An Efficient Technique for Synthesis and Optimization of Polynomials in GF(2m)

Abstract

This paper presents an efficient technique for synthesis and optimization of 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 mis a non-zero positive integer. The technique is based on a graph-based decomposition and factorization of polynomials over GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ), followed by efficient network factorization and optimization. A technique for efficiently computing coefficients 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 all the 51 fields in GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ), k = 2... 8 in 0.18 micron CMOS technology with the help of the Synopsysreg design compiler. It has also been applied to minimize combinational exponentiation circuits, 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 amount