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 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 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 Ζp, 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. curve arithmetic can be employed to develop a variety of 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.
Highlights
Cryptography is an important field of science developed to facilitate secret communication and safeguard information
To illustrate the situation created if an interrupter interrupt and corrupt the message sent by Alice, two new separate graphical user interface (GUI)’s were designed for Bob and Jane
When illustrating the communication process using Elliptic Curve Cryptography (ECC), the program was designed to generate a cyclic subgroup based on the elliptic curve chosen to perform the communication
Summary
Cryptography is an important field of science developed to facilitate secret communication and safeguard information. In Private key cryptography a single key is used for both encryption and decryption of messages which renders the inconvenience of having to agree on a common key by the communicating parties prior to the communication. The two communicating parties, say Alice and Bob, first have to generate each a pair of keys, the Private key and the Public key. If Alice needs to send Bob a message she should encrypt it using Bob’s Public key which can only be decrypted using Bob’s Private key and vice versa. An elliptic curve is a curve given by an equation of the form, y2 = x3 + ax + b with the requirement that the discriminent is non-zero. The points on an elliptic curve naturally form an abelian group and the group law can be constructed geometrically. To perform cryptography it is necessary to obtain a cyclic subgroup of this abelian group
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