Representation of Symmetric Shift Registers

TL;DR

This paper develops a mathematical framework for symmetric shift registers over GF(2), representing them via nonlinear difference equations to analyze their structure and sequence periods.

Contribution

It introduces a new approach using nonlinear difference equations and arithmetical progressions to study symmetric shift registers, enhancing understanding of their properties.

Findings

Clarifies the structure of symmetric shift registers.

Provides methods to determine minimal periods of generated sequences.

Includes open-source algorithms and an interactive web application.

Abstract

The objective of this work is to establish a mathematical framework for the study of symmetric shift registers over the field GF(2). The present paper gives a new approach where the symmetric shift registers are represented by associated systems of nonlinear difference equations. Arithmetical progressions will play a central part. This approach clarifies the underlying structures and makes it easier to determine the minimal periods of the sequences generated by the symmetric shift registers. Key words: Shift registers, nonlinear difference equations, periods, arithmetical progressions, GF(2). An open-source implementation of the algorithms presented in this paper is available on GitHub (https://github.com/paalsoreng/symmetric-shift-register). In addition, an interactive web application is provided for experimenting with and evaluating the algorithms in practice (https://paalsoreng.github.io).