Very simple method to find the minimum polynomial of an arbitrary nonzero element of a finite field

Abstract

A method is described of finding the minimum polynomial of an arbitrary, nonzero element of GF(2^m). This method requires finite arithmetic on only a single element of the field, and not on each of the k+1 coefficients over GF(2^m), where k is the degree of the minimum polynomial. Applications include the design of error-correcting codes and of apparatus to process them, as well as cryptographic systems.