Elliptic Curve Cryptography Final report for a project in computer security

An Elliptic Curve can be roughly described as the set of solutions of an equation of the form y2 = x3 + ax+ b over some field (e.g. C,R,Q or some finite field Fpn). The importance of elliptic curves stems from their rich structure: there is a rather simple addition law definable on elliptic curves which makes them into an abelian group. Studying the emerging structure of elliptic curves over various fields has been a major theme in the mathematics of the 20th century, and elliptic curves were connected to many famous problems and results, most notably the proof of Fermat’s last theorem. See [12] for an introduction to the subject via the solution of a specific problem (the congruent number problem). A more detailed and through treatment is presented in [20] which has become the standard text on the subject.