A note on cyclotomic polynomials and Linear Feedback Shift Registers

TL;DR

This paper explores the properties of cyclotomic polynomials and their relation to Linear Feedback Shift Registers, providing classifications based on symmetry and roots over finite fields and rationals.

Contribution

It characterizes LFRS with symmetry properties and classifies polynomials with roots satisfying specific power relations, depending on the coefficient field.

Findings

Classification of polynomials with roots satisfying f(a^n)=0

Characterization of symmetric LFRS

Dependence on the field of coefficients

Abstract

Linear Feedback Shift Registers (LFRS) are tools commonly used in cryptography in many different context, for example as pseudo-random numbers generators. In this paper we characterize LFRS with certain symmetry properties. Related to this question we also classify polynomials f of degree n satisfying the property that if a is a root of f then $f(a^n)=0$. The classification heavily depends on the choice of the fields of coefficients of the polynomial; we consider the cases $K=\mathbb{F}_p$ and $K=\mathbb{Q}$.