Abstract:
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. It is an
application of different disciplines such as Algebra, Number Theory, Graph Theory etc. 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. For an elliptic curve defined over a field, the points on an elliptic curve
naturally form an abelian group. Elliptic curve arithmetic can be employed to develop a variety of Elliptic
curve cryptographic schemes such as key exchange, encryption, digital signatures and specific construction
of a keyed-Hash Message Authentication Code (HMAC) which are illustrated through this study. 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.