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-rw-r--r--crypto/curve.go363
1 files changed, 363 insertions, 0 deletions
diff --git a/crypto/curve.go b/crypto/curve.go
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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
+}