Experiments on Implementation of Elliptic Curve Arithmetic over Complex Fields Using java BigInteger Class

Abstract

Elliptic curve cryptosystems are nowadays widely used in public communication channels for network security. Their security depends on the complexity of solving the elliptic curve discrete logarithm problem. But, there are several general attacks in them. Elliptic curve arithmetic is implemented over complex fields to improve the security of elliptic curve cryptosystems. This paper begins the characteristics of elliptic curve cryptosystems and their security services. Then we discuss finite field arithmetic and its properties, prime field arithmetic, binary field arithmetic and complex number arithmetic, and elliptic curve arithmetic over prime field and binary field. This paper proposes how to implement complex number arithmetic under prime field and binary field using java BigInteger class and we implement elliptic curve arithmetic and elliptic curve cryptosystems using complex numbers over prime field and binary field and discuss the experiments that got from our implementations.