Elliptic Curve arithmetic over extension field to intensify security and privacy

In modern days security issues play a vital and integral part of information technology and its related applications such as ATM and smart cards. Cryptography has evolved as a important element of security process. Elliptic curve cryptography is used to develop a variety of scheme for security purpose. Point addition and point doubling arithmetic used in elliptic curve is applicable for prime field. When same arithmetic performs over extension field it increases the complexity of the application but enhances the security in the field of communication technology. This paper communicates Weierstrass equation as the root of elliptic curve. Also described the group operation on elliptic curve that is point addition and point doubling used for encryption application based on prime field Fp. The proposed work is based on elliptic curve group operations solved in extension field F p 2 . This new approach is verified using System for Algebra and Geometry and Experimentation (SAGE) open source software.