Elliptic Curves of Prime Order over Optimal Extension Fields for Use in Cryptography

Abstract

We present an algorithm for generating elliptic curves of prime order over Optimal Extension Fields suitable for use in cryptography. The algorithm is based on the theory of Complex Multiplication. Furthermore, we demonstrate the efficiency of the algorithm in practice by giving practical running times. In addition, we present statistics on the number of cryptographically strong elliptic curves of prime order for Optimal Extension Fields of cardinality (232 + c)5 with c < 0. We conclude that there are sufficiently many curves in this case.