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