diff --git a/src/ec.js b/src/ec.js index 2981817..894c53e 100644 --- a/src/ec.js +++ b/src/ec.js @@ -9,113 +9,117 @@ var BigInteger = require('bigi') var THREE = BigInteger.valueOf(3) function ECFieldElementFp(q,x) { - this.x = x; + this.x = x // TODO if (x.compareTo(q) >= 0) error - this.q = q; + this.q = q } function feFpEquals(other) { - if (other == this) return true; - return (this.q.equals(other.q) && this.x.equals(other.x)); + if (other == this) return true + return (this.q.equals(other.q) && this.x.equals(other.x)) } function feFpToBigInteger() { - return this.x; + return this.x } function feFpNegate() { - return new ECFieldElementFp(this.q, this.x.negate().mod(this.q)); + return new ECFieldElementFp(this.q, this.x.negate().mod(this.q)) } function feFpAdd(b) { - return new ECFieldElementFp(this.q, this.x.add(b.toBigInteger()).mod(this.q)); + return new ECFieldElementFp(this.q, this.x.add(b.toBigInteger()).mod(this.q)) } function feFpSubtract(b) { - return new ECFieldElementFp(this.q, this.x.subtract(b.toBigInteger()).mod(this.q)); + return new ECFieldElementFp(this.q, this.x.subtract(b.toBigInteger()).mod(this.q)) } function feFpMultiply(b) { - return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger()).mod(this.q)); + return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger()).mod(this.q)) } function feFpSquare() { - return new ECFieldElementFp(this.q, this.x.square().mod(this.q)); + return new ECFieldElementFp(this.q, this.x.square().mod(this.q)) } function feFpDivide(b) { - return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger().modInverse(this.q)).mod(this.q)); + return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger().modInverse(this.q)).mod(this.q)) } -ECFieldElementFp.prototype.equals = feFpEquals; -ECFieldElementFp.prototype.toBigInteger = feFpToBigInteger; -ECFieldElementFp.prototype.negate = feFpNegate; -ECFieldElementFp.prototype.add = feFpAdd; -ECFieldElementFp.prototype.subtract = feFpSubtract; -ECFieldElementFp.prototype.multiply = feFpMultiply; -ECFieldElementFp.prototype.square = feFpSquare; -ECFieldElementFp.prototype.divide = feFpDivide; +ECFieldElementFp.prototype.equals = feFpEquals +ECFieldElementFp.prototype.toBigInteger = feFpToBigInteger +ECFieldElementFp.prototype.negate = feFpNegate +ECFieldElementFp.prototype.add = feFpAdd +ECFieldElementFp.prototype.subtract = feFpSubtract +ECFieldElementFp.prototype.multiply = feFpMultiply +ECFieldElementFp.prototype.square = feFpSquare +ECFieldElementFp.prototype.divide = feFpDivide // ---------------- // ECPointFp // constructor function ECPointFp(curve,x,y,z) { - this.curve = curve; - this.x = x; - this.y = y; + this.curve = curve + this.x = x + this.y = y + // Projective coordinates: either zinv == null or z * zinv == 1 // z and zinv are just BigIntegers, not fieldElements if (z == null) { - this.z = BigInteger.ONE; - } - else { - this.z = z; + this.z = BigInteger.ONE + + } else { + this.z = z } - this.zinv = null; + this.zinv = null + //TODO: compression flag } function pointFpGetX() { if (this.zinv == null) { - this.zinv = this.z.modInverse(this.curve.q); + this.zinv = this.z.modInverse(this.curve.q) } - return this.curve.fromBigInteger(this.x.toBigInteger().multiply(this.zinv).mod(this.curve.q)); + + return this.curve.fromBigInteger(this.x.toBigInteger().multiply(this.zinv).mod(this.curve.q)) } function pointFpGetY() { if (this.zinv == null) { - this.zinv = this.z.modInverse(this.curve.q); + this.zinv = this.z.modInverse(this.curve.q) } - return this.curve.fromBigInteger(this.y.toBigInteger().multiply(this.zinv).mod(this.curve.q)); + + return this.curve.fromBigInteger(this.y.toBigInteger().multiply(this.zinv).mod(this.curve.q)) } function pointFpEquals(other) { - if (other == this) return true; - if (this.isInfinity()) return other.isInfinity(); - if (other.isInfinity()) return this.isInfinity(); + if (other == this) return true + if (this.isInfinity()) return other.isInfinity() + if (other.isInfinity()) return this.isInfinity() - var u, v; // u = Y2 * Z1 - Y1 * Z2 - u = other.y.toBigInteger().multiply(this.z).subtract(this.y.toBigInteger().multiply(other.z)).mod(this.curve.q); - if (u.signum() !