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

TL;DR

This paper investigates primitive transformation shift registers of order two over binary fields, proving their existence, providing search algorithms, and exploring bounds and conjectures related to primitive TSRs over finite fields.

Contribution

It establishes the existence of primitive TSRs of order two over binary fields and introduces algorithms and bounds for primitive TSRs of odd order over finite fields.

Findings

Primitive TSRs of order two exist over binary field extensions.

A general search algorithm for primitive TSRs of odd order is proposed.

Bounds on the number of primitive TSRs are provided in specific cases.

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..