Properties and constructions of coincident functions

TL;DR

This paper explores the properties and construction methods of coincident Boolean functions, which are invariant under the Mobius transform, offering new insights and techniques for analyzing their structure and characteristics.

Contribution

It introduces a novel approach to handle the Mobius transform, enabling composition of Boolean functions and leveraging Shannon or Reed-Muller decompositions, advancing understanding of coincident functions.

Findings

Coincident functions exhibit diverse features similar to general Boolean functions.

New properties of coincident functions are identified through the proposed methods.

Experimental results show coincident functions' features align with those of typical Boolean functions.

Abstract

Extensive studies of Boolean functions are carried in many fields. The Mobius transform is often involved for these studies. In particular, it plays a central role in coincident functions, the class of Boolean functions invariant by this transformation. This class -- which has been recently introduced -- has interesting properties, in particular if we want to control both the Hamming weight and the degree. We propose an innovative way to handle the Mobius transform which allows the composition between several Boolean functions and the use of Shannon or Reed-Muller decompositions. Thus we benefit from a better knowledge of coin-cident functions and introduce new properties. We show experimentally that for many features, coincident functions look like any Boolean functions.