Subversion Repositories openrisc

// 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 returns true if 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, scalar []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(scalar []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

}

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

}

// affineFromJacobian reverses the Jacobian transform. See the comment at the

// top of the file.

func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *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) {

        z := new(big.Int).SetInt64(1)

        return curve.affineFromJacobian(curve.addJacobian(x1, y1, z, x2, y2, z))

}

// 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

        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)

        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)

        }

        r.Lsh(r, 1)

        v := new(big.Int).Mul(u1, i)

        x3 := new(big.Int).Set(r)

        x3.Mul(x3, x3)

        x3.Sub(x3, j)

        x3.Sub(x3, v)

        x3.Sub(x3, v)

        x3.Mod(x3, curve.P)

        y3 := new(big.Int).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 := new(big.Int).Add(z1, z2)

        z3.Mul(z3, z3)

        z3.Sub(z3, z1z1)

        if z3.Sign() == -1 {

                z3.Add(z3, curve.P)

        }

        z3.Sub(z3, z2z2)

        if z3.Sign() == -1 {

                z3.Add(z3, curve.P)

        }

        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 := new(big.Int).SetInt64(1)

        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) {

        // We have a slight problem in that the identity of the group (the

        // point at infinity) cannot be represented in (x, y) form on a finite

        // machine. Thus the standard add/double algorithm has to be tweaked

        // slightly: our initial state is not the identity, but x, and we

        // ignore the first true bit in |k|.  If we don't find any true bits in

        // |k|, then we return nil, nil, because we cannot return the identity

        // element.

        Bz := new(big.Int).SetInt64(1)

        x := Bx

        y := By

        z := Bz

        seenFirstTrue := false

        for _, byte := range k {

                for bitNum := 0; bitNum < 8; bitNum++ {

                        if seenFirstTrue {

                                x, y, z = curve.doubleJacobian(x, y, z)

                        }

                        if byte&0x80 == 0x80 {

                                if !seenFirstTrue {

                                        seenFirstTrue = true

                                } else {

                                        x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)

                                }

                        }

                        byte <<= 1

                }

        }

        if !seenFirstTrue {

                return nil, nil

        }

        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) {

        bitSize := curve.Params().BitSize

        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

                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. 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:])

        return

}

var initonce sync.Once

var p256 *CurveParams

var p384 *CurveParams

var p521 *CurveParams

func initAll() {

        initP224()

        initP256()

        initP384()

        initP521()

}

func initP256() {

        // See FIPS 186-3, section D.2.3

        p256 = new(CurveParams)

        p256.P, _ = new(big.Int).SetString("115792089210356248762697446949407573530086143415290314195533631308867097853951", 10)

        p256.N, _ = new(big.Int).SetString("115792089210356248762697446949407573529996955224135760342422259061068512044369", 10)

        p256.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", 16)

        p256.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", 16)

        p256.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5", 16)

        p256.BitSize = 256

}

func initP384() {

        // See FIPS 186-3, section D.2.4

        p384 = new(CurveParams)

        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 = new(CurveParams)

        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)

func P256() Curve {

        initonce.Do(initAll)

        return p256

}

// P384 returns a Curve which implements P-384 (see FIPS 186-3, section D.2.4)

func P384() Curve {

        initonce.Do(initAll)

        return p384

}

// P256 returns a Curve which implements P-521 (see FIPS 186-3, section D.2.5)

func P521() Curve {

        initonce.Do(initAll)

        return p521

}