Classification of Boolean Functions where Affine Functions are Uniformly Distributed

TL;DR

This paper proposes two methods for classifying all n-variable Boolean functions into classes with exactly one affine function, revealing symmetry and size properties of these classes.

Contribution

It introduces a systematic classification approach for Boolean functions, utilizing recursive and invariant-based methods to distinguish classes with unique affine functions.

Findings

Two classification methods successfully partition Boolean functions into classes with one affine function

Invariant bit positions reveal size and symmetry properties of classes

Classification enhances understanding of Boolean function structure

Abstract

Classification of Non-linear Boolean functions is a long-standing problem in the area of theoretical computer science. In this paper, effort has been made to achieve a systematic classification of all n-variable Boolean functions, where only one affine Boolean function belongs to each class. Two different methods are proposed to achieve this classification. The first method is a recursive procedure that uses the Cartesian product of sets starting from the set of 1-variable Boolean function and in the second method classification is achieved through a set of invariant bit positions with respect to an affine function belonging to that class. The invariant bit positions also provide information concerning the size and symmetry properties of the classes/sub-classes, such that the members of classes/sub-classes satisfy certain similar properties.