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 // 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. // // Note that the point at infinity (0, 0) is not considered on the curve, and // although it can be returned by Add, Double, ScalarMult, or ScalarBaseMult, it // can't be marshaled or unmarshaled, and IsOnCurve will return false for it. 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) } // 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 } // 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 { // 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) { 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) { 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) { 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) { 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 // section 4.3.6 of ANSI X9.62. 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 section 4.3.6 of ANSI X9.62. 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 or is not on the curve. // 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 or is not on the curve. // 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 var p384 *CurveParams var p521 *CurveParams func initAll() { initP224() initP256() initP384() initP521() } func initP384() { // See FIPS 186-3, section D.2.4 p384 = &CurveParams{Name: "P-384"} 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 = &CurveParams{Name: "P-521"} 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 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. // // The cryptographic operations 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 do not use 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 do not use constant-time algorithms. func P521() Curve { initonce.Do(initAll) return p521 }