Fast inversion algorithm in GF(2m) suitable for implementation with a polynomial multiply instruction on GF(2)

Abstract

The authors propose a fast inversion algorithm in Galois field GF(2m). In this algorithm, the operations required in several contiguous iterations of a previously reported algorithm based on the extended Euclid's algorithm are represented as a matrix. These operations are performed at once through the matrix by means of a polynomial multiply instruction on GF(2). When the word size of a processor is 32 or 64 and m is larger than 233 for National Institute of Standards and Technology (NIST)-recommended irreducible polynomials, the proposed algorithm computes inversion with less polynomial multiply instructions on GF(2) and exclusive-OR instructions required by previously reported inversion algorithms on an average.