Exponentiation in finite fields using dual basis multiplier

Abstract

Implementing finite fields arithmetic is very important, when realizing error control systems and cryptosystems. Recenty several algorithms for implementing multiplication in GF(2^m) have been proposed. When using the polynomial (or standard) basis representation, it is also important that efficient squaring algorithm is improved.

In this paper we present an efficient bit-serial squarer in polynomial basis representation for GF(2^m). First, we give an interesting relation between exponentiation and maximum length feedback shift register sequences(m-sequences) in GF(q ^m). Secondly, we present an efficient sequarer in GF(2^m) based upon Berlekamp's bit-serial multiplier (also called dual basis multiplier) architecture. The squarer has very simple structure and can compute the square in [m/2] steps.