In this paper, the Hermite polynomial representation is adapted as a new way to represent certain finite fields of characteristic three. We give the multiplication method to multiply two elements of F-3n in the Hermite polynomial representation with subquadratic computational complexity by using a divide-and-conquer idea. We show that in some cases there is a set of irreducible binomials in the Hermite polynomial representation to obtain modular reduction with a lower addition complexity than the standard polynomial representation. We also investigate the matrix vector product method for the multiplication of the field elements represented by Hermite polynomials. (C) 2013 Elsevier B.V. All rights reserved.