[6], &p1[1], t[5]) // y1 * j
-	g.f.double(t[6], t[6])
+	fp2DoubleAssign(t[6])       // 2 * y1 * j
-	g.f.mul(t[0], t[0], t[4])
+	g.f.mulAssign(t[0], t[4])   // r * (v - x3)
-	g.f.sub(&r[1], t[0], t[6])
+	fp2Sub(&r[1], t[0], t[6])   // y3 = r * (v - x3) - (2 * y1 * j)
-	g.f.add(t[0], &p1[2], &p2[2])
+	fp2Add(t[0], &p1[2], t[1])  // z1 + h
-	g.f.square(t[0], t[0])
+	g.f.squareAssign(t[0])      // (z1 + h)^2
-	g.f.sub(t[0], t[0], t[7])
+	fp2SubAssign(t[0], t[7])    // (z1 + h)^2 - z1z1
-	g.f.sub(t[0], t[0], t[8])
+	fp2Sub(&r[2], t[0], t[2])   // z3 = (z1 + z2)^2 - z1z1 - hh
-	g.f.mul(&r[2], t[0], t[1])
 	return r
 }
 
 // Double doubles a G2 point p and assigns the result to the point at first argument.
 func (g *G2) Double(r, p *PointG2) *PointG2 {
-	// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
+	// http://www.hyperelliptic.org/EFD/gp/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
 	if g.IsZero(p) {
 		return r.Set(p)
 	}
 	t := g.t
-	g.f.square(t[0], &p[0])
+	g.f.square(t[0], &p[0])     // a = x^2
-	g.f.square(t[1], &p[1])
+	g.f.square(t[1], &p[1])     // b = y^2
-	g.f.square(t[2], t[1])
+	g.f.square(t[2], t[1])      // c = b^2
-	g.f.add(t[1], &p[0], t[1])
+	fp2AddAssign(t[1], &p[0])   // b + x1
-	g.f.square(t[1], t[1])
+	g.f.squareAssign(t[1])      // (b + x1)^2
-	g.f.sub(t[1], t[1], t[0])
+	fp2SubAssign(t[1], t[0])    // (b + x1)^2 - a
-	g.f.sub(t[1], t[1], t[2])
+	fp2SubAssign(t[1], t[2])    // (b + x1)^2 - a - c
-	g.f.double(t[1], t[1])
+	fp2DoubleAssign(t[1])       // d = 2((b+x1)^2 - a - c)
-	g.f.double(t[3], t[0])
+	fp2Double(t[3], t[0])       // 2a
-	g.f.add(t[0], t[3], t[0])
+	fp2AddAssign(t[0], t[3])    // e = 3a
-	g.f.square(t[4], t[0])
+	g.f.square(t[4], t[0])      // f = e^2
-	g.f.double(t[3], t[1])
+	fp2Double(t[3], t[1])       // 2d
-	g.f.sub(&r[0], t[4], t[3])
+	fp2Sub(&r[0], t[4], t[3])   // x3 = f - 2d
-	g.f.sub(t[1], t[1], &r[0])
+	fp2SubAssign(t[1], &r[0])   // d-x3
-	g.f.double(t[2], t[2])
+	fp2DoubleAssign(t[2])       //
-	g.f.double(t[2], t[2])
+	fp2DoubleAssign(t[2])       //
-	g.f.double(t[2], t[2])
+	fp2DoubleAssign(t[2])       // 8c
-	g.f.mul(t[0], t[0], t[1])
+	g.f.mulAssign(t[0], t[1])   // e * (d - x3)
-	g.f.sub(t[1], t[0], t[2])
+	fp2Sub(t[1], t[0], t[2])    // x3 = e * (d - x3) - 8c
-	g.f.mul(t[0], &p[1], &p[2])
+	g.f.mul(t[0], &p[1], &p[2]) // y1 * z1
-	r[1].set(t[1])
+	r[1].set(t[1])              //
-	g.f.double(&r[2], t[0])
+	fp2Double(&r[2], t[0])      // z3 = 2(y1 * z1)
 	return r
 }
 
 // Neg negates a G2 point p and assigns the result to the point at first argument.
 func (g *G2) Neg(r, p *PointG2) *PointG2 {
 	r[0].set(&p[0])
-	g.f.neg(&r[1], &p[1])
+	fp2Neg(&r[1], &p[1])
 	r[2].set(&p[2])
 	return r
 }
@@ -361,8 +498,29 @@ func (g *G2) Sub(c, a, b *PointG2) *PointG2 {
 	return c
 }
 
-// MulScalar multiplies a point by given scalar value in big.Int and assigns the result to point at first argument.
+// MulScalar multiplies a point by given scalar value and assigns the result to point at first argument.
-func (g *G2) MulScalar(c, p *PointG2, e *big.Int) *PointG2 {
+func (g *G2) MulScalar(r, p *PointG2, e *Fr) *PointG2 {
+	return g.glvMulFr(r, p, e)
+}
+
+// MulScalarBig multiplies a point by given scalar value in big.Int and assigns the result to point at first argument.
+func (g *G2) MulScalarBig(r, p *PointG2, e *big.Int) *PointG2 {
+	return g.glvMulBig(r, p, e)
+}
+
+func (g *G2) mulScalar(c, p *PointG2, e *Fr) *PointG2 {
+	q, n := &PointG2{}, &PointG2{}
+	n.Set(p)
+	for i := 0; i < frBitSize; i++ {
+		if e.Bit(i) {
+			g.Add(q, q, n)
+		}
+		g.Double(n, n)
+	}
+	return c.Set(q)
+}
+
+func (g *G2) mulScalarBig(c, p *PointG2, e *big.Int) *PointG2 {
 	q, n := &PointG2{}, &PointG2{}
 	n.Set(p)
 	l := e.BitLen()
@@ -375,67 +533,299 @@ func (g *G2) MulScalar(c, p *PointG2, e *big.Int) *PointG2 {
 	return c.Set(q)
 }
 
-// ClearCofactor maps given a G2 point to correct subgroup
+func (g *G2) wnafMulFr(r, p *PointG2, e *Fr) *PointG2 {
-func (g *G2) ClearCofactor(p *PointG2) {
+	wnaf := e.toWNAF(wnafMulWindowG2)
-	g.MulScalar(p, p, cofactorEFFG2)
+	return g.wnafMul(r, p, wnaf)
 }
 
-// MultiExp calculates multi exponentiation. Given pairs of G2 point and scalar values
+func (g *G2) wnafMulBig(r, p *PointG2, e *big.Int) *PointG2 {
-// (P_0, e_0), (P_1, e_1), ... (P_n, e_n) calculates r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n
+	wnaf := bigToWNAF(e, wnafMulWindowG2)
-// Length of points and scalars are expected to be equal, otherwise an error is returned.
+	return g.wnafMul(r, p, wnaf)
-// Result is assigned to point at first argument.
+}
-func (g *G2) MultiExp(r *PointG2, points []*PointG2, powers []*big.Int) (*PointG2, error) {
+
-	if len(points) != len(powers) {
+func (g *G2) wnafMul(c, p *PointG2, wnaf nafNumber) *PointG2 {
-		return nil, errors.New("point and scalar vectors should be in same length")
+
+	l := (1 << (wnafMulWindowG2 - 1))
+
+	twoP, acc := g.New(), new(PointG2).Set(p)
+	g.Double(twoP, p)
+	g.Affine(twoP)
+
+	// table = {p, 3p, 5p, ..., -p, -3p, -5p}
+	table := make([]*PointG2, l*2)
+	table[0], table[l] = g.New(), g.New()
+	table[0].Set(p)
+	g.Neg(table[l], table[0])
+
+	for i := 1; i < l; i++ {
+		g.AddMixed(acc, acc, twoP)
+		table[i], table[i+l] = g.New(), g.New()
+		table[i].Set(acc)
+		g.Neg(table[i+l], table[i])
 	}
-	var c uint32 = 3
+
-	if len(powers) >= 32 {
+	q := g.Zero()
-		c = uint32(math.Ceil(math.Log10(float64(len(powers)))))
+	for i := len(wnaf) - 1; i >= 0; i-- {
+		if wnaf[i] > 0 {
+			g.Add(q, q, table[wnaf[i]>>1])
+		} else if wnaf[i] < 0 {
+			g.Add(q, q, table[((-wnaf[i])>>1)+l])
 		}
-	bucketSize, numBits := (1<<c)-1, uint32(g.Q().BitLen())
+		if i != 0 {
-	windows := make([]*PointG2, numBits/c+1)
+			g.Double(q, q)
-	bucket := make([]*PointG2, bucketSize)
-	acc, sum := g.New(), g.New()
-	for i := 0; i < bucketSize; i++ {
-		bucket[i] = g.New()
 		}
-	mask := (uint64(1) << c) - 1
-	j := 0
-	var cur uint32
-	for cur <= numBits {
-		acc.Zero()
-		bucket = make([]*PointG2, (1<<c)-1)
-		for i := 0; i < len(bucket); i++ {
-			bucket[i] = g.New()
 	}
-		for i := 0; i < len(powers); i++ {
+	return c.Set(q)
-			s0 := powers[i].Uint64()
+}
-			index := uint(s0 & mask)
+
-			if index != 0 {
+func (g *G2) glvMulFr(r, p *PointG2, e *Fr) *PointG2 {
-				g.Add(bucket[index-1], bucket[index-1], points[i])
+	return g.glvMul(r, p, new(glvVectorFr).new(e))
+}
+
+func (g *G2) glvMulBig(r, p *PointG2, e *big.Int) *PointG2 {
+	return g.glvMul(r, p, new(glvVectorBig).new(e))
+}
+
+func (g *G2) glvMul(r, p0 *PointG2, v glvVector) *PointG2 {
+
+	w := glvMulWindowG2
+	l := 1 << (w - 1)
+
+	// prepare tables
+	// tableK1 = {P, 3P, 5P, ...}
+	// tableK2 = {λP, 3λP, 5λP, ...}
+	tableK1, tableK2 := make([]*PointG2, l), make([]*PointG2, l)
+	double := g.New()
+	g.Double(double, p0)
+	g.affine(double, double)
+	tableK1[0] = new(PointG2)
+	tableK1[0].Set(p0)
+	for i := 1; i < l; i++ {
+		tableK1[i] = new(PointG2)
+		g.AddMixed(tableK1[i], tableK1[i-1], double)
 	}
-			powers[i] = new(big.Int).Rsh(powers[i], uint(c))
+	g.AffineBatch(tableK1)
+	for i := 0; i < l; i++ {
+		tableK2[i] = new(PointG2)
+		g.glvEndomorphism(tableK2[i], tableK1[i])
 	}
-		sum.Zero()
+
-		for i := len(bucket) - 1; i >= 0; i-- {
+	// recode small scalars
-			g.Add(sum, sum, bucket[i])
+	naf1, naf2 := v.wnaf(w)
-			g.Add(acc, acc, sum)
+	lenNAF1, lenNAF2 := len(naf1), len(naf2)
+	lenNAF := lenNAF1
+	if lenNAF2 > lenNAF {
+		lenNAF = lenNAF2
 	}
-		windows[j] = g.New()
+
-		windows[j].Set(acc)
+	acc, p1 := g.New(), g.New()
-		j++
+
-		cur += c
+	// function for naf addition
+	add := func(table []*PointG2, naf int) {
+		if naf != 0 {
+			nafAbs := naf
+			if nafAbs < 0 {
+				nafAbs = -nafAbs
 			}
-	acc.Zero()
+			p1.Set(table[nafAbs>>1])
-	for i := len(windows) - 1; i >= 0; i-- {
+			if naf < 0 {
-		for j := uint32(0); j < c; j++ {
+				g.Neg(p1, p1)
+			}
+			g.AddMixed(acc, acc, p1)
+		}
+	}
+
+	// sliding
+	for i := lenNAF - 1; i >= 0; i-- {
+		if i < lenNAF1 {
+			add(tableK1, naf1[i])
+		}
+		if i < lenNAF2 {
+			add(tableK2, naf2[i])
+		}
+		if i != 0 {
 			g.Double(acc, acc)
 		}
-		g.Add(acc, acc, windows[i])
+	}
+	return r.Set(acc)
+}
+
+// MultiExpBig calculates multi exponentiation. Scalar values are received as big.Int type.
+// Given pairs of G2 point and scalar values `(P_0, e_0), (P_1, e_1), ... (P_n, e_n)`,
+// calculates `r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n`.
+// Length of points and scalars are expected to be equal, otherwise an error is returned.
+// Result is assigned to point at first argument.
+func (g *G2) MultiExpBig(r *PointG2, points []*PointG2, scalars []*big.Int) (*PointG2, error) {
+	if len(points) != len(scalars) {
+		return nil, errors.New("point and scalar vectors should be in same length")
+	}
+
+	c := 3
+	if len(scalars) >= 32 {
+		c = int(math.Ceil(math.Log(float64(len(scalars)))))
+	}
+
+	bucketSize := (1 << c) - 1
+	windows := make([]PointG2, 255/c+1)
+	bucket := make([]PointG2, bucketSize)
+
+	for j := 0; j < len(windows); j++ {
+
+		for i := 0; i < bucketSize; i++ {
+			bucket[i].Zero()
+		}
+
+		for i := 0; i < len(scalars); i++ {
+			index := bucketSize & int(new(big.Int).Rsh(scalars[i], uint(c*j)).Int64())
+			if index != 0 {
+				g.Add(&bucket[index-1], &bucket[index-1], points[i])
+			}
+		}
+
+		acc, sum := g.New(), g.New()
+		for i := bucketSize - 1; i >= 0; i-- {
+			g.Add(sum, sum, &bucket[i])
+			g.Add(acc, acc, sum)
+		}
+		windows[j].Set(acc)
+	}
+
+	acc := g.New()
+	for i := len(windows) - 1; i >= 0; i-- {
+		for j := 0; j < c; j++ {
+			g.Double(acc, acc)
+		}
+		g.Add(acc, acc, &windows[i])
 	}
 	return r.Set(acc), nil
 }
 
