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Combinatorial Designs and Cellular Automata: A Survey

Luca Manzoni, Luca Mariot, Giuliamaria Menara

TL;DR

Problem: Understand how combinatorial designs, especially Latin squares and orthogonal arrays, can be generated from cellular automata and leveraged for cryptographic primitives. Approach: interpret one-dimensional bipermutive CA as algebraic systems through the block transformation, characterize linear MOCA via coprime polynomials, and explore nonlinear MOCA with combinatorial and optimization methods, linking to OA/MOLS and Hadamard/bent function constructions. Contributions: comprehensive survey of linear MOCA theory, partial results for nonlinear MOCA, and connections to threshold secret sharing, PRNGs, bent functions, and correlation-immunity, plus a roadmap of open problems. Significance: clarifies when CA yield provable orthogonal designs and illustrates practical cryptographic applications, with avenues for generalization to higher-dimensional designs and diffusion layers.

Abstract

Cellular Automata (CA) are commonly investigated as a particular type of dynamical systems, defined by shift-invariant local rules. In this paper, we consider instead CA as algebraic systems, focusing on the combinatorial designs induced by their short-term behavior. Specifically, we review the main results published in the literature concerning the construction of mutually orthogonal Latin squares via bipermutive CA, considering both the linear and nonlinear cases. We then survey some significant applications of these results to cryptography, and conclude with a discussion of open problems to be addressed in future research on CA-based combinatorial designs.

Combinatorial Designs and Cellular Automata: A Survey

TL;DR

Problem: Understand how combinatorial designs, especially Latin squares and orthogonal arrays, can be generated from cellular automata and leveraged for cryptographic primitives. Approach: interpret one-dimensional bipermutive CA as algebraic systems through the block transformation, characterize linear MOCA via coprime polynomials, and explore nonlinear MOCA with combinatorial and optimization methods, linking to OA/MOLS and Hadamard/bent function constructions. Contributions: comprehensive survey of linear MOCA theory, partial results for nonlinear MOCA, and connections to threshold secret sharing, PRNGs, bent functions, and correlation-immunity, plus a roadmap of open problems. Significance: clarifies when CA yield provable orthogonal designs and illustrates practical cryptographic applications, with avenues for generalization to higher-dimensional designs and diffusion layers.

Abstract

Cellular Automata (CA) are commonly investigated as a particular type of dynamical systems, defined by shift-invariant local rules. In this paper, we consider instead CA as algebraic systems, focusing on the combinatorial designs induced by their short-term behavior. Specifically, we review the main results published in the literature concerning the construction of mutually orthogonal Latin squares via bipermutive CA, considering both the linear and nonlinear cases. We then survey some significant applications of these results to cryptography, and conclude with a discussion of open problems to be addressed in future research on CA-based combinatorial designs.

Paper Structure

This paper contains 21 sections, 7 theorems, 18 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Cellular Automata
  4. Combinatorial Designs
  5. CA as Algebraic Systems
  6. The Block Transformation
  7. Preimage Computation for Permutive CA
  8. Latin Squares from Bipermutive CA
  9. Mutually Orthogonal Cellular Automata (MOCA)
  10. The Linear Case and the Connection with Coprime Polynomials
  11. The Nonlinear Case
  12. Applications to Cryptography
  13. Threshold Secret Sharing Schemes
  14. Pseudorandom Number Generators
  15. Boolean Functions
  16. ...and 6 more sections

Key Result

Lemma 1

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 $N=q^{d-1}$, where $q = |\Sigma|$. Equivalently, the algebraic structure $\langle \hat{\Sigma}, F \rangle$ is a quasigroup of o

Figures (8)

  • Figure 1: Example of NBCA defined by rule 150, together with its truth table and its de Bruijn graph representations.
  • Figure 2: Orthogonal Latin squares of order $N=4$, and their superposition.
  • Figure 3: Example of block transformation for a CA of diameter $d=5$. The original local rule is defined as $f(x_1,x_2,x_3,x_4,x_5) = x_1 \oplus x_3 \oplus x_5$.
  • Figure 4: Preimage computation for $c=(1,0,0,1,1,0) \in \mathbb{F}_2^6$ using rule 150.
  • Figure 5: Setup phase of the SSS scheme from ml-acri-2014 with two copies of the secret $S$.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1: e-nonlin-1993mgfl-desi-2020
  • Theorem 1: mgfl-desi-2020
  • Theorem 2: mgfl-desi-2020
  • Theorem 3: mgfl-desi-2020
  • ...and 4 more