Signed digit representation with NAF and balanced ternary form and efficient exponentiation in GF(qn) using a Gaussian normal basis of type II

Abstract

We present an efficient exponentiation algorithm for a finite field GF(qn) with small characteristic determined by a Gaussian normal basis of type II using signed digit representation of the exponents. Our signed digit representation uses a nonadjacent form (NAF) for GF(2n) and the balanced ternary number system for GF(3n). It is generally believed that a signed digit representation is hard to use when a normal basis is given because the inversion of a normal element requires quite a computational delay. On the other hand, the method of a signed digit representation is easily applicable to the fields with polynomial bases. However our result shows that a special normal basis called a Gaussian normal basis of type II or an optimal normal basis (ONB) of type II has a nice property which admits an effective exponentiation using signed digit representations of the exponents. © Springer-Verlag Berlin Heidelberg 2004.