Paper 2023/879

On cubic-like bent Boolean functions

Abstract

Cubic bent Boolean functions (i.e. bent functions of algebraic degree at most 3) have the property that, for every nonzero element $a$ of $\mathbb{F}_2^n$, the derivative $D_af(x)=f(x)+f(x+a)$ of $f$ admits at least one derivative $D_bD_af(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b)$ that is equal to constant function 1. We study the general class of those Boolean functions having this property, which we call cubic-like bent. We study the properties of such functions and the structure of their constant second-order derivatives. We characterize them by means of their Walsh transform (that is, by their duals), by the Walsh transform of their derivatives and by other means. We study them within the Maiorana-McFarland class of bent functions, providing characterizations and constructions and showing the existence of cubic-like bent functions of any algebraic degree between 2 and $\frac n2$.