Fast Parallel GF(2^m) Polynomial Multiplication for All Degrees

Numerous works have addressed efficient parallel GF(2^m) multiplication based on polynomial basis or some of its variants. For those field degrees where neither irreducible trinomials nor Equally Spaced Polynomials (EPSs) exist, the best area/time performance has been achieved for special-type irreducible pentanomials, which however do not exist for all degrees. In other words, no multiplier architecture has been proposed so far achieving the best performance and, at the same time, being general enough to support any field degrees. In this paper, we propose a new representation, based on what we called Generalized Polynomial Bases (GPBs), covering polynomial bases and the so-called Shifted Polynomial Bases (SPBs) as special cases. In order to study the new representation, we introduce a novel formulation for polynomial basis and its variants, which is able to express concisely all implementation aspects of interest, i.e., gate count, subexpression sharing, and time delay. The methodology enabled by the new formulation is completely general and repetitive in its application, allowing the development of an ad-hoc software tool to derive proofs for area complexity and time delays automatically. As the central contribution of this paper, we introduce some new types of irreducible pentanomials and an associated GPB. Based on the above formulation, we prove that carefully chosen GPBs yield multiplier architectures matching, or even outperforming, the best special-type pentanomials from both the area and time point of view. Most importantly, the proposed GPB architectures require pentanomials existing for all degrees of practical interest. A list of suitable irreducible pentanomials for all degrees less than 1,000 is given in the appendix (Fig. 5 and Tables 4-11 are provided in a separate file containing the body of Appendix, which can be found on the Computer Society Digital Library at >http://doi.ieeecomputersociety.org/10.1109/TC.2012.63).