+// MultiExp calculates multi exponentiation. Given pairs of G2 point and scalar values `(P_0, e_0), (P_1, e_1), ... (P_n, e_n)`,
+// calculates `r = e_0 * P_0 + e_1 * P_1 + ... + e_n * P_n`. Length of points and scalars are expected to be equal,
+// otherwise an error is returned. Result is assigned to point at first argument.
+func (g *G2) MultiExp(r *PointG2, points []*PointG2, scalars []*Fr) (*PointG2, error) {
+	if len(points) != len(scalars) {
+		return nil, errors.New("point and scalar vectors should be in same length")
+	}
+
+	g.AffineBatch(points)
+
+	c := 3
+	if len(scalars) >= 32 {
+		c = int(math.Ceil(math.Log(float64(len(scalars)))))
+	}
+
+	bucketSize := (1 << c) - 1
+	windows := make([]*PointG2, 255/c+1)
+	bucket := make([]PointG2, bucketSize)
+
+	for j := 0; j < len(windows); j++ {
+
+		for i := 0; i < bucketSize; i++ {
+			bucket[i].Zero()
+		}
+
+		for i := 0; i < len(scalars); i++ {
+			index := bucketSize & int(scalars[i].sliceUint64(c*j))
+			if index != 0 {
+				g.AddMixed(&bucket[index-1], &bucket[index-1], points[i])
+			}
+		}
+
+		acc, sum := g.New(), g.New()
+		for i := bucketSize - 1; i >= 0; i-- {
+			g.Add(sum, sum, &bucket[i])
+			g.Add(acc, acc, sum)
+		}
+		windows[j] = g.New().Set(acc)
+	}
+
+	g.AffineBatch(windows)
+
+	acc := g.New()
+	for i := len(windows) - 1; i >= 0; i-- {
+		for j := 0; j < c; j++ {
+			g.Double(acc, acc)
+		}
+		g.AddMixed(acc, acc, windows[i])
+	}
+	return r.Set(acc), nil
+}
+
+// InCorrectSubgroup checks whether given point is in correct subgroup.
+func (g *G2) InCorrectSubgroup(p *PointG2) bool {
+
+	// Faster Subgroup Checks for BLS12-381
+	// S. Bowe
+	// https://eprint.iacr.org/2019/814.pdf
+
+	// [z]ψ^3(P) − ψ^2(P) + P = O
+	t0, t1 := g.New().Set(p), g.New()
+
+	g.psi(t0)
+	g.psi(t0)
+	g.Neg(t1, t0) // - ψ^2(P)
+	g.psi(t0)     // ψ^3(P)
+	g.mulX(t0)    // - x ψ^3(P)
+	g.Neg(t0, t0)
+
+	g.Add(t0, t0, t1)
+	g.Add(t0, t0, p)
+
+	return g.IsZero(t0)
+}
+
+// ClearCofactor maps given a G2 point to correct subgroup
+func (g *G2) ClearCofactor(p *PointG2) *PointG2 {
+
+	// Efficient hash maps to G2 on BLS curves
+	// A. Budroni, F. Pintore
+	// https://eprint.iacr.org/2017/419.pdf
+
+	// [h(ψ)]P = [x^2 − x − 1]P + [x − 1]ψ(P) + ψ^2(2P)
+	t0, t1, t2, t3 := g.New().Set(p), g.New().Set(p), g.New().Set(p), g.New()
+
+	g.Double(t0, t0)
+	g.psi(t0)
+	g.psi(t0)  // P2 = ψ^2(2P)
+	g.psi(t2)  // P1 = ψ(P)
+	g.mulX(t1) // -xP0
+
+	g.Sub(t3, t1, t2) // -xP0 - P1
+	g.mulX(t3)        // (x^2)P0 + xP1
+	g.Sub(t1, t1, p)  // (-x-1)P0
+	g.Add(t3, t3, t1) // (x^2-x-1)P0 + xP1
+	g.Sub(t3, t3, t2) // (x^2-x-1)P0 + (x-1)P1
+	g.Add(t3, t3, t0) // (x^2-x-1)P0 + (x-1)P1 + P2
+	return p.Set(t3)
+}
+
+func (g *G2) psi(p *PointG2) {
+	fp2Conjugate(&p[0], &p[0])
+	fp2Conjugate(&p[1], &p[1])
+	fp2Conjugate(&p[2], &p[2])
+	g.f.mul(&p[0], &p[0], &psix)
+	g.f.mul(&p[1], &p[1], &psiy)
+}
+
+func (g *G2) mulX(p *PointG2) {
+
+	chain := func(p0 *PointG2, n int, p1 *PointG2) {
+		g.Add(p0, p0, p1)
+		for i := 0; i < n; i++ {
+			g.Double(p0, p0)
+		}
+	}
+
+	t := g.New().Set(p)
+	g.Double(p, t)
+	chain(p, 2, t)
+	chain(p, 3, t)
+	chain(p, 9, t)
+	chain(p, 32, t)
+	chain(p, 16, t)
+}
+
 // MapToCurve given a byte slice returns a valid G2 point.
 // This mapping function implements the Simplified Shallue-van de Woestijne-Ulas method.
 // https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-05#section-6.6.2
@@ -453,3 +843,45 @@ func (g *G2) MapToCurve(in []byte) (*PointG2, error) {
 	g.ClearCofactor(q)
 	return g.Affine(q), nil
 }
+
+// EncodeToCurve given a message and domain seperator tag returns the hash result
+// which is a valid curve point.
+// Implementation follows BLS12381G1_XMD:SHA-256_SSWU_NU_ suite at
+// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06
+func (g *G2) EncodeToCurve(msg, domain []byte) (*PointG2, error) {
+	hashRes, err := hashToFpXMDSHA256(msg, domain, 2)
+	if err != nil {
+		return nil, err
+	}
+	fp2 := g.f
+	u := &fe2{*hashRes[0], *hashRes[1]}
+	x, y := swuMapG2(fp2, u)
+	isogenyMapG2(fp2, x, y)
+	z := new(fe2).one()
+	q := &PointG2{*x, *y, *z}
+	g.ClearCofactor(q)
+	return g.Affine(q), nil
+}
+
+// HashToCurve given a message and domain seperator tag returns the hash result
+// which is a valid curve point.
+// Implementation follows BLS12381G1_XMD:SHA-256_SSWU_RO_ suite at
+// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06
+func (g *G2) HashToCurve(msg, domain []byte) (*PointG2, error) {
+	hashRes, err := hashToFpXMDSHA256(msg, domain, 4)
+	if err != nil {
+		return nil, err
+	}
+	fp2 := g.f
+	u0, u1 := &fe2{*hashRes[0], *hashRes[1]}, &fe2{*hashRes[2], *hashRes[3]}
+	x0, y0 := swuMapG2(fp2, u0)
+	x1, y1 := swuMapG2(fp2, u1)
+	z0 := new(fe2).one()
+	z1 := new(fe2).one()
+	p0, p1 := &PointG2{*x0, *y0, *z0}, &PointG2{*x1, *y1, *z1}
+	g.Add(p0, p0, p1)
+	g.Affine(p0)
+	isogenyMapG2(fp2, &p0[0], &p0[1])
+	g.ClearCofactor(p0)
+	return g.Affine(p0), nil
+}

crypto/bls12381/g2_test.go

@@ -3,54 +3,117 @@ package bls12381
 import (
 	"bytes"
 	"crypto/rand"
+	"fmt"
+	"io/ioutil"
 	"math/big"
 	"testing"
-
-	"github.com/ethereum/go-ethereum/common"
 )
 
 func (g *G2) one() *PointG2 {
-	one, _ := g.fromBytesUnchecked(
+	return g.New().Set(&g2One)
-		common.FromHex("" +
-			"13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e" +
-			"024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8" +
-			"0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be" +
-			"0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801",
-		),
-	)
-	return one
 }
 
 func (g *G2) rand() *PointG2 {
-	k, err := rand.Int(rand.Reader, q)
+	p := &PointG2{}
-	if err != nil {
+	z, _ := new(fe2).rand(rand.Reader)
-		panic(err)
+	z6, bz6 := new(fe2), new(fe2)
+	g.f.square(z6, z)
+	g.f.square(z6, z6)
+	g.f.mul(z6, z6, z)
+	g.f.mul(z6, z6, z)
+	g.f.mul(bz6, z6, b2)
+	for {
+		x, _ := new(fe2).rand(rand.Reader)
+		y := new(fe2)
+		g.f.square(y, x)
+		g.f.mul(y, y, x)
+		fp2Add(y, y, bz6)
+		if g.f.sqrt(y, y) {
+			p.Set(&PointG2{*x, *y, *z})
+			break
 		}
-	return g.MulScalar(&PointG2{}, g.one(), k)
+	}
+	if !g.IsOnCurve(p) {
+		panic("rand point must be on curve")
+	}
+	if g.InCorrectSubgroup(p) {
+		panic("rand point must be out of correct subgroup")
+	}
+	return p
+}
+
+func (g *G2) randCorrect() *PointG2 {
+	p := g.ClearCofactor(g.rand())
+	if !g.InCorrectSubgroup(p) {
+		panic("must be in correct subgroup")
+	}
+	return p
+}
+
+func (g *G2) randAffine() *PointG2 {
+	return g.Affine(g.randCorrect())
+}
+
+func (g *G2) new() *PointG2 {
+	return g.Zero()
 }
 
 func TestG2Serialization(t *testing.T) {
+	var err error
 	g2 := NewG2()
+	zero := g2.Zero()
+	b0 := g2.ToUncompressed(zero)
+	p0, err := g2.FromUncompressed(b0)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !g2.IsZero(p0) {
+		t.Fatal("infinity serialization failed")
+	}
+	b0 = g2.ToCompressed(zero)
+	p0, err = g2.FromCompressed(b0)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !g2.IsZero(p0) {
+		t.Fatal("infinity serialization failed")
+	}
+	b0 = g2.ToBytes(zero)
+	p0, err = g2.FromBytes(b0)
+	if err != nil {
+		t.Fatal(err)
+	}
+	if !g2.IsZero(p0) {
+		t.Fatal("infinity serialization failed")
+	}
 	for i := 0; i < fuz; i++ {
-		a := g2.rand()
+		a := g2.randAffine()
-		buf := g2.ToBytes(a)
+		uncompressed := g2.ToUncompressed(a)
-		b, err := g2.FromBytes(buf)
+		b, err := g2.FromUncompressed(uncompressed)
 		if err != nil {
 			t.Fatal(err)
 		}
 		if !g2.Equal(a, b) {
-			t.Fatal("bad serialization from/to")
+			t.Fatal("serialization failed")
 		}
-	}
+		compressed := g2.ToCompressed(b)
-	for i := 0; i < fuz; i++ {
+		a, err = g2.FromCompressed(compressed)
-		a := g2.rand()
-		encoded := g2.EncodePoint(a)
-		b, err := g2.DecodePoint(encoded)
 		if err != nil {
 			t.Fatal(err)
 		}
 		if !g2.Equal(a, b) {
-			t.Fatal("bad serialization encode/decode")
+			t.Fatal("serialization failed")
+		}
+	}
+	for i := 0; i < fuz; i++ {
+		a := g2.rand()
+		uncompressed := g2.ToBytes(a)
+		b, err := g2.FromBytes(uncompressed)
+		if err != nil {
+			t.Fatal(err)
+		}
+		if !g2.Equal(a, b) {
+			t.Fatal("serialization failed")
 		}
 	}
 }
@@ -68,6 +131,26 @@ func TestG2IsOnCurve(t *testing.T) {
 	}
 }
 
+func TestG2BatchAffine(t *testing.T) {
+	n := 20
+	g := NewG2()
+	points0 := make([]*PointG2, n)
+	points1 := make([]*PointG2, n)
+	for i := 0; i < n; i++ {
+		points0[i] = g.rand()
+		points1[i] = g.New().Set(points0[i])
+		if g.IsAffine(points0[i]) {
+			t.Fatal("expect non affine point")
+		}
+	}
+	g.AffineBatch(points0)
+	for i := 0; i < n; i++ {
+		if !g.Equal(points0[i], points1[i]) {
+			t.Fatal("batch affine failed")
+		}
+	}
+}
+
 func TestG2AdditiveProperties(t *testing.T) {
 	g := NewG2()
 	t0, t1 := g.New(), g.New()
@@ -138,14 +221,71 @@ func TestG2AdditiveProperties(t *testing.T) {
 	}
 }
 
+func TestG2MixedAdd(t *testing.T) {
+	g := NewG2()
+	for i := 0; i < fuz; i++ {
+		a, b := g.rand(), g.rand()
+		if g.IsAffine(a) || g.IsAffine(b) {
+			t.Fatal("expect non affine points")
+		}
+		bAffine := g.New().Set(b)
+		g.Affine(bAffine)
+		r0, r1 := g.New(), g.New()
+		g.Add(r0, a, b)
+		g.AddMixed(r1, a, bAffine)
+		if !g.Equal(r0, r1) {
+			t.Fatal("mixed addition failed")
+		}
+		aAffine := g.New().Set(a)
+		g.Affine(aAffine)
+		g.AddMixed(r0, a, aAffine)
+		g.Double(r1, a)
+		if !g.Equal(r0, r1) {
+			t.Fatal("mixed addition must double where points are equal")
+		}
+	}
+}
+
+func TestG2MultiplicationCross(t *testing.T) {
+	g := NewG2()
+	for i := 0; i < fuz; i++ {
+
+		a := g.randCorrect()
+		s, _ := new(Fr).Rand(rand.Reader)
+		sBig := s.ToBig()
+		res0, res1, res2, res3, res4 := g.New(), g.New(), g.New(), g.New(), g.New()
+
+		g.mulScalar(res0, a, s)
+		g.glvMulFr(res1, a, s)
+		g.glvMulBig(res2, a, sBig)
+		g.wnafMulFr(res3, a, s)
+		g.wnafMulBig(res4, a, sBig)
+
+		if !g.Equal(res0, res1) {
+			t.Fatal("cross multiplication failed (glv, fr)", i)
+		}
+		if !g.Equal(res0, res2) {
+			t.Fatal("cross multiplication failed (glv, big)", i)
+		}
+		if !g.Equal(res0, res3) {
+			t.Fatal("cross multiplication failed (wnaf, fr)", i)
+		}
+		if !g.Equal(res0, res4) {
+			t.Fatal("cross multiplication failed (wnaf, big)", i)
+		}
+	}
+}
+
 func TestG2MultiplicativeProperties(t *testing.T) {
 	g := NewG2()
 	t0, t1 := g.New(), g.New()
 	zero := g.Zero()
 	for i := 0; i < fuz; i++ {
-		a := g.rand()
+		a := g.randCorrect()
-		s1, s2, s3 := randScalar(q), randScalar(q), randScalar(q)
+		s1, _ := new(Fr).Rand(rand.Reader)
-		sone := big.NewInt(1)
+		s2, _ := new(Fr).Rand(rand.Reader)
+		s3, _ := new(Fr).Rand(rand.Reader)
+		sone := &Fr{1}
 		g.MulScalar(t0, zero, s1)
 		if !g.Equal(t0, zero) {
 			t.Fatal(" 0 ^ s == 0")
@@ -163,7 +303,7 @@ func TestG2MultiplicativeProperties(t *testing.T) {
 		s3.Mul(s1, s2)
 		g.MulScalar(t1, a, s3)
 		if !g.Equal(t0, t1) {
-			t.Errorf(" (a ^ s1) ^ s2 == a ^ (s1 * s2)")
+			t.Fatal(" (a ^ s1) ^ s2 == a ^ (s1 * s2)")
 		}
 		g.MulScalar(t0, a, s1)
 		g.MulScalar(t1, a, s2)
@@ -171,12 +311,71 @@ func TestG2MultiplicativeProperties(t *testing.T) {
 		s3.Add(s1, s2)
 		g.MulScalar(t1, a, s3)
 		if !g.Equal(t0, t1) {
-			t.Errorf(" (a ^ s1) + (a ^ s2) == a ^ (s1 + s2)")
+			t.Fatal(" (a ^ s1) + (a ^ s2) == a ^ (s1 + s2)")
 		}
 	}
 }
 
+func TestZKCryptoVectorsG2UncompressedValid(t *testing.T) {
+	data, err := ioutil.ReadFile("tests/g2_uncompressed_valid_test_vectors.dat")
+	if err != nil {
+		panic(err)
+	}
+	g := NewG2()
+	p1 := g.Zero()
+	for i := 0; i < 1000; i++ {
+		vector := data[i*192 : (i+1)*192]
+		p2, err := g.FromUncompressed(vector)
+		if err != nil {
+			t.Fatal("decoing fails", err, i)
+		}
+		uncompressed := g.ToUncompressed(p2)
+		if !bytes.Equal(vector, uncompressed) || !g.Equal(p1, p2) {
+			t.Fatal("serialization failed")
+		}
+		g.Add(p1, p1, &g2One)
+	}
+}
+
+func TestZKCryptoVectorsG2CompressedValid(t *testing.T) {
+	data, err := ioutil.ReadFile("tests/g2_compressed_valid_test_vectors.dat")
+	if err != nil {
+		panic(err)
+	}
+	g := NewG2()
+	p1 := g.Zero()
+	for i := 0; i < 1000; i++ {
+		vector := data[i*2*fpByteSize : (i+1)*2*fpByteSize]
+		p2, err := g.FromCompressed(vector)
+		if err != nil {
+			t.Fatal("decoing fails", err, i)
+		}
+		compressed := g.ToCompressed(p2)
+		if !bytes.Equal(vector, compressed) || !g.Equal(p1, p2) {
+			t.Fatal("serialization failed")
+		}
+
+		g.Add(p1, p1, &g2One)
+	}
+}
+
 func TestG2MultiExpExpected(t *testing.T) {
+	g := NewG2()
+	one := g.one()
+	var scalars [2]*Fr
+	var bases [2]*PointG2
+	scalars[0] = &Fr{2}
+	scalars[1] = &Fr{3}
+	bases[0], bases[1] = new(PointG2).Set(one), new(PointG2).Set(one)
+	expected, result := g.New(), g.New()
+	g.mulScalar(expected, one, &Fr{5})
+	_, _ = g.MultiExp(result, bases[:], scalars[:])
+	if !g.Equal(expected, result) {
+		t.Fatal("multi-exponentiation failed")
+	}
+}
+
+func TestG2MultiExpBigExpected(t *testing.T) {
 	g := NewG2()
 	one := g.one()
 	var scalars [2]*big.Int
@@ -185,36 +384,76 @@ func TestG2MultiExpExpected(t *testing.T) {
 	scalars[1] = big.NewInt(3)
 	bases[0], bases[1] = new(PointG2).Set(one), new(PointG2).Set(one)
 	expected, result := g.New(), g.New()
-	g.MulScalar(expected, one, big.NewInt(5))
+	g.mulScalarBig(expected, one, big.NewInt(5))
-	_, _ = g.MultiExp(result, bases[:], scalars[:])
+	_, _ = g.MultiExpBig(result, bases[:], scalars[:])
 	if !g.Equal(expected, result) {
-		t.Fatal("bad multi-exponentiation")
+		t.Fatal("multi-exponentiation failed")
 	}
 }
 
