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From the earliest days of history, the requirement for methods of secret communication and protection of information had been present. Cryptography is such an important field of science developed to facilitate secret communication and safeguard information. Cryptography is based on mathematics. Private key cryptography and Public key cryptography are the two main types of cryptography. Public key cryptosystems offer more security and convenience for the users. The main objective of this study is to explore the possibilities of further improvement of Elliptic Curve Cryptography (ECC) by studying the mathematical aspects behind the "Elliptic Curve Cryptosystem" which is one of the latest of this kind and develop a computer program to generate the cyclic subgroup of a given elliptic curve defined over a finite field Zp, where p is a prime, which is the major requirement to perform ECC and then use the same to illustrate how data security is achieved from this. Moreover this study proposes an improvement for the encryption of a message through utilization of a concept in "Coding Theory" of Abstract algebra which offers an additional shield for the transmitted message.