It is interesting to analyze the
group law of the elliptic curve, defining the addition and doubling formulas (because the
simple power analysis and
differential power analysis attacks are based on the running time of these operations). In general, the group law is defined in the following way: if three points lies in the same line then they sum up to zero. So, by this property, the
explicit formulas for the group law depend on the curve shape. Let be a point; its inverse is then in the plane. In projective coordinates, let be a point; then is its inverse. Furthermore, the
neutral element in affine plane is , and in projective coordinates it is . In some applications of elliptic curves for
cryptography and
integer factorization, it is necessary to compute
scalar multiples of , say for some
integer , and they are based on the
double-and-add method, so the addition and doubling formulas are needed. Using affine coordinates, the addition and doubling formulas for this
elliptic curve are as follows.
Addition formulas Let and ; then, , where x_3=\frac{x_1-y_1^2\cdot x_2\cdot y_2}{a\cdot x_1\cdot y_1\cdot x_2^2-y_2} y_3=\frac{y_1\cdot y_2^2 -a\cdot x_1^2\cdot x_2}{a\cdot x_1\cdot y_1\cdot x_2^2-y_2}
Doubling formulas Let ; then , where x_1=\frac{x-y^3\cdot x}{a\cdot y\cdot x^3-y} y_1=\frac{y^3-a\cdot x^3}{a\cdot y\cdot x^3-y} ==Algorithms and examples==