g1.go

// 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 consensys/gnark-crypto DO NOT EDIT

package bw6761

import (
	"github.com/consensys/gnark-crypto/ecc"
	"github.com/consensys/gnark-crypto/ecc/bw6-761/fp"
	"github.com/consensys/gnark-crypto/ecc/bw6-761/fr"
	"github.com/consensys/gnark-crypto/internal/parallel"
	"math/big"
	"runtime"
)

// G1Affine point in affine coordinates
type G1Affine struct {
	X, Y fp.Element
}

// G1Jac is a point with fp.Element coordinates
type G1Jac struct {
	X, Y, Z fp.Element
}

// g1JacExtended parameterized Jacobian coordinates (x=X/ZZ, y=Y/ZZZ, ZZ³=ZZZ²)
type g1JacExtended struct {
	X, Y, ZZ, ZZZ fp.Element
}

// g1Proj point in projective coordinates
type g1Proj struct {
	x, y, z fp.Element
}

// -------------------------------------------------------------------------------------------------
// Affine

// Set sets p to the provided point
func (p *G1Affine) Set(a *G1Affine) *G1Affine {
	p.X, p.Y = a.X, a.Y
	return p
}

// setInfinity sets p to O
func (p *G1Affine) setInfinity() *G1Affine {
	p.X.SetZero()
	p.Y.SetZero()
	return p
}

// ScalarMultiplication computes and returns p = a ⋅ s
func (p *G1Affine) ScalarMultiplication(a *G1Affine, s *big.Int) *G1Affine {
	var _p G1Jac
	_p.FromAffine(a)
	_p.mulGLV(&_p, s)
	p.FromJacobian(&_p)
	return p
}

// ScalarMultiplicationAffine computes and returns p = a ⋅ s
// Takes an affine point and returns a Jacobian point (useful for KZG)
func (p *G1Jac) ScalarMultiplicationAffine(a *G1Affine, s *big.Int) *G1Jac {
	p.FromAffine(a)
	p.mulGLV(p, s)
	return p
}

// ScalarMultiplicationBase computes and returns p = g ⋅ s where g is the prime subgroup generator
func (p *G1Affine) ScalarMultiplicationBase(s *big.Int) *G1Affine {
	var _p G1Jac
	_p.mulGLV(&g1Gen, s)
	p.FromJacobian(&_p)
	return p
}

// Add adds two point in affine coordinates.
// This should rarely be used as it is very inefficient compared to Jacobian
func (p *G1Affine) Add(a, b *G1Affine) *G1Affine {
	var p1, p2 G1Jac
	p1.FromAffine(a)
	p2.FromAffine(b)
	p1.AddAssign(&p2)
	p.FromJacobian(&p1)
	return p
}

// Sub subs two point in affine coordinates.
// This should rarely be used as it is very inefficient compared to Jacobian
func (p *G1Affine) Sub(a, b *G1Affine) *G1Affine {
	var p1, p2 G1Jac
	p1.FromAffine(a)
	p2.FromAffine(b)
	p1.SubAssign(&p2)
	p.FromJacobian(&p1)
	return p
}

// Equal tests if two points (in Affine coordinates) are equal
func (p *G1Affine) Equal(a *G1Affine) bool {
	return p.X.Equal(&a.X) && p.Y.Equal(&a.Y)
}

// Neg computes -G
func (p *G1Affine) Neg(a *G1Affine) *G1Affine {
	p.X = a.X
	p.Y.Neg(&a.Y)
	return p
}

// FromJacobian rescales a point in Jacobian coord in z=1 plane
func (p *G1Affine) FromJacobian(p1 *G1Jac) *G1Affine {

	var a, b fp.Element

	if p1.Z.IsZero() {
		p.X.SetZero()
		p.Y.SetZero()
		return p
	}

	a.Inverse(&p1.Z)
	b.Square(&a)
	p.X.Mul(&p1.X, &b)
	p.Y.Mul(&p1.Y, &b).Mul(&p.Y, &a)

	return p
}

// String returns the string representation of the point or "O" if it is infinity
func (p *G1Affine) String() string {
	if p.IsInfinity() {
		return "O"
	}
	return "E([" + p.X.String() + "," + p.Y.String() + "])"
}

// IsInfinity checks if the point is infinity
// in affine, it's encoded as (0,0)
// (0,0) is never on the curve for j=0 curves
func (p *G1Affine) IsInfinity() bool {
	return p.X.IsZero() && p.Y.IsZero()
}

// IsOnCurve returns true if p in on the curve
func (p *G1Affine) IsOnCurve() bool {
	var point G1Jac
	point.FromAffine(p)
	return point.IsOnCurve() // call this function to handle infinity point
}

// IsInSubGroup returns true if p is in the correct subgroup, false otherwise
func (p *G1Affine) IsInSubGroup() bool {
	var _p G1Jac
	_p.FromAffine(p)
	return _p.IsInSubGroup()
}

// -------------------------------------------------------------------------------------------------
// Jacobian

