````// Copyright 2010 The Go Authors. All rights reserved.`
`// Use of this source code is governed by a BSD-style`
`// license that can be found in the LICENSE file.`

`// Package elliptic implements several standard elliptic curves over prime`
`// fields.``package elliptic`

`// This package operates, internally, on Jacobian coordinates. For a given`
`// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)`
`// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole`
`// calculation can be performed within the transform (as in ScalarMult and`
`// ScalarBaseMult). But even for Add and Double, it's faster to apply and`
`// reverse the transform than to operate in affine coordinates.`

`import (`
`	"io"`
`	"math/big"`
`	"sync"`
`)`

`// A Curve represents a short-form Weierstrass curve with a=-3.`
`//`
`// Note that the point at infinity (0, 0) is not considered on the curve, and`
`// although it can be returned by Add, Double, ScalarMult, or ScalarBaseMult, it`
`// can't be marshaled or unmarshaled, and IsOnCurve will return false for it.`
`type Curve interface {`
`	// Params returns the parameters for the curve.`
`	Params() *CurveParams`
`	// IsOnCurve reports whether the given (x,y) lies on the curve.`
`	IsOnCurve(x, y *big.Int) bool`
`	// Add returns the sum of (x1,y1) and (x2,y2)`
`	Add(x1, y1, x2, y2 *big.Int) (x, y *big.Int)`
`	// Double returns 2*(x,y)`
`	Double(x1, y1 *big.Int) (x, y *big.Int)`
`	// ScalarMult returns k*(Bx,By) where k is a number in big-endian form.`
`	ScalarMult(x1, y1 *big.Int, k []byte) (x, y *big.Int)`
`	// ScalarBaseMult returns k*G, where G is the base point of the group`
`	// and k is an integer in big-endian form.`
`	ScalarBaseMult(k []byte) (x, y *big.Int)`
`}`

`func matchesSpecificCurve(params *CurveParams, available ...Curve) (Curve, bool) {`
`	for _, c := range available {`
`		if params == c.Params() {`
`			return c, true`
`		}`
`	}`
`	return nil, false`
`}`

`// CurveParams contains the parameters of an elliptic curve and also provides`
`// a generic, non-constant time implementation of Curve.`
`type CurveParams struct {`
`	P       *big.Int // the order of the underlying field`
`	N       *big.Int // the order of the base point`
`	B       *big.Int // the constant of the curve equation`
`	Gx, Gy  *big.Int // (x,y) of the base point`
`	BitSize int      // the size of the underlying field`
`	Name    string   // the canonical name of the curve`
`}`

`func (curve *CurveParams) Params() *CurveParams {`
`	return curve`
`}`

`// polynomial returns x³ - 3x + b.`
`func (curve *CurveParams) polynomial(x *big.Int) *big.Int {`
`	x3 := new(big.Int).Mul(x, x)`
`	x3.Mul(x3, x)`

`	threeX := new(big.Int).Lsh(x, 1)`
`	threeX.Add(threeX, x)`

`	x3.Sub(x3, threeX)`
`	x3.Add(x3, curve.B)`
`	x3.Mod(x3, curve.P)`

`	return x3`
`}`

`func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {`
`	// If there is a dedicated constant-time implementation for this curve operation,`
`	// use that instead of the generic one.`
`	if specific, ok := matchesSpecificCurve(curve, p224, p521); ok {`
`		return specific.IsOnCurve(x, y)`
`	}`

`	// y² = x³ - 3x + b`
`	y2 := new(big.Int).Mul(y, y)`
`	y2.Mod(y2, curve.P)`

`	return curve.polynomial(x).Cmp(y2) == 0`
`}`

`// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and`
`// y are zero, it assumes that they represent the point at infinity because (0,`
`// 0) is not on the any of the curves handled here.`
`func zForAffine(x, y *big.Int) *big.Int {`
`	z := new(big.Int)`
`	if x.Sign() != 0 || y.Sign() != 0 {`
`		z.SetInt64(1)`
`	}`
`	return z`
`}`

`// affineFromJacobian reverses the Jacobian transform. See the comment at the`
