package bitcurves // 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 { Name string 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 } // Params returns the parameters of the given BitCurve (see BitCurve struct) func (bitCurve *BitCurve) Params() (cp *elliptic.CurveParams) { cp = new(elliptic.CurveParams) cp.Name = bitCurve.Name cp.P = bitCurve.P cp.N = bitCurve.N cp.Gx = bitCurve.Gx cp.Gy = bitCurve.Gy cp.BitSize = bitCurve.BitSize return cp } // IsOnCurve 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 } // 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) { if z.Cmp(big.NewInt(0)) == 0 { panic("bitcurve: Can't convert to affine with Jacobian Z = 0") } // x = YZ^2 mod P 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) // y = YZ^3 mod P zinvsq.Mul(zinvsq, zinv) yOut = new(big.Int).Mul(y, zinvsq) yOut.Mod(yOut, bitCurve.P) return xOut, yOut } // 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) x, y, z := bitCurve.addJacobian(x1, y1, z, x2, y2, z) return bitCurve.affineFromJacobian(x, y, 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 secp160k1 *BitCurve var secp192k1 *BitCurve var secp224k1 *BitCurve var secp256k1 *BitCurve func initAll() { initS160() initS192() initS224() initS256() } func initS160() { // See SEC 2 section 2.4.1 secp160k1 = new(BitCurve) secp160k1.Name = "secp160k1" secp160k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFAC73", 16) secp160k1.N, _ = new(big.Int).SetString("0100000000000000000001B8FA16DFAB9ACA16B6B3", 16) secp160k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000007", 16) secp160k1.Gx, _ = new(big.Int).SetString("3B4C382CE37AA192A4019E763036F4F5DD4D7EBB", 16) secp160k1.Gy, _ = new(big.Int).SetString("938CF935318FDCED6BC28286531733C3F03C4FEE", 16) secp160k1.BitSize = 160 } func initS192() { // See SEC 2 section 2.5.1 secp192k1 = new(BitCurve) secp192k1.Name = "secp192k1" secp192k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFEE37", 16) secp192k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFE26F2FC170F69466A74DEFD8D", 16) secp192k1.B, _ = new(big.Int).SetString("000000000000000000000000000000000000000000000003", 16) secp192k1.Gx, _ = new(big.Int).SetString("DB4FF10EC057E9AE26B07D0280B7F4341DA5D1B1EAE06C7D", 16) secp192k1.Gy, _ = new(big.Int).SetString("9B2F2F6D9C5628A7844163D015BE86344082AA88D95E2F9D", 16) secp192k1.BitSize = 192 } func initS224() { // See SEC 2 section 2.6.1 secp224k1 = new(BitCurve) secp224k1.Name = "secp224k1" secp224k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFE56D", 16) secp224k1.N, _ = new(big.Int).SetString("010000000000000000000000000001DCE8D2EC6184CAF0A971769FB1F7", 16) secp224k1.B, _ = new(big.Int).SetString("00000000000000000000000000000000000000000000000000000005", 16) secp224k1.Gx, _ = new(big.Int).SetString("A1455B334DF099DF30FC28A169A467E9E47075A90F7E650EB6B7A45C", 16) secp224k1.Gy, _ = new(big.Int).SetString("7E089FED7FBA344282CAFBD6F7E319F7C0B0BD59E2CA4BDB556D61A5", 16) secp224k1.BitSize = 224 } func initS256() { // See SEC 2 section 2.7.1 secp256k1 = new(BitCurve) secp256k1.Name = "secp256k1" secp256k1.P, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F", 16) secp256k1.N, _ = new(big.Int).SetString("FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141", 16) secp256k1.B, _ = new(big.Int).SetString("0000000000000000000000000000000000000000000000000000000000000007", 16) secp256k1.Gx, _ = new(big.Int).SetString("79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798", 16) secp256k1.Gy, _ = new(big.Int).SetString("483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8", 16) secp256k1.BitSize = 256 } // S160 returns a BitCurve which implements secp160k1 (see SEC 2 section 2.4.1) func S160() *BitCurve { initonce.Do(initAll) return secp160k1 } // S192 returns a BitCurve which implements secp192k1 (see SEC 2 section 2.5.1) func S192() *BitCurve { initonce.Do(initAll) return secp192k1 } // S224 returns a BitCurve which implements secp224k1 (see SEC 2 section 2.6.1) func S224() *BitCurve { initonce.Do(initAll) return secp224k1 } // S256 returns a BitCurve which implements bitcurves (see SEC 2 section 2.7.1) func S256() *BitCurve { initonce.Do(initAll) return secp256k1 }