== 0) return false; + u = other.y.toBigInteger().multiply(this.z).subtract(this.y.toBigInteger().multiply(other.z)).mod(this.curve.q) + if (u.signum() !== 0) return false + // v = X2 * Z1 - X1 * Z2 - v = other.x.toBigInteger().multiply(this.z).subtract(this.x.toBigInteger().multiply(other.z)).mod(this.curve.q); - return v.signum() === 0; + v = other.x.toBigInteger().multiply(this.z).subtract(this.x.toBigInteger().multiply(other.z)).mod(this.curve.q) + return v.signum() === 0 } function pointFpIsInfinity() { - if ((this.x == null) && (this.y == null)) return true; - return this.z.signum() === 0 && this.y.toBigInteger().signum() !== 0; + if ((this.x == null) && (this.y == null)) return true + return this.z.signum() === 0 && this.y.toBigInteger().signum() !== 0 } function pointFpNegate() { - return new ECPointFp(this.curve, this.x, this.y.negate(), this.z); + return new ECPointFp(this.curve, this.x, this.y.negate(), this.z) } function pointFpAdd(b) { - if (this.isInfinity()) return b; - if (b.isInfinity()) return this; + if (this.isInfinity()) return b + if (b.isInfinity()) return this var x1 = this.x.toBigInteger() var y1 = this.y.toBigInteger() @@ -129,171 +133,170 @@ function pointFpAdd(b) { if (v.signum() === 0) { if (u.signum() === 0) { - return this.twice(); // this == b, so double + return this.twice() // this == b, so double } - return this.curve.getInfinity(); // this = -b, so infinity + return this.curve.getInfinity() // this = -b, so infinity } - var v2 = v.square(); - var v3 = v2.multiply(v); - var x1v2 = x1.multiply(v2); - var zu2 = u.square().multiply(this.z); + var v2 = v.square() + var v3 = v2.multiply(v) + var x1v2 = x1.multiply(v2) + var zu2 = u.square().multiply(this.z) // x3 = v * (z2 * (z1 * u^2 - 2 * x1 * v^2) - v^3) - var x3 = zu2.subtract(x1v2.shiftLeft(1)).multiply(b.z).subtract(v3).multiply(v).mod(this.curve.q); + var x3 = zu2.subtract(x1v2.shiftLeft(1)).multiply(b.z).subtract(v3).multiply(v).mod(this.curve.q) // y3 = z2 * (3 * x1 * u * v^2 - y1 * v^3 - z1 * u^3) + u * v^3 - var y3 = x1v2.multiply(THREE).multiply(u).subtract(y1.multiply(v3)).subtract(zu2.multiply(u)).multiply(b.z).add(u.multiply(v3)).mod(this.curve.q); + var y3 = x1v2.multiply(THREE).multiply(u).subtract(y1.multiply(v3)).subtract(zu2.multiply(u)).multiply(b.z).add(u.multiply(v3)).mod(this.curve.q) // z3 = v^3 * z1 * z2 - var z3 = v3.multiply(this.z).multiply(b.z).mod(this.curve.q); + var z3 = v3.multiply(this.z).multiply(b.z).mod(this.curve.q) - return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3); + return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3) } function pointFpTwice() { - if (this.isInfinity()) return this; - if (this.y.toBigInteger().signum() === 0) return this.curve.getInfinity(); + if (this.isInfinity()) return this + if (this.y.toBigInteger().signum() === 0) return this.curve.getInfinity() - var x1 = this.x.toBigInteger(); - var y1 = this.y.toBigInteger(); + var x1 = this.x.toBigInteger() + var y1 = this.y.toBigInteger() - var y1z1 = y1.multiply(this.z); - var y1sqz1 = y1z1.multiply(y1).mod(this.curve.q); - var a = this.curve.a.toBigInteger(); + var y1z1 = y1.multiply(this.z) + var y1sqz1 = y1z1.multiply(y1).mod(this.curve.q) + var a = this.curve.a.toBigInteger() // w = 3 * x1^2 + a * z1^2 - var w = x1.square().multiply(THREE); + var w = x1.square().multiply(THREE) if (a.signum() !