Point zero on an elliptic curve

Question

I am working on a library that allows me to work with elliptic curves. It is still at embryonic state, and so far it only consists of two classes, EllipticCurve and Point .

Right now I'm implementing the basic operations of existence, belonging, sum, reflection, etc.

Unfortunately, I'm stuck now that I have to implement the concept of zero, ie, the point of the elliptic curve E crossing the line through P and -P, where P = (x,y) and -P = (x,-y). So, my question could be rephrased as "how do I implement a point at infinity?"

Here's part of the Point class so far:

public class Point implements Comparable<Point> {
    private static final BigDecimal MINUSONE = new BigDecimal(-1);

    private BigDecimal x;
    private BigDecimal y;
    private EllipticCurve e;

    public Point(BigDecimal x, BigDecimal y, EllipticCurve e) {
        if(x != null && y != null && e != null) {
            if(liesOn(x,y,e)) {
                this.x = x;
                this.y = y;
                this.e = e;
            }
        }
    }

    public Point reflect() {
        return new Point(x,y.multiply(MINUSONE),e);
    }

    public Point add(Point o) {
        if(this.e == o.getE()) {
            if(this == o) {
                return this.multiply(2);
            }
            if(o == this.reflect()) {
                return /*THE INFAMOUS ZERO POINT*/;
            }

            BigDecimal a;
            BigDecimal b;

            /*
             * computation I still haven't implemented
             */

            return new Point(a,b,e);
        }
    }
    /*
     * other methods
     */
}

PS: I am aware of the existence of java.security.spec.EllipticCurve, but since I'm going to use this class mostly for mathematical purposes, I felt the need to create my personal library ex novo .

Answer 1

There is no way to represent infinity per se using BigDecimal . The only way in Java I know is:

Double.POSITIVE_INFINITY;

or Integer , Float etc. You also can't take valueOf from the above, since it will throw a NumberFormatException .

You can use a workaround, ie a very large value that you know will always be larger than any other.