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Edit File : ec.js \| Size : 14.96 KB
// 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 thisj + xk (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 Back

Edit File : ec.js | Size : 14.96 KB

// 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;

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;
    }
}

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

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