-func TestG2MultiExpBatch(t *testing.T) {
+func TestG2MultiExp(t *testing.T) {
 	g := NewG2()
-	one := g.one()
+	for n := 1; n < 1024+1; n = n * 2 {
-	n := 1000
 		bases := make([]*PointG2, n)
-	scalars := make([]*big.Int, n)
+		scalars := make([]*Fr, n)
-	// scalars: [s0,s1 ... s(n-1)]
+		var err error
-	// bases: [P0,P1,..P(n-1)] = [s(n-1)*G, s(n-2)*G ... s0*G]
+		for i := 0; i < n; i++ {
-	for i, j := 0, n-1; i < n; i, j = i+1, j-1 {
+			scalars[i], err = new(Fr).Rand(rand.Reader)
-		scalars[j], _ = rand.Int(rand.Reader, big.NewInt(100000))
+			if err != nil {
-		bases[i] = g.New()
+				t.Fatal(err)
-		g.MulScalar(bases[i], one, scalars[j])
+			}
+			bases[i] = g.rand()
 		}
-	// expected: s(n-1)*P0 + s(n-2)*P1 + s0*P(n-1)
 		expected, tmp := g.New(), g.New()
 		for i := 0; i < n; i++ {
-		g.MulScalar(tmp, bases[i], scalars[i])
+			g.mulScalar(tmp, bases[i], scalars[i])
 			g.Add(expected, expected, tmp)
 		}
 		result := g.New()
 		_, _ = g.MultiExp(result, bases, scalars)
 		if !g.Equal(expected, result) {
-		t.Fatal("bad multi-exponentiation")
+			t.Fatal("multi-exponentiation failed")
+		}
+	}
+}
+
+func TestG2MultiExpBig(t *testing.T) {
+	g := NewG2()
+	for n := 1; n < 1024+1; n = n * 2 {
+		bases := make([]*PointG2, n)
+		scalars := make([]*big.Int, n)
+		var err error
+		for i := 0; i < n; i++ {
+			scalars[i], err = rand.Int(rand.Reader, qBig)
+			if err != nil {
+				t.Fatal(err)
+			}
+			bases[i] = g.rand()
+		}
+		expected, tmp := g.New(), g.New()
+		for i := 0; i < n; i++ {
+			g.mulScalarBig(tmp, bases[i], scalars[i])
+			g.Add(expected, expected, tmp)
+		}
+		result := g.New()
+		_, _ = g.MultiExpBig(result, bases, scalars)
+		if !g.Equal(expected, result) {
+			t.Fatal("multi-exponentiation failed")
+		}
+	}
+}
+
+func TestG2ClearCofactor(t *testing.T) {
+	g := NewG2()
+	for i := 0; i < fuz; i++ {
+		p0 := g.rand()
+		if g.InCorrectSubgroup(p0) {
+			t.Fatal("rand point should be out of correct subgroup")
+		}
+		g.ClearCofactor(p0)
+		if !g.InCorrectSubgroup(p0) {
+			t.Fatal("cofactor clearing is failed")
+		}
 	}
 }
 
@@ -224,24 +463,60 @@ func TestG2MapToCurve(t *testing.T) {
 		expected []byte
 	}{
 		{
-			u:        make([]byte, 96),
+			u: make([]byte, 2*fpByteSize),
-			expected: common.FromHex("0a67d12118b5a35bb02d2e86b3ebfa7e23410db93de39fb06d7025fa95e96ffa428a7a27c3ae4dd4b40bd251ac658892" + "018320896ec9eef9d5e619848dc29ce266f413d02dd31d9b9d44ec0c79cd61f18b075ddba6d7bd20b7ff27a4b324bfce" + "04c69777a43f0bda07679d5805e63f18cf4e0e7c6112ac7f70266d199b4f76ae27c6269a3ceebdae30806e9a76aadf5c" + "0260e03644d1a2c321256b3246bad2b895cad13890cbe6f85df55106a0d334604fb143c7a042d878006271865bc35941"),
+			expected: fromHex(-1, "0a67d12118b5a35bb02d2e86b3ebfa7e23410db93de39fb06d7025fa95e96ffa428a7a27c3ae4dd4b40bd251ac658892",
+				"018320896ec9eef9d5e619848dc29ce266f413d02dd31d9b9d44ec0c79cd61f18b075ddba6d7bd20b7ff27a4b324bfce",
+				"04c69777a43f0bda07679d5805e63f18cf4e0e7c6112ac7f70266d199b4f76ae27c6269a3ceebdae30806e9a76aadf5c",
+				"0260e03644d1a2c321256b3246bad2b895cad13890cbe6f85df55106a0d334604fb143c7a042d878006271865bc35941",
+			),
 		},
 		{
-			u:        common.FromHex("025fbc07711ba267b7e70c82caa70a16fbb1d470ae24ceef307f5e2000751677820b7013ad4e25492dcf30052d3e5eca" + "0e775d7827adf385b83e20e4445bd3fab21d7b4498426daf3c1d608b9d41e9edb5eda0df022e753b8bb4bc3bb7db4914"),
+			u: fromHex(-1,
-			expected: common.FromHex("0d4333b77becbf9f9dfa3ca928002233d1ecc854b1447e5a71f751c9042d000f42db91c1d6649a5e0ad22bd7bf7398b8" + "027e4bfada0b47f9f07e04aec463c7371e68f2fd0c738cd517932ea3801a35acf09db018deda57387b0f270f7a219e4d" + "0cc76dc777ea0d447e02a41004f37a0a7b1fafb6746884e8d9fc276716ccf47e4e0899548a2ec71c2bdf1a2a50e876db" + "053674cba9ef516ddc218fedb37324e6c47de27f88ab7ef123b006127d738293c0277187f7e2f80a299a24d84ed03da7"),
+				"025fbc07711ba267b7e70c82caa70a16fbb1d470ae24ceef307f5e2000751677820b7013ad4e25492dcf30052d3e5eca",
+				"0e775d7827adf385b83e20e4445bd3fab21d7b4498426daf3c1d608b9d41e9edb5eda0df022e753b8bb4bc3bb7db4914",
+			),
+			expected: fromHex(-1,
+				"0d4333b77becbf9f9dfa3ca928002233d1ecc854b1447e5a71f751c9042d000f42db91c1d6649a5e0ad22bd7bf7398b8",
+				"027e4bfada0b47f9f07e04aec463c7371e68f2fd0c738cd517932ea3801a35acf09db018deda57387b0f270f7a219e4d",
+				"0cc76dc777ea0d447e02a41004f37a0a7b1fafb6746884e8d9fc276716ccf47e4e0899548a2ec71c2bdf1a2a50e876db",
+				"053674cba9ef516ddc218fedb37324e6c47de27f88ab7ef123b006127d738293c0277187f7e2f80a299a24d84ed03da7",
+			),
 		},
 		{
-			u:        common.FromHex("1870a7dbfd2a1deb74015a3546b20f598041bf5d5202997956a94a368d30d3f70f18cdaa1d33ce970a4e16af961cbdcb" + "045ab31ce4b5a8ba7c4b2851b64f063a66cd1223d3c85005b78e1beee65e33c90ceef0244e45fc45a5e1d6eab6644fdb"),
+			u: fromHex(-1,
-			expected: common.FromHex("18f0f87b40af67c056915dbaf48534c592524e82c1c2b50c3734d02c0172c80df780a60b5683759298a3303c5d942778" + "09349f1cb5b2e55489dcd45a38545343451cc30a1681c57acd4fb0a6db125f8352c09f4a67eb7d1d8242cb7d3405f97b" + "10a2ba341bc689ab947b7941ce6ef39be17acaab067bd32bd652b471ab0792c53a2bd03bdac47f96aaafe96e441f63c0" + "02f2d9deb2c7742512f5b8230bf0fd83ea42279d7d39779543c1a43b61c885982b611f6a7a24b514995e8a098496b811"),
+				"1870a7dbfd2a1deb74015a3546b20f598041bf5d5202997956a94a368d30d3f70f18cdaa1d33ce970a4e16af961cbdcb",
+				"045ab31ce4b5a8ba7c4b2851b64f063a66cd1223d3c85005b78e1beee65e33c90ceef0244e45fc45a5e1d6eab6644fdb",
+			),
+			expected: fromHex(-1,
+				"18f0f87b40af67c056915dbaf48534c592524e82c1c2b50c3734d02c0172c80df780a60b5683759298a3303c5d942778",
+				"09349f1cb5b2e55489dcd45a38545343451cc30a1681c57acd4fb0a6db125f8352c09f4a67eb7d1d8242cb7d3405f97b",
+				"10a2ba341bc689ab947b7941ce6ef39be17acaab067bd32bd652b471ab0792c53a2bd03bdac47f96aaafe96e441f63c0",
+				"02f2d9deb2c7742512f5b8230bf0fd83ea42279d7d39779543c1a43b61c885982b611f6a7a24b514995e8a098496b811",
+			),
 		},
 		{
-			u:        common.FromHex("088fe329b054db8a6474f21a7fbfdf17b4c18044db299d9007af582c3d5f17d00e56d99921d4b5640fce44b05219b5de" + "0b6e6135a4cd31ba980ddbd115ac48abef7ec60e226f264d7befe002c165f3a496f36f76dd524efd75d17422558d10b4"),
+			u: fromHex(-1,
-			expected: common.FromHex("19808ec5930a53c7cf5912ccce1cc33f1b3dcff24a53ce1cc4cba41fd6996dbed4843ccdd2eaf6a0cd801e562718d163" + "149fe43777d34f0d25430dea463889bd9393bdfb4932946db23671727081c629ebb98a89604f3433fba1c67d356a4af7" + "04783e391c30c83f805ca271e353582fdf19d159f6a4c39b73acbb637a9b8ac820cfbe2738d683368a7c07ad020e3e33" + "04c0d6793a766233b2982087b5f4a254f261003ccb3262ea7c50903eecef3e871d1502c293f9e063d7d293f6384f4551"),
+				"088fe329b054db8a6474f21a7fbfdf17b4c18044db299d9007af582c3d5f17d00e56d99921d4b5640fce44b05219b5de",
+				"0b6e6135a4cd31ba980ddbd115ac48abef7ec60e226f264d7befe002c165f3a496f36f76dd524efd75d17422558d10b4",
+			),
+			expected: fromHex(-1,
+				"19808ec5930a53c7cf5912ccce1cc33f1b3dcff24a53ce1cc4cba41fd6996dbed4843ccdd2eaf6a0cd801e562718d163",
+				"149fe43777d34f0d25430dea463889bd9393bdfb4932946db23671727081c629ebb98a89604f3433fba1c67d356a4af7",
+				"04783e391c30c83f805ca271e353582fdf19d159f6a4c39b73acbb637a9b8ac820cfbe2738d683368a7c07ad020e3e33",
+				"04c0d6793a766233b2982087b5f4a254f261003ccb3262ea7c50903eecef3e871d1502c293f9e063d7d293f6384f4551",
+			),
 		},
 		{
-			u:        common.FromHex("03df16a66a05e4c1188c234788f43896e0565bfb64ac49b9639e6b284cc47dad73c47bb4ea7e677db8d496beb907fbb6" + "0f45b50647d67485295aa9eb2d91a877b44813677c67c8d35b2173ff3ba95f7bd0806f9ca8a1436b8b9d14ee81da4d7e"),
+			u: fromHex(-1,
-			expected: common.FromHex("0b8e0094c886487870372eb6264613a6a087c7eb9804fab789be4e47a57b29eb19b1983a51165a1b5eb025865e9fc63a" + "0804152cbf8474669ad7d1796ab92d7ca21f32d8bed70898a748ed4e4e0ec557069003732fc86866d938538a2ae95552" + "14c80f068ece15a3936bb00c3c883966f75b4e8d9ddde809c11f781ab92d23a2d1d103ad48f6f3bb158bf3e3a4063449" + "09e5c8242dd7281ad32c03fe4af3f19167770016255fb25ad9b67ec51d62fade31a1af101e8f6172ec2ee8857662be3a"),
+				"03df16a66a05e4c1188c234788f43896e0565bfb64ac49b9639e6b284cc47dad73c47bb4ea7e677db8d496beb907fbb6",
+				"0f45b50647d67485295aa9eb2d91a877b44813677c67c8d35b2173ff3ba95f7bd0806f9ca8a1436b8b9d14ee81da4d7e",
+			),
+			expected: fromHex(-1,
+				"0b8e0094c886487870372eb6264613a6a087c7eb9804fab789be4e47a57b29eb19b1983a51165a1b5eb025865e9fc63a",
+				"0804152cbf8474669ad7d1796ab92d7ca21f32d8bed70898a748ed4e4e0ec557069003732fc86866d938538a2ae95552",
+				"14c80f068ece15a3936bb00c3c883966f75b4e8d9ddde809c11f781ab92d23a2d1d103ad48f6f3bb158bf3e3a4063449",
+				"09e5c8242dd7281ad32c03fe4af3f19167770016255fb25ad9b67ec51d62fade31a1af101e8f6172ec2ee8857662be3a",
+			),
 		},
 	} {
 		g := NewG2()
@@ -255,6 +530,114 @@ func TestG2MapToCurve(t *testing.T) {
 	}
 }
 
