Constructing $2m$-variable Boolean functions with optimal algebraic immunity based on polar decomposition of $\mathbb{F}_{2^{2m}}^*$

TL;DR

This paper introduces a novel method for constructing $2m$-variable Boolean functions with optimal algebraic immunity using polar decomposition, resulting in balanced functions with high nonlinearity and resistance to algebraic attacks.

Contribution

It proposes a new construction approach based on polar decomposition of finite fields, improving upon previous methods for optimal algebraic immunity in Boolean functions.

Findings

Constructed Boolean functions with optimal algebraic immunity.

Achieved balanced functions with optimal algebraic degree and high nonlinearity.

Functions show strong resistance to fast algebraic attacks.

Abstract

Constructing $2m$-variable Boolean functions with optimal algebraic immunity based on decomposition of additive group of the finite field $\mathbb{F}_{2^{2m}}$ seems to be a promising approach since Tu and Deng's work. In this paper, we consider the same problem in a new way. Based on polar decomposition of the multiplicative group of $\mathbb{F}_{2^{2m}}$, we propose a new construction of Boolean functions with optimal algebraic immunity. By a slight modification of it, we obtain a class of balanced Boolean functions achieving optimal algebraic immunity, which also have optimal algebraic degree and high nonlinearity. Computer investigations imply that this class of functions also behave well against fast algebraic attacks.