// Set sets p to the provided point
func (p *G1Jac) Set(a *G1Jac) *G1Jac {
	p.X, p.Y, p.Z = a.X, a.Y, a.Z
	return p
}

// Equal tests if two points (in Jacobian coordinates) are equal
func (p *G1Jac) Equal(a *G1Jac) bool {

	if p.Z.IsZero() && a.Z.IsZero() {
		return true
	}
	_p := G1Affine{}
	_p.FromJacobian(p)

	_a := G1Affine{}
	_a.FromJacobian(a)

	return _p.X.Equal(&_a.X) && _p.Y.Equal(&_a.Y)
}

// Neg computes -G
func (p *G1Jac) Neg(a *G1Jac) *G1Jac {
	*p = *a
	p.Y.Neg(&a.Y)
	return p
}

// SubAssign subtracts two points on the curve
func (p *G1Jac) SubAssign(a *G1Jac) *G1Jac {
	var tmp G1Jac
	tmp.Set(a)
	tmp.Y.Neg(&tmp.Y)
	p.AddAssign(&tmp)
	return p
}

// AddAssign point addition in montgomery form
// https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
func (p *G1Jac) AddAssign(a *G1Jac) *G1Jac {

	// p is infinity, return a
	if p.Z.IsZero() {
		p.Set(a)
		return p
	}

	// a is infinity, return p
	if a.Z.IsZero() {
		return p
	}

	var Z1Z1, Z2Z2, U1, U2, S1, S2, H, I, J, r, V fp.Element
	Z1Z1.Square(&a.Z)
	Z2Z2.Square(&p.Z)
	U1.Mul(&a.X, &Z2Z2)
	U2.Mul(&p.X, &Z1Z1)
	S1.Mul(&a.Y, &p.Z).
		Mul(&S1, &Z2Z2)
	S2.Mul(&p.Y, &a.Z).
		Mul(&S2, &Z1Z1)

	// if p == a, we double instead
	if U1.Equal(&U2) && S1.Equal(&S2) {
		return p.DoubleAssign()
	}

	H.Sub(&U2, &U1)
	I.Double(&H).
		Square(&I)
	J.Mul(&H, &I)
	r.Sub(&S2, &S1).Double(&r)
	V.Mul(&U1, &I)
	p.X.Square(&r).
		Sub(&p.X, &J).
		Sub(&p.X, &V).
		Sub(&p.X, &V)
	p.Y.Sub(&V, &p.X).
		Mul(&p.Y, &r)
	S1.Mul(&S1, &J).Double(&S1)
	p.Y.Sub(&p.Y, &S1)
	p.Z.Add(&p.Z, &a.Z)
	p.Z.Square(&p.Z).
		Sub(&p.Z, &Z1Z1).
		Sub(&p.Z, &Z2Z2).
		Mul(&p.Z, &H)

	return p
}

// AddMixed point addition
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-madd-2007-bl
func (p *G1Jac) AddMixed(a *G1Affine) *G1Jac {

	//if a is infinity return p
	if a.IsInfinity() {
		return p
	}
	// p is infinity, return a
	if p.Z.IsZero() {
		p.X = a.X
		p.Y = a.Y
		p.Z.SetOne()
		return p
	}

	var Z1Z1, U2, S2, H, HH, I, J, r, V fp.Element
	Z1Z1.Square(&p.Z)
	U2.Mul(&a.X, &Z1Z1)
	S2.Mul(&a.Y, &p.Z).
		Mul(&S2, &Z1Z1)

	// if p == a, we double instead
	if U2.Equal(&p.X) && S2.Equal(&p.Y) {
		return p.DoubleAssign()
	}

	H.Sub(&U2, &p.X)
	HH.Square(&H)
	I.Double(&HH).Double(&I)
	J.Mul(&H, &I)
	r.Sub(&S2, &p.Y).Double(&r)
	V.Mul(&p.X, &I)
	p.X.Square(&r).
		Sub(&p.X, &J).
		Sub(&p.X, &V).
		Sub(&p.X, &V)
	J.Mul(&J, &p.Y).Double(&J)
	p.Y.Sub(&V, &p.X).
		Mul(&p.Y, &r)
	p.Y.Sub(&p.Y, &J)
	p.Z.Add(&p.Z, &H)
	p.Z.Square(&p.Z).
		Sub(&p.Z, &Z1Z1).
		Sub(&p.Z, &HH)

	return p
}

// Double doubles a point in Jacobian coordinates
// https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2007-bl
func (p *G1Jac) Double(q *G1Jac) *G1Jac {
	p.Set(q)
	p.DoubleAssign()
	return p
}

