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

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.