Simpler proof for nonlinearity of majority function

TL;DR

This paper presents a simplified proof for the nonlinearity of the majority function, a key Boolean function in cryptography, avoiding complex polynomial calculations.

Contribution

It provides a more straightforward proof for the nonlinearity of the majority function, replacing complex polynomial-based methods with simpler reasoning.

Findings

Simplified proof for the majority function's nonlinearity

Avoids complex Krawtchouk polynomial calculations

Enhances understanding of cryptographic Boolean functions

Abstract

Given a Boolean function f, the (Hamming) weight wt(f) and the nonlinearity N(f) are well known to be important in designing functions that are useful in cryptography. The nonlinearity is expensive to compute, in general, so any shortcuts for doing that for particular functions f are significant. The well known majority function has been extensively studied in a cryptographic context for the last dozen years or so, and there is a formula for its nonlinearity. The known proofs for this formula rely on many detailed results for the Krawtchouk polynomials. This paper gives a much simpler proof.