+func TestG2EncodeToCurve(t *testing.T) {
+	domain := []byte("BLS12381G2_XMD:SHA-256_SSWU_NU_TESTGEN")
+	for i, v := range []struct {
+		msg      []byte
+		expected []byte
+	}{
+		{
+			msg: []byte(""),
+			expected: fromHex(-1,
+				"0d4333b77becbf9f9dfa3ca928002233d1ecc854b1447e5a71f751c9042d000f42db91c1d6649a5e0ad22bd7bf7398b8",
+				"027e4bfada0b47f9f07e04aec463c7371e68f2fd0c738cd517932ea3801a35acf09db018deda57387b0f270f7a219e4d",
+				"0cc76dc777ea0d447e02a41004f37a0a7b1fafb6746884e8d9fc276716ccf47e4e0899548a2ec71c2bdf1a2a50e876db",
+				"053674cba9ef516ddc218fedb37324e6c47de27f88ab7ef123b006127d738293c0277187f7e2f80a299a24d84ed03da7",
+			),
+		},
+		{
+			msg: []byte("abc"),
+			expected: fromHex(-1,
+				"18f0f87b40af67c056915dbaf48534c592524e82c1c2b50c3734d02c0172c80df780a60b5683759298a3303c5d942778",
+				"09349f1cb5b2e55489dcd45a38545343451cc30a1681c57acd4fb0a6db125f8352c09f4a67eb7d1d8242cb7d3405f97b",
+				"10a2ba341bc689ab947b7941ce6ef39be17acaab067bd32bd652b471ab0792c53a2bd03bdac47f96aaafe96e441f63c0",
+				"02f2d9deb2c7742512f5b8230bf0fd83ea42279d7d39779543c1a43b61c885982b611f6a7a24b514995e8a098496b811",
+			),
+		},
+		{
+			msg: []byte("abcdef0123456789"),
+			expected: fromHex(-1,
+				"19808ec5930a53c7cf5912ccce1cc33f1b3dcff24a53ce1cc4cba41fd6996dbed4843ccdd2eaf6a0cd801e562718d163",
+				"149fe43777d34f0d25430dea463889bd9393bdfb4932946db23671727081c629ebb98a89604f3433fba1c67d356a4af7",
+				"04783e391c30c83f805ca271e353582fdf19d159f6a4c39b73acbb637a9b8ac820cfbe2738d683368a7c07ad020e3e33",
+				"04c0d6793a766233b2982087b5f4a254f261003ccb3262ea7c50903eecef3e871d1502c293f9e063d7d293f6384f4551",
+			),
+		},
+		{
+			msg: []byte("a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"),
+			expected: fromHex(-1,
+				"0b8e0094c886487870372eb6264613a6a087c7eb9804fab789be4e47a57b29eb19b1983a51165a1b5eb025865e9fc63a",
+				"0804152cbf8474669ad7d1796ab92d7ca21f32d8bed70898a748ed4e4e0ec557069003732fc86866d938538a2ae95552",
+				"14c80f068ece15a3936bb00c3c883966f75b4e8d9ddde809c11f781ab92d23a2d1d103ad48f6f3bb158bf3e3a4063449",
+				"09e5c8242dd7281ad32c03fe4af3f19167770016255fb25ad9b67ec51d62fade31a1af101e8f6172ec2ee8857662be3a",
+			),
+		},
+	} {
+		g := NewG2()
+		p0, err := g.EncodeToCurve(v.msg, domain)
+		if err != nil {
+			t.Fatal("encode to point fails", i, err)
+		}
+		if !bytes.Equal(g.ToBytes(p0), v.expected) {
+			t.Fatal("encode to point fails x", i)
+		}
+	}
+}
+
+func TestG2HashToCurve(t *testing.T) {
+	domain := []byte("BLS12381G2_XMD:SHA-256_SSWU_RO_TESTGEN")
+	for i, v := range []struct {
+		msg      []byte
+		expected []byte
+	}{
+		{
+			msg: []byte(""),
+			expected: fromHex(-1,
+				"0fbdae26f9f9586a46d4b0b70390d09064ef2afe5c99348438a3c7d9756471e015cb534204c1b6824617a85024c772dc",
+				"0a650bd36ae7455cb3fe5d8bb1310594551456f5c6593aec9ee0c03d2f6cb693bd2c5e99d4e23cbaec767609314f51d3",
+				"02e5cf8f9b7348428cc9e66b9a9b36fe45ba0b0a146290c3a68d92895b1af0e1f2d9f889fb412670ae8478d8abd4c5aa",
+				"0d8d49e7737d8f9fc5cef7c4b8817633103faf2613016cb86a1f3fc29968fe2413e232d9208d2d74a89bf7a48ac36f83",
+			),
+		},
+		{
+			msg: []byte("abc"),
+			expected: fromHex(-1,
+				"03578447618463deb106b60e609c6f7cc446dc6035f84a72801ba17c94cd800583b493b948eff0033f09086fdd7f6175",
+				"1953ce6d4267939c7360756d9cca8eb34aac4633ef35369a7dc249445069888e7d1b3f9d2e75fbd468fbcbba7110ea02",
+				"0184d26779ae9d4670aca9b267dbd4d3b30443ad05b8546d36a195686e1ccc3a59194aea05ed5bce7c3144a29ec047c4",
+				"0882ab045b8fe4d7d557ebb59a63a35ac9f3d312581b509af0f8eaa2960cbc5e1e36bb969b6e22980b5cbdd0787fcf4e",
+			),
+		},
+		{
+			msg: []byte("abcdef0123456789"),
+			expected: fromHex(-1,
+				"195fad48982e186ce3c5c82133aefc9b26d55979b6f530992a8849d4263ec5d57f7a181553c8799bcc83da44847bdc8d",
+				"17b461fc3b96a30c2408958cbfa5f5927b6063a8ad199d5ebf2d7cdeffa9c20c85487204804fab53f950b2f87db365aa",
+				"005cdf3d984e3391e7e969276fb4bc02323c5924a4449af167030d855acc2600cf3d4fab025432c6d868c79571a95bef",
+				"174a3473a3af2d0302b9065e895ca4adba4ece6ce0b41148ba597001abb152f852dd9a96fb45c9de0a43d944746f833e",
+			),
+		},
+		{
+			msg: []byte("a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa"),
+			expected: fromHex(-1,
+				"123b6bd9feeba26dd4ad00f8bfda2718c9700dc093ea5287d7711844644eb981848316d3f3f57d5d3a652c6cdc816aca",
+				"0a162306f3b0f2bb326f0c4fb0e1fea020019c3af796dcd1d7264f50ddae94cacf3cade74603834d44b9ab3d5d0a6c98",
+				"05483f3b96d9252dd4fc0868344dfaf3c9d145e3387db23fa8e449304fab6a7b6ec9c15f05c0a1ea66ff0efcc03e001a",
+				"15c1d4f1a685bb63ee67ca1fd96155e3d091e852a684b78d085fd34f6091e5249ddddbdcf2e7ec82ce6c04c63647eeb7",
+			),
+		},
+	} {
+		g := NewG2()
+		p0, err := g.HashToCurve(v.msg, domain)
+		if err != nil {
+			t.Fatal("encode to point fails", i, err)
+		}
+		if !bytes.Equal(g.ToBytes(p0), v.expected) {
+			t.Fatal("encode to point fails x", i)
+		}
+	}
+}
+
 func BenchmarkG2Add(t *testing.B) {
 	g2 := NewG2()
 	a, b, c := g2.rand(), g2.rand(), PointG2{}
@@ -264,18 +647,112 @@ func BenchmarkG2Add(t *testing.B) {
 	}
 }
 
-func BenchmarkG2Mul(t *testing.B) {
+func BenchmarkG2MulWNAF(t *testing.B) {
-	worstCaseScalar, _ := new(big.Int).SetString("ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", 16)
+	g := NewG2()
-	g2 := NewG2()
+	p := new(PointG2).Set(&g2One)
-	a, e, c := g2.rand(), worstCaseScalar, PointG2{}
+	s, _ := new(Fr).Rand(rand.Reader)
+	sBig := s.ToBig()
+	res := new(PointG2)
+	t.Run("Naive", func(t *testing.B) {
 		t.ResetTimer()
 		for i := 0; i < t.N; i++ {
-		g2.MulScalar(&c, a, e)
+			g.mulScalar(res, p, s)
+		}
+	})
+	for i := 1; i < 8; i++ {
+		wnafMulWindowG2 = uint(i)
+		t.Run(fmt.Sprintf("Fr, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.wnafMulFr(res, p, s)
+			}
+		})
+		t.Run(fmt.Sprintf("Big, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.wnafMulBig(res, p, sBig)
+			}
+		})
+	}
+}
+
+func BenchmarkG2MulGLV(t *testing.B) {
+
+	g := NewG2()
+	p := new(PointG2).Set(&g2One)
+	s, _ := new(Fr).Rand(rand.Reader)
+	sBig := s.ToBig()
+	res := new(PointG2)
+	t.Run("Naive", func(t *testing.B) {
+		t.ResetTimer()
+		for i := 0; i < t.N; i++ {
+			g.mulScalar(res, p, s)
+		}
+	})
+	for i := 1; i < 8; i++ {
+		glvMulWindowG2 = uint(i)
+		t.Run(fmt.Sprintf("Fr, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.glvMulFr(res, p, s)
+			}
+		})
+		t.Run(fmt.Sprintf("Big, window: %d", i), func(t *testing.B) {
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				g.glvMulBig(res, p, sBig)
+			}
+		})
+	}
+}
+
+func BenchmarkG2MultiExp(t *testing.B) {
+	g := NewG2()
+	v := func(n int) ([]*PointG2, []*Fr) {
+		bases := make([]*PointG2, n)
+		scalars := make([]*Fr, n)
+		var err error
+		for i := 0; i < n; i++ {
+			scalars[i], err = new(Fr).Rand(rand.Reader)
+			if err != nil {
+				t.Fatal(err)
+			}
+			bases[i] = g.randAffine()
+		}
+		return bases, scalars
+	}
+	for _, i := range []int{2, 10, 100, 1000} {
+		t.Run(fmt.Sprint(i), func(t *testing.B) {
+			bases, scalars := v(i)
+			result := g.New()
+			t.ResetTimer()
+			for i := 0; i < t.N; i++ {
+				_, _ = g.MultiExp(result, bases, scalars)
+			}
+		})
+	}
+}
+
+func BenchmarkG2ClearCofactor(t *testing.B) {
+	g2 := NewG2()
+	a := g2.rand()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		g2.ClearCofactor(a)
+	}
+}
+
+func BenchmarkG2SubgroupCheck(t *testing.B) {
+	g2 := NewG2()
+	a := g2.rand()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		g2.InCorrectSubgroup(a)
 	}
 }
 
 func BenchmarkG2SWUMap(t *testing.B) {
-	a := make([]byte, 96)
+	a := fromHex(2*fpByteSize, "0x1234")
 	g2 := NewG2()
 	t.ResetTimer()
 	for i := 0; i < t.N; i++ {

crypto/bls12381/glv.go

@@ -0,0 +1,200 @@
+package bls12381
+
+import (
+	"math/big"
+)
+
+// Guide to Pairing Based Cryptography
+// 6.3.2. Decompositions for the k = 12 BLS Family
+
+// glvQ1 = x^2 * R / q
+var glvQ1 = &Fr{0x63f6e522f6cfee30, 0x7c6becf1e01faadd, 0x1, 0}
+var glvQ1Big = bigFromHex("0x017c6becf1e01faadd63f6e522f6cfee30")
+
+// glvQ2 = R / q = 2
+var glvQ2 = &Fr{0x02, 0, 0, 0}
+var glvQ2Big = bigFromHex("0x02")
+
+// glvB1 = x^2 - 1 = 0xac45a4010001a40200000000ffffffff
+var glvB1 = &Fr{0x00000000ffffffff, 0xac45a4010001a402, 0, 0}
+var glvB1Big = bigFromHex("0xac45a4010001a40200000000ffffffff")
+
+// glvB2 = x^2 = 0xac45a4010001a4020000000100000000
+var glvB2 = &Fr{0x0000000100000000, 0xac45a4010001a402, 0, 0}
+var glvB2Big = bigFromHex("0xac45a4010001a4020000000100000000")
+
+// glvLambdaA = x^2 - 1
+var glvLambda = &Fr{0x00000000ffffffff, 0xac45a4010001a402, 0, 0}
+var glvLambdaBig = bigFromHex("0xac45a4010001a40200000000ffffffff")
+
+// halfR = 2**256 / 2
+var halfR = &wideFr{0, 0, 0, 0x8000000000000000, 0, 0, 0}
+var halfRBig = bigFromHex("0x8000000000000000000000000000000000000000000000000000000000000000")
+
+// r128 = 2**128 - 1
+var r128 = &Fr{0xffffffffffffffff, 0xffffffffffffffff, 0, 0}
+
+// glvPhi1 ^ 3 = 1
+var glvPhi1 = &fe{0xcd03c9e48671f071, 0x5dab22461fcda5d2, 0x587042afd3851b95, 0x8eb60ebe01bacb9e, 0x03f97d6e83d050d2, 0x18f0206554638741}
+
+// glvPhi2 ^ 3 = 1
+var glvPhi2 = &fe{0x30f1361b798a64e8, 0xf3b8ddab7ece5a2a, 0x16a8ca3ac61577f7, 0xc26a2ff874fd029b, 0x3636b76660701c6e, 0x051ba4ab241b6160}
+
+var glvMulWindowG1 uint = 4
+var glvMulWindowG2 uint = 4
+
+type glvVector interface {
+	wnaf(w uint) (nafNumber, nafNumber)
+}
+
+type glvVectorFr struct {
+	k1   *Fr
+	k2   *Fr
+	neg1 bool
+	neg2 bool
+}
+
+type glvVectorBig struct {
+	k1 *big.Int
+	k2 *big.Int
+}
+
+func (v *glvVectorFr) wnaf(w uint) (nafNumber, nafNumber) {
+	naf1 := v.k1.toWNAF(w)
+	naf2 := v.k2.toWNAF(w)
+	if v.neg1 {
+		naf1.neg()
+	}
+	if !v.neg2 {
+		naf2.neg()
+	}
+	return naf1, naf2
+}
+
+func (v *glvVectorBig) wnaf(w uint) (nafNumber, nafNumber) {
+	naf1, naf2 := bigToWNAF(v.k1, w), bigToWNAF(v.k2, w)
+	zero := new(big.Int)
+	if v.k1.Cmp(zero) < 0 {
+		naf1.neg()
+	}
+	if v.k2.Cmp(zero) > 0 {
+		naf2.neg()
+	}
+	return naf1, naf2
+}
+
+func (v *glvVectorFr) new(m *Fr) *glvVectorFr {
+	// Guide to Pairing Based Cryptography
+	// 6.3.2. Decompositions for the k = 12 BLS Family
+
+	// alpha1 = round(x^2 * m  / r)
+	alpha1 := alpha1(m)
+	// alpha2 = round(m / r)
+	alpha2 := alpha2(m)
+
+	z1, z2 := new(Fr), new(Fr)
+
+	// z1 = (x^2 - 1) * round(x^2 * m  / r)
+	z1.Mul(alpha1, glvB1)
+	// z2 = x^2 * round(m / r)
+	z2.Mul(alpha2, glvB2)
+
+	k1, k2 := new(Fr), new(Fr)
+	// k1 = m - z1 - alpha2
+	k1.Sub(m, z1)
+	k1.Sub(k1, alpha2)
+
+	// k2 = z2 - alpha1
+	k2.Sub(z2, alpha1)
+
+	if k1.Cmp(r128) == 1 {
+		k1.Neg(k1)
+		v.neg1 = true
+	}
+	v.k1 = new(Fr).Set(k1)
+	if k2.Cmp(r128) == 1 {
+		k2.Neg(k2)
+		v.neg2 = true
+	}
+	v.k2 = new(Fr).Set(k2)
+	return v
+}
+
+func (v *glvVectorBig) new(m *big.Int) *glvVectorBig {
+	// Guide to Pairing Based Cryptography
+	// 6.3.2. Decompositions for the k = 12 BLS Family
+
+	// alpha1 = round(x^2 * m  / r)
+	alpha1 := new(big.Int).Mul(m, glvQ1Big)
+	alpha1.Add(alpha1, halfRBig)
+	alpha1.Rsh(alpha1, fourWordBitSize)
+
+	// alpha2 = round(m / r)
+	alpha2 := new(big.Int).Mul(m, glvQ2Big)
+	alpha2.Add(alpha2, halfRBig)
+	alpha2.Rsh(alpha2, fourWordBitSize)
+
+	z1, z2 := new(big.Int), new(big.Int)
+	// z1 = (x^2 - 1) * round(x^2 * m  / r)
+	z1.Mul(alpha1, glvB1Big).Mod(z1, qBig)
+	// z2 = x^2 * round(m / r)
+	z2.Mul(alpha2, glvB2Big).Mod(z2, qBig)
+
+	k1, k2 := new(big.Int), new(big.Int)
+
+	// k1 = m - z1 - alpha2
+	k1.Sub(m, z1)
+	k1.Sub(k1, alpha2)
+
+	// k2 = z2 - alpha1
+	k2.Sub(z2, alpha1)
+
+	v.k1 = new(big.Int).Set(k1)
+	v.k2 = new(big.Int).Set(k2)
+	return v
+}
+
+// round(x^2 * m / q)
+func alpha1(m *Fr) *Fr {
+	a := new(wideFr)
+	a.mul(m, glvQ1)
+	return a.round()
+}
+
+// round(m / q)
+func alpha2(m *Fr) *Fr {
+	a := new(wideFr)
+	a.mul(m, glvQ2)
+	return a.round()
+}
+
+func phi(a, b *fe) {
+	mul(a, b, glvPhi1)
+}
+
+func (e *fp2) phi(a, b *fe2) {
+	mul(&a[0], &b[0], glvPhi2)
+	mul(&a[1], &b[1], glvPhi2)
+}
+
+func (g *G1) glvEndomorphism(r, p *PointG1) {
+	t := g.Affine(p)
+	if g.IsZero(p) {
+		r.Zero()
+		return
+	}
+	r[1].set(&t[1])
+	phi(&r[0], &t[0])
+	r[2].one()
+}
+
+func (g *G2) glvEndomorphism(r, p *PointG2) {
+	t := g.Affine(p)
+	if g.IsZero(p) {
+		r.Zero()
+		return
+	}
+	r[1].set(&t[1])
+	g.f.phi(&r[0], &t[0])
+	r[2].one()
+}

