Symmetric Boolean Function with Maximum Algebraic Immunity on Odd Number of Variables

TL;DR

This paper proves the uniqueness of trivial balanced symmetric Boolean functions with maximum algebraic immunity for odd variables and provides a necessary condition for their algebraic normal form.

Contribution

It establishes the exact count of such functions and derives a necessary condition for their algebraic normal form, advancing understanding of symmetric Boolean functions in cryptography.

Findings

Exactly one trivial balanced symmetric Boolean function achieves maximum AI for each odd n.

A necessary condition for the algebraic normal form of these functions is obtained.

The results deepen the understanding of symmetric Boolean functions' resistance to algebraic attacks.

Abstract

To resist algebraic attack, a Boolean function should possess good algebraic immunity (AI). Several papers constructed symmetric functions with the maximum algebraic immunity $\lceil \frac{n}{2}\rceil $. In this correspondence we prove that for each odd $n$, there is exactly one trivial balanced $n$-variable symmetric Boolean function achieving the algebraic immunity $\lceil \frac{n}{2}\rceil $. And we also obtain a necessary condition for the algebraic normal form of a symmetric Boolean function with maximum algebraic immunity.