Study of Finite Field over Elliptic Curve: Arithmetic Means

Public key cryptography systems are based on sound mathematical foundations that are designed to make the problem hard for an intruder to break into the system. Number theory and algebraic geometry, namely the theory of elliptic curves defined over finite fields, has found applications in cryptology. The basic reason for this is that elliptic curves over finite fields provide an inexhaustible supply of finite abelian groups which, even when large, are amenable to computation because of their rich structure. The first level is the mathematical background concerning the needed tools from algebraic geometry and arithmetic. This paper introduces the elementary algebraic structures and the basic facts on number theory in finite fields. It includes the minimal amount of mathematical background necessary to understand the applications to cryptology. Elliptic curves are intimately connected with the theory of modular forms, in more than one ways. The paper gives a brief introduction to modular arithmetic, which is the core arithmetic of almost all public key algorithms. . The ultimate goal of the paper is to completely understand the structure of the points on the elliptic curve over any field F and being able to find them.