crypto/bls12381/glv_test.go

@@ -0,0 +1,134 @@
+package bls12381
+
+import (
+	"crypto/rand"
+	"math/big"
+	"testing"
+)
+
+func TestGLVConstruction(t *testing.T) {
+	t.Run("Parameters", func(t *testing.T) {
+		t0, t1 := new(Fr), new(Fr)
+		one := new(Fr).setUint64(1)
+		t0.Square(glvLambda)
+		t0.Add(t0, glvLambda)
+		t1.Sub(&q, one)
+		if !t0.Equal(t1) {
+			t.Fatal("lambda1^2 + lambda1 + 1 = 0")
+		}
+		c0 := new(fe)
+		square(c0, glvPhi1)
+		mul(c0, c0, glvPhi1)
+		if !c0.isOne() {
+			t.Fatal("phi1^3 = 1")
+		}
+		square(c0, glvPhi2)
+		mul(c0, c0, glvPhi2)
+		if !c0.isOne() {
+			t.Fatal("phi2^3 = 1")
+		}
+	})
+	t.Run("Endomorphism G1", func(t *testing.T) {
+		g := NewG1()
+		{
+			p0, p1 := g.randAffine(), g.New()
+			g.MulScalar(p1, p0, glvLambda)
+			g.Affine(p1)
+			r := g.New()
+			g.glvEndomorphism(r, p0)
+			if !g.Equal(r, p1) {
+				t.Fatal("f(x, y) = (phi * x, y)")
+			}
+		}
+	})
+	t.Run("Endomorphism G2", func(t *testing.T) {
+		g := NewG2()
+		{
+			p0, p1 := g.randAffine(), g.New()
+			g.MulScalar(p1, p0, glvLambda)
+			g.Affine(p1)
+			r := g.New()
+			g.glvEndomorphism(r, p0)
+			if !g.Equal(r, p1) {
+				t.Fatal("f(x, y) = (phi * x, y)")
+			}
+		}
+	})
+	t.Run("Scalar Decomposition", func(t *testing.T) {
+		for i := 0; i < fuz; i++ {
+			m, err := new(Fr).Rand(rand.Reader)
+			if err != nil {
+				t.Fatal(err)
+			}
+			mBig := m.ToBig()
+			var vFr *glvVectorFr
+			var vBig *glvVectorBig
+			{
+				vFr = new(glvVectorFr).new(m)
+				v := vFr
+
+				if v.k1.Cmp(r128) >= 0 {
+					t.Fatal("bad scalar component, k1")
+				}
+				if v.k2.Cmp(r128) >= 0 {
+					t.Fatal("bad scalar component, k2")
+				}
+
+				k := new(Fr)
+				if v.neg1 && v.neg2 {
+					k.Mul(glvLambda, v.k2)
+					k.Sub(k, v.k1)
+				} else if v.neg1 {
+					k.Mul(glvLambda, v.k2)
+					k.Add(k, v.k1)
+					k.Neg(k)
+				} else if v.neg2 {
+					k.Mul(glvLambda, v.k2)
+					k.Add(v.k1, k)
+				} else {
+					k.Mul(glvLambda, v.k2)
+					k.Sub(v.k1, k)
+				}
+
+				if !k.Equal(m) {
+					t.Fatal("scalar decomposing failed")
+				}
+			}
+
+			r128Big := r128.ToBig()
+			{
+				vBig = new(glvVectorBig).new(mBig)
+
+				if new(big.Int).Abs(vBig.k1).Cmp(r128Big) >= 0 {
+					t.Fatal("bad scalar component, big k1")
+				}
+				if new(big.Int).Abs(vBig.k2).Cmp(r128Big) >= 0 {
+					t.Fatal("bad scalar component, big k2")
+				}
+
+				k := new(big.Int)
+				k.Mul(glvLambdaBig, vBig.k2)
+				k.Sub(vBig.k1, k).Mod(k, qBig)
+				if k.Cmp(mBig) != 0 {
+					t.Fatal("scalar decomposing with big.Int failed", i)
+				}
+			}
+
+			zeroBig := new(big.Int)
+			k1Abs, k2Abs := new(big.Int).Abs(vBig.k1), new(big.Int).Abs(vBig.k2)
+
+			if vFr.neg1 != (vBig.k1.Cmp(zeroBig) == -1) {
+				t.Fatal("cross: scalar decomposing with failed neg1")
+			}
+			if vFr.neg2 != (vBig.k2.Cmp(zeroBig) == -1) {
+				t.Fatal("cross: scalar decomposing with failed neg2")
+			}
+			if k1Abs.Cmp(vFr.k1.ToBig()) != 0 {
+				t.Fatal("cross: scalar decomposing with failed k1", i)
+			}
+			if k2Abs.Cmp(vFr.k2.ToBig()) != 0 {
+				t.Fatal("cross: scalar decomposing with failed k2", i)
+			}
+		}
+	})
+}

crypto/bls12381/gt.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
@@ -29,6 +13,7 @@ type GT struct {
 	fp12 *fp12
 }
 
+// Set copies given value into the destination
 func (e *E) Set(e2 *E) *E {
 	return e.set(e2)
 }
@@ -57,7 +42,7 @@ func NewGT() *GT {
 
 // Q returns group order in big.Int.
 func (g *GT) Q() *big.Int {
-	return new(big.Int).Set(q)
+	return new(big.Int).Set(qBig)
 }
 
 // FromBytes expects 576 byte input and returns target group element
@@ -81,7 +66,7 @@ func (g *GT) ToBytes(e *E) []byte {
 // IsValid checks whether given target group element is in correct subgroup.
 func (g *GT) IsValid(e *E) bool {
 	r := g.New()
-	g.fp12.exp(r, e, q)
+	g.fp12.exp(r, e, qBig)
 	return r.isOne()
 }
 
@@ -92,12 +77,12 @@ func (g *GT) New() *E {
 
 // Add adds two field element `a` and `b` and assigns the result to the element in first argument.
 func (g *GT) Add(c, a, b *E) {
-	g.fp12.add(c, a, b)
+	fp12Add(c, a, b)
 }
 
 // Sub subtracts two field element `a` and `b`, and assigns the result to the element in first argument.
 func (g *GT) Sub(c, a, b *E) {
-	g.fp12.sub(c, a, b)
+	fp12Sub(c, a, b)
 }
 
 // Mul multiplies two field element `a` and `b` and assigns the result to the element in first argument.
@@ -107,7 +92,8 @@ func (g *GT) Mul(c, a, b *E) {
 
 // Square squares an element `a` and assigns the result to the element in first argument.
 func (g *GT) Square(c, a *E) {
-	g.fp12.cyclotomicSquare(c, a)
+	c.set(a)
+	g.fp12.cyclotomicSquare(c)
 }
 
 // Exp exponents an element `a` by a scalar `s` and assigns the result to the element in first argument.

crypto/bls12381/hash_to_field.go

@@ -0,0 +1,70 @@
+package bls12381
+
+import (
+	"crypto/sha256"
+	"errors"
+)
+
+func hashToFpXMDSHA256(msg []byte, domain []byte, count int) ([]*fe, error) {
+	randBytes, err := expandMsgSHA256XMD(msg, domain, count*64)
+	if err != nil {
+		return nil, err
+	}
+	els := make([]*fe, count)
+	for i := 0; i < count; i++ {
+		els[i], err = from64Bytes(randBytes[i*64 : (i+1)*64])
+		if err != nil {
+			return nil, err
+		}
+	}
+	return els, nil
+}
+
+func expandMsgSHA256XMD(msg []byte, domain []byte, outLen int) ([]byte, error) {
+	h := sha256.New()
+	domainLen := uint8(len(domain))
+	if domainLen > 255 {
+		return nil, errors.New("invalid domain length")
+	}
+	// DST_prime = DST || I2OSP(len(DST), 1)
+	// b_0 = H(Z_pad || msg || l_i_b_str || I2OSP(0, 1) || DST_prime)
+	_, _ = h.Write(make([]byte, h.BlockSize()))
+	_, _ = h.Write(msg)
+	_, _ = h.Write([]byte{uint8(outLen >> 8), uint8(outLen)})
+	_, _ = h.Write([]byte{0})
+	_, _ = h.Write(domain)
+	_, _ = h.Write([]byte{domainLen})
+	b0 := h.Sum(nil)
+
+	// b_1 = H(b_0 || I2OSP(1, 1) || DST_prime)
+	h.Reset()
+	_, _ = h.Write(b0)
+	_, _ = h.Write([]byte{1})
+	_, _ = h.Write(domain)
+	_, _ = h.Write([]byte{domainLen})
+	b1 := h.Sum(nil)
+
+	// b_i = H(strxor(b_0, b_(i - 1)) || I2OSP(i, 1) || DST_prime)
+	ell := (outLen + h.Size() - 1) / h.Size()
+	bi := b1
+	out := make([]byte, outLen)
+	for i := 1; i < ell; i++ {
+		h.Reset()
+		// b_i = H(strxor(b_0, b_(i - 1)) || I2OSP(i, 1) || DST_prime)
+		tmp := make([]byte, h.Size())
+		for j := 0; j < h.Size(); j++ {
+			tmp[j] = b0[j] ^ bi[j]
+		}
+		_, _ = h.Write(tmp)
+		_, _ = h.Write([]byte{1 + uint8(i)})
+		_, _ = h.Write(domain)
+		_, _ = h.Write([]byte{domainLen})
+
+		// b_1 || ... || b_(ell - 1)
+		copy(out[(i-1)*h.Size():i*h.Size()], bi[:])
+		bi = h.Sum(nil)
+	}
+	// b_ell
+	copy(out[(ell-1)*h.Size():], bi[:])
+	return out[:outLen], nil
+}

crypto/bls12381/isogeny.go

@@ -1,82 +1,70 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 // isogenyMapG1 applies 11-isogeny map for BLS12-381 G1 defined at draft-irtf-cfrg-hash-to-curve-06.
 func isogenyMapG1(x, y *fe) {
-	// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06#appendix-C.2
-	params := isogenyConstantsG1
-	degree := 15
 	xNum, xDen, yNum, yDen := new(fe), new(fe), new(fe), new(fe)
-	xNum.set(params[0][degree])
+	xNum.set(isogenyConstansG1[0][15])
-	xDen.set(params[1][degree])
+	xDen.set(isogenyConstansG1[1][15])
-	yNum.set(params[2][degree])
+	yNum.set(isogenyConstansG1[2][15])
-	yDen.set(params[3][degree])
+	yDen.set(isogenyConstansG1[3][15])
-	for i := degree - 1; i >= 0; i-- {
+	for i := 14; i > -1; i-- {
 		mul(xNum, xNum, x)
 		mul(xDen, xDen, x)
 		mul(yNum, yNum, x)
 		mul(yDen, yDen, x)
-		add(xNum, xNum, params[0][i])
+		addAssign(xNum, isogenyConstansG1[0][i])
-		add(xDen, xDen, params[1][i])
+		addAssign(xDen, isogenyConstansG1[1][i])
-		add(yNum, yNum, params[2][i])
+		addAssign(yNum, isogenyConstansG1[2][i])
-		add(yDen, yDen, params[3][i])
+		addAssign(yDen, isogenyConstansG1[3][i])
 	}
 	inverse(xDen, xDen)
 	inverse(yDen, yDen)
-	mul(xNum, xNum, xDen)
+	mul(x, xNum, xDen)
 	mul(yNum, yNum, yDen)
-	mul(yNum, yNum, y)
+	mul(y, y, yNum)
-	x.set(xNum)
-	y.set(yNum)
 }
 
-// isogenyMapG2 applies 11-isogeny map for BLS12-381 G1 defined at draft-irtf-cfrg-hash-to-curve-06.
+// isogenyMapG2 applies 3-isogeny map for BLS12-381 G2 defined at draft-irtf-cfrg-hash-to-curve-06.
 func isogenyMapG2(e *fp2, x, y *fe2) {
 	if e == nil {
 		e = newFp2()
 	}
-	// https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-06#appendix-C.2
+	xNum := new(fe2).set(isogenyConstantsG2[0][3])
-	params := isogenyConstantsG2
+	xDen := new(fe2).set(x)
-	degree := 3
+	yNum := new(fe2).set(isogenyConstantsG2[2][3])
-	xNum := new(fe2).set(params[0][degree])
+	yDen := new(fe2).set(x)
-	xDen := new(fe2).set(params[1][degree])
+
-	yNum := new(fe2).set(params[2][degree])
+	e.mulAssign(xNum, x)
-	yDen := new(fe2).set(params[3][degree])
+	e.mulAssign(yNum, x)
-	for i := degree - 1; i >= 0; i-- {
+	fp2AddAssign(xNum, isogenyConstantsG2[0][2])
-		e.mul(xNum, xNum, x)
+	fp2AddAssign(yNum, isogenyConstantsG2[2][2])
-		e.mul(xDen, xDen, x)
+	fp2AddAssign(yDen, isogenyConstantsG2[3][2])
-		e.mul(yNum, yNum, x)
+
-		e.mul(yDen, yDen, x)
+	e.mulAssign(xNum, x)
-		e.add(xNum, xNum, params[0][i])
+	e.mulAssign(yNum, x)
-		e.add(xDen, xDen, params[1][i])
+	e.mulAssign(yDen, x)
-		e.add(yNum, yNum, params[2][i])
+	fp2AddAssign(xNum, isogenyConstantsG2[0][1])
-		e.add(yDen, yDen, params[3][i])
+	fp2AddAssign(xDen, isogenyConstantsG2[1][1])
-	}
+	fp2AddAssign(yNum, isogenyConstantsG2[2][1])
+	fp2AddAssign(yDen, isogenyConstantsG2[3][1])
+
+	e.mulAssign(xNum, x)
+	e.mulAssign(xDen, x)
+	e.mulAssign(yNum, x)
+	e.mulAssign(yDen, x)
+	fp2AddAssign(xNum, isogenyConstantsG2[0][0])
+	fp2AddAssign(xDen, isogenyConstantsG2[1][0])
+	fp2AddAssign(yNum, isogenyConstantsG2[2][0])
+	fp2AddAssign(yDen, isogenyConstantsG2[3][0])
+
 	e.inverse(xDen, xDen)
 	e.inverse(yDen, yDen)
-	e.mul(xNum, xNum, xDen)
+	e.mul(x, xNum, xDen)
-	e.mul(yNum, yNum, yDen)
+	e.mulAssign(yNum, yDen)
-	e.mul(yNum, yNum, y)
+	e.mulAssign(y, yNum)
-	x.set(xNum)
-	y.set(yNum)
 }
 
