Constructing Orthogonal Latin Squares from Linear Cellular Automata

TL;DR

This paper explores how linear cellular automata can generate orthogonal Latin squares, which are useful in cryptography, by analyzing polynomial conditions for orthogonality.

Contribution

It introduces a method to construct orthogonal Latin squares from bipermutive linear cellular automata using polynomial properties.

Findings

Linear CA can generate Latin squares via bipermutive rules.

Orthogonality of Latin squares from CA is characterized by polynomial coprimality.

The approach links cellular automata properties with combinatorial design theory.

Abstract

We undertake an investigation of combinatorial designs engendered by cellular automata (CA), focusing in particular on orthogonal Latin squares and orthogonal arrays. The motivation is of cryptographic nature. Indeed, we consider the problem of employing CA to define threshold secret sharing schemes via orthogonal Latin squares. We first show how to generate Latin squares through bipermutive CA. Then, using a characterization based on Sylvester matrices, we prove that two linear CA induce a pair of orthogonal Latin squares if and only if the polynomials associated to their local rules are relatively prime.