Nonlinearity measures of random Boolean functions

TL;DR

This paper investigates the behavior of r-th order nonlinearity in random Boolean functions, showing strong convergence results for all r ≥ 1, and provides elementary combinatorial proofs that simplify previous findings.

Contribution

It extends existing results on nonlinearity of Boolean functions to all r ≥ 1 and introduces simpler combinatorial proofs for these convergence properties.

Findings

Strong convergence of normalized r-th order nonlinearity for all r ≥ 1

Extension of previous results by Rodier and Dib to higher orders

Elementary combinatorial methods used for proofs

Abstract

The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order nonlinearity of a random Boolean function converges strongly for all r\ge 1. This extends results by Rodier for r=1 and by Dib for r=2. The methods in the present paper are mostly of elementary combinatorial nature and also lead to simpler proofs in the cases that r=1 or 2.