A Note on "On the Construction of Boolean Functions with Optimal Algebraic Immunity"

TL;DR

This paper explores the construction of Boolean functions with optimal algebraic immunity using the basis exchange method and Mobius inversion, providing new insights and exact counts for functions with maximum immunity.

Contribution

It introduces a novel approach using Mobius inversion to analyze Boolean functions and offers exact counts and necessary conditions for maximum algebraic immunity.

Findings

Matrix $S_1(f)S_0(f)^{-1}$ has a specific form for majority functions

Exact counting of Boolean functions with maximum algebraic immunity

Necessary conditions based on weight distribution for high immunity

Abstract

In this note, we go further on the "basis exchange" idea presented in \cite{LiNa1} by using Mobious inversion. We show that the matrix $S_1(f)S_0(f)^{-1}$ has a nice form when $f$ is chosen to be the majority function, where $S_1(f)$ is the matrix with row vectors $\upsilon_k(\alpha)$ for all $\alpha \in 1_f$ and $S_0(f)=S_1(f\oplus1)$. And an exact counting for Boolean functions with maximum algebraic immunity by exchanging one point in on-set with one point in off-set of the majority function is given. Furthermore, we present a necessary condition according to weight distribution for Boolean functions to achieve algebraic immunity not less than a given number.