Linear Recurrences over a Finite Field with Exactly Two Periods

TL;DR

This paper investigates the periodicity of linear recurring sequences over finite fields, establishing conditions under which such sequences have exactly two possible periods, one trivial and one non-trivial.

Contribution

It provides necessary and sufficient conditions for a characteristic polynomial to generate sequences with precisely two periods over finite fields.

Findings

Characterization of polynomials with exactly two periods

Conditions for the period structure of linear recurring sequences

Insights into the periodicity behavior over finite fields

Abstract

In this paper, we study the periodicity structure of finite field linear recurring sequences whose period is not necessarily maximal and determine necessary and sufficient conditions for the characteristic polynomial~\(f\) to have exactly two periods in the sense that the period of any sequence generated by~\(f\) is either one or a unique integer greater than one.