-var isogenyConstantsG1 = [4][16]*fe{
+var isogenyConstansG1 = [4][16]*fe{
 	{
 		{0x4d18b6f3af00131c, 0x19fa219793fee28c, 0x3f2885f1467f19ae, 0x23dcea34f2ffb304, 0xd15b58d2ffc00054, 0x0913be200a20bef4},
 		{0x898985385cdbbd8b, 0x3c79e43cc7d966aa, 0x1597e193f4cd233a, 0x8637ef1e4d6623ad, 0x11b22deed20d827b, 0x07097bc5998784ad},
@@ -154,74 +142,74 @@ var isogenyConstantsG1 = [4][16]*fe{
 var isogenyConstantsG2 = [4][4]*fe2{
 	{
 		{
-			fe{0x47f671c71ce05e62, 0x06dd57071206393e, 0x7c80cd2af3fd71a2, 0x048103ea9e6cd062, 0xc54516acc8d037f6, 0x13808f550920ea41},
+			{0x47f671c71ce05e62, 0x06dd57071206393e, 0x7c80cd2af3fd71a2, 0x048103ea9e6cd062, 0xc54516acc8d037f6, 0x13808f550920ea41},
-			fe{0x47f671c71ce05e62, 0x06dd57071206393e, 0x7c80cd2af3fd71a2, 0x048103ea9e6cd062, 0xc54516acc8d037f6, 0x13808f550920ea41},
+			{0x47f671c71ce05e62, 0x06dd57071206393e, 0x7c80cd2af3fd71a2, 0x048103ea9e6cd062, 0xc54516acc8d037f6, 0x13808f550920ea41},
 		},
 		{
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
-			fe{0x5fe55555554c71d0, 0x873fffdd236aaaa3, 0x6a6b4619b26ef918, 0x21c2888408874945, 0x2836cda7028cabc5, 0x0ac73310a7fd5abd},
+			{0x5fe55555554c71d0, 0x873fffdd236aaaa3, 0x6a6b4619b26ef918, 0x21c2888408874945, 0x2836cda7028cabc5, 0x0ac73310a7fd5abd},
 		},
 		{
-			fe{0x0a0c5555555971c3, 0xdb0c00101f9eaaae, 0xb1fb2f941d797997, 0xd3960742ef416e1c, 0xb70040e2c20556f4, 0x149d7861e581393b},
+			{0x0a0c5555555971c3, 0xdb0c00101f9eaaae, 0xb1fb2f941d797997, 0xd3960742ef416e1c, 0xb70040e2c20556f4, 0x149d7861e581393b},
-			fe{0xaff2aaaaaaa638e8, 0x439fffee91b55551, 0xb535a30cd9377c8c, 0x90e144420443a4a2, 0x941b66d3814655e2, 0x0563998853fead5e},
+			{0xaff2aaaaaaa638e8, 0x439fffee91b55551, 0xb535a30cd9377c8c, 0x90e144420443a4a2, 0x941b66d3814655e2, 0x0563998853fead5e},
 		},
 		{
-			fe{0x40aac71c71c725ed, 0x190955557a84e38e, 0xd817050a8f41abc3, 0xd86485d4c87f6fb1, 0x696eb479f885d059, 0x198e1a74328002d2},
+			{0x40aac71c71c725ed, 0x190955557a84e38e, 0xd817050a8f41abc3, 0xd86485d4c87f6fb1, 0x696eb479f885d059, 0x198e1a74328002d2},
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
 		},
 	},
 	{
 		{
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
-			fe{0x1f3affffff13ab97, 0xf25bfc611da3ff3e, 0xca3757cb3819b208, 0x3e6427366f8cec18, 0x03977bc86095b089, 0x04f69db13f39a952},
+			{0x1f3affffff13ab97, 0xf25bfc611da3ff3e, 0xca3757cb3819b208, 0x3e6427366f8cec18, 0x03977bc86095b089, 0x04f69db13f39a952},
 		},
 		{
-			fe{0x447600000027552e, 0xdcb8009a43480020, 0x6f7ee9ce4a6e8b59, 0xb10330b7c0a95bc6, 0x6140b1fcfb1e54b7, 0x0381be097f0bb4e1},
+			{0x447600000027552e, 0xdcb8009a43480020, 0x6f7ee9ce4a6e8b59, 0xb10330b7c0a95bc6, 0x6140b1fcfb1e54b7, 0x0381be097f0bb4e1},
-			fe{0x7588ffffffd8557d, 0x41f3ff646e0bffdf, 0xf7b1e8d2ac426aca, 0xb3741acd32dbb6f8, 0xe9daf5b9482d581f, 0x167f53e0ba7431b8},
+			{0x7588ffffffd8557d, 0x41f3ff646e0bffdf, 0xf7b1e8d2ac426aca, 0xb3741acd32dbb6f8, 0xe9daf5b9482d581f, 0x167f53e0ba7431b8},
 		},
 		{
-			fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+			{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
 		},
 		{
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
 		},
 	},
 	{
 		{
-			fe{0x96d8f684bdfc77be, 0xb530e4f43b66d0e2, 0x184a88ff379652fd, 0x57cb23ecfae804e1, 0x0fd2e39eada3eba9, 0x08c8055e31c5d5c3},
+			{0x96d8f684bdfc77be, 0xb530e4f43b66d0e2, 0x184a88ff379652fd, 0x57cb23ecfae804e1, 0x0fd2e39eada3eba9, 0x08c8055e31c5d5c3},
-			fe{0x96d8f684bdfc77be, 0xb530e4f43b66d0e2, 0x184a88ff379652fd, 0x57cb23ecfae804e1, 0x0fd2e39eada3eba9, 0x08c8055e31c5d5c3},
+			{0x96d8f684bdfc77be, 0xb530e4f43b66d0e2, 0x184a88ff379652fd, 0x57cb23ecfae804e1, 0x0fd2e39eada3eba9, 0x08c8055e31c5d5c3},
 		},
 		{
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
-			fe{0xbf0a71c71c91b406, 0x4d6d55d28b7638fd, 0x9d82f98e5f205aee, 0xa27aa27b1d1a18d5, 0x02c3b2b2d2938e86, 0x0c7d13420b09807f},
+			{0xbf0a71c71c91b406, 0x4d6d55d28b7638fd, 0x9d82f98e5f205aee, 0xa27aa27b1d1a18d5, 0x02c3b2b2d2938e86, 0x0c7d13420b09807f},
 		},
 		{
-			fe{0xd7f9555555531c74, 0x21cffff748daaaa8, 0x5a9ad1866c9bbe46, 0x4870a2210221d251, 0x4a0db369c0a32af1, 0x02b1ccc429ff56af},
+			{0xd7f9555555531c74, 0x21cffff748daaaa8, 0x5a9ad1866c9bbe46, 0x4870a2210221d251, 0x4a0db369c0a32af1, 0x02b1ccc429ff56af},
-			fe{0xe205aaaaaaac8e37, 0xfcdc000768795556, 0x0c96011a8a1537dd, 0x1c06a963f163406e, 0x010df44c82a881e6, 0x174f45260f808feb},
+			{0xe205aaaaaaac8e37, 0xfcdc000768795556, 0x0c96011a8a1537dd, 0x1c06a963f163406e, 0x010df44c82a881e6, 0x174f45260f808feb},
 		},
 		{
-			fe{0xa470bda12f67f35c, 0xc0fe38e23327b425, 0xc9d3d0f2c6f0678d, 0x1c55c9935b5a982e, 0x27f6c0e2f0746764, 0x117c5e6e28aa9054},
+			{0xa470bda12f67f35c, 0xc0fe38e23327b425, 0xc9d3d0f2c6f0678d, 0x1c55c9935b5a982e, 0x27f6c0e2f0746764, 0x117c5e6e28aa9054},
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
 		},
 	},
 	{
 		{
-			fe{0x0162fffffa765adf, 0x8f7bea480083fb75, 0x561b3c2259e93611, 0x11e19fc1a9c875d5, 0xca713efc00367660, 0x03c6a03d41da1151},
+			{0x0162fffffa765adf, 0x8f7bea480083fb75, 0x561b3c2259e93611, 0x11e19fc1a9c875d5, 0xca713efc00367660, 0x03c6a03d41da1151},
-			fe{0x0162fffffa765adf, 0x8f7bea480083fb75, 0x561b3c2259e93611, 0x11e19fc1a9c875d5, 0xca713efc00367660, 0x03c6a03d41da1151},
+			{0x0162fffffa765adf, 0x8f7bea480083fb75, 0x561b3c2259e93611, 0x11e19fc1a9c875d5, 0xca713efc00367660, 0x03c6a03d41da1151},
 		},
 		{
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
-			fe{0x5db0fffffd3b02c5, 0xd713f52358ebfdba, 0x5ea60761a84d161a, 0xbb2c75a34ea6c44a, 0x0ac6735921c1119b, 0x0ee3d913bdacfbf6},
+			{0x5db0fffffd3b02c5, 0xd713f52358ebfdba, 0x5ea60761a84d161a, 0xbb2c75a34ea6c44a, 0x0ac6735921c1119b, 0x0ee3d913bdacfbf6},
 		},
 		{
-			fe{0x66b10000003affc5, 0xcb1400e764ec0030, 0xa73e5eb56fa5d106, 0x8984c913a0fe09a9, 0x11e10afb78ad7f13, 0x05429d0e3e918f52},
+			{0x66b10000003affc5, 0xcb1400e764ec0030, 0xa73e5eb56fa5d106, 0x8984c913a0fe09a9, 0x11e10afb78ad7f13, 0x05429d0e3e918f52},
-			fe{0x534dffffffc4aae6, 0x5397ff174c67ffcf, 0xbff273eb870b251d, 0xdaf2827152870915, 0x393a9cbaca9e2dc3, 0x14be74dbfaee5748},
+			{0x534dffffffc4aae6, 0x5397ff174c67ffcf, 0xbff273eb870b251d, 0xdaf2827152870915, 0x393a9cbaca9e2dc3, 0x14be74dbfaee5748},
 		},
 		{
-			fe{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
+			{0x760900000002fffd, 0xebf4000bc40c0002, 0x5f48985753c758ba, 0x77ce585370525745, 0x5c071a97a256ec6d, 0x15f65ec3fa80e493},
-			fe{0, 0, 0, 0, 0, 0},
+			{0, 0, 0, 0, 0, 0},
 		},
 	},
 }

crypto/bls12381/pairing.go

@@ -1,19 +1,3 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 type pair struct {
@@ -35,8 +19,8 @@ type Engine struct {
 	pairs []pair
 }
 
-// NewPairingEngine creates new pairing engine instance.
+// NewEngine creates new pairing engine insteace.
-func NewPairingEngine() *Engine {
+func NewEngine() *Engine {
 	fp2 := newFp2()
 	fp6 := newFp6(fp2)
 	fp12 := newFp12(fp6)
@@ -52,24 +36,25 @@ func NewPairingEngine() *Engine {
 }
 
 type pairingEngineTemp struct {
-	t2  [10]*fe2
+	t2  [9]*fe2
-	t12 [9]fe12
+	t12 [3]fe12
 }
 
 func newEngineTemp() pairingEngineTemp {
-	t2 := [10]*fe2{}
+	t2 := [9]*fe2{}
-	for i := 0; i < 10; i++ {
+	for i := 0; i < len(t2); i++ {
 		t2[i] = &fe2{}
 	}
-	t12 := [9]fe12{}
+	t12 := [3]fe12{}
 	return pairingEngineTemp{t2, t12}
 }
 
 // AddPair adds a g1, g2 point pair to pairing engine
 func (e *Engine) AddPair(g1 *PointG1, g2 *PointG2) *Engine {
 	p := newPair(g1, g2)
-	if !e.isZero(p) {
+	if !(e.G1.IsZero(p.g1) || e.G2.IsZero(p.g2)) {
-		e.affine(p)
+		e.G1.Affine(p.g1)
+		e.G2.Affine(p.g2)
 		e.pairs = append(e.pairs, p)
 	}
 	return e
@@ -77,8 +62,9 @@ func (e *Engine) AddPair(g1 *PointG1, g2 *PointG2) *Engine {
 
 // AddPairInv adds a G1, G2 point pair to pairing engine. G1 point is negated.
 func (e *Engine) AddPairInv(g1 *PointG1, g2 *PointG2) *Engine {
-	e.G1.Neg(g1, g1)
+	ng1 := e.G1.New().Set(g1)
-	e.AddPair(g1, g2)
+	e.G1.Neg(ng1, g1)
+	e.AddPair(ng1, g2)
 	return e
 }
 
@@ -88,170 +74,246 @@ func (e *Engine) Reset() *Engine {
 	return e
 }
 
-func (e *Engine) isZero(p pair) bool {
+func (e *Engine) double(f *fe12, r *PointG2, k int) {
-	return e.G1.IsZero(p.g1) || e.G2.IsZero(p.g2)
+	fp2, t := e.fp2, e.t2
-}
 
-func (e *Engine) affine(p pair) {
-	e.G1.Affine(p.g1)
-	e.G2.Affine(p.g2)
-}
-
-func (e *Engine) doublingStep(coeff *[3]fe2, r *PointG2) {
-	// Adaptation of Formula 3 in https://eprint.iacr.org/2010/526.pdf
-	fp2 := e.fp2
-	t := e.t2
 	fp2.mul(t[0], &r[0], &r[1])
-	fp2.mulByFq(t[0], t[0], twoInv)
+	fp2.mul0(t[0], t[0], twoInv)
 	fp2.square(t[1], &r[1])
 	fp2.square(t[2], &r[2])
-	fp2.double(t[7], t[2])
+	fp2Double(t[7], t[2])
-	fp2.add(t[7], t[7], t[2])
+	fp2AddAssign(t[7], t[2])
 	fp2.mulByB(t[3], t[7])
-	fp2.double(t[4], t[3])
+	fp2Double(t[4], t[3])
-	fp2.add(t[4], t[4], t[3])
+	fp2AddAssign(t[4], t[3])
-	fp2.add(t[5], t[1], t[4])
+	fp2Add(t[5], t[1], t[4])
-	fp2.mulByFq(t[5], t[5], twoInv)
+	fp2.mul0(t[5], t[5], twoInv)
-	fp2.add(t[6], &r[1], &r[2])
+	fp2Add(t[6], &r[1], &r[2])
-	fp2.square(t[6], t[6])
+	fp2.squareAssign(t[6])
-	fp2.add(t[7], t[2], t[1])
+	fp2Add(t[7], t[2], t[1])
-	fp2.sub(t[6], t[6], t[7])
+	fp2SubAssign(t[6], t[7])
-	fp2.sub(&coeff[0], t[3], t[1])
+
+	fp2Sub(t[8], t[3], t[1])
+
 	fp2.square(t[7], &r[0])
-	fp2.sub(t[4], t[1], t[4])
+	fp2Sub(t[4], t[1], t[4])
 	fp2.mul(&r[0], t[4], t[0])
 	fp2.square(t[2], t[3])
-	fp2.double(t[3], t[2])
+	fp2Double(t[3], t[2])
-	fp2.add(t[3], t[3], t[2])
+	fp2AddAssign(t[3], t[2])
-	fp2.square(t[5], t[5])
+	fp2.squareAssign(t[5])
-	fp2.sub(&r[1], t[5], t[3])
+	fp2Sub(&r[1], t[5], t[3])
 	fp2.mul(&r[2], t[1], t[6])
-	fp2.double(t[0], t[7])
+	fp2Double(t[0], t[7])
-	fp2.add(&coeff[1], t[0], t[7])
+
-	fp2.neg(&coeff[2], t[6])
+	fp2AddAssign(t[0], t[7])
+	fp2Neg(t[6], t[6])
+
+	// line eval
+	e.fp2.mul0Assign(t[6], &e.pairs[k].g1[1])
+	e.fp2.mul0Assign(t[0], &e.pairs[k].g1[0])
+	e.fp12.mul014(f, t[8], t[0], t[6])
+
 }
 
-func (e *Engine) additionStep(coeff *[3]fe2, r, q *PointG2) {
+func (e *Engine) add(f *fe12, r *PointG2, k int) {
-	// Algorithm 12 in https://eprint.iacr.org/2010/526.pdf
+	fp2, t := e.fp2, e.t2
-	fp2 := e.fp2
+
-	t := e.t2
+	fp2.mul(t[0], &e.pairs[k].g2[1], &r[2])
-	fp2.mul(t[0], &q[1], &r[2])
+	fp2Neg(t[0], t[0])
-	fp2.neg(t[0], t[0])
+	fp2AddAssign(t[0], &r[1])
-	fp2.add(t[0], t[0], &r[1])
+	fp2.mul(t[1], &e.pairs[k].g2[0], &r[2])
-	fp2.mul(t[1], &q[0], &r[2])
+	fp2Neg(t[1], t[1])
-	fp2.neg(t[1], t[1])
+	fp2AddAssign(t[1], &r[0])
-	fp2.add(t[1], t[1], &r[0])
 	fp2.square(t[2], t[0])
 	fp2.square(t[3], t[1])
 	fp2.mul(t[4], t[1], t[3])
 	fp2.mul(t[2], &r[2], t[2])
-	fp2.mul(t[3], &r[0], t[3])
+	fp2.mulAssign(t[3], &r[0])
-	fp2.double(t[5], t[3])
+	fp2Double(t[5], t[3])
-	fp2.sub(t[5], t[4], t[5])
+	fp2Sub(t[5], t[4], t[5])
-	fp2.add(t[5], t[5], t[2])
+	fp2AddAssign(t[5], t[2])
 	fp2.mul(&r[0], t[1], t[5])
-	fp2.sub(t[2], t[3], t[5])
+	fp2SubAssign(t[3], t[5])
-	fp2.mul(t[2], t[2], t[0])
+	fp2.mulAssign(t[3], t[0])
-	fp2.mul(t[3], &r[1], t[4])
+	fp2.mul(t[2], &r[1], t[4])
-	fp2.sub(&r[1], t[2], t[3])
+	fp2Sub(&r[1], t[3], t[2])
-	fp2.mul(&r[2], &r[2], t[4])
+	fp2.mulAssign(&r[2], t[4])
-	fp2.mul(t[2], t[1], &q[1])
+	fp2.mul(t[2], t[1], &e.pairs[k].g2[1])
-	fp2.mul(t[3], t[0], &q[0])
+	fp2.mul(t[3], t[0], &e.pairs[k].g2[0])
-	fp2.sub(&coeff[0], t[3], t[2])
+
-	fp2.neg(&coeff[1], t[0])
+	fp2SubAssign(t[3], t[2])
-	coeff[2].set(t[1])
+	fp2Neg(t[0], t[0])
+
+	// line eval
+	e.fp2.mul0Assign(t[1], &e.pairs[k].g1[1])
+	e.fp2.mul0Assign(t[0], &e.pairs[k].g1[0])
+	e.fp12.mul014(f, t[3], t[0], t[1])
 }
 
