A lower bound on the number of bent squares

TL;DR

This paper establishes a lower bound on the number of bent functions in n variables, showing they grow exponentially with a specific bound related to Walsh spectra representations.

Contribution

It provides a new lower bound on the count of bent functions using the representation as bent squares, advancing understanding of their combinatorial complexity.

Findings

Lower bound on bent functions: at least 2^{n * 2^{n/2} * (1 + O(1/n))} for even n

Representation of bent functions as bent squares

Growth rate of bent functions in relation to n

Abstract

Bent functions are Boolean functions that are maximally nonlinear. They can be represented as bent squares, i.e., square matrices for which each row and each column is the Walsh spectrum of a Boolean function. Using this representation, it is shown in this note that the number of bent functions in $n$ variables is at least $2^{n \cdot 2^{\frac{n}{2}} \left(1 + O\left(\frac{1}{n}\right)\right)}$ for even integers $n$.