// DoubleAssign doubles a point in Jacobian coordinates
// https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2007-bl
func (p *G1Jac) DoubleAssign() *G1Jac {

	var XX, YY, YYYY, ZZ, S, M, T fp.Element

	XX.Square(&p.X)
	YY.Square(&p.Y)
	YYYY.Square(&YY)
	ZZ.Square(&p.Z)
	S.Add(&p.X, &YY)
	S.Square(&S).
		Sub(&S, &XX).
		Sub(&S, &YYYY).
		Double(&S)
	M.Double(&XX).Add(&M, &XX)
	p.Z.Add(&p.Z, &p.Y).
		Square(&p.Z).
		Sub(&p.Z, &YY).
		Sub(&p.Z, &ZZ)
	T.Square(&M)
	p.X = T
	T.Double(&S)
	p.X.Sub(&p.X, &T)
	p.Y.Sub(&S, &p.X).
		Mul(&p.Y, &M)
	YYYY.Double(&YYYY).Double(&YYYY).Double(&YYYY)
	p.Y.Sub(&p.Y, &YYYY)

	return p
}

// ScalarMultiplication computes and returns p = a ⋅ s
// see https://www.iacr.org/archive/crypto2001/21390189.pdf
func (p *G1Jac) ScalarMultiplication(a *G1Jac, s *big.Int) *G1Jac {
	return p.mulGLV(a, s)
}

// String returns canonical representation of the point in affine coordinates
func (p *G1Jac) String() string {
	_p := G1Affine{}
	_p.FromJacobian(p)
	return _p.String()
}

// FromAffine sets p = Q, p in Jacobian, Q in affine
func (p *G1Jac) FromAffine(Q *G1Affine) *G1Jac {
	if Q.IsInfinity() {
		p.Z.SetZero()
		p.X.SetOne()
		p.Y.SetOne()
		return p
	}
	p.Z.SetOne()
	p.X.Set(&Q.X)
	p.Y.Set(&Q.Y)
	return p
}

// IsOnCurve returns true if p in on the curve
func (p *G1Jac) IsOnCurve() bool {
	var left, right, tmp fp.Element
	left.Square(&p.Y)
	right.Square(&p.X).Mul(&right, &p.X)
	tmp.Square(&p.Z).
		Square(&tmp).
		Mul(&tmp, &p.Z).
		Mul(&tmp, &p.Z).
		Mul(&tmp, &bCurveCoeff)
	right.Add(&right, &tmp)
	return left.Equal(&right)
}

// IsInSubGroup returns true if p is on the r-torsion, false otherwise.

// Z[r,0]+Z[-lambdaG1Affine, 1] is the kernel
// of (u,v)->u+lambdaG1Affinev mod r. Expressing r, lambdaG1Affine as
// polynomials in x, a short vector of this Zmodule is
// (x+1), (x³-x²+1). So we check that (x+1)p+(x³-x²+1)ϕ(p)
// is the infinity.
func (p *G1Jac) IsInSubGroup() bool {

	var res, phip G1Jac
	phip.phi(p)
	res.ScalarMultiplication(&phip, &xGen).
		SubAssign(&phip).
		ScalarMultiplication(&res, &xGen).
		ScalarMultiplication(&res, &xGen).
		AddAssign(&phip)

	phip.ScalarMultiplication(p, &xGen).AddAssign(p).AddAssign(&res)

	return phip.IsOnCurve() && phip.Z.IsZero()

}

// mulWindowed computes a 2-bits windowed scalar multiplication
func (p *G1Jac) mulWindowed(a *G1Jac, s *big.Int) *G1Jac {

	var res G1Jac
	var ops [3]G1Jac

	res.Set(&g1Infinity)
	ops[0].Set(a)
	ops[1].Double(&ops[0])
	ops[2].Set(&ops[0]).AddAssign(&ops[1])

	b := s.Bytes()
	for i := range b {
		w := b[i]
		mask := byte(0xc0)
		for j := 0; j < 4; j++ {
			res.DoubleAssign().DoubleAssign()
			c := (w & mask) >> (6 - 2*j)
			if c != 0 {
				res.AddAssign(&ops[c-1])
			}
			mask = mask >> 2
		}
	}
	p.Set(&res)

	return p

}

// ϕ assigns p to ϕ(a) where ϕ: (x,y) → (w x,y), and returns p
// where w is a third root of unity in 𝔽p
func (p *G1Jac) phi(a *G1Jac) *G1Jac {
	p.Set(a)
	p.X.Mul(&p.X, &thirdRootOneG1)
	return p
}

// mulGLV computes the scalar multiplication using a windowed-GLV method
// see https://www.iacr.org/archive/crypto2001/21390189.pdf
func (p *G1Jac) mulGLV(a *G1Jac, s *big.Int) *G1Jac {

	var table [15]G1Jac
	var res G1Jac
	var k1, k2 fr.Element

	res.Set(&g1Infinity)

	// table[b3b2b1b0-1] = b3b2 ⋅ ϕ(a) + b1b0*a
	table[0].Set(a)
	table[3].phi(a)

	// split the scalar, modifies ±a, ϕ(a) accordingly
	k := ecc.SplitScalar(s, &glvBasis)

	if k[0].Sign() == -1 {
		k[0].Neg(&k[0])
		table[0].Neg(&table[0])
	}
	if k[1].Sign() == -1 {
		k[1].Neg(&k[1])
		table[3].Neg(&table[3])
	}

