diff options
Diffstat (limited to 'crypto')
| -rw-r--r-- | crypto/curve.go | 363 |
1 files changed, 363 insertions, 0 deletions
diff --git a/crypto/curve.go b/crypto/curve.go new file mode 100644 index 000000000..131a0dd2f --- /dev/null +++ b/crypto/curve.go @@ -0,0 +1,363 @@ +package crypto + +// Copyright 2010 The Go Authors. All rights reserved. +// Copyright 2011 ThePiachu. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +// Package bitelliptic implements several Koblitz elliptic curves over prime +// fields. + +// 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 ( + "crypto/elliptic" + "io" + "math/big" + "sync" +) + +// A BitCurve represents a Koblitz Curve with a=0. +// See http://www.hyperelliptic.org/EFD/g1p/auto-shortw.html +type BitCurve 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 BitCurve equation + Gx, Gy *big.Int // (x,y) of the base point + BitSize int // the size of the underlying field +} + +func (BitCurve *BitCurve) Params() *elliptic.CurveParams { + return &elliptic.CurveParams{BitCurve.P, BitCurve.N, BitCurve.B, BitCurve.Gx, BitCurve.Gy, BitCurve.BitSize} +} + +// IsOnBitCurve returns true if the given (x,y) lies on the BitCurve. +func (BitCurve *BitCurve) IsOnCurve(x, y *big.Int) bool { + // y² = x³ + b + y2 := new(big.Int).Mul(y, y) //y² + y2.Mod(y2, BitCurve.P) //y²%P + + x3 := new(big.Int).Mul(x, x) //x² + x3.Mul(x3, x) //x³ + + x3.Add(x3, BitCurve.B) //x³+B + x3.Mod(x3, BitCurve.P) //(x³+B)%P + + return x3.Cmp(y2) == 0 +} + +//TODO: double check if the function is okay +// affineFromJacobian reverses the Jacobian transform. See the comment at the +// top of the file. +func (BitCurve *BitCurve) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) { + zinv := new(big.Int).ModInverse(z, BitCurve.P) + zinvsq := new(big.Int).Mul(zinv, zinv) + + xOut = new(big.Int).Mul(x, zinvsq) + xOut.Mod(xOut, BitCurve.P) + zinvsq.Mul(zinvsq, zinv) + yOut = new(big.Int).Mul(y, zinvsq) + yOut.Mod(yOut, BitCurve.P) + return +} + +// Add returns the sum of (x1,y1) and (x2,y2) +func (BitCurve *BitCurve) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) { + z := new(big.Int).SetInt64(1) + return BitCurve.affineFromJacobian(BitCurve.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 (BitCurve *BitCurve) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) { + // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl + z1z1 := new(big.Int).Mul(z1, z1) + z1z1.Mod(z1z1, BitCurve.P) + z2z2 := new(big.Int).Mul(z2, z2) + z2z2.Mod(z2z2, BitCurve.P) + + u1 := new(big.Int).Mul(x1, z2z2) + u1.Mod(u1, BitCurve.P) + u2 := new(big.Int).Mul(x2, z1z1) + u2.Mod(u2, BitCurve.P) + h := new(big.Int).Sub(u2, u1) + if h.Sign() == -1 { + h.Add(h, BitCurve.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, BitCurve.P) + s2 := new(big.Int).Mul(y2, z1) + s2.Mul(s2, z1z1) + s2.Mod(s2, BitCurve.P) + r := new(big.Int).Sub(s2, s1) + if r.Sign() == -1 { + r.Add(r, BitCurve.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, BitCurve.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, BitCurve.P) + + z3 := new(big.Int).Add(z1, z2) + z3.Mul(z3, z3) + z3.Sub(z3, z1z1) + if z3.Sign() == -1 { + z3.Add(z3, BitCurve.P) + } + z3.Sub(z3, z2z2) + if z3.Sign() == -1 { + z3.Add(z3, BitCurve.P) + } + z3.Mul(z3, h) + z3.Mod(z3, BitCurve.P) + + return x3, y3, z3 +} + +// Double returns 2*(x,y) +func (BitCurve *BitCurve) Double(x1, y1 *big.Int) (*big.Int, *big.Int) { + z1 := new(big.Int).SetInt64(1) + return BitCurve.affineFromJacobian(BitCurve.doubleJacobian(x1, y1, z1)) +} + +// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and +// returns its double, also in Jacobian form. +func (BitCurve *BitCurve) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) { + // See http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l + + a := new(big.Int).Mul(x, x) //X1² + b := new(big.Int).Mul(y, y) //Y1² + c := new(big.Int).Mul(b, b) //B² + + d := new(big.Int).Add(x, b) //X1+B + d.Mul(d, d) //(X1+B)² + d.Sub(d, a) //(X1+B)²-A + d.Sub(d, c) //(X1+B)²-A-C + d.Mul(d, big.NewInt(2)) //2*((X1+B)²-A-C) + + e := new(big.Int).Mul(big.NewInt(3), a) //3*A + f := new(big.Int).Mul(e, e) //E² + + x3 := new(big.Int).Mul(big.NewInt(2), d) //2*D + x3.Sub(f, x3) //F-2*D + x3.Mod(x3, BitCurve.P) + + y3 := new(big.Int).Sub(d, x3) //D-X3 + y3.Mul(e, y3) //E*(D-X3) + y3.Sub(y3, new(big.Int).Mul(big.NewInt(8), c)) //E*(D-X3)-8*C + y3.Mod(y3, BitCurve.P) + + z3 := new(big.Int).Mul(y, z) //Y1*Z1 + z3.Mul(big.NewInt(2), z3) //3*Y1*Z1 + z3.Mod(z3, BitCurve.P) + + return x3, y3, z3 +} + +//TODO: double check if it is okay +// ScalarMult returns k*(Bx,By) where k is a number in big-endian form. +func (BitCurve *BitCurve) 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 = BitCurve.doubleJacobian(x, y, z) + } + if byte&0x80 == 0x80 { + if !seenFirstTrue { + seenFirstTrue = true + } else { + x, y, z = BitCurve.addJacobian(Bx, By, Bz, x, y, z) + } + } + byte <<= 1 + } + } + + if !seenFirstTrue { + return nil, nil + } + + return BitCurve.affineFromJacobian(x, y, z) +} + +// ScalarBaseMult returns k*G, where G is the base point of the group and k is +// an integer in big-endian form. +func (BitCurve *BitCurve) ScalarBaseMult(k []byte) (*big.Int, *big.Int) { + return BitCurve.ScalarMult(BitCurve.Gx, BitCurve.Gy, k) +} + +var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f} + +//TODO: double check if it is okay +// GenerateKey returns a public/private key pair. The private key is generated +// using the given reader, which must return random data. +func (BitCurve *BitCurve) GenerateKey(rand io.Reader) (priv []byte, x, y *big.Int, err error) { + byteLen := (BitCurve.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[BitCurve.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 = BitCurve.ScalarBaseMult(priv) + } + return +} + +// Marshal converts a point into the form specified in section 4.3.6 of ANSI +// X9.62. +func (BitCurve *BitCurve) Marshal(x, y *big.Int) []byte { + byteLen := (BitCurve.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, serialised by Marshal, into an x, y pair. On +// error, x = nil. +func (BitCurve *BitCurve) Unmarshal(data []byte) (x, y *big.Int) { + byteLen := (BitCurve.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 +} + +//curve parameters taken from: +//http://www.secg.org/collateral/sec2_final.pdf + +var initonce sync.Once +var ecp160k1 *BitCurve +var ecp192k1 *BitCurve +var ecp224k1 *BitCurve +var ecp256k1 *BitCurve + +func initAll() { + initS160() + initS192() + initS224() + initS256() +} + +func initS160() { + // See SEC 2 section 2.4.1 + ecp160k1 = new(BitCurve) + ecp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16) + ecp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16) + ecp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16) + ecp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16) + ecp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16) + ecp160k1.BitSize = 160 +} + +func initS192() { + // See SEC 2 section 2.5.1 + ecp192k1 = new(BitCurve) + ecp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16) + ecp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16) + ecp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16) + ecp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16) + ecp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16) + ecp192k1.BitSize = 192 +} + +func initS224() { + // See SEC 2 section 2.6.1 + ecp224k1 = new(BitCurve) + ecp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16) + ecp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16) + ecp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16) + ecp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16) + ecp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16) + ecp224k1.BitSize = 224 +} + +func initS256() { + // See SEC 2 section 2.7.1 + ecp256k1 = new(BitCurve) + ecp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16) + ecp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16) + ecp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16) + ecp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16) + ecp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16) + ecp256k1.BitSize = 256 +} + +// S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1) +func S160() *BitCurve { + initonce.Do(initAll) + return ecp160k1 +} + +// S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1) +func S192() *BitCurve { + initonce.Do(initAll) + return ecp192k1 +} + +// S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1) +func S224() *BitCurve { + initonce.Do(initAll) + return ecp224k1 +} + +// S256 returns a BitCurve which implements secp256k1 (see SEC 2 section 2.7.1) +func S256() *BitCurve { + initonce.Do(initAll) + return ecp256k1 +} |