Primitive transformation shift registers of order two over fields of characteristic two

Abstract

We consider the problem of enumeration of primitive TSRs of order n over any finite field. Here we prove the existence of primitive TSRs of order two over binary field extensions. Moreover we give a general search algorithm for primitive TSRs of odd order over any finite field and in particular of order two over fields of characteristic 2. We also give certain bounds on the number of primitive TSRs in special cases and propose a conjecture regarding the existence of certain special type of primitive polynomials, which answers the existence of primitive TSRs of odd order n over Fqm and primitive TSRs of order greater than 3 over binary field extensions..