	// precompute table (2 bits sliding window)
	// table[b3b2b1b0-1] = b3b2 ⋅ ϕ(a) + b1b0 ⋅ a if b3b2b1b0 != 0
	table[1].Double(&table[0])
	table[2].Set(&table[1]).AddAssign(&table[0])
	table[4].Set(&table[3]).AddAssign(&table[0])
	table[5].Set(&table[3]).AddAssign(&table[1])
	table[6].Set(&table[3]).AddAssign(&table[2])
	table[7].Double(&table[3])
	table[8].Set(&table[7]).AddAssign(&table[0])
	table[9].Set(&table[7]).AddAssign(&table[1])
	table[10].Set(&table[7]).AddAssign(&table[2])
	table[11].Set(&table[7]).AddAssign(&table[3])
	table[12].Set(&table[11]).AddAssign(&table[0])
	table[13].Set(&table[11]).AddAssign(&table[1])
	table[14].Set(&table[11]).AddAssign(&table[2])

	// bounds on the lattice base vectors guarantee that k1, k2 are len(r)/2 or len(r)/2+1 bits long max
	// this is because we use a probabilistic scalar decomposition that replaces a division by a right-shift
	k1 = k1.SetBigInt(&k[0]).Bits()
	k2 = k2.SetBigInt(&k[1]).Bits()

	// we don't target constant-timeness so we check first if we increase the bounds or not
	maxBit := k1.BitLen()
	if k2.BitLen() > maxBit {
		maxBit = k2.BitLen()
	}
	hiWordIndex := (maxBit - 1) / 64

	// loop starts from len(k1)/2 or len(k1)/2+1 due to the bounds
	for i := hiWordIndex; i >= 0; i-- {
		mask := uint64(3) << 62
		for j := 0; j < 32; j++ {
			res.Double(&res).Double(&res)
			b1 := (k1[i] & mask) >> (62 - 2*j)
			b2 := (k2[i] & mask) >> (62 - 2*j)
			if b1|b2 != 0 {
				s := (b2<<2 | b1)
				res.AddAssign(&table[s-1])
			}
			mask = mask >> 2
		}
	}

	p.Set(&res)
	return p
}

// ClearCofactor maps a point in curve to r-torsion
func (p *G1Affine) ClearCofactor(a *G1Affine) *G1Affine {
	var _p G1Jac
	_p.FromAffine(a)
	_p.ClearCofactor(&_p)
	p.FromJacobian(&_p)
	return p
}

// ClearCofactor maps a point in E(Fp) to E(Fp)[r]
func (p *G1Jac) ClearCofactor(a *G1Jac) *G1Jac {

	// https://eprint.iacr.org/2020/351.pdf
	var points [4]G1Jac
	points[0].Set(a)
	points[1].ScalarMultiplication(a, &xGen)
	points[2].ScalarMultiplication(&points[1], &xGen)
	points[3].ScalarMultiplication(&points[2], &xGen)

	var scalars [7]big.Int
	scalars[0].SetInt64(103)
	scalars[1].SetInt64(83)
	scalars[2].SetInt64(40)
	scalars[3].SetInt64(136)

	scalars[4].SetInt64(7)
	scalars[5].SetInt64(89)
	scalars[6].SetInt64(130)

	var p1, p2, tmp G1Jac
	p1.ScalarMultiplication(&points[3], &scalars[0])
	tmp.ScalarMultiplication(&points[2], &scalars[1]).Neg(&tmp)
	p1.AddAssign(&tmp)
	tmp.ScalarMultiplication(&points[1], &scalars[2]).Neg(&tmp)
	p1.AddAssign(&tmp)
	tmp.ScalarMultiplication(&points[0], &scalars[3])
	p1.AddAssign(&tmp)

	p2.ScalarMultiplication(&points[2], &scalars[4])
	tmp.ScalarMultiplication(&points[1], &scalars[5])
	p2.AddAssign(&tmp)
	tmp.ScalarMultiplication(&points[0], &scalars[6])
	p2.AddAssign(&tmp)
	p2.phi(&p2)

	p.Set(&p1).AddAssign(&p2)

	return p

}

// JointScalarMultiplicationBase computes [s1]g+[s2]a using Straus-Shamir technique
// where g is the prime subgroup generator
func (p *G1Jac) JointScalarMultiplicationBase(a *G1Affine, s1, s2 *big.Int) *G1Jac {

	var res, p1, p2 G1Jac
	res.Set(&g1Infinity)
	p1.Set(&g1Gen)
	p2.FromAffine(a)

	var table [15]G1Jac

	var k1, k2 big.Int
	if s1.Sign() == -1 {
		k1.Neg(s1)
		table[0].Neg(&p1)
	} else {
		k1.Set(s1)
		table[0].Set(&p1)
	}
	if s2.Sign() == -1 {
		k2.Neg(s2)
		table[3].Neg(&p2)
	} else {
		k2.Set(s2)
		table[3].Set(&p2)
	}

	// precompute table (2 bits sliding window)
	table[1].Double(&table[0])
	table[2].Set(&table[1]).AddAssign(&table[0])
	table[4].Set(&table[3]).AddAssign(&table[0])
	table[5].Set(&table[3]).AddAssign(&table[1])
	table[6].Set(&table[3]).AddAssign(&table[2])
	table[7].Double(&table[3])
	table[8].Set(&table[7]).AddAssign(&table[0])
	table[9].Set(&table[7]).AddAssign(&table[1])
	table[10].Set(&table[7]).AddAssign(&table[2])
	table[11].Set(&table[7]).AddAssign(&table[3])
	table[12].Set(&table[11]).AddAssign(&table[0])
	table[13].Set(&table[11]).AddAssign(&table[1])
	table[14].Set(&table[11]).AddAssign(&table[2])

	var s [2]fr.Element
	s[0] = s[0].SetBigInt(&k1).Bits()
	s[1] = s[1].SetBigInt(&k2).Bits()

	maxBit := k1.BitLen()
	if k2.BitLen() > maxBit {
		maxBit = k2.BitLen()
	}
	hiWordIndex := (maxBit - 1) / 64

	for i := hiWordIndex; i >= 0; i-- {
		mask := uint64(3) << 62
		for j := 0; j < 32; j++ {
			res.Double(&res).Double(&res)
			b1 := (s[0][i] & mask) >> (62 - 2*j)
			b2 := (s[1][i] & mask) >> (62 - 2*j)
			if b1|b2 != 0 {
				s := (b2<<2 | b1)
				res.AddAssign(&table[s-1])
			}
			mask = mask >> 2
		}
	}

	p.Set(&res)
	return p

}

// -------------------------------------------------------------------------------------------------
// Jacobian extended

// Set sets p to the provided point
func (p *g1JacExtended) Set(a *g1JacExtended) *g1JacExtended {
	p.X, p.Y, p.ZZ, p.ZZZ = a.X, a.Y, a.ZZ, a.ZZZ
	return p
}

// setInfinity sets p to O
func (p *g1JacExtended) setInfinity() *g1JacExtended {
	p.X.SetOne()
	p.Y.SetOne()
	p.ZZ = fp.Element{}
	p.ZZZ = fp.Element{}
	return p
}

// fromJacExtended sets Q in affine coordinates
func (p *G1Affine) fromJacExtended(Q *g1JacExtended) *G1Affine {
	if Q.ZZ.IsZero() {
		p.X = fp.Element{}
		p.Y = fp.Element{}
		return p
	}
	p.X.Inverse(&Q.ZZ).Mul(&p.X, &Q.X)
	p.Y.Inverse(&Q.ZZZ).Mul(&p.Y, &Q.Y)
	return p
}

// fromJacExtended sets Q in Jacobian coordinates
func (p *G1Jac) fromJacExtended(Q *g1JacExtended) *G1Jac {
	if Q.ZZ.IsZero() {
		p.Set(&g1Infinity)
		return p
	}
	p.X.Mul(&Q.ZZ, &Q.X).Mul(&p.X, &Q.ZZ)
	p.Y.Mul(&Q.ZZZ, &Q.Y).Mul(&p.Y, &Q.ZZZ)
	p.Z.Set(&Q.ZZZ)
	return p
}

// unsafeFromJacExtended sets p in Jacobian coordinates, but don't check for infinity
func (p *G1Jac) unsafeFromJacExtended(Q *g1JacExtended) *G1Jac {
	p.X.Square(&Q.ZZ).Mul(&p.X, &Q.X)
	p.Y.Square(&Q.ZZZ).Mul(&p.Y, &Q.Y)
	p.Z = Q.ZZZ
	return p
}

// add point in Jacobian extended coordinates
// https://www.hyperelliptic.org/EFD/g1p/auto-shortw-xyzz.html#addition-add-2008-s
func (p *g1JacExtended) add(q *g1JacExtended) *g1JacExtended {
	//if q is infinity return p
	if q.ZZ.IsZero() {
		return p
	}
	// p is infinity, return q
	if p.ZZ.IsZero() {
		p.Set(q)
		return p
	}

	var A, B, U1, U2, S1, S2 fp.Element

	// p2: q, p1: p
	U2.Mul(&q.X, &p.ZZ)
	U1.Mul(&p.X, &q.ZZ)
	A.Sub(&U2, &U1)
	S2.Mul(&q.Y, &p.ZZZ)
	S1.Mul(&p.Y, &q.ZZZ)
	B.Sub(&S2, &S1)

	if A.IsZero() {
		if B.IsZero() {
			return p.double(q)

