STUDY OF ELLIPTIC CURVES IN PROJECTIVE PLANES RELATED TO CRYPTOGRAPHY

Abstract

Elliptic curves are of great importance in present-day cryptography. In the past two decades, much progress has been made to make elliptic curves ready for prime-time use. Especially in the past few years a lot of research has been undertaken in order to find alternative applications of elliptic curves and to make existing applications more efficient by arithmetic speedups in this expository thesis we study elliptic curves and their role in cryptography. In doing so we examine an intersection of linear algebra abstract algebra number theory and algebraic Geometry all of which combined provides the necessary background. First we present Background information on classification of fields as well as construction of finite fields and Computation in finite fields. We next explore logarithms in finite fields and introduce the Diffie-Hellman key exchange system. Subsequently we take a look at the Projective and Affine planes and we examine the action of the general linear group of degree 3 (over K) of the points of the projective plane  (K). We then explore the geometry of the projective Plane .we study forms of degree 3 and we are able to explore cubics and the group law on an elliptic curve which leads us to our ultimate goal of examining the role of elliptic curves in cryptography