Schoof's algorithm: Point counting on elliptic curves

Elliptic curves are smooth projective algebraic curves of genus 1. The points of an elliptic curve form a group; over a finite field F_p this group is finite. Elliptic curves are widely used in cryptography; these crypto systems are based on the difficulty of the discrete logarithm problem (DLP) for the group consisting of all rational points of elliptic curves defined over the field F_p. Determining the size of this group is an important step in testing the difficulty of elliptic curve DLP. Rene Schoof in 1985 introduced an algorithm, which counts points on elliptic curves over finite fields. In this thesis, we will go over the general theory of elliptic curves and discuss how this knowledge is incorporated into Schoof's algorithm.