Elliptic Curves and Elliptic Curve Cryptography

Abstract

Koblitz [7, 1987] and independently Miller [10, 1986] suggested the use of the group of points of an elliptic curve de ned over a nite eld Fq. The main incentive for this suggestion was based on the fact that the discrete logarithm problem over this group appeared to be quite hard to solve. The main objective of this dissertation was to understand how the theory of elliptic curves is applied in the design of cryptosystems. The rst chapter is an overview of cryptography - the basic fundamental aspects and de nitions that would be used in later sections. Chapter two is about the theory of elliptic curves - from the Weierstrass equations to elliptic curves over nite elds. In chapter three, we discuss about the applications of the theory of elliptic curves in public key cryptography before nally discussing about the bilinear pairings and the MOV attack on the elliptic curve discrete logarithm problem.