International Journal of Information and Electronics Engineering
this paper is explained about elliptic curve exponentiation over prime field (Fp) which are Affine, projective, Jacobian, Chunnovsky-Jacobian and the modified Jacobian coordinate systems that how to implement elliptic curve exponentiation over prime field (Fp) into Elliptic Curve (EC) encryption and digital signature in the global smart cards. These coordinate systems are explained the underlying arithmetic operations with formulae and geometrical diagrams. Point addition and doubling of coordinate systems over prime field Fp are mentioned for every coordinate system. Thus, the authors proposed mixed coordinate systems over prime field (Fp) in the EC encryption and digital signature of the global smart cards.