src/crypto/elliptic/elliptic.go - go - Git at Google

 // 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.
 // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
 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)
 }

 // 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
 }

 func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
 	// y² = x³ - 3x + b
 	y2 := new(big.Int).Mul(y, y)
 	y2.Mod(y2, curve.P)

 	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.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) {
 	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 http://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) {
 	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 http://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)
 	x3.Sub(x3, beta8)
 	for 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) {
 	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) {
 	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) >> 3
 	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 into the 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) >> 3

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

 	xBytes := x.Bytes()
 	copy(ret[1+byteLen-len(xBytes):], xBytes)
 	yBytes := y.Bytes()
 	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
 	return ret
 }

 // Unmarshal converts a point, serialized by Marshal, into an x, y pair.
 // It is an error if the point is not on the curve. On error, x = nil.
 func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {
 	byteLen := (curve.Params().BitSize + 7) >> 3
 	if len(data) != 1+2*byteLen {
 		return
 	}
 	if data[0] != 4 { // uncompressed form
 		return
 	}
 	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
 	y = new(big.Int).SetBytes(data[1+byteLen:])
 	if !curve.IsOnCurve(x, y) {
 		x, y = nil, nil
 	}
 	return
 }

 var initonce sync.Once
 var p384 *CurveParams
 var p521 *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
 }

 func initP521() {
 	// See FIPS 186-3, section D.2.5
 	p521 = &CurveParams{Name: "P-521"}
 	p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
 	p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
 	p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
 	p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
 	p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
 	p521.BitSize = 521
 }

 // P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3)
 //
 // The cryptographic operations are implemented using constant-time algorithms.
 func P256() Curve {
 	initonce.Do(initAll)
 	return p256
 }

 // P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)
 //
 // The cryptographic operations do not use constant-time algorithms.
 func P384() Curve {
 	initonce.Do(initAll)
 	return p384
 }

 // P521 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)
 //
 // The cryptographic operations do not use constant-time algorithms.
 func P521() Curve {
 	initonce.Do(initAll)
 	return p521
 }
	// 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.
	// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html
	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)
	}

	// 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
	}

	func (curve CurveParams) IsOnCurve(x, y big.Int) bool {
	// y² = x³ - 3x + b
	y2 := new(big.Int).Mul(y, y)
	y2.Mod(y2, curve.P)

	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.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) {
	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 http://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) {
	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 http://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)
	x3.Sub(x3, beta8)
	for 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) {
	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) {
	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) >> 3
	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 into the 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) >> 3

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

	xBytes := x.Bytes()
	copy(ret[1+byteLen-len(xBytes):], xBytes)
	yBytes := y.Bytes()
	copy(ret[1+2*byteLen-len(yBytes):], yBytes)
	return ret
	}

	// Unmarshal converts a point, serialized by Marshal, into an x, y pair.
	// It is an error if the point is not on the curve. On error, x = nil.
	func Unmarshal(curve Curve, data []byte) (x, y *big.Int) {
	byteLen := (curve.Params().BitSize + 7) >> 3
	if len(data) != 1+2*byteLen {
	return
	}
	if data[0] != 4 { // uncompressed form
	return
	}
	x = new(big.Int).SetBytes(data[1 : 1+byteLen])
	y = new(big.Int).SetBytes(data[1+byteLen:])
	if !curve.IsOnCurve(x, y) {
	x, y = nil, nil
	}
	return
	}

	var initonce sync.Once
	var p384 *CurveParams
	var p521 *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
	}

	func initP521() {
	// See FIPS 186-3, section D.2.5
	p521 = &CurveParams{Name: "P-521"}
	p521.P, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151", 10)
	p521.N, _ = new(big.Int).SetString("6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449", 10)
	p521.B, _ = new(big.Int).SetString("051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00", 16)
	p521.Gx, _ = new(big.Int).SetString("c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66", 16)
	p521.Gy, _ = new(big.Int).SetString("11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650", 16)
	p521.BitSize = 521
	}

	// P256 returns a Curve which implements P-256 (see FIPS 186-3, section D.2.3)
	//
	// The cryptographic operations are implemented using constant-time algorithms.
	func P256() Curve {
	initonce.Do(initAll)
	return p256
	}

	// P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)
	//
	// The cryptographic operations do not use constant-time algorithms.
	func P384() Curve {
	initonce.Do(initAll)
	return p384
	}

	// P521 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)
	//
	// The cryptographic operations do not use constant-time algorithms.
	func P521() Curve {
	initonce.Do(initAll)
	return p521
	}