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562 lines
15 KiB
562 lines
15 KiB
// Basic Javascript Elliptic Curve implementation
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// Ported loosely from BouncyCastle's Java EC code
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// Only Fp curves implemented for now
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// Requires jsbn.js and jsbn2.js
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var BigInteger = require('jsbn').BigInteger
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var Barrett = BigInteger.prototype.Barrett
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// ----------------
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// ECFieldElementFp
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// constructor
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function ECFieldElementFp(q,x) {
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this.x = x;
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// TODO if(x.compareTo(q) >= 0) error
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this.q = q;
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}
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function feFpEquals(other) {
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if(other == this) return true;
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return (this.q.equals(other.q) && this.x.equals(other.x));
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}
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function feFpToBigInteger() {
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return this.x;
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}
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function feFpNegate() {
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return new ECFieldElementFp(this.q, this.x.negate().mod(this.q));
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}
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function feFpAdd(b) {
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return new ECFieldElementFp(this.q, this.x.add(b.toBigInteger()).mod(this.q));
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}
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function feFpSubtract(b) {
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return new ECFieldElementFp(this.q, this.x.subtract(b.toBigInteger()).mod(this.q));
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}
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function feFpMultiply(b) {
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return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger()).mod(this.q));
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}
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function feFpSquare() {
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return new ECFieldElementFp(this.q, this.x.square().mod(this.q));
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}
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function feFpDivide(b) {
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return new ECFieldElementFp(this.q, this.x.multiply(b.toBigInteger().modInverse(this.q)).mod(this.q));
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}
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ECFieldElementFp.prototype.equals = feFpEquals;
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ECFieldElementFp.prototype.toBigInteger = feFpToBigInteger;
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ECFieldElementFp.prototype.negate = feFpNegate;
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ECFieldElementFp.prototype.add = feFpAdd;
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ECFieldElementFp.prototype.subtract = feFpSubtract;
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ECFieldElementFp.prototype.multiply = feFpMultiply;
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ECFieldElementFp.prototype.square = feFpSquare;
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ECFieldElementFp.prototype.divide = feFpDivide;
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// ----------------
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// ECPointFp
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// constructor
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function ECPointFp(curve,x,y,z) {
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this.curve = curve;
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this.x = x;
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this.y = y;
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// Projective coordinates: either zinv == null or z * zinv == 1
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// z and zinv are just BigIntegers, not fieldElements
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if(z == null) {
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this.z = BigInteger.ONE;
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}
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else {
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this.z = z;
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}
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this.zinv = null;
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//TODO: compression flag
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}
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function pointFpGetX() {
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if(this.zinv == null) {
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this.zinv = this.z.modInverse(this.curve.q);
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}
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var r = this.x.toBigInteger().multiply(this.zinv);
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this.curve.reduce(r);
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return this.curve.fromBigInteger(r);
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}
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function pointFpGetY() {
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if(this.zinv == null) {
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this.zinv = this.z.modInverse(this.curve.q);
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}
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var r = this.y.toBigInteger().multiply(this.zinv);
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this.curve.reduce(r);
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return this.curve.fromBigInteger(r);
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}
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function pointFpEquals(other) {
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if(other == this) return true;
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if(this.isInfinity()) return other.isInfinity();
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if(other.isInfinity()) return this.isInfinity();
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var u, v;