`// top of the file. If the point is ∞ it returns 0, 0.`
`func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {`
`	if z.Sign() == 0 {`
`		return new(big.Int), new(big.Int)`
`	}`

`	zinv := new(big.Int).ModInverse(z, curve.P)`
`	zinvsq := new(big.Int).Mul(zinv, zinv)`

`	xOut = new(big.Int).Mul(x, zinvsq)`
`	xOut.Mod(xOut, curve.P)`
`	zinvsq.Mul(zinvsq, zinv)`
`	yOut = new(big.Int).Mul(y, zinvsq)`
`	yOut.Mod(yOut, curve.P)`
`	return`
`}`

`func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {`
`	// If there is a dedicated constant-time implementation for this curve operation,`
`	// use that instead of the generic one.`
`	if specific, ok := matchesSpecificCurve(curve, p224, p521); ok {`
`		return specific.Add(x1, y1, x2, y2)`
`	}`

`	z1 := zForAffine(x1, y1)`
`	z2 := zForAffine(x2, y2)`
`	return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))`
`}`

`// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and`
`// (x2, y2, z2) and returns their sum, also in Jacobian form.`
`func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {`
`	// See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl`
`	x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)`
`	if z1.Sign() == 0 {`
`		x3.Set(x2)`
`		y3.Set(y2)`
`		z3.Set(z2)`
`		return x3, y3, z3`
`	}`
`	if z2.Sign() == 0 {`
`		x3.Set(x1)`
`		y3.Set(y1)`
`		z3.Set(z1)`
`		return x3, y3, z3`
`	}`

`	z1z1 := new(big.Int).Mul(z1, z1)`
`	z1z1.Mod(z1z1, curve.P)`
`	z2z2 := new(big.Int).Mul(z2, z2)`
`	z2z2.Mod(z2z2, curve.P)`

`	u1 := new(big.Int).Mul(x1, z2z2)`
`	u1.Mod(u1, curve.P)`
`	u2 := new(big.Int).Mul(x2, z1z1)`
`	u2.Mod(u2, curve.P)`
`	h := new(big.Int).Sub(u2, u1)`
`	xEqual := h.Sign() == 0`
`	if h.Sign() == -1 {`
`		h.Add(h, curve.P)`
`	}`
`	i := new(big.Int).Lsh(h, 1)`
`	i.Mul(i, i)`
`	j := new(big.Int).Mul(h, i)`

`	s1 := new(big.Int).Mul(y1, z2)`
`	s1.Mul(s1, z2z2)`
`	s1.Mod(s1, curve.P)`
`	s2 := new(big.Int).Mul(y2, z1)`
`	s2.Mul(s2, z1z1)`
`	s2.Mod(s2, curve.P)`
`	r := new(big.Int).Sub(s2, s1)`
`	if r.Sign() == -1 {`
`		r.Add(r, curve.P)`
`	}`
`	yEqual := r.Sign() == 0`
`	if xEqual && yEqual {`
`		return curve.doubleJacobian(x1, y1, z1)`
`	}`
`	r.Lsh(r, 1)`
`	v := new(big.Int).Mul(u1, i)`

`	x3.Set(r)`
`	x3.Mul(x3, x3)`
`	x3.Sub(x3, j)`
`	x3.Sub(x3, v)`
`	x3.Sub(x3, v)`
`	x3.Mod(x3, curve.P)`

`	y3.Set(r)`
`	v.Sub(v, x3)`
`	y3.Mul(y3, v)`
`	s1.Mul(s1, j)`
`	s1.Lsh(s1, 1)`
`	y3.Sub(y3, s1)`
`	y3.Mod(y3, curve.P)`

`	z3.Add(z1, z2)`
`	z3.Mul(z3, z3)`
`	z3.Sub(z3, z1z1)`
`	z3.Sub(z3, z2z2)`
`	z3.Mul(z3, h)`
`	z3.Mod(z3, curve.P)`

`	return x3, y3, z3`
`}`

`func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {`
`	// If there is a dedicated constant-time implementation for this curve operation,`
`	// use that instead of the generic one.`
`	if specific, ok := matchesSpecificCurve(curve, p224, p521); ok {`
`		return specific.Double(x1, y1)`
`	}`

`	z1 := zForAffine(x1, y1)`
`	return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))`
`}`

`// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and`
`// returns its double, also in Jacobian form.`
`func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {`
`	// See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b`
`	delta := new(big.Int).Mul(z, z)`
`	delta.Mod(delta, curve.P)`
`	gamma := new(big.Int).Mul(y, y)`
`	gamma.Mod(gamma, curve.P)`
`	alpha := new(big.Int).Sub(x, delta)`
`	if alpha.Sign() == -1 {`
`		alpha.Add(alpha, curve.P)`
`	}`
`	alpha2 := new(big.Int).Add(x, delta)`
`	alpha.Mul(alpha, alpha2)`
`	alpha2.Set(alpha)`
`	alpha.Lsh(alpha, 1)`
`	alpha.Add(alpha, alpha2)`

