On the normality of Boolean quartics

TL;DR

This paper investigates the normality of Boolean quartic functions, providing numerical analysis that confirms all 8-variable bent functions are normal or weakly normal, and conjectures this extends to all degree 4 functions in 8 variables.

Contribution

It offers the first detailed numerical exploration of the normality of 8-variable bent functions and extends previous results to all such functions.

Findings

All 8-variable bent functions are normal or weakly normal.

Numerical methods support the conjecture for degree 4 Boolean functions in 8 variables.

Completes classification results for dimensions up to 7.

Abstract

In the BFA 2023 conference paper, A. Polujan, L. Mariot and S. Picek exhibited the first example of a non-normal but weakly normal bent function in dimension 8. In this note, we present numerical approaches based on the classification of Boolean spaces to explore in detail the normality of bent functions of 8 variables and we complete S. Dubuc s results for dimensions less or equal to 7. Based on our investigations, we show that all bent functions in 8 variables are normal or weakly normal. Finally, we conjecture that more generally all Boolean functions of degree at most 4 in 8 variables are normal or weakly normal.