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// u = Y2 * Z1 - Y1 * Z2
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u = other.y.toBigInteger().multiply(this.z).subtract(this.y.toBigInteger().multiply(other.z)).mod(this.curve.q);
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if(!u.equals(BigInteger.ZERO)) return false;
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// v = X2 * Z1 - X1 * Z2
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v = other.x.toBigInteger().multiply(this.z).subtract(this.x.toBigInteger().multiply(other.z)).mod(this.curve.q);
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return v.equals(BigInteger.ZERO);
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}
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function pointFpIsInfinity() {
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if((this.x == null) && (this.y == null)) return true;
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return this.z.equals(BigInteger.ZERO) && !this.y.toBigInteger().equals(BigInteger.ZERO);
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}
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function pointFpNegate() {
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return new ECPointFp(this.curve, this.x, this.y.negate(), this.z);
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}
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function pointFpAdd(b) {
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if(this.isInfinity()) return b;
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if(b.isInfinity()) return this;
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// u = Y2 * Z1 - Y1 * Z2
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var u = b.y.toBigInteger().multiply(this.z).subtract(this.y.toBigInteger().multiply(b.z)).mod(this.curve.q);
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// v = X2 * Z1 - X1 * Z2
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var v = b.x.toBigInteger().multiply(this.z).subtract(this.x.toBigInteger().multiply(b.z)).mod(this.curve.q);
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if(BigInteger.ZERO.equals(v)) {
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if(BigInteger.ZERO.equals(u)) {
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return this.twice(); // this == b, so double
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}
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return this.curve.getInfinity(); // this = -b, so infinity
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}
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var THREE = new BigInteger("3");
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var x1 = this.x.toBigInteger();
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var y1 = this.y.toBigInteger();
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var x2 = b.x.toBigInteger();
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var y2 = b.y.toBigInteger();
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var v2 = v.square();
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var v3 = v2.multiply(v);
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var x1v2 = x1.multiply(v2);
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var zu2 = u.square().multiply(this.z);
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// x3 = v * (z2 * (z1 * u^2 - 2 * x1 * v^2) - v^3)
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var x3 = zu2.subtract(x1v2.shiftLeft(1)).multiply(b.z).subtract(v3).multiply(v).mod(this.curve.q);
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// y3 = z2 * (3 * x1 * u * v^2 - y1 * v^3 - z1 * u^3) + u * v^3
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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);
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// z3 = v^3 * z1 * z2
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var z3 = v3.multiply(this.z).multiply(b.z).mod(this.curve.q);
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return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3);
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}
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function pointFpTwice() {
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if(this.isInfinity()) return this;
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if(this.y.toBigInteger().signum() == 0) return this.curve.getInfinity();
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// TODO: optimized handling of constants
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var THREE = new BigInteger("3");
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var x1 = this.x.toBigInteger();
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var y1 = this.y.toBigInteger();
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var y1z1 = y1.multiply(this.z);
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var y1sqz1 = y1z1.multiply(y1).mod(this.curve.q);
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var a = this.curve.a.toBigInteger();
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// w = 3 * x1^2 + a * z1^2
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var w = x1.square().multiply(THREE);
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if(!BigInteger.ZERO.equals(a)) {
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w = w.add(this.z.square().multiply(a));
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}
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w = w.mod(this.curve.q);
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//this.curve.reduce(w);
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// x3 = 2 * y1 * z1 * (w^2 - 8 * x1 * y1^2 * z1)
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var x3 = w.square().subtract(x1.shiftLeft(3).multiply(y1sqz1)).shiftLeft(1).multiply(y1z1).mod(this.curve.q);
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// y3 = 4 * y1^2 * z1 * (3 * w * x1 - 2 * y1^2 * z1) - w^3
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var y3 = w.multiply(THREE).multiply(x1).subtract(y1sqz1.shiftLeft(1)).shiftLeft(2).multiply(y1sqz1).subtract(w.square().multiply(w)).mod(this.curve.q);
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// z3 = 8 * (y1 * z1)^3
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var z3 = y1z1.square().multiply(y1z1).shiftLeft(3).mod(this.curve.q);
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return new ECPointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3);
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}
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// Simple NAF (Non-Adjacent Form) multiplication algorithm
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// TODO: modularize the multiplication algorithm
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function pointFpMultiply(k) {
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if(this.isInfinity()) return this;
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if(k.signum() == 0) return this.curve.getInfinity();