`	beta := alpha2.Mul(x, gamma)`

`	x3 := new(big.Int).Mul(alpha, alpha)`
`	beta8 := new(big.Int).Lsh(beta, 3)`
`	beta8.Mod(beta8, curve.P)`
`	x3.Sub(x3, beta8)`
`	if x3.Sign() == -1 {`
`		x3.Add(x3, curve.P)`
`	}`
`	x3.Mod(x3, curve.P)`

`	z3 := new(big.Int).Add(y, z)`
`	z3.Mul(z3, z3)`
`	z3.Sub(z3, gamma)`
`	if z3.Sign() == -1 {`
`		z3.Add(z3, curve.P)`
`	}`
`	z3.Sub(z3, delta)`
`	if z3.Sign() == -1 {`
`		z3.Add(z3, curve.P)`
`	}`
`	z3.Mod(z3, curve.P)`

`	beta.Lsh(beta, 2)`
`	beta.Sub(beta, x3)`
`	if beta.Sign() == -1 {`
`		beta.Add(beta, curve.P)`
`	}`
`	y3 := alpha.Mul(alpha, beta)`

`	gamma.Mul(gamma, gamma)`
`	gamma.Lsh(gamma, 3)`
`	gamma.Mod(gamma, curve.P)`

`	y3.Sub(y3, gamma)`
`	if y3.Sign() == -1 {`
`		y3.Add(y3, curve.P)`
`	}`
`	y3.Mod(y3, curve.P)`

`	return x3, y3, z3`
`}`

`func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {`
`	// If there is a dedicated constant-time implementation for this curve operation,`
`	// use that instead of the generic one.`
`	if specific, ok := matchesSpecificCurve(curve, p224, p256, p521); ok {`
`		return specific.ScalarMult(Bx, By, k)`
`	}`

`	Bz := new(big.Int).SetInt64(1)`
`	x, y, z := new(big.Int), new(big.Int), new(big.Int)`

`	for _, byte := range k {`
`		for bitNum := 0; bitNum < 8; bitNum++ {`
`			x, y, z = curve.doubleJacobian(x, y, z)`
`			if byte&0x80 == 0x80 {`
`				x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)`
`			}`
`			byte <<= 1`
`		}`
`	}`

`	return curve.affineFromJacobian(x, y, z)`
`}`

`func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {`
`	// If there is a dedicated constant-time implementation for this curve operation,`
`	// use that instead of the generic one.`
`	if specific, ok := matchesSpecificCurve(curve, p224, p256, p521); ok {`
`		return specific.ScalarBaseMult(k)`
`	}`

`	return curve.ScalarMult(curve.Gx, curve.Gy, k)`
`}`

`var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}`

`// GenerateKey returns a public/private key pair. The private key is`
`// generated using the given reader, which must return random data.`
`func GenerateKey(curve Curve, rand io.Reader) (priv []byte, x, y *big.Int, err error) {`
`	N := curve.Params().N`
`	bitSize := N.BitLen()`
`	byteLen := (bitSize + 7) / 8`
`	priv = make([]byte, byteLen)`

`	for x == nil {`
`		_, err = io.ReadFull(rand, priv)`
`		if err != nil {`
`			return`
`		}`
`		// We have to mask off any excess bits in the case that the size of the`
`		// underlying field is not a whole number of bytes.`
`		priv[0] &= mask[bitSize%8]`
`		// This is because, in tests, rand will return all zeros and we don't`
`		// want to get the point at infinity and loop forever.`
`		priv[1] ^= 0x42`

`		// If the scalar is out of range, sample another random number.`
`		if new(big.Int).SetBytes(priv).Cmp(N) >= 0 {`
`			continue`
`		}`

`		x, y = curve.ScalarBaseMult(priv)`
`	}`
`	return`
`}`

`// Marshal converts a point on the curve into the uncompressed form specified in`
`// section 4.3.6 of ANSI X9.62.`
`func Marshal(curve Curve, x, y *big.Int) []byte {`
`	byteLen := (curve.Params().BitSize + 7) / 8`

`	ret := make([]byte, 1+2*byteLen)`
`	ret[0] = 4 // uncompressed point`

`	x.FillBytes(ret[1 : 1+byteLen])`
`	y.FillBytes(ret[1+byteLen : 1+2*byteLen])`

`	return ret`
`}`

`// MarshalCompressed converts a point on the curve into the compressed form`
`// specified in section 4.3.6 of ANSI X9.62.`
`func MarshalCompressed(curve Curve, x, y *big.Int) []byte {`
`	byteLen := (curve.Params().BitSize + 7) / 8`
`	compressed := make([]byte, 1+byteLen)`
`	compressed[0] = byte(y.Bit(0)) | 2`
`	x.FillBytes(compressed[1:])`
`	return compressed`
`}`

