Wreath products and cascaded FSRs

TL;DR

This paper introduces an algebraic framework using wreath products to analyze the periods of cascaded finite state registers (FSRs), providing bounds and explicit characterizations for specific cases like De Bruijn sequences.

Contribution

It presents a novel algebraic approach to study cascaded FSRs using wreath products, offering new bounds and detailed period analysis.

Findings

Established a connection between cascaded FSRs and wreath products.

Derived a general upper bound on the maximum period of cascaded FSRs.

Provided explicit period characterizations for cascaded De Bruijn sequences.

Abstract

We show that the transition function of the cascaded connection of two FSRs can be viewed as a wreath product element. This allows us to study periods of cascaded connections with algebraic methods, obtaining both a general, nontrivial upper bound on the maximum period of a cascaded connection and a complete, explicit understanding of the periods in the important case of the cascaded connection of an $n$-dimensional De Bruijn sequence into an $m$-dimensional linear FSR.