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On the normality of Boolean quartics

Valérie Gillot, Philippe Langevin, Alexandr Polujan

TL;DR

The paper investigates normality and its relative-degree generalization for Boolean functions, focusing on dimensions up to $8$ and aiming to complete Dubuc's results. It introduces a classification-based numerical framework under $AGL(m,2)$ to compute $deg_r(f)$ and the invariants $D_r(k,m)$, applying it to small dimensions. In dimension $8$, it shows that all bent functions are normal or weakly normal, confirms that all 8-bit cubics are normal or weakly normal, and provides strong evidence that all Boolean quartics in $B(8)$ share this property, including a conjecture. The authors also deliver large-scale computational results via a precomputed set of $3$-spaces and a $355{,}073{,}617$-function cover set, highlighting the structural constraints on bent functions within 8-variable APN components.

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.

On the normality of Boolean quartics

TL;DR

The paper investigates normality and its relative-degree generalization for Boolean functions, focusing on dimensions up to and aiming to complete Dubuc's results. It introduces a classification-based numerical framework under to compute and the invariants , applying it to small dimensions. In dimension , it shows that all bent functions are normal or weakly normal, confirms that all 8-bit cubics are normal or weakly normal, and provides strong evidence that all Boolean quartics in share this property, including a conjecture. The authors also deliver large-scale computational results via a precomputed set of -spaces and a -function cover set, highlighting the structural constraints on bent functions within 8-variable APN components.

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.
Paper Structure (3 sections, 1 theorem, 7 equations, 3 tables)

This paper contains 3 sections, 1 theorem, 7 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Normality and Relative degree
  3. Normality in dimension 8

Key Result

Proposition 1

All the cubics in $B(8)$ are normal or weakly normal.

Theorems & Definitions (7)

  • Example 1
  • Conjecture 1
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Conjecture 2