GfXpress: A Technique for Synthesis and Optimization of $\hbox{GF}(2^{m})$ Polynomials

Abstract

This paper presents an efficient technique for synthesis and optimization of the polynomials over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), where to is a nonzero positive integer. The technique is based on a graph-based decomposition and factorization of the polynomials, followed by efficient network factorization and optimization. A technique for efficiently computing the coefficients of the polynomials over GF(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), where p is a prime number, is first presented. The coefficients are stored as polynomial graphs over GF(p <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ). The synthesis and optimization is initiated from this graph-based representation. The technique has been applied to minimize multipliers over the fields GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ), where k = 2,...,8, generated with all the 51 primitive polynomials in the 0.18-mum CMOS technology with the help of the Synopsys design compiler. It has also been applied to minimize combinational exponentiation circuits, parallel integer adders and multipliers, and other multivariate bit- as well as word-level polynomials. The experimental results suggest that the proposed technique can reduce area, delay, and power by significant amounts. We also observed that the technique is capable of producing 100% testable circuits for stuck-at faults.