Gauss period, sparse polynomial, redundant basis, and efficient exponentiation for a class of finite fields with small characteristic

Abstract

We present an efficient exponentiation algorithm in a finite field GF(qn) using a Gauss period of type (n, 1). Though the Gauss period a of type (n, 1) in GF(qn) is never primitive, a computational evidence says that there always exists a sparse polynomial (especially, a trinomial) of a which is a primitive element in GF(qn). Our idea is easily generalized to the field determined by a root of unity over GF(q) with redundant basis technique. Consequently, we find primitive elements which yield a fast exponentiation algorithm for many finite fields GF(qn), where a Gauss period of type (n, k) exists only for larger values of k or the existing Gauss period is not primitive and has large index in the multiplicative group GF(gn)x. © Springer-Verlag Berlin Heidelberg 2003.