# Search Space Reduction of Asynchrony Immune Cellular Automata by Center Permutivity

## TL;DR

This paper investigates conditions for asynchrony immunity in cellular automata, focusing on center permutivity, and uses theoretical insights to efficiently identify immune CA rules with small neighborhoods.

## Contribution

It establishes necessary conditions for asynchrony immunity, especially center permutivity, and employs these to significantly reduce the search space for immune CA rules.

## Key findings

- Center permutivity is necessary for asynchrony immunity.
- Theoretical conditions enable efficient exhaustive search.
- Identified all immune CA rules with neighborhood size up to 5.

## Abstract

We continue the study of asynchrony immunity in cellular automata (CA), which can be considered as a weaker version of correlation immunity in the context of vectorial Boolean functions. The property could have applications as a countermeasure for side-channel attacks in CA-based cryptographic primitives, such as S-boxes and pseudorandom number generators. We first give some theoretical results on the necessary conditions that a CA rule must satisfy in order to meet asynchrony immunity, the most important one being center permutivity. Next, we perform an exhaustive search of all asynchrony immune CA rules of neighborhood size up to $5$, leveraging on the discovered theoretical properties to greatly reduce the size of the search space.