Efficient Exponentiation in GF(pm) Using the Frobenius Map

The problem of exponentiation over a finite field is to compute Ae for a field element A and a positive integer e. This problem has many useful applications in cryptography and information security. In this paper, we present an efficient exponentiation algorithm in optimal extension field (OEF) GF(pm), which uses the fact that the Frobenius map, i.e., the p-th powering operation is very efficient in OEFs. Our analysis shows that the new algorithm is twice as fast as the conventional square-and-multiply exponentiation. One of the important applications of our new algorithm is random generation of a base point for elliptic curve cryptography, which is an attractive public-key mechanism for resource-constrained devices. We present a further optimized exponentiation algorithm for this application. Our experimental results show that the new technique accelerates the generation process by factors of 1.62–6.55 over various practical elliptic curves.