Cubic bent functions outside the completed Maiorana-McFarland class

TL;DR

This paper demonstrates that for all sufficiently large even numbers of variables, cubic bent functions cannot be fully characterized by the Maiorana-McFarland class, revealing new structural insights.

Contribution

It proves that the Maiorana-McFarland class does not encompass all cubic bent functions for all even n ≥ 10, highlighting the existence of more complex structures.

Findings

Maiorana-McFarland class is incomplete for n ≥ 10

Existence of cubic bent functions outside the class for large n

Almost all such functions are homogeneous with no affine derivatives

Abstract

In this paper we prove that in opposite to the cases of 6 and 8 variables, the Maiorana-McFarland construction does not describe the whole class of cubic bent functions in $n$ variables for all $n\ge 10$. Moreover, we show that for almost all values of $n$, these functions can simultaneously be homogeneous and have no affine derivatives.