== 0) { - w = w.add(this.z.square().multiply(a)); + w = w.add(this.z.square().multiply(a)) } - w = w.mod(this.curve.q); + w = w.mod(this.curve.q) // x3 = 2 * y1 * z1 * (w^2 - 8 * x1 * y1^2 * z1) - var x3 = w.square().subtract(x1.shiftLeft(3).multiply(y1sqz1)).shiftLeft(1).multiply(y1z1).mod(this.curve.q); + var x3 = w.square().subtract(x1.shiftLeft(3).multiply(y1sqz1)).shiftLeft(1).multiply(y1z1).mod(this.curve.q) // y3 = 4 * y1^2 * z1 * (3 * w * x1 - 2 * y1^2 * z1) - w^3 - var y3 = w.multiply(THREE).multiply(x1).subtract(y1sqz1.shiftLeft(1)).shiftLeft(2).multiply(y1sqz1).subtract(w.pow(3)).mod(this.curve.q); + var y3 = w.multiply(THREE).multiply(x1).subtract(y1sqz1.shiftLeft(1)).shiftLeft(2).multiply(y1sqz1).subtract(w.pow(3)).mod(this.curve.q) // z3 = 8 * (y1 * z1)^3 - var z3 = y1z1.pow(3).shiftLeft(3).mod(this.curve.q); + var z3 = y1z1.pow(3).shiftLeft(3).mod(this.curve.q) - return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3); + return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3) } // Simple NAF (Non-Adjacent Form) multiplication algorithm // TODO: modularize the multiplication algorithm function pointFpMultiply(k) { - if (this.isInfinity()) return this; + if (this.isInfinity()) return this if (k.signum() === 0) return this.curve.getInfinity() - var e = k; + var e = k var h = e.multiply(THREE) - var neg = this.negate(); - var R = this; + var neg = this.negate() + var R = this - var i; - for(i = h.bitLength() - 2; i > 0; --i) { - R = R.twice(); + for (var i = h.bitLength() - 2; i > 0; --i) { + R = R.twice() - var hBit = h.testBit(i); - var eBit = e.testBit(i); + var hBit = h.testBit(i) + var eBit = e.testBit(i) if (hBit != eBit) { - R = R.add(hBit ? this : neg); + R = R.add(hBit ? this : neg) } } - return R; + return R } // Compute this*j + x*k (simultaneous multiplication) function pointFpMultiplyTwo(j,x,k) { - var i; + var i if (j.bitLength() > k.bitLength()) - i = j.bitLength() - 1; + i = j.bitLength() - 1 else - i = k.bitLength() - 1; + i = k.bitLength() - 1 - var R = this.curve.getInfinity(); - var both = this.add(x); - while(i >= 0) { - R = R.twice(); + var R = this.curve.getInfinity() + var both = this.add(x) + while (i >= 0) { + R = R.twice() if (j.testBit(i)) { if (k.testBit(i)) { - R = R.add(both); + R = R.add(both) } else { - R = R.add(this); + R = R.add(this) } } else { if (k.testBit(i)) { - R = R.add(x); + R = R.add(x) } } - --i; + --i } - return R; + return R } -ECPointFp.prototype.getX = pointFpGetX; -ECPointFp.prototype.getY = pointFpGetY; -ECPointFp.prototype.equals = pointFpEquals; -ECPointFp.prototype.isInfinity = pointFpIsInfinity; -ECPointFp.prototype.negate = pointFpNegate; -ECPointFp.prototype.add = pointFpAdd; -ECPointFp.prototype.twice = pointFpTwice; -ECPointFp.prototype.multiply = pointFpMultiply; -ECPointFp.prototype.multiplyTwo = pointFpMultiplyTwo; +ECPointFp.prototype.getX = pointFpGetX +ECPointFp.prototype.getY = pointFpGetY +ECPointFp.prototype.equals = pointFpEquals +ECPointFp.prototype.isInfinity = pointFpIsInfinity +ECPointFp.prototype.negate = pointFpNegate +ECPointFp.prototype.add = pointFpAdd +ECPointFp.prototype.twice = pointFpTwice +ECPointFp.prototype.multiply = pointFpMultiply +ECPointFp.prototype.multiplyTwo = pointFpMultiplyTwo // ---------------- // ECCurveFp // constructor function ECCurveFp(q,a,b) { - this.q = q; - this.a = this.fromBigInteger(a); - this.b = this.fromBigInteger(b); - this.infinity = new ECPointFp(this, null, null); + this.q = q + this.a = this.fromBigInteger(a) + this.b = this.fromBigInteger(b) + this.infinity = new ECPointFp(this, null, null) } function curveFpGetQ() { - return this.q; + return this.q } function curveFpGetA() { - return