-func (e *Engine) preCompute(ellCoeffs *[68][3]fe2, twistPoint *PointG2) {
+func (e *Engine) nDoubleAdd(f *fe12, r []PointG2, n int) {
-	// Algorithm 5 in  https://eprint.iacr.org/2019/077.pdf
+	for i := 0; i < n; i++ {
-	if e.G2.IsZero(twistPoint) {
+		e.fp12.squareAssign(f)
-		return
+		for j := 0; j < len(e.pairs); j++ {
+			e.double(f, &r[j], j)
 		}
-	r := new(PointG2).Set(twistPoint)
-	j := 0
-	for i := x.BitLen() - 2; i >= 0; i-- {
-		e.doublingStep(&ellCoeffs[j], r)
-		if x.Bit(i) != 0 {
-			j++
-			ellCoeffs[j] = fe6{}
-			e.additionStep(&ellCoeffs[j], r, twistPoint)
 	}
-		j++
+	for j := 0; j < len(e.pairs); j++ {
+		e.add(f, &r[j], j)
+	}
+}
+
+func (e *Engine) nDouble(f *fe12, r []PointG2, n int) {
+	for i := 0; i < n; i++ {
+		e.fp12.squareAssign(f)
+		for j := 0; j < len(e.pairs); j++ {
+			e.double(f, &r[j], j)
+		}
 	}
 }
 
 func (e *Engine) millerLoop(f *fe12) {
-	pairs := e.pairs
-	ellCoeffs := make([][68][3]fe2, len(pairs))
-	for i := 0; i < len(pairs); i++ {
-		e.preCompute(&ellCoeffs[i], pairs[i].g2)
-	}
-	fp12, fp2 := e.fp12, e.fp2
-	t := e.t2
 	f.one()
-	j := 0
+
-	for i := 62; /* x.BitLen() - 2 */ i >= 0; i-- {
+	r := make([]PointG2, len(e.pairs))
-		if i != 62 {
+	for i := 0; i < len(e.pairs); i++ {
-			fp12.square(f, f)
+		r[i].Set(e.pairs[i].g2)
 	}
-		for i := 0; i <= len(pairs)-1; i++ {
+
-			fp2.mulByFq(t[0], &ellCoeffs[i][j][2], &pairs[i].g1[1])
+	for j := 0; j < len(e.pairs); j++ {
-			fp2.mulByFq(t[1], &ellCoeffs[i][j][1], &pairs[i].g1[0])
+		e.double(f, &r[j], j)
-			fp12.mulBy014Assign(f, &ellCoeffs[i][j][0], t[1], t[0])
 	}
-		if x.Bit(i) != 0 {
+	for j := 0; j < len(e.pairs); j++ {
-			j++
+		e.add(f, &r[j], j)
-			for i := 0; i <= len(pairs)-1; i++ {
-				fp2.mulByFq(t[0], &ellCoeffs[i][j][2], &pairs[i].g1[1])
-				fp2.mulByFq(t[1], &ellCoeffs[i][j][1], &pairs[i].g1[0])
-				fp12.mulBy014Assign(f, &ellCoeffs[i][j][0], t[1], t[0])
 	}
-		}
+
-		j++
+	e.nDoubleAdd(f, r, 2)
-	}
+	e.nDoubleAdd(f, r, 3)
-	fp12.conjugate(f, f)
+	e.nDoubleAdd(f, r, 9)
+	e.nDoubleAdd(f, r, 32)
+	e.nDouble(f, r, 16)
+
+	fp12Conjugate(f, f)
 }
 
+// exp raises element by x = -15132376222941642752
 func (e *Engine) exp(c, a *fe12) {
-	fp12 := e.fp12
+	c.set(a)
-	fp12.cyclotomicExp(c, a, x)
+	e.fp12.cyclotomicSquare(c) // (a ^ 2)
-	fp12.conjugate(c, c)
+
+	// (a ^ (2 + 1)) ^ (2 ^ 2) = a ^ 12
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+
+	// (a ^ (12 + 1)) ^ (2 ^ 3) = a ^ 104
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+
+	// (a ^ (104 + 1)) ^ (2 ^ 9) = a ^ 53760
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	// (a ^ (53760 + 1)) ^ (2 ^ 32) = a ^ 230901736800256
+	e.fp12.mulAssign(c, a)
+	for i := 0; i < 32; i++ {
+		e.fp12.cyclotomicSquare(c)
+	}
+
+	// (a ^ (230901736800256 + 1)) ^ (2 ^ 16) = a ^ 15132376222941642752
+	e.fp12.mulAssign(c, a)
+	for i := 0; i < 16; i++ {
+		e.fp12.cyclotomicSquare(c)
+	}
+	// invert chain result since x is negative
+	fp12Conjugate(c, c)
+}
+
+// expDrop raises element by x = -15132376222941642752 / 2
+func (e *Engine) expDrop(c, a *fe12) {
+	c.set(a)
+	e.fp12.cyclotomicSquare(c) // (a ^ 2)
+
+	// (a ^ (2 + 1)) ^ (2 ^ 2) = a ^ 12
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+
+	// (a ^ (12 + 1)) ^ (2 ^ 3) = a ^ 104
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+
+	// (a ^ (104 + 1)) ^ (2 ^ 9) = a ^ 53760
+	e.fp12.mulAssign(c, a)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	e.fp12.cyclotomicSquare(c)
+	// (a ^ (53760 + 1)) ^ (2 ^ 32) = a ^ 230901736800256
+	e.fp12.mulAssign(c, a)
+	for i := 0; i < 32; i++ {
+		e.fp12.cyclotomicSquare(c)
+	}
+
+	// (a ^ (230901736800256 + 1)) ^ (2 ^ 16) = a ^ 15132376222941642752
+	e.fp12.mulAssign(c, a)
+	for i := 0; i < 15; i++ {
+		e.fp12.cyclotomicSquare(c)
+	}
+	// invert chain result since x is negative
+	fp12Conjugate(c, c)
 }
 
 func (e *Engine) finalExp(f *fe12) {
-	fp12 := e.fp12
 	t := e.t12
+	// Efficient Final Exponentiation via Cyclotomic Structure for Pairings over Families of Elliptic Curves
+	// https: //eprint.iacr.org/2020/875.pdf
+
 	// easy part
-	fp12.frobeniusMap(&t[0], f, 6)
+	fp12Conjugate(&t[0], f)
-	fp12.inverse(&t[1], f)
+	e.fp12.inverse(f, f)
-	fp12.mul(&t[2], &t[0], &t[1])
+	e.fp12.mulAssign(&t[0], f)
-	t[1].set(&t[2])
+	f.set(&t[0])
-	fp12.frobeniusMapAssign(&t[2], 2)
+	e.fp12.frobeniusMap2(f)
-	fp12.mulAssign(&t[2], &t[1])
+	e.fp12.mulAssign(f, &t[0])
-	fp12.cyclotomicSquare(&t[1], &t[2])
+
-	fp12.conjugate(&t[1], &t[1])
 	// hard part
-	e.exp(&t[3], &t[2])
+	t[0].set(f)
-	fp12.cyclotomicSquare(&t[4], &t[3])
+	e.fp12.cyclotomicSquare(&t[0])
-	fp12.mul(&t[5], &t[1], &t[3])
+	e.expDrop(&t[1], &t[0])
-	e.exp(&t[1], &t[5])
+	fp12Conjugate(&t[2], f)
+	e.fp12.mulAssign(&t[1], &t[2])
+	e.exp(&t[2], &t[1])
+	fp12Conjugate(&t[1], &t[1])
+	e.fp12.mulAssign(&t[1], &t[2])
+	e.exp(&t[2], &t[1])
+	e.fp12.frobeniusMap1(&t[1])
+	e.fp12.mulAssign(&t[1], &t[2])
+	e.fp12.mulAssign(f, &t[0])
 	e.exp(&t[0], &t[1])
-	e.exp(&t[6], &t[0])
+	e.exp(&t[2], &t[0])
-	fp12.mulAssign(&t[6], &t[4])
+	t[0].set(&t[1])
-	e.exp(&t[4], &t[6])
+	e.fp12.frobeniusMap2(&t[0])
-	fp12.conjugate(&t[5], &t[5])
+	fp12Conjugate(&t[1], &t[1])
-	fp12.mulAssign(&t[4], &t[5])
+	e.fp12.mulAssign(&t[1], &t[2])
-	fp12.mulAssign(&t[4], &t[2])
+	e.fp12.mulAssign(&t[1], &t[0])
-	fp12.conjugate(&t[5], &t[2])
+	e.fp12.mulAssign(f, &t[1])
-	fp12.mulAssign(&t[1], &t[2])
-	fp12.frobeniusMapAssign(&t[1], 3)
-	fp12.mulAssign(&t[6], &t[5])
-	fp12.frobeniusMapAssign(&t[6], 1)
-	fp12.mulAssign(&t[3], &t[0])
-	fp12.frobeniusMapAssign(&t[3], 2)
-	fp12.mulAssign(&t[3], &t[1])
-	fp12.mulAssign(&t[3], &t[6])
-	fp12.mul(f, &t[3], &t[4])
 }
 
 func (e *Engine) calculate() *fe12 {

crypto/bls12381/pairing_test.go

@@ -3,28 +3,27 @@ package bls12381
 import (
 	"math/big"
 	"testing"
-
-	"github.com/ethereum/go-ethereum/common"
 )
 
 func TestPairingExpected(t *testing.T) {
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	G1, G2 := bls.G1, bls.G2
 	GT := bls.GT()
 	expected, err := GT.FromBytes(
-		common.FromHex("" +
+		fromHex(
-			"0f41e58663bf08cf068672cbd01a7ec73baca4d72ca93544deff686bfd6df543d48eaa24afe47e1efde449383b676631" +
+			fpByteSize,
-			"04c581234d086a9902249b64728ffd21a189e87935a954051c7cdba7b3872629a4fafc05066245cb9108f0242d0fe3ef" +
+			"0x0f41e58663bf08cf068672cbd01a7ec73baca4d72ca93544deff686bfd6df543d48eaa24afe47e1efde449383b676631",
-			"03350f55a7aefcd3c31b4fcb6ce5771cc6a0e9786ab5973320c806ad360829107ba810c5a09ffdd9be2291a0c25a99a2" +
+			"0x04c581234d086a9902249b64728ffd21a189e87935a954051c7cdba7b3872629a4fafc05066245cb9108f0242d0fe3ef",
-			"11b8b424cd48bf38fcef68083b0b0ec5c81a93b330ee1a677d0d15ff7b984e8978ef48881e32fac91b93b47333e2ba57" +
+			"0x03350f55a7aefcd3c31b4fcb6ce5771cc6a0e9786ab5973320c806ad360829107ba810c5a09ffdd9be2291a0c25a99a2",
-			"06fba23eb7c5af0d9f80940ca771b6ffd5857baaf222eb95a7d2809d61bfe02e1bfd1b68ff02f0b8102ae1c2d5d5ab1a" +
+			"0x11b8b424cd48bf38fcef68083b0b0ec5c81a93b330ee1a677d0d15ff7b984e8978ef48881e32fac91b93b47333e2ba57",
-			"19f26337d205fb469cd6bd15c3d5a04dc88784fbb3d0b2dbdea54d43b2b73f2cbb12d58386a8703e0f948226e47ee89d" +
+			"0x06fba23eb7c5af0d9f80940ca771b6ffd5857baaf222eb95a7d2809d61bfe02e1bfd1b68ff02f0b8102ae1c2d5d5ab1a",
-			"018107154f25a764bd3c79937a45b84546da634b8f6be14a8061e55cceba478b23f7dacaa35c8ca78beae9624045b4b6" +
+			"0x19f26337d205fb469cd6bd15c3d5a04dc88784fbb3d0b2dbdea54d43b2b73f2cbb12d58386a8703e0f948226e47ee89d",
-			"01b2f522473d171391125ba84dc4007cfbf2f8da752f7c74185203fcca589ac719c34dffbbaad8431dad1c1fb597aaa5" +
+			"0x018107154f25a764bd3c79937a45b84546da634b8f6be14a8061e55cceba478b23f7dacaa35c8ca78beae9624045b4b6",
-			"193502b86edb8857c273fa075a50512937e0794e1e65a7617c90d8bd66065b1fffe51d7a579973b1315021ec3c19934f" +
+			"0x01b2f522473d171391125ba84dc4007cfbf2f8da752f7c74185203fcca589ac719c34dffbbaad8431dad1c1fb597aaa5",
-			"1368bb445c7c2d209703f239689ce34c0378a68e72a6b3b216da0e22a5031b54ddff57309396b38c881c4c849ec23e87" +
+			"0x193502b86edb8857c273fa075a50512937e0794e1e65a7617c90d8bd66065b1fffe51d7a579973b1315021ec3c19934f",
-			"089a1c5b46e5110b86750ec6a532348868a84045483c92b7af5af689452eafabf1a8943e50439f1d59882a98eaa0170f" +
+			"0x1368bb445c7c2d209703f239689ce34c0378a68e72a6b3b216da0e22a5031b54ddff57309396b38c881c4c849ec23e87",
-			"1250ebd871fc0a92a7b2d83168d0d727272d441befa15c503dd8e90ce98db3e7b6d194f60839c508a84305aaca1789b6",
+			"0x089a1c5b46e5110b86750ec6a532348868a84045483c92b7af5af689452eafabf1a8943e50439f1d59882a98eaa0170f",
+			"0x1250ebd871fc0a92a7b2d83168d0d727272d441befa15c503dd8e90ce98db3e7b6d194f60839c508a84305aaca1789b6",
 		),
 	)
 	if err != nil {
@@ -32,7 +31,7 @@ func TestPairingExpected(t *testing.T) {
 	}
 	r := bls.AddPair(G1.One(), G2.One()).Result()
 	if !r.Equal(expected) {
-		t.Fatal("bad pairing")
+		t.Fatal("expected pairing failed")
 	}
 	if !GT.IsValid(r) {
 		t.Fatal("element is not in correct subgroup")
@@ -40,7 +39,7 @@ func TestPairingExpected(t *testing.T) {
 }
 
 func TestPairingNonDegeneracy(t *testing.T) {
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	G1, G2 := bls.G1, bls.G2
 	g1Zero, g2Zero, g1One, g2One := G1.Zero(), G2.Zero(), G1.One(), G2.One()
 	GT := bls.GT()
@@ -89,19 +88,20 @@ func TestPairingNonDegeneracy(t *testing.T) {
 	bls.Reset()
 	{
 		expected, err := GT.FromBytes(
-			common.FromHex("" +
+			fromHex(
-				"0f41e58663bf08cf068672cbd01a7ec73baca4d72ca93544deff686bfd6df543d48eaa24afe47e1efde449383b676631" +
+				fpByteSize,
-				"04c581234d086a9902249b64728ffd21a189e87935a954051c7cdba7b3872629a4fafc05066245cb9108f0242d0fe3ef" +
+				"0x0f41e58663bf08cf068672cbd01a7ec73baca4d72ca93544deff686bfd6df543d48eaa24afe47e1efde449383b676631",
-				"03350f55a7aefcd3c31b4fcb6ce5771cc6a0e9786ab5973320c806ad360829107ba810c5a09ffdd9be2291a0c25a99a2" +
+				"0x04c581234d086a9902249b64728ffd21a189e87935a954051c7cdba7b3872629a4fafc05066245cb9108f0242d0fe3ef",
-				"11b8b424cd48bf38fcef68083b0b0ec5c81a93b330ee1a677d0d15ff7b984e8978ef48881e32fac91b93b47333e2ba57" +
+				"0x03350f55a7aefcd3c31b4fcb6ce5771cc6a0e9786ab5973320c806ad360829107ba810c5a09ffdd9be2291a0c25a99a2",
-				"06fba23eb7c5af0d9f80940ca771b6ffd5857baaf222eb95a7d2809d61bfe02e1bfd1b68ff02f0b8102ae1c2d5d5ab1a" +
+				"0x11b8b424cd48bf38fcef68083b0b0ec5c81a93b330ee1a677d0d15ff7b984e8978ef48881e32fac91b93b47333e2ba57",
-				"19f26337d205fb469cd6bd15c3d5a04dc88784fbb3d0b2dbdea54d43b2b73f2cbb12d58386a8703e0f948226e47ee89d" +
+				"0x06fba23eb7c5af0d9f80940ca771b6ffd5857baaf222eb95a7d2809d61bfe02e1bfd1b68ff02f0b8102ae1c2d5d5ab1a",
-				"018107154f25a764bd3c79937a45b84546da634b8f6be14a8061e55cceba478b23f7dacaa35c8ca78beae9624045b4b6" +
+				"0x19f26337d205fb469cd6bd15c3d5a04dc88784fbb3d0b2dbdea54d43b2b73f2cbb12d58386a8703e0f948226e47ee89d",
-				"01b2f522473d171391125ba84dc4007cfbf2f8da752f7c74185203fcca589ac719c34dffbbaad8431dad1c1fb597aaa5" +
+				"0x018107154f25a764bd3c79937a45b84546da634b8f6be14a8061e55cceba478b23f7dacaa35c8ca78beae9624045b4b6",
-				"193502b86edb8857c273fa075a50512937e0794e1e65a7617c90d8bd66065b1fffe51d7a579973b1315021ec3c19934f" +
+				"0x01b2f522473d171391125ba84dc4007cfbf2f8da752f7c74185203fcca589ac719c34dffbbaad8431dad1c1fb597aaa5",
-				"1368bb445c7c2d209703f239689ce34c0378a68e72a6b3b216da0e22a5031b54ddff57309396b38c881c4c849ec23e87" +
+				"0x193502b86edb8857c273fa075a50512937e0794e1e65a7617c90d8bd66065b1fffe51d7a579973b1315021ec3c19934f",
-				"089a1c5b46e5110b86750ec6a532348868a84045483c92b7af5af689452eafabf1a8943e50439f1d59882a98eaa0170f" +
+				"0x1368bb445c7c2d209703f239689ce34c0378a68e72a6b3b216da0e22a5031b54ddff57309396b38c881c4c849ec23e87",
-				"1250ebd871fc0a92a7b2d83168d0d727272d441befa15c503dd8e90ce98db3e7b6d194f60839c508a84305aaca1789b6",
+				"0x089a1c5b46e5110b86750ec6a532348868a84045483c92b7af5af689452eafabf1a8943e50439f1d59882a98eaa0170f",
+				"0x1250ebd871fc0a92a7b2d83168d0d727272d441befa15c503dd8e90ce98db3e7b6d194f60839c508a84305aaca1789b6",
 			),
 		)
 		if err != nil {
@@ -113,13 +113,13 @@ func TestPairingNonDegeneracy(t *testing.T) {
 		bls.AddPair(g1One, g2One)
 		e := bls.Result()
 		if !e.Equal(expected) {
-			t.Fatal("bad pairing")
+			t.Fatal("pairing failed")
 		}
 	}
 }
 
