How are elliptic curves used?
The crucial property of an elliptic curve is that we can define a rule for “adding” two points which are on the curve, to obtain a 3rd point which is also on the curve. This addition rule satisfies the normal properties of addition. In math jargon, the points and the addition law form a finite Abelian group. The equations for the addition rule are given in (7) and (8). For addition to be well defined for any two points, we need to include an extra ‘zero’ point O, which does not satisfy the elliptic curve equation. This ‘zero’ point is taken to be a fully paid up point of the curve. The order of the curve is the number of distinct points on the curve, including the zero point. Having defined addition of two points, we can also define multiplication k*P where k is a positive integer and P is a point as the sum of k copies of P. Thus 2*P = P+P 3*P = P+P+P etc. This is analagous to how we define “powers” in normal arithmetic, where x^2 = x.x x^3 = x.x.x etc. Now we are in a position to do
