// Copyright 2021 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 import "math/big" // 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 } // CurveParams 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. // 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 { // If there is a dedicated constant-time implementation for this curve operation, // use that instead of the generic one. if specific, ok := matchesSpecificCurve(curve); ok { return specific.IsOnCurve(x, y) } if x.Sign() < 0 || x.Cmp(curve.P) >= 0 || y.Sign() < 0 || y.Cmp(curve.P) >= 0 { return false } // 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) { // If there is a dedicated constant-time implementation for this curve operation, // use that instead of the generic one. if specific, ok := matchesSpecificCurve(curve); ok { return specific.Add(x1, y1, x2, y2) } panicIfNotOnCurve(curve, x1, y1) panicIfNotOnCurve(curve, x2, y2) 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) { // If there is a dedicated constant-time implementation for this curve operation, // use that instead of the generic one. if specific, ok := matchesSpecificCurve(curve); ok { return specific.Double(x1, y1) } panicIfNotOnCurve(curve, x1, y1) 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) { // If there is a dedicated constant-time implementation for this curve operation, // use that instead of the generic one. if specific, ok := matchesSpecificCurve(curve); ok { return specific.ScalarMult(Bx, By, k) } panicIfNotOnCurve(curve, Bx, By) 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) { // If there is a dedicated constant-time implementation for this curve operation, // use that instead of the generic one. if specific, ok := matchesSpecificCurve(curve); ok { return specific.ScalarBaseMult(k) } return curve.ScalarMult(curve.Gx, curve.Gy, k) } func matchesSpecificCurve(params *CurveParams) (Curve, bool) { for _, c := range []Curve{p224, p256, p384, p521} { if params == c.Params() { return c, true } } return nil, false }