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var e = k;
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var h = e.multiply(new BigInteger("3"));
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var neg = this.negate();
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var R = this;
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var i;
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for(i = h.bitLength() - 2; i > 0; --i) {
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R = R.twice();
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var hBit = h.testBit(i);
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var eBit = e.testBit(i);
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if (hBit != eBit) {
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R = R.add(hBit ? this : neg);
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}
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}
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return R;
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}
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// Compute this*j + x*k (simultaneous multiplication)
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function pointFpMultiplyTwo(j,x,k) {
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var i;
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if(j.bitLength() > k.bitLength())
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i = j.bitLength() - 1;
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else
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i = k.bitLength() - 1;
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var R = this.curve.getInfinity();
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var both = this.add(x);
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while(i >= 0) {
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R = R.twice();
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if(j.testBit(i)) {
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if(k.testBit(i)) {
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R = R.add(both);
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}
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else {
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R = R.add(this);
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}
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}
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else {
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if(k.testBit(i)) {
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R = R.add(x);
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}
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}
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--i;
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}
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return R;
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}
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ECPointFp.prototype.getX = pointFpGetX;
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ECPointFp.prototype.getY = pointFpGetY;
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ECPointFp.prototype.equals = pointFpEquals;
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ECPointFp.prototype.isInfinity = pointFpIsInfinity;
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ECPointFp.prototype.negate = pointFpNegate;
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ECPointFp.prototype.add = pointFpAdd;
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ECPointFp.prototype.twice = pointFpTwice;
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ECPointFp.prototype.multiply = pointFpMultiply;
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ECPointFp.prototype.multiplyTwo = pointFpMultiplyTwo;
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// ----------------
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// ECCurveFp
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// constructor
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function ECCurveFp(q,a,b) {
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this.q = q;
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this.a = this.fromBigInteger(a);
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this.b = this.fromBigInteger(b);
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this.infinity = new ECPointFp(this, null, null);
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this.reducer = new Barrett(this.q);
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}
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function curveFpGetQ() {
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return this.q;
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}
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function curveFpGetA() {
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return this.a;
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}
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function curveFpGetB() {
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return this.b;
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}
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function curveFpEquals(other) {
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if(other == this) return true;
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return(this.q.equals(other.q) && this.a.equals(other.a) && this.b.equals(other.b));
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}
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function curveFpGetInfinity() {
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return this.infinity;
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}
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function curveFpFromBigInteger(x) {
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return new ECFieldElementFp(this.q, x);
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}
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function curveReduce(x) {
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this.reducer.reduce(x);
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}
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// for now, work with hex strings because they're easier in JS
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function curveFpDecodePointHex(s) {
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switch(parseInt(s.substr(0,2), 16)) { // first byte
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case 0:
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return this.infinity;
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case 2:
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case 3:
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// point compression not supported yet
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return null;
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case 4:
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case 6:
