Hessian Form of An Elliptic Curve - Group Law

Group Law

It is interesting to analyze the group law of the elliptic curve, defining the addition and doubling formulas (because the SPA and DPA attacks are based on the running time of these operations). Furthermore, in this case, we only need to use the same procedure to compute the addition, doubling or subtraction of points to get efficient results, as said above. In general, the group law is defined in the following way: if three points lie in the same line then they sum up to zero. So, by this property, the group laws are different for every curve.

In this case, the correct way is to use the Cauchy-Desboves´ formulas, obtaining the point at infinity = ( 1 : -1: 0), that is, the neutral element (the inverse of is again). Let P=(x₁,y₁) be a point on the curve. The line contains the point and the point at infinity . Therefore, -P is the third point of the intersection of this line with the curve. Intersecting the elliptic curve with the line, the following condition is obtained

Since is non zero (because is distinct to 1), the x-coordinate of is and the y-coordinate of is, i.e., or in projective coordinates .

In some application of elliptic curve cryptography and the elliptic curve method of factorization (ECM) it is necessary to compute the scalar multiplications of P, say P for some integer n, and they are based on the double-and-add method; these operations need the addition and dobling formulas.

Doubling

Now, if is a point on the elliptic curve, it is possible to define a "doubling" operation using Cauchy-Desboves´ formulae:

Addition

In the same way, for two different points, say and, it is possible to define the addition formula. Let denote the sum of these points, then its coordinates are given by: