An Example on Elliptic Curve Cryptography

Cryptography on Elliptic curve is one of the most important public key encryption systems, whose security depends on difficulty of solving the discrete logarithm problem. The reason of importance is that elliptic curves provide security equivalent to classical systems while using fewer bits. To encrypt a plain message with El-Gamal algorithm, we embed the message to the point on the elliptic curve using the Koblitz technique. In this paper, we explore the conditions that encrypt the message without mapping the point on the elliptic curve.