On Secure and Fast Elliptic Curve Cryptosystems over F p

From a practical point of view, a cryptosystem should require a small key size and less running time. For this purpose, we often select its definition field in such a way that the arithmetic can be implemented fast. But it often brings attacks which depend on the definition field. In this paper, we investigate the definition field F_p on which elliptic curve cryptosystems can be implemented fast, while maintaining the security. The expected running time on a general construction of many elliptic curves with a given number of rational points is also discussed.