Elliptic curve

The elliptic curve equation is: y² = x³ + ax + b

An elliptic curve over some field F is the set of pairs of elements (x,y) in F satisfying the above equation, for a given a and b in F.

If we add a point at infinity, an elliptic curve forms an abelian group (as long as a certain constraint is met on the values of a and b, which ensures that the polynomial x³ + ax + b does not have a multiple zero and the curve does not have a singularity). Each point on the curve is an element of the group, and a geometrical construction allows us to define addition of points in a consistent manner.

Specifically, to add distinct points we draw a straight line through them, and find the unique third point which this line meets. The reflection of this line in the x-axis is the sum of the two original points.

Elliptic curves on finite fields are used in some cryptographic applications, see elliptic curve cryptography.