		}
		p.ZZ = fp.Element{}
		p.ZZZ = fp.Element{}
		return p
	}

	var P, R, PP, PPP, Q, V fp.Element
	P.Sub(&U2, &U1)
	R.Sub(&S2, &S1)
	PP.Square(&P)
	PPP.Mul(&P, &PP)
	Q.Mul(&U1, &PP)
	V.Mul(&S1, &PPP)

	p.X.Square(&R).
		Sub(&p.X, &PPP).
		Sub(&p.X, &Q).
		Sub(&p.X, &Q)
	p.Y.Sub(&Q, &p.X).
		Mul(&p.Y, &R).
		Sub(&p.Y, &V)
	p.ZZ.Mul(&p.ZZ, &q.ZZ).
		Mul(&p.ZZ, &PP)
	p.ZZZ.Mul(&p.ZZZ, &q.ZZZ).
		Mul(&p.ZZZ, &PPP)

	return p
}

// double point in Jacobian extended coordinates
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-xyzz.html#doubling-dbl-2008-s-1
// since we consider any point on Z=0 as the point at infinity
// this doubling formula works for infinity points as well
func (p *g1JacExtended) double(q *g1JacExtended) *g1JacExtended {
	var U, V, W, S, XX, M fp.Element

	U.Double(&q.Y)
	V.Square(&U)
	W.Mul(&U, &V)
	S.Mul(&q.X, &V)
	XX.Square(&q.X)
	M.Double(&XX).
		Add(&M, &XX) // -> + a, but a=0 here
	U.Mul(&W, &q.Y)

	p.X.Square(&M).
		Sub(&p.X, &S).
		Sub(&p.X, &S)
	p.Y.Sub(&S, &p.X).
		Mul(&p.Y, &M).
		Sub(&p.Y, &U)
	p.ZZ.Mul(&V, &q.ZZ)
	p.ZZZ.Mul(&W, &q.ZZZ)

	return p
}

// subMixed same as addMixed, but will negate a.Y
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-xyzz.html#addition-madd-2008-s
func (p *g1JacExtended) subMixed(a *G1Affine) *g1JacExtended {

	//if a is infinity return p
	if a.IsInfinity() {
		return p
	}
	// p is infinity, return a
	if p.ZZ.IsZero() {
		p.X = a.X
		p.Y.Neg(&a.Y)
		p.ZZ.SetOne()
		p.ZZZ.SetOne()
		return p
	}

	var P, R fp.Element

	// p2: a, p1: p
	P.Mul(&a.X, &p.ZZ)
	P.Sub(&P, &p.X)

	R.Mul(&a.Y, &p.ZZZ)
	R.Neg(&R)
	R.Sub(&R, &p.Y)

	if P.IsZero() {
		if R.IsZero() {
			return p.doubleNegMixed(a)

		}
		p.ZZ = fp.Element{}
		p.ZZZ = fp.Element{}
		return p
	}

	var PP, PPP, Q, Q2, RR, X3, Y3 fp.Element

	PP.Square(&P)
	PPP.Mul(&P, &PP)
	Q.Mul(&p.X, &PP)
	RR.Square(&R)
	X3.Sub(&RR, &PPP)
	Q2.Double(&Q)
	p.X.Sub(&X3, &Q2)
	Y3.Sub(&Q, &p.X).Mul(&Y3, &R)
	R.Mul(&p.Y, &PPP)
	p.Y.Sub(&Y3, &R)
	p.ZZ.Mul(&p.ZZ, &PP)
	p.ZZZ.Mul(&p.ZZZ, &PPP)

	return p

}

// addMixed
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-xyzz.html#addition-madd-2008-s
func (p *g1JacExtended) addMixed(a *G1Affine) *g1JacExtended {

	//if a is infinity return p
	if a.IsInfinity() {
		return p
	}
	// p is infinity, return a
	if p.ZZ.IsZero() {
		p.X = a.X
		p.Y = a.Y
		p.ZZ.SetOne()
		p.ZZZ.SetOne()
		return p
	}

	var P, R fp.Element

	// p2: a, p1: p
	P.Mul(&a.X, &p.ZZ)
	P.Sub(&P, &p.X)

	R.Mul(&a.Y, &p.ZZZ)
	R.Sub(&R, &p.Y)

	if P.IsZero() {
		if R.IsZero() {
			return p.doubleMixed(a)

		}
		p.ZZ = fp.Element{}
		p.ZZZ = fp.Element{}
		return p
	}

	var PP, PPP, Q, Q2, RR, X3, Y3 fp.Element

	PP.Square(&P)
	PPP.Mul(&P, &PP)
	Q.Mul(&p.X, &PP)
	RR.Square(&R)
	X3.Sub(&RR, &PPP)
	Q2.Double(&Q)
	p.X.Sub(&X3, &Q2)
	Y3.Sub(&Q, &p.X).Mul(&Y3, &R)
	R.Mul(&p.Y, &PPP)
	p.Y.Sub(&Y3, &R)
	p.ZZ.Mul(&p.ZZ, &PP)
	p.ZZZ.Mul(&p.ZZZ, &PPP)

	return p

}

// doubleNegMixed same as double, but will negate q.Y
func (p *g1JacExtended) doubleNegMixed(q *G1Affine) *g1JacExtended {

