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Self-Orthogonal Cellular Automata

Luca Mariot, Federico Mazzone

TL;DR

The paper investigates when the Latin square generated by a bipermutive CA is self-orthogonal, introducing an algebraic criterion based on the invertibility of a stacked transition matrix. By exploiting the circulant structure of this matrix, it reduces self-orthogonality to gcd tests between the local-rule polynomial $p_f(X)$ and $X^{2(d-1)}-1$, with a binary-specific simplification to gcd$(p_f(X),X^{d-1}+1)=1$; irreducibility of $p_f$ suffices in the binary case, and a parity criterion applies when the stacked size is a power of two. Empirical exploration up to diameter $d=6$ shows no nonlinear self-orthogonal CA and highlights the prominence of linear SOCA, while the theoretical results provide a complete linear-characterization and a clear pathway to further questions. The findings link CA-based Latin squares to circulant-matrix invertibility and gcd computations, with potential implications for combinatorial designs, secret sharing, and cryptographic constructions.

Abstract

It is known that no-boundary Cellular Automata (CA) defined by bipermutive local rules give rise to Latin squares. In this paper, we study under which conditions the Latin square generated by a bipermutive CA is self-orthogonal, i.e. orthogonal to its transpose. We first enumerate all bipermutive CA over the binary alphabet up to diameter $d=6$, remarking that only some linear rules give rise to self-orthogonal Latin squares. We then give a full theoretical characterization of self-orthogonal linear CA, by considering the square matrix obtained by stacking the transition matrices of the CA and of its transpose, and determining when it is invertible. Interestingly, the stacked matrix turns out to have a circulant structure, for which there exists an extensive body of results to characterize its invertibility. Further, for the case of the binary alphabet we prove that irreducibility is a sufficient condition for self-orthogonality, and we derive a simpler characterization which boils down to computing the parity of the central coefficients of the local rule.

Self-Orthogonal Cellular Automata

TL;DR

The paper investigates when the Latin square generated by a bipermutive CA is self-orthogonal, introducing an algebraic criterion based on the invertibility of a stacked transition matrix. By exploiting the circulant structure of this matrix, it reduces self-orthogonality to gcd tests between the local-rule polynomial and , with a binary-specific simplification to gcd; irreducibility of suffices in the binary case, and a parity criterion applies when the stacked size is a power of two. Empirical exploration up to diameter shows no nonlinear self-orthogonal CA and highlights the prominence of linear SOCA, while the theoretical results provide a complete linear-characterization and a clear pathway to further questions. The findings link CA-based Latin squares to circulant-matrix invertibility and gcd computations, with potential implications for combinatorial designs, secret sharing, and cryptographic constructions.

Abstract

It is known that no-boundary Cellular Automata (CA) defined by bipermutive local rules give rise to Latin squares. In this paper, we study under which conditions the Latin square generated by a bipermutive CA is self-orthogonal, i.e. orthogonal to its transpose. We first enumerate all bipermutive CA over the binary alphabet up to diameter , remarking that only some linear rules give rise to self-orthogonal Latin squares. We then give a full theoretical characterization of self-orthogonal linear CA, by considering the square matrix obtained by stacking the transition matrices of the CA and of its transpose, and determining when it is invertible. Interestingly, the stacked matrix turns out to have a circulant structure, for which there exists an extensive body of results to characterize its invertibility. Further, for the case of the binary alphabet we prove that irreducibility is a sufficient condition for self-orthogonality, and we derive a simpler characterization which boils down to computing the parity of the central coefficients of the local rule.

Paper Structure

This paper contains 9 sections, 9 theorems, 17 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Cellular Automata
  4. Latin Squares
  5. Empirical Search of Self-Orthogonal CA
  6. Characterization of the Linear Case
  7. General Results for the Alphabet $\Sigma = \mathbb{F}_q$
  8. The Special Case $\Sigma = \mathbb{F}_2$
  9. Conclusions and Open Problems

Key Result

Lemma 1

Let $\Sigma$ be a finite alphabet of size $q$. The Cayley table $C_F$ of a CA $F: \Sigma^{2(d-1)} \rightarrow \Sigma^{d-1}$ defined by a bipermutive local rule $f:\Sigma^{d}\rightarrow \Sigma$ is a Latin square of order $N=q^{d-1}$.

Figures (3)

  • Figure 1: Examples of NBCA and PBCA evaluations under local rule 150.
  • Figure 2: Latin square of order $4$ generated by rule $150$.
  • Figure 3: Self-superposition of rule 90 and 150. Only rule 150 is self-orthogonal, since all 16 pairs in the Cartesian product of $[4]\times [4]$ occur exactly once.

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Lemma 1
  • Theorem 1
  • Definition 5
  • Lemma 2
  • proof
  • Theorem 2: lidl94
  • ...and 7 more