A Computational Search for Cubic-Like Bent Functions

Abstract

Boolean functions are a central topic in computer science. A subset of Boolean functions, Bent Boolean functions, provide optimal resistance to various cryptographical attack vectors, making them an interesting subject for cryptography, as well as many other branches of mathematics and computer science. In this work, we search for cubic Bent Boolean functions using a novel characterization presented by Carlet & Villa in [CV23]. We implement a tool for the search of Bent Boolean functions and cubic-like Bent Boolean functions, allowing for constraints to be set on the form of the ANF of Boolean functions generated by the tool; reducing the search space required for an exhaustive search. The tool guarantees efficient traversal of the search space without redundancies. We use this tool to perform an exhaustive search for cubic-like Bent Boolean functions in dimension 6. This search proves unfeasible for dimension 8 and higher. We further attempt to find novel instances of Bent functions that are not Maioarana-McFarland in dimension 10 but fail to find any interesting results. We conclude that the proposed characterization does not yield a significant enough reduction of the search space to make the classification of cubic Bent Boolean functions of dimensions 8 or higher viable; nor could we use it to produce new instances of cubic Bent Boolean functions in 10 variables.