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case 7:
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var len = (s.length - 2) / 2;
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var xHex = s.substr(2, len);
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var yHex = s.substr(len+2, len);
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return new ECPointFp(this,
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this.fromBigInteger(new BigInteger(xHex, 16)),
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this.fromBigInteger(new BigInteger(yHex, 16)));
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default: // unsupported
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return null;
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}
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}
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function curveFpEncodePointHex(p) {
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if (p.isInfinity()) return "00";
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var xHex = p.getX().toBigInteger().toString(16);
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var yHex = p.getY().toBigInteger().toString(16);
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var oLen = this.getQ().toString(16).length;
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if ((oLen % 2) != 0) oLen++;
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while (xHex.length < oLen) {
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xHex = "0" + xHex;
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}
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while (yHex.length < oLen) {
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yHex = "0" + yHex;
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}
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return "04" + xHex + yHex;
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}
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ECCurveFp.prototype.getQ = curveFpGetQ;
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ECCurveFp.prototype.getA = curveFpGetA;
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ECCurveFp.prototype.getB = curveFpGetB;
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ECCurveFp.prototype.equals = curveFpEquals;
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ECCurveFp.prototype.getInfinity = curveFpGetInfinity;
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ECCurveFp.prototype.fromBigInteger = curveFpFromBigInteger;
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ECCurveFp.prototype.reduce = curveReduce;
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//ECCurveFp.prototype.decodePointHex = curveFpDecodePointHex;
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ECCurveFp.prototype.encodePointHex = curveFpEncodePointHex;
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// from: https://github.com/kaielvin/jsbn-ec-point-compression
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ECCurveFp.prototype.decodePointHex = function(s)
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{
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var yIsEven;
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switch(parseInt(s.substr(0,2), 16)) { // first byte
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case 0:
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return this.infinity;
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case 2:
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yIsEven = false;
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case 3:
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if(yIsEven == undefined) yIsEven = true;
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var len = s.length - 2;
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var xHex = s.substr(2, len);
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var x = this.fromBigInteger(new BigInteger(xHex,16));
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var alpha = x.multiply(x.square().add(this.getA())).add(this.getB());
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var beta = alpha.sqrt();
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if (beta == null) throw "Invalid point compression";
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var betaValue = beta.toBigInteger();
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if (betaValue.testBit(0) != yIsEven)
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{
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// Use the other root
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beta = this.fromBigInteger(this.getQ().subtract(betaValue));
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}
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return new ECPointFp(this,x,beta);
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case 4:
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case 6:
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case 7:
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var len = (s.length - 2) / 2;
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var xHex = s.substr(2, len);
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var yHex = s.substr(len+2, len);
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return new ECPointFp(this,
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this.fromBigInteger(new BigInteger(xHex, 16)),
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this.fromBigInteger(new BigInteger(yHex, 16)));
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default: // unsupported
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return null;
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}
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}
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ECCurveFp.prototype.encodeCompressedPointHex = function(p)
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{
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if (p.isInfinity()) return "00";
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var xHex = p.getX().toBigInteger().toString(16);
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var oLen = this.getQ().toString(16).length;
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if ((oLen % 2) != 0) oLen++;
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while (xHex.length < oLen)
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xHex = "0" + xHex;
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var yPrefix;
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if(p.getY().toBigInteger().isEven()) yPrefix = "02";
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else yPrefix = "03";
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return yPrefix + xHex;
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}
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ECFieldElementFp.prototype.getR = function()
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{
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if(this.r != undefined) return this.r;