`// Unmarshal converts a point, serialized by Marshal, into an x, y pair.`
`// It is an error if the point is not in uncompressed form or is not on the curve.`
`// On error, x = nil.`
`func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {`
`	byteLen := (curve.Params().BitSize + 7) / 8`
`	if len(data) != 1+2*byteLen {`
`		return nil, nil`
`	}`
`	if data[0] != 4 { // uncompressed form`
`		return nil, nil`
`	}`
`	p := curve.Params().P`
`	x = new(big.Int).SetBytes(data[1 : 1+byteLen])`
`	y = new(big.Int).SetBytes(data[1+byteLen:])`
`	if x.Cmp(p) >= 0 || y.Cmp(p) >= 0 {`
`		return nil, nil`
`	}`
`	if !curve.IsOnCurve(x, y) {`
`		return nil, nil`
`	}`
`	return`
`}`

`// UnmarshalCompressed converts a point, serialized by MarshalCompressed, into an x, y pair.`
`// It is an error if the point is not in compressed form or is not on the curve.`
`// On error, x = nil.`
`func UnmarshalCompressed(curve Curve, data []byte) (x, y *big.Int) {`
`	byteLen := (curve.Params().BitSize + 7) / 8`
`	if len(data) != 1+byteLen {`
`		return nil, nil`
`	}`
`	if data[0] != 2 && data[0] != 3 { // compressed form`
`		return nil, nil`
`	}`
`	p := curve.Params().P`
`	x = new(big.Int).SetBytes(data[1:])`
`	if x.Cmp(p) >= 0 {`
`		return nil, nil`
`	}`
`	// y² = x³ - 3x + b`
`	y = curve.Params().polynomial(x)`
`	y = y.ModSqrt(y, p)`
`	if y == nil {`
`		return nil, nil`
`	}`
`	if byte(y.Bit(0)) != data[0]&1 {`
`		y.Neg(y).Mod(y, p)`
`	}`
`	if !curve.IsOnCurve(x, y) {`
`		return nil, nil`
`	}`
`	return`
`}`

`var initonce sync.Once`
`var p384 *CurveParams`

`func initAll() {`
`	initP224()`
`	initP256()`
`	initP384()`
`	initP521()`
`}`

`func initP384() {`
`	// See FIPS 186-3, section D.2.4`
`	p384 = &CurveParams{Name: "P-384"}`
`	p384.P, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319", 10)`
`	p384.N, _ = new(big.Int).SetString("39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643", 10)`
`	p384.B, _ = new(big.Int).SetString("b3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef", 16)`
`	p384.Gx, _ = new(big.Int).SetString("aa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7", 16)`
`	p384.Gy, _ = new(big.Int).SetString("3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f", 16)`
`	p384.BitSize = 384`
`}`

`// P256 returns a Curve which implements NIST P-256 (FIPS 186-3, section D.2.3),`
`// also known as secp256r1 or prime256v1. The CurveParams.Name of this Curve is`
`// "P-256".`
`//`
`// Multiple invocations of this function will return the same value, so it can`
`// be used for equality checks and switch statements.`
`//`
`// ScalarMult and ScalarBaseMult are implemented using constant-time algorithms.`
`func P256() Curve {`
`	initonce.Do(initAll)`
`	return p256`
`}`

`// P384 returns a Curve which implements NIST P-384 (FIPS 186-3, section D.2.4),`
`// also known as secp384r1. The CurveParams.Name of this Curve is "P-384".`
`//`
`// Multiple invocations of this function will return the same value, so it can`
`// be used for equality checks and switch statements.`
`//`
`// The cryptographic operations do not use constant-time algorithms.`
`func P384() Curve {`
`	initonce.Do(initAll)`
`	return p384`
`}`

`// P521 returns a Curve which implements NIST P-521 (FIPS 186-3, section D.2.5),`
`// also known as secp521r1. The CurveParams.Name of this Curve is "P-521".`
`//`
`// Multiple invocations of this function will return the same value, so it can`
`// be used for equality checks and switch statements.`
`//`
`// The cryptographic operations are implemented using constant-time algorithms.`
`func P521() Curve {`
`	initonce.Do(initAll)`
`	return p521`
`}`
```