A Study Of Elliptic Curve Cryptography

This paper basically two themes. One is the study of existing "Trapdoor One-way functions based on elliptic curves over Z n^ prime prime and other is "Trapdooring Discrete logarithms on elliptic curves over Rings". In view of a "Trapdoor one-way functions" based on elliptic curves over a ring Z_{s} whose security is based on the difficulty of factoring n. Also we propose a new public key cryptosystem based on the elliptic curves over a ring Z_{n} The security of the proposed scheme is based on the factoring composite numbers.