Establishement of elliptic curve cryptosystem

The cryptosystem of elliptic curve has gained more and more researchers´ attention from home and abroad in recent years as it owns a short key and a fine security. This paper elaborates the definition of elliptic curve which is used to establish cryptosystem and also makes some explanation to the realization of the Add Point Operation on the elliptic curve together with the fast Add Point Operation of its corresponding coordinate system. In addition, the security of the elliptic curve cryptosystem is also analyzed starting from the theory of the elliptic curve in this paper. And the issue of production of the safe elliptic curve is discussed. Finally, the elliptic curve cryptosystem has aroused great attention of the researchers as the public key system of the next generation. The production arithmetic of the safe elliptic curve, the security analysis of the elliptic curve cryptosystem together with the fast arithmetic on the elliptic curve will be the hotspot and the issues that are remained to be solved in its future development.