	var U, V, W, S, XX, M, S2, L fp.Element

	U.Double(&q.Y)
	U.Neg(&U)
	V.Square(&U)
	W.Mul(&U, &V)
	S.Mul(&q.X, &V)
	XX.Square(&q.X)
	M.Double(&XX).
		Add(&M, &XX) // -> + a, but a=0 here
	S2.Double(&S)
	L.Mul(&W, &q.Y)

	p.X.Square(&M).
		Sub(&p.X, &S2)
	p.Y.Sub(&S, &p.X).
		Mul(&p.Y, &M).
		Add(&p.Y, &L)
	p.ZZ.Set(&V)
	p.ZZZ.Set(&W)

	return p
}

// doubleMixed point in Jacobian extended coordinates
// http://www.hyperelliptic.org/EFD/g1p/auto-shortw-xyzz.html#doubling-dbl-2008-s-1
func (p *g1JacExtended) doubleMixed(q *G1Affine) *g1JacExtended {

	var U, V, W, S, XX, M, S2, L fp.Element

	U.Double(&q.Y)
	V.Square(&U)
	W.Mul(&U, &V)
	S.Mul(&q.X, &V)
	XX.Square(&q.X)
	M.Double(&XX).
		Add(&M, &XX) // -> + a, but a=0 here
	S2.Double(&S)
	L.Mul(&W, &q.Y)

	p.X.Square(&M).
		Sub(&p.X, &S2)
	p.Y.Sub(&S, &p.X).
		Mul(&p.Y, &M).
		Sub(&p.Y, &L)
	p.ZZ.Set(&V)
	p.ZZZ.Set(&W)

	return p
}

// -------------------------------------------------------------------------------------------------
// Homogenous projective

// Set sets p to the provided point
func (p *g1Proj) Set(a *g1Proj) *g1Proj {
	p.x, p.y, p.z = a.x, a.y, a.z
	return p
}

// Neg computes -G
func (p *g1Proj) Neg(a *g1Proj) *g1Proj {
	*p = *a
	p.y.Neg(&a.y)
	return p
}

// FromAffine sets p = Q, p in homogenous projective, Q in affine
func (p *g1Proj) FromAffine(Q *G1Affine) *g1Proj {
	if Q.X.IsZero() && Q.Y.IsZero() {
		p.z.SetZero()
		p.x.SetOne()
		p.y.SetOne()
		return p
	}
	p.z.SetOne()
	p.x.Set(&Q.X)
	p.y.Set(&Q.Y)
	return p
}

// BatchProjectiveToAffineG1 converts points in Projective coordinates to Affine coordinates
// performing a single field inversion (Montgomery batch inversion trick).
func BatchProjectiveToAffineG1(points []g1Proj) []G1Affine {
	result := make([]G1Affine, len(points))
	zeroes := make([]bool, len(points))
	accumulator := fp.One()

	// batch invert all points[].Z coordinates with Montgomery batch inversion trick
	// (stores points[].Z^-1 in result[i].X to avoid allocating a slice of fr.Elements)
	for i := 0; i < len(points); i++ {
		if points[i].z.IsZero() {
			zeroes[i] = true
			continue
		}
		result[i].X = accumulator
		accumulator.Mul(&accumulator, &points[i].z)
	}

	var accInverse fp.Element
	accInverse.Inverse(&accumulator)

	for i := len(points) - 1; i >= 0; i-- {
		if zeroes[i] {
			// do nothing, (X=0, Y=0) is infinity point in affine
			continue
		}
		result[i].X.Mul(&result[i].X, &accInverse)
		accInverse.Mul(&accInverse, &points[i].z)
	}

	// batch convert to affine.
	parallel.Execute(len(points), func(start, end int) {
		for i := start; i < end; i++ {
			if zeroes[i] {
				// do nothing, (X=0, Y=0) is infinity point in affine
				continue
			}
			a := result[i].X
			result[i].X.Mul(&points[i].x, &a)
			result[i].Y.Mul(&points[i].y, &a)
		}
	})
	return result
}

// BatchJacobianToAffineG1 converts points in Jacobian coordinates to Affine coordinates
// performing a single field inversion (Montgomery batch inversion trick).
func BatchJacobianToAffineG1(points []G1Jac) []G1Affine {
	result := make([]G1Affine, len(points))
	zeroes := make([]bool, len(points))
	accumulator := fp.One()

	// batch invert all points[].Z coordinates with Montgomery batch inversion trick
	// (stores points[].Z^-1 in result[i].X to avoid allocating a slice of fr.Elements)
	for i := 0; i < len(points); i++ {
		if points[i].Z.IsZero() {
			zeroes[i] = true
			continue
		}
		result[i].X = accumulator
		accumulator.Mul(&accumulator, &points[i].Z)
	}

	var accInverse fp.Element
	accInverse.Inverse(&accumulator)

	for i := len(points) - 1; i >= 0; i-- {
		if zeroes[i] {
			// do nothing, (X=0, Y=0) is infinity point in affine
			continue
		}
		result[i].X.Mul(&result[i].X, &accInverse)
		accInverse.Mul(&accInverse, &points[i].Z)
	}

