In this report, we provide an elementary exposition of elliptic curve cryptography (ECC), which was invented around 1985 independently by Miller and Koblitz. Since then there has been extensive research on it and recently it is being used in commercial cryptosystems. In order to see where elliptic curves are used in cryptography, we begin by reviewing the abstract version of the ElGamal scheme, which can be used for arbitrary groups. Then we introduce groups based on elliptic curves. Additionally, we discuss how to encode and decode between text and points on an elliptic curve. Then we present an efficient method of performing iterated group operations. We briefly discuss how to choose the elliptic curves so that the resulting cryptosystem is secure. Finally we briefly describe the routines in the program written in Maple, that implements the algorithm.