Primitive transformation shift registers over finite fields

TL;DR

This paper investigates the existence, enumeration, and construction of primitive transformation shift registers over finite fields, establishing new theoretical results and algorithms for their generation, especially over fields of characteristic two.

Contribution

It proves the existence of primitive TSRs of order two over characteristic two fields, links them to special primitive polynomials, and provides algorithms for their enumeration and construction.

Findings

Existence of primitive TSRs of order two over characteristic two fields.

An equivalence between primitive TSRs and certain primitive polynomials.

A general search algorithm for primitive TSRs of odd order.

Abstract

We consider the problem of existence and enumeration of primitive TSRs of order n over any finite field. Here we prove the existence of primitive TSRs of order two over finite fields of characteristic two and establish an equivalence between primitive TSRs and primitive polynomials of special form. A conjecture regarding the existence of these special type of primitive polynomials is submitted by us along with some experimental verification. Further we have attempted to enumerate primitive TSRs of order two over finite fields of characteristic two. Finally 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 two.