Exploring Semi-bent Boolean Functions Arising from Cellular Automata

TL;DR

This paper explores a novel method for generating semi-bent Boolean functions using cellular automata, focusing on quadratic rules and their cryptographic properties, including balancedness and linear structures.

Contribution

It introduces a CA-based construction for semi-bent functions, along with an exhaustive algorithm for quadratic rules up to 6 variables, revealing properties of the generated functions.

Findings

Successfully enumerated quadratic rules producing semi-bent functions

Generated semi-bent functions with constant linear structures

Filtered functions based on balancedness and cryptographic suitability

Abstract

Semi-bent Boolean functions are interesting from a cryptographic standpoint, since they possess several desirable properties such as having a low and flat Walsh spectrum, which is useful to resist linear cryptanalysis. In this paper, we consider the search of semi-bent functions through a construction based on cellular automata (CA). In particular, the construction defines a Boolean function by computing the XOR of all output cells in the CA. Since the resulting Boolean functions have the same algebraic degree of the CA local rule, we devise a combinatorial algorithm to enumerate all quadratic Boolean functions. We then apply this algorithm to exhaustively explore the space of quadratic rules of up to 6 variables, selecting only those for which our CA-based construction always yields semi-bent functions of up to 20 variables. Finally, we filter the obtained rules with respect to their balancedness, and remark that the semi-bent functions generated through our construction by the remaining rules have a constant number of linear structures.