this.a; + return this.a } function curveFpGetB() { - return this.b; + return this.b } function curveFpEquals(other) { - if (other == this) return true; - return(this.q.equals(other.q) && this.a.equals(other.a) && this.b.equals(other.b)); + if (other == this) return true + return(this.q.equals(other.q) && this.a.equals(other.a) && this.b.equals(other.b)) } function curveFpGetInfinity() { - return this.infinity; + return this.infinity } function curveFpFromBigInteger(x) { - return new ECFieldElementFp(this.q, x); + return new ECFieldElementFp(this.q, x) } -ECCurveFp.prototype.getQ = curveFpGetQ; -ECCurveFp.prototype.getA = curveFpGetA; -ECCurveFp.prototype.getB = curveFpGetB; -ECCurveFp.prototype.equals = curveFpEquals; -ECCurveFp.prototype.getInfinity = curveFpGetInfinity; -ECCurveFp.prototype.fromBigInteger = curveFpFromBigInteger; +ECCurveFp.prototype.getQ = curveFpGetQ +ECCurveFp.prototype.getA = curveFpGetA +ECCurveFp.prototype.getB = curveFpGetB +ECCurveFp.prototype.equals = curveFpEquals +ECCurveFp.prototype.getInfinity = curveFpGetInfinity +ECCurveFp.prototype.fromBigInteger = curveFpFromBigInteger ECFieldElementFp.prototype.getByteLength = function () { - return Math.floor((this.toBigInteger().bitLength() + 7) / 8); -}; + return Math.floor((this.toBigInteger().bitLength() + 7) / 8) +} ECPointFp.prototype.getEncoded = function(compressed) { var x = this.getX().toBigInteger() @@ -361,20 +364,20 @@ ECPointFp.decodeFrom = function (curve, buffer) { } ECPointFp.prototype.isOnCurve = function () { - var x = this.getX().toBigInteger(); - var y = this.getY().toBigInteger(); - var a = this.curve.getA().toBigInteger(); - var b = this.curve.getB().toBigInteger(); + var x = this.getX().toBigInteger() + var y = this.getY().toBigInteger() + var a = this.curve.getA().toBigInteger() + var b = this.curve.getB().toBigInteger() var p = this.curve.getQ() var lhs = y.square().mod(p) var rhs = x.pow(3).add(a.multiply(x)).add(b).mod(p) - return lhs.equals(rhs); -}; + return lhs.equals(rhs) +} ECPointFp.prototype.toString = function () { return '('+this.getX().toBigInteger().toString()+','+ - this.getY().toBigInteger().toString()+')'; -}; + this.getY().toBigInteger().toString()+')' +} /** * Validate an elliptic curve point. @@ -382,38 +385,38 @@ ECPointFp.prototype.toString = function () { * See SEC 1, section 3.2.2.1: Elliptic Curve Public Key Validation Primitive */ ECPointFp.prototype.validate = function () { - var n = this.curve.getQ(); + var n = this.curve.getQ() // Check Q != O if (this.isInfinity()) { - throw new Error("Point is at infinity."); + throw new Error("Point is at infinity.") } // Check coordinate bounds - var x = this.getX().toBigInteger(); - var y = this.getY().toBigInteger(); + var x = this.getX().toBigInteger() + var y = this.getY().toBigInteger() if (x.compareTo(BigInteger.ONE) < 0 || x.compareTo(n.subtract(BigInteger.ONE)) > 0) { - throw new Error('x coordinate out of bounds'); + throw new Error('x coordinate out of bounds') } if (y.compareTo(BigInteger.ONE) < 0 || y.compareTo(n.subtract(BigInteger.ONE)) > 0) { - throw new Error('y coordinate out of bounds'); + throw new Error('y coordinate out of bounds') } // Check y^2 = x^3 + ax + b (mod n) if (!this.isOnCurve()) { - throw new Error("Point is not on the curve."); + throw new Error("Point is not on the curve.") } // Check nQ = 0 (Q is a scalar multiple of G) if (this.multiply(n).isInfinity()) { // TODO: This check doesn't work - fix. - throw new Error("Point is not a scalar multiple of G."); + throw new Error("Point is not a scalar multiple of G.") } - return true; -}; + return true +} -module.exports = ECCurveFp; -module.exports.ECPointFp = ECPointFp; +module.exports = ECCurveFp +module.exports.ECPointFp = ECPointFp