	// batch convert to affine.
	parallel.Execute(len(points), func(start, end int) {
		for i := start; i < end; i++ {
			if zeroes[i] {
				// do nothing, (X=0, Y=0) is infinity point in affine
				continue
			}
			var a, b fp.Element
			a = result[i].X
			b.Square(&a)
			result[i].X.Mul(&points[i].X, &b)
			result[i].Y.Mul(&points[i].Y, &b).
				Mul(&result[i].Y, &a)
		}
	})

	return result
}

// BatchScalarMultiplicationG1 multiplies the same base by all scalars
// and return resulting points in affine coordinates
// uses a simple windowed-NAF like exponentiation algorithm
func BatchScalarMultiplicationG1(base *G1Affine, scalars []fr.Element) []G1Affine {
	// approximate cost in group ops is
	// cost = 2^{c-1} + n(scalar.nbBits+nbChunks)

	nbPoints := uint64(len(scalars))
	min := ^uint64(0)
	bestC := 0
	for c := 2; c <= 16; c++ {
		cost := uint64(1 << (c - 1)) // pre compute the table
		nbChunks := computeNbChunks(uint64(c))
		cost += nbPoints * (uint64(c) + 1) * nbChunks // doublings + point add
		if cost < min {
			min = cost
			bestC = c
		}
	}
	c := uint64(bestC) // window size
	nbChunks := int(computeNbChunks(c))

	// last window may be slightly larger than c; in which case we need to compute one
	// extra element in the baseTable
	maxC := lastC(c)
	if c > maxC {
		maxC = c
	}

	// precompute all powers of base for our window
	// note here that if performance is critical, we can implement as in the msmX methods
	// this allocation to be on the stack
	baseTable := make([]G1Jac, (1 << (maxC - 1)))
	baseTable[0].FromAffine(base)
	for i := 1; i < len(baseTable); i++ {
		baseTable[i] = baseTable[i-1]
		baseTable[i].AddMixed(base)
	}
	// convert our base exp table into affine to use AddMixed
	baseTableAff := BatchJacobianToAffineG1(baseTable)
	toReturn := make([]G1Jac, len(scalars))

	// partition the scalars into digits
	digits, _ := partitionScalars(scalars, c, runtime.NumCPU())

	// for each digit, take value in the base table, double it c time, voilà.
	parallel.Execute(len(scalars), func(start, end int) {
		var p G1Jac
		for i := start; i < end; i++ {
			p.Set(&g1Infinity)
			for chunk := nbChunks - 1; chunk >= 0; chunk-- {
				if chunk != nbChunks-1 {
					for j := uint64(0); j < c; j++ {
						p.DoubleAssign()
					}
				}
				offset := chunk * len(scalars)
				digit := digits[i+offset]

				if digit == 0 {
					continue
				}

				// if msbWindow bit is set, we need to substract
				if digit&1 == 0 {
					// add
					p.AddMixed(&baseTableAff[(digit>>1)-1])
				} else {
					// sub
					t := baseTableAff[digit>>1]
					t.Neg(&t)
					p.AddMixed(&t)
				}
			}

			// set our result point
			toReturn[i] = p

		}
	})
	toReturnAff := BatchJacobianToAffineG1(toReturn)
	return toReturnAff
}

// batch add affine coordinates
// using batch inversion
// special cases (doubling, infinity) must be filtered out before this call
func batchAddG1Affine[TP pG1Affine, TPP ppG1Affine, TC cG1Affine](R *TPP, P *TP, batchSize int) {
	var lambda, lambdain TC

	// add part
	for j := 0; j < batchSize; j++ {
		lambdain[j].Sub(&(*P)[j].X, &(*R)[j].X)
	}

	// invert denominator using montgomery batch invert technique
	{
		var accumulator fp.Element
		lambda[0].SetOne()
		accumulator.Set(&lambdain[0])

		for i := 1; i < batchSize; i++ {
			lambda[i] = accumulator
			accumulator.Mul(&accumulator, &lambdain[i])
		}

		accumulator.Inverse(&accumulator)

		for i := batchSize - 1; i > 0; i-- {
			lambda[i].Mul(&lambda[i], &accumulator)
			accumulator.Mul(&accumulator, &lambdain[i])
		}
		lambda[0].Set(&accumulator)
	}

	var d fp.Element
	var rr G1Affine

	// add part
	for j := 0; j < batchSize; j++ {
		// computa lambda
		d.Sub(&(*P)[j].Y, &(*R)[j].Y)
		lambda[j].Mul(&lambda[j], &d)

		// compute X, Y
		rr.X.Square(&lambda[j])
		rr.X.Sub(&rr.X, &(*R)[j].X)
		rr.X.Sub(&rr.X, &(*P)[j].X)
		d.Sub(&(*R)[j].X, &rr.X)
		rr.Y.Mul(&lambda[j], &d)
		rr.Y.Sub(&rr.Y, &(*R)[j].Y)
		(*R)[j].Set(&rr)
	}
}