Source File
	elliptic.go

Belonging Package
	crypto/elliptic

// 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 the standard NIST P-224, P-256, P-384, and P-521
// elliptic curves over prime fields.
package elliptic

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

// A Curve represents a short-form Weierstrass curve with a=-3.
//
// The behavior of Add, Double, and ScalarMult when the input is not a point on
// the curve is undefined.
//
// Note that the conventional point at infinity (0, 0) is not considered on the
// curve, although it can be returned by Add, Double, ScalarMult, or
// ScalarBaseMult (but not the Unmarshal or UnmarshalCompressed functions).
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
}

// CurveParams 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.

// 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, p384, p521); ok {
		return specific.IsOnCurve(x, y)
	}

	if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
		y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
		return false
	}

	// 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, p384, 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, p384, 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, p384, 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, p384, 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
// SEC 1, Version 2.0, Section 2.3.3. If the point is not on the curve (or is
// the conventional point at infinity), the behavior is undefined.
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 SEC 1, Version 2.0, Section 2.3.3. If the point is not on the
// curve (or is the conventional point at infinity), the behavior is undefined.
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, is not on the curve, or is
// the point at infinity. 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, is not
// on the curve, or is the point at infinity. 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

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

// P224 returns a Curve which implements NIST P-224 (FIPS 186-3, section D.2.2),
// also known as secp224r1. The CurveParams.Name of this Curve is "P-224".
//
// 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 P224() Curve {
	initonce.Do(initAll)
	return p224
}

// 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 are implemented using 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
}

The pages are generated with Golds v0.4.9. (GOOS=linux GOARCH=amd64)