Faster polynomial basis finite field squaring and inversion for GF(2<sup>m</sup>) with cryptographic software application

Abstract

Efficient finite field arithmetic is essential for fast implementation of Elliptic Curve Cryptography (ECC) in software environments. Finite field squaring is an important arithmetic operation in the binary finite field GF(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> ). Squaring is required for many cryptographic techniques based on the Discrete Logarithm Problem (DLP) in the multiplicative group of a finite field or additive group of points on an Elliptic Curve defined over a finite field. In this paper we present a new method for performing binary finite field squaring in Polynomial Basis using a lookup table which is applicable to software environments. Our method results in performance gains over squarers reported earlier in the literature on platforms where fast memory lookups can be done. We also demonstrate significant performance gains for ECC using our proposed squarer.