Combinatorial Designs and Cellular Automata: A Survey

2025-03-13

AI Summary

This work models cellular automata (CA) as algebraic systems, focusing on combinatorial design problems induced by their short-term dynamics—particularly the construction of mutually orthogonal Latin squares (MOLS) via bipermutive CA. Methodologically, it unifies the treatment of both linear and nonlinear bipermutive CA, establishing a rigorous correspondence among CA local rules, finite field algebraic structures, and the constructibility of MOLS. It provides the first systematic characterization of necessary and sufficient algebraic conditions for CA to generate high-order MOLS, along with efficient constructive algorithms. The framework deepens theoretical connections between CA and combinatorial design, while also yielding provably secure design principles and analytical tools for lightweight cryptographic primitives—including S-boxes, authentication codes, and stream cipher components. Finally, the paper identifies several open problems to advance interdisciplinary research at the intersection of algebraic combinatorics, CA theory, and cryptography.

Technology Category

• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Explores Cellular Automata as algebraic systems
• Focuses on combinatorial designs from short-term CA behavior
• Reviews construction of orthogonal Latin squares via CA

Innovation

• Cellular Automata as algebraic systems
• Construction of orthogonal Latin squares
• Applications in cryptography using CA