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this.r = null;
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var bitLength = this.q.bitLength();
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if (bitLength > 128)
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{
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var firstWord = this.q.shiftRight(bitLength - 64);
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if (firstWord.intValue() == -1)
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{
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this.r = BigInteger.ONE.shiftLeft(bitLength).subtract(this.q);
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}
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}
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return this.r;
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}
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ECFieldElementFp.prototype.modMult = function(x1,x2)
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{
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return this.modReduce(x1.multiply(x2));
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}
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ECFieldElementFp.prototype.modReduce = function(x)
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{
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if (this.getR() != null)
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{
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var qLen = q.bitLength();
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while (x.bitLength() > (qLen + 1))
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{
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var u = x.shiftRight(qLen);
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var v = x.subtract(u.shiftLeft(qLen));
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if (!this.getR().equals(BigInteger.ONE))
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{
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u = u.multiply(this.getR());
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}
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x = u.add(v);
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}
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while (x.compareTo(q) >= 0)
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{
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x = x.subtract(q);
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}
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}
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else
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{
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x = x.mod(q);
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}
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return x;
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}
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ECFieldElementFp.prototype.sqrt = function()
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{
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if (!this.q.testBit(0)) throw "unsupported";
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// p mod 4 == 3
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if (this.q.testBit(1))
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{
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var z = new ECFieldElementFp(this.q,this.x.modPow(this.q.shiftRight(2).add(BigInteger.ONE),this.q));
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return z.square().equals(this) ? z : null;
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}
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// p mod 4 == 1
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var qMinusOne = this.q.subtract(BigInteger.ONE);
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var legendreExponent = qMinusOne.shiftRight(1);
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if (!(this.x.modPow(legendreExponent, this.q).equals(BigInteger.ONE)))
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{
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return null;
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}
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var u = qMinusOne.shiftRight(2);
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var k = u.shiftLeft(1).add(BigInteger.ONE);
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var Q = this.x;
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var fourQ = modDouble(modDouble(Q));
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var U, V;
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do
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{
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var P;
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do
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{
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P = new BigInteger(this.q.bitLength(), new SecureRandom());
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}
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while (P.compareTo(this.q) >= 0
|
|
|| !(P.multiply(P).subtract(fourQ).modPow(legendreExponent, this.q).equals(qMinusOne)));
|
|
|
|
var result = this.lucasSequence(P, Q, k);
|
|
U = result[0];
|
|
V = result[1];
|
|
|
|
if (this.modMult(V, V).equals(fourQ))
|
|
{
|
|
// Integer division by 2, mod q
|
|
if (V.testBit(0))
|
|
{
|
|
V = V.add(q);
|
|
}
|
|
|
|
V = V.shiftRight(1);
|
|
|
|
return new ECFieldElementFp(q,V);
|
|
}
|
|
}
|
|
while (U.equals(BigInteger.ONE) || U.equals(qMinusOne));
|
|
|
|
return null;
|
|
}
|
|
ECFieldElementFp.prototype.lucasSequence = function(P,Q,k)
|
|
{
|
|
var n = k.bitLength();
|
|
var s = k.getLowestSetBit();
|
|
|
|
var Uh = BigInteger.ONE;
|
|
var Vl = BigInteger.TWO;
|
|
var Vh = P;
|
|
var Ql = BigInteger.ONE;
|
|
var Qh = BigInteger.ONE;
|
|
|
|
for (var j = n - 1; j >= s + 1; --j)
|
|
{
|
|
Ql = this.modMult(Ql, Qh);
|
|
|
|
if (k.testBit(j))
|
|
{
|
|
Qh = this.modMult(Ql, Q);
|
|
Uh = this.modMult(Uh, Vh);
|
|
Vl = this.modReduce(Vh.multiply(Vl).subtract(P.multiply(Ql)));
|
|
Vh = this.modReduce(Vh.multiply(Vh).subtract(Qh.shiftLeft(1)));
|
|
}
|
|
else
|
|
{
|
|
Qh = Ql;
|
|
Uh = this.modReduce(Uh.multiply(Vl).subtract(Ql));
|
|
Vh = this.modReduce(Vh.multiply(Vl).subtract(P.multiply(Ql)));
|
|
Vl = this.modReduce(Vl.multiply(Vl).subtract(Ql.shiftLeft(1)));
|
|
}
|
|
}
|
|
|
|
Ql = this.modMult(Ql, Qh);
|
|
Qh = this.modMult(Ql, Q);
|
|
Uh = this.modReduce(Uh.multiply(Vl).subtract(Ql));
|
|
Vl = this.modReduce(Vh.multiply(Vl).subtract(P.multiply(Ql)));
|
|
Ql = this.modMult(Ql, Qh);
|
|
|
|
for (var j = 1; j <= s; ++j)
|
|
{
|
|
Uh = this.modMult(Uh, Vl);
|
|
Vl = this.modReduce(Vl.multiply(Vl).subtract(Ql.shiftLeft(1)));
|
|
Ql = this.modMult(Ql, Ql);
|
|
}
|
|
|
|
return [ Uh, Vl ];
|
|
}
|
|
|
|
var exports = {
|
|
ECCurveFp: ECCurveFp,
|
|
ECPointFp: ECPointFp,
|
|
ECFieldElementFp: ECFieldElementFp
|
|
}
|
|
|
|
module.exports = exports
|