For easy reference, let me quote the relevant section of the code:
ec.PointFp = function (curve, x, y, z, compressed) {
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;
// compression flag
this.compressed = !!compressed;
};
ec.PointFp.prototype.getX = function () {
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);
};
ec.PointFp.prototype.getY = function () {
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);
};
ec.PointFp.prototype.equals = function (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);
};
ec.PointFp.prototype.isInfinity = function () {
if ((this.x == null) && (this.y == null)) return true;
return this.z.equals(BigInteger.ZERO) && !this.y.toBigInteger().equals(BigInteger.ZERO);
};
ec.PointFp.prototype.negate = function () {
return new ec.PointFp(this.curve, this.x, this.y.negate(), this.z);
};
ec.PointFp.prototype.add = function (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 ec.PointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3);
};
ec.PointFp.prototype.twice = function () {
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 ec.PointFp(this.curve, this.curve.fromBigInteger(x3), this.curve.fromBigInteger(y3), z3);
};
// Simple NAF (Non-Adjacent Form) multiplication algorithm
// TODO: modularize the multiplication algorithm
ec.PointFp.prototype.multiply = function (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)
ec.PointFp.prototype.multiplyTwo = function (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;
};
// patched by bitaddress.org and Casascius for use with Bitcoin.ECKey
// patched by coretechs to support compressed public keys
ec.PointFp.prototype.getEncoded = function (compressed) {
var x = this.getX().toBigInteger();
var y = this.getY().toBigInteger();
var len = 32; // integerToBytes will zero pad if integer is less than 32 bytes. 32 bytes length is required by the Bitcoin protocol.
var enc = ec.integerToBytes(x, len);
// when compressed prepend byte depending if y point is even or odd
if (compressed) {
if (y.isEven()) {
enc.unshift(0x02);
}
else {
enc.unshift(0x03);
}
}
else {
enc.unshift(0x04);
enc = enc.concat(ec.integerToBytes(y, len)); // uncompressed public key appends the bytes of the y point
}
return enc;
};
ec.PointFp.decodeFrom = function (curve, enc) {
var type = enc[0];
var dataLen = enc.length - 1;
// Extract x and y as byte arrays
var xBa = enc.slice(1, 1 + dataLen / 2);
var yBa = enc.slice(1 + dataLen / 2, 1 + dataLen);
// Prepend zero byte to prevent interpretation as negative integer
xBa.unshift(0);
yBa.unshift(0);
// Convert to BigIntegers
var x = new BigInteger(xBa);
var y = new BigInteger(yBa);
// Return point
return new ec.PointFp(curve, curve.fromBigInteger(x), curve.fromBigInteger(y));
};
ec.PointFp.prototype.add2D = function (b) {
if (this.isInfinity()) return b;
if (b.isInfinity()) return this;
if (this.x.equals(b.x)) {
if (this.y.equals(b.y)) {
// this = b, i.e. this must be doubled
return this.twice();
}
// this = -b, i.e. the result is the point at infinity
return this.curve.getInfinity();
}
var x_x = b.x.subtract(this.x);
var y_y = b.y.subtract(this.y);
var gamma = y_y.divide(x_x);
var x3 = gamma.square().subtract(this.x).subtract(b.x);
var y3 = gamma.multiply(this.x.subtract(x3)).subtract(this.y);
return new ec.PointFp(this.curve, x3, y3);
};
ec.PointFp.prototype.twice2D = function () {
if (this.isInfinity()) return this;
if (this.y.toBigInteger().signum() == 0) {
// if y1 == 0, then (x1, y1) == (x1, -y1)
// and hence this = -this and thus 2(x1, y1) == infinity
return this.curve.getInfinity();
}
var TWO = this.curve.fromBigInteger(BigInteger.valueOf(2));
var THREE = this.curve.fromBigInteger(BigInteger.valueOf(3));
var gamma = this.x.square().multiply(THREE).add(this.curve.a).divide(this.y.multiply(TWO));
var x3 = gamma.square().subtract(this.x.multiply(TWO));
var y3 = gamma.multiply(this.x.subtract(x3)).subtract(this.y);
return new ec.PointFp(this.curve, x3, y3);
};
ec.PointFp.prototype.multiply2D = function (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.add2D(hBit ? this : neg);
}
}
return R;
};
ec.PointFp.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 n = this.curve.getQ();
var lhs = y.multiply(y).mod(n);
var rhs = x.multiply(x).multiply(x).add(a.multiply(x)).add(b).mod(n);
return lhs.equals(rhs);
};
ec.PointFp.prototype.toString = function () {
return '(' + this.getX().toBigInteger().toString() + ',' + this.getY().toBigInteger().toString() + ')';
};
/**
* Validate an elliptic curve point.
*
* See SEC 1, section 3.2.2.1: Elliptic Curve Public Key Validation Primitive
*/
ec.PointFp.prototype.validate = function () {
var n = this.curve.getQ();
// Check Q != O
if (this.isInfinity()) {
throw new Error("Point is at infinity.");
}
// Check coordinate bounds
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');
}
if (y.compareTo(BigInteger.ONE) < 0 || y.compareTo(n.subtract(BigInteger.ONE)) > 0) {
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.");
}
// 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.");
}
return true;
};