Heuristic Search of (Semi-)Bent Functions based on Cellular Automata

TL;DR

This paper explores a cellular automata-based method for extending known bent and semi-bent Boolean functions, preserving algebraic degree and classifying results, with a focus on quadratic functions and heuristic optimization.

Contribution

It introduces a new secondary construction method using cellular automata for (semi-)bent functions, analyzing algebraic degree preservation and classification of extended functions.

Findings

Construction preserves algebraic degree of local rule.

Number of equivalence classes grows rapidly with CA diameter.

Heuristic optimization effectively extends functions.

Abstract

An interesting thread in the research of Boolean functions for cryptography and coding theory is the study of secondary constructions: given a known function with a good cryptographic profile, the aim is to extend it to a (usually larger) function possessing analogous properties. In this work, we continue the investigation of a secondary construction based on cellular automata, focusing on the classes of bent and semi-bent functions. We prove that our construction preserves the algebraic degree of the local rule, and we narrow our attention to the subclass of quadratic functions, performing several experiments based on exhaustive combinatorial search and heuristic optimization through Evolutionary Strategies (ES). Finally, we classify the obtained results up to permutation equivalence, remarking that the number of equivalence classes that our CA-XOR construction can successfully extend grows very quickly with respect to the CA diameter.