 func TestPairingBilinearity(t *testing.T) {
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	g1, g2 := bls.G1, bls.G2
 	gt := bls.GT()
 	// e(a*G1, b*G2) = e(G1, G2)^c
@@ -129,50 +129,50 @@ func TestPairingBilinearity(t *testing.T) {
 		G1, G2 := g1.One(), g2.One()
 		e0 := bls.AddPair(G1, G2).Result()
 		P1, P2 := g1.New(), g2.New()
-		g1.MulScalar(P1, G1, a)
+		g1.MulScalarBig(P1, G1, a)
-		g2.MulScalar(P2, G2, b)
+		g2.MulScalarBig(P2, G2, b)
 		e1 := bls.AddPair(P1, P2).Result()
 		gt.Exp(e0, e0, c)
 		if !e0.Equal(e1) {
-			t.Fatal("bad pairing, 1")
+			t.Fatal("pairing failed")
 		}
 	}
-	// e(a * G1, b * G2) = e((a + b) * G1, G2)
+	// e(a * G1, b * G2) = e((a * b) * G1, G2)
 	{
 		// scalars
 		a, b := big.NewInt(17), big.NewInt(117)
 		c := new(big.Int).Mul(a, b)
 		// LHS
 		G1, G2 := g1.One(), g2.One()
-		g1.MulScalar(G1, G1, c)
+		g1.MulScalarBig(G1, G1, c)
 		bls.AddPair(G1, G2)
 		// RHS
 		P1, P2 := g1.One(), g2.One()
-		g1.MulScalar(P1, P1, a)
+		g1.MulScalarBig(P1, P1, a)
-		g2.MulScalar(P2, P2, b)
+		g2.MulScalarBig(P2, P2, b)
 		bls.AddPairInv(P1, P2)
 		// should be one
 		if !bls.Check() {
-			t.Fatal("bad pairing, 2")
+			t.Fatal("pairing failed")
 		}
 	}
-	// e(a * G1, b * G2) = e((a + b) * G1, G2)
+	// e(a * G1, b * G2) = e(G1, (a * b) * G2)
 	{
 		// scalars
 		a, b := big.NewInt(17), big.NewInt(117)
 		c := new(big.Int).Mul(a, b)
 		// LHS
 		G1, G2 := g1.One(), g2.One()
-		g2.MulScalar(G2, G2, c)
+		g2.MulScalarBig(G2, G2, c)
 		bls.AddPair(G1, G2)
 		// RHS
 		H1, H2 := g1.One(), g2.One()
-		g1.MulScalar(H1, H1, a)
+		g1.MulScalarBig(H1, H1, a)
-		g2.MulScalar(H2, H2, b)
+		g2.MulScalarBig(H2, H2, b)
 		bls.AddPairInv(H1, H2)
 		// should be one
 		if !bls.Check() {
-			t.Fatal("bad pairing, 3")
+			t.Fatal("pairing failed")
 		}
 	}
 }
@@ -180,17 +180,17 @@ func TestPairingBilinearity(t *testing.T) {
 func TestPairingMulti(t *testing.T) {
 	// e(G1, G2) ^ t == e(a01 * G1, a02 * G2) * e(a11 * G1, a12 * G2) * ... * e(an1 * G1, an2 * G2)
 	// where t = sum(ai1 * ai2)
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	g1, g2 := bls.G1, bls.G2
 	numOfPair := 100
 	targetExp := new(big.Int)
 	// RHS
 	for i := 0; i < numOfPair; i++ {
 		// (ai1 * G1, ai2 * G2)
-		a1, a2 := randScalar(q), randScalar(q)
+		a1, a2 := randScalar(qBig), randScalar(qBig)
 		P1, P2 := g1.One(), g2.One()
-		g1.MulScalar(P1, P1, a1)
+		g1.MulScalarBig(P1, P1, a1)
-		g2.MulScalar(P2, P2, a2)
+		g2.MulScalarBig(P2, P2, a2)
 		bls.AddPair(P1, P2)
 		// accumulate targetExp
 		// t += (ai1 * ai2)
@@ -200,7 +200,7 @@ func TestPairingMulti(t *testing.T) {
 	// LHS
 	// e(t * G1, G2)
 	T1, T2 := g1.One(), g2.One()
-	g1.MulScalar(T1, T1, targetExp)
+	g1.MulScalarBig(T1, T1, targetExp)
 	bls.AddPairInv(T1, T2)
 	if !bls.Check() {
 		t.Fatal("fail multi pairing")
@@ -208,7 +208,7 @@ func TestPairingMulti(t *testing.T) {
 }
 
 func TestPairingEmpty(t *testing.T) {
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	if !bls.Check() {
 		t.Fatal("empty check should be accepted")
 	}
@@ -218,7 +218,7 @@ func TestPairingEmpty(t *testing.T) {
 }
 
 func BenchmarkPairing(t *testing.B) {
-	bls := NewPairingEngine()
+	bls := NewEngine()
 	g1, g2, gt := bls.G1, bls.G2, bls.GT()
 	bls.AddPair(g1.One(), g2.One())
 	e := gt.New()
@@ -228,3 +228,25 @@ func BenchmarkPairing(t *testing.B) {
 	}
 	_ = e
 }
+
+func BenchmarkMillerLoop(t *testing.B) {
+	bls := NewEngine()
+	g1, g2, gt := bls.G1, bls.G2, bls.GT()
+	bls.AddPair(g1.One(), g2.One())
+	f := gt.New().one()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		bls.millerLoop(f)
+	}
+}
+
+func BenchmarkFinalExp(t *testing.B) {
+	bls := NewEngine()
+	g1, g2, gt := bls.G1, bls.G2, bls.GT()
+	bls.AddPair(g1.One(), g2.One())
+	f := gt.New().one()
+	t.ResetTimer()
+	for i := 0; i < t.N; i++ {
+		bls.finalExp(f)
+	}
+}

crypto/bls12381/swu.go

@@ -1,23 +1,7 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 // swuMapG1 is implementation of Simplified Shallue-van de Woestijne-Ulas Method
-// follows the implementation at draft-irtf-cfrg-hash-to-curve-06.
+// follows the implmentation at draft-irtf-cfrg-hash-to-curve-06.
 func swuMapG1(u *fe) (*fe, *fe) {
 	var params = swuParamsForG1
 	var tv [4]*fe
@@ -79,19 +63,19 @@ func swuMapG2(e *fp2, u *fe2) (*fe2, *fe2) {
 	e.mul(tv[0], tv[0], params.z)
 	e.square(tv[1], tv[0])
 	x1 := e.new()
-	e.add(x1, tv[0], tv[1])
+	fp2Add(x1, tv[0], tv[1])
 	e.inverse(x1, x1)
 	e1 := x1.isZero()
-	e.add(x1, x1, e.one())
+	fp2Add(x1, x1, e.one())
 	if e1 {
 		x1.set(params.zInv)
 	}
 	e.mul(x1, x1, params.minusBOverA)
 	gx1 := e.new()
 	e.square(gx1, x1)
-	e.add(gx1, gx1, params.a)
+	fp2Add(gx1, gx1, params.a)
 	e.mul(gx1, gx1, x1)
-	e.add(gx1, gx1, params.b)
+	fp2Add(gx1, gx1, params.b)
 	x2 := e.new()
 	e.mul(x2, tv[0], x1)
 	e.mul(tv[1], tv[0], tv[1])
@@ -107,9 +91,9 @@ func swuMapG2(e *fp2, u *fe2) (*fe2, *fe2) {
 		y2.set(gx2)
 	}
 	y := e.new()
-	e.sqrt(y, y2)
+	e.sqrtBLST(y, y2)
 	if y.sign() != u.sign() {
-		e.neg(y, y)
+		fp2Neg(y, y)
 	}
 	return x, y
 }

crypto/bls12381/utils.go

@@ -1,45 +1,13 @@
-// Copyright 2020 The go-ethereum Authors
-// This file is part of the go-ethereum library.
-//
-// The go-ethereum library is free software: you can redistribute it and/or modify
-// it under the terms of the GNU Lesser General Public License as published by
-// the Free Software Foundation, either version 3 of the License, or
-// (at your option) any later version.
-//
-// The go-ethereum library is distributed in the hope that it will be useful,
-// but WITHOUT ANY WARRANTY; without even the implied warranty of
-// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
-// GNU Lesser General Public License for more details.
-//
-// You should have received a copy of the GNU Lesser General Public License
-// along with the go-ethereum library. If not, see <http://www.gnu.org/licenses/>.
-
 package bls12381
 
 import (
-	"errors"
 	"math/big"
-
-	"github.com/ethereum/go-ethereum/common"
 )
 
 func bigFromHex(hex string) *big.Int {
-	return new(big.Int).SetBytes(common.FromHex(hex))
+	if len(hex) > 1 && hex[:2] == "0x" {
-}
+		hex = hex[2:]
-
+	}
-// decodeFieldElement expects 64 byte input with zero top 16 bytes,
+	n, _ := new(big.Int).SetString(hex, 16)
-// returns lower 48 bytes.
+	return n
-func decodeFieldElement(in []byte) ([]byte, error) {
-	if len(in) != 64 {
-		return nil, errors.New("invalid field element length")
-	}
-	// check top bytes
-	for i := 0; i < 16; i++ {
-		if in[i] != byte(0x00) {
-			return nil, errors.New("invalid field element top bytes")
-		}
-	}
-	out := make([]byte, 48)
-	copy(out[:], in[16:])
-	return out, nil
 }

crypto/bls12381/wnaf.go

@@ -0,0 +1,110 @@
+package bls12381
+
+import (
+	"math/big"
+)
+
+type nafNumber []int
+
+func (n nafNumber) neg() {
+	for i := 0; i < len(n); i++ {
+		n[i] = -n[i]
+	}
+}
+
+var bigZero = big.NewInt(0)
+var bigOne = big.NewInt(1)
+
+func (e *Fr) toWNAF(w uint) nafNumber {
+	naf := nafNumber{}
+	if w == 0 {
+		return naf
+	}
+	windowSize, halfSize, mask := 1<<(w+1), 1<<w, (1<<(w+1))-1
+	ee := new(Fr).Set(e)
+	z := new(Fr)
+	for !ee.IsZero() {
+		if !ee.isEven() {
+			nafSign := int(ee[0]) & mask
+			if nafSign >= halfSize {
+				nafSign = nafSign - windowSize
+			}
+			naf = append(naf, int(nafSign))
+			if nafSign < 0 {
+				laddAssignFR(ee, z.setUint64(uint64(-nafSign)))
+			} else {
+				lsubAssignFR(ee, z.setUint64(uint64(nafSign)))
+			}
+		} else {
+			naf = append(naf, 0)
+		}
+		ee.div2()
+	}
+
+	return naf
+}
+
+func (e *Fr) fromWNAF(naf nafNumber, w uint) *Fr {
+	if w == 0 {
+		return e
+	}
+	l := (1 << (w - 1))
+	table := make([]*Fr, l)
+	table[0] = new(Fr).One()
+	two := new(Fr).setUint64(2)
+	for i := 1; i < l; i++ {
+		table[i] = new(Fr)
+		table[i].Add(table[i-1], two)
+	}
+	acc := new(Fr).Zero()
+	for i := len(naf) - 1; i >= 0; i-- {
+		if naf[i] < 0 {
+			acc.Sub(acc, table[-naf[i]>>1])
+		} else if naf[i] > 0 {
+			acc.Add(acc, table[naf[i]>>1])
+		}
+		if i != 0 {
+			acc.Double(acc)
+		}
+	}
+	return e.Set(acc)
+}
+
+// caution: does not cover negative case
+func bigToWNAF(e *big.Int, w uint) nafNumber {
+	naf := nafNumber{}
+	if w == 0 {
+		return naf
+	}
+	windowSize := new(big.Int).Lsh(bigOne, uint(w+1))
+	halfSize := new(big.Int).Rsh(windowSize, 1)
+	ee := new(big.Int).Abs(e)
+	for ee.Cmp(bigZero) != 0 {
+		if ee.Bit(0) == 1 {
+			nafSign := new(big.Int)
+			nafSign.Mod(ee, windowSize)
+			if nafSign.Cmp(halfSize) >= 0 {
+				nafSign.Sub(nafSign, windowSize)
+			}
+			naf = append(naf, int(nafSign.Int64()))
+			ee.Sub(ee, nafSign)
+		} else {
+			naf = append(naf, 0)
+		}
+		ee.Rsh(ee, 1)
+	}
+	return naf
+}
+
+func bigFromWNAF(naf nafNumber) *big.Int {
+	acc := new(big.Int)
+	k := new(big.Int).Set(bigOne)
+	for i := 0; i < len(naf); i++ {
+		if naf[i] != 0 {
+			z := new(big.Int).Mul(k, big.NewInt(int64(naf[i])))
+			acc.Add(acc, z)
+		}
+		k.Lsh(k, 1)
+	}
+	return acc
+}

crypto/bls12381/wnaf_test.go

@@ -0,0 +1,67 @@
+package bls12381
+
+import (
+	"crypto/rand"
+	"math/big"
+	"testing"
+)
+
+var maxWindowSize uint = 9
+
+func TestWNAFBig(t *testing.T) {
+	var w uint
+	for w = 1; w <= maxWindowSize; w++ {
+		for i := 0; i < fuz; i++ {
+			e0, err := rand.Int(rand.Reader, new(big.Int).SetUint64(100))
+			if err != nil {
+				t.Fatal(err)
+			}
+			n0 := bigToWNAF(e0, w)
+			e1 := bigFromWNAF(n0)
+			if e0.Cmp(e1) != 0 {
+				t.Fatal("wnaf conversion failed")
+			}
+		}
+	}
+}
+
+func TestFrWNAF(t *testing.T) {
+	var w uint
+	for w = 1; w <= maxWindowSize; w++ {
+		for i := 0; i < fuz; i++ {
+			a0, _ := new(Fr).Rand(rand.Reader)
+			naf := a0.toWNAF(w)
+			a1 := new(Fr).fromWNAF(naf, w)
+			if !a0.Equal(a1) {
+				t.Fatal("wnaf conversion failed")
+			}
+			naf.neg()
+			a1.fromWNAF(naf, w)
+			a0.Neg(a0)
+			if !a0.Equal(a1) {
+				t.Fatal("negated wnaf conversion failed")
+			}
+		}
+	}
+}
+
+func TestFrWNAFCrossAgainstBig(t *testing.T) {
+	var maxWindowSize uint = 20
+	var w uint
+	for w = 1; w <= maxWindowSize; w++ {
+		for i := 0; i < fuz; i++ {
+			a, _ := new(Fr).Rand(rand.Reader)
+			aBig := a.ToBig()
+			naf1 := a.toWNAF(w)
+			naf2 := bigToWNAF(aBig, w)
+			if len(naf1) != len(naf2) {
+				t.Fatal("naf conversion failed", len(naf1), len(naf2))
+			}
+			for i := 0; i < len(naf1); i++ {
+				if naf1[i] != naf2[i] {
+					t.Fatal("naf conversion failed", i)
+				}
+			}
+		}
+	}
+}