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In the study of Diophantine equations elliptic curves play a vital role due to its application in proof of famous Fermat's Las t Theorem (FLT) and were further developed with its applications in factoring and primality. In this paper we define elliptic curve E(K) over the field K and describe the arithmetic on elliptic curve and give the group law with respect to the characteristic of K.. Due to its group structure and its analogue nature to multiplicative group of a finite field , elliptic curves find their way in enormous applications in cryptography. In this paper we also describe the group law for elliptic curve over a finite ring and propose a public key encryption with elliptic curve over the ring Z pq for p, q are primes.

Encryption on Elliptic Curves over Zpq with Arithmetic on E(Zpq) via E(Zp ) and E(Zq )