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On the minimum Hamming distance between vectorial Boolean and affine functions

Gabor P. Nagy

TL;DR

This work advances understanding of how far vectorial Boolean functions are from affine mappings by precisely determining the distance d_H to affine functions for three infinite classes of $(2m,t)$-bent functions derived from pre-quasifields, and by deriving tight bounds for particular monomial APN functions such as Nyberg's inverse and a Gold-type monomial. It leverages Sidon-set structures, gerbera configurations, and detailed Walsh-spectrum analysis to connect nonlinearity, differential uniformity, and distance to affine functions, providing evidence for the Liu-Mesnager-Chen conjecture in specific bent contexts and delivering exact distances in several small-dimension APN families. The results have cryptographic relevance in S-box design due to refined characterizations of nonlinearity and differential resistance, and they open avenues to extend these exact-distance analyses to broader classes. Overall, the paper integrates algebraic, combinatorial, and computational techniques to sharpen bounds and exact values for d_H(f,\mathcal{A}) across bent and APN function families.

Abstract

In this paper, we study the Hamming distance between vectorial Boolean functions and affine functions. This parameter is known to be related to the non-linearity and differential uniformity of vectorial functions, while the calculation of it is in general difficult. In 2017, Liu, Mesnager and Chen conjectured an upper bound for this metric. We prove this bound for two classes of vectorial bent functions, obtained from finite quasigroups in characteristic two, and we improve the known bounds for two classes of monomial functions of differential uniformity two or four. For many of the known APN functions of dimension at most nine, we compute the exact distance to affine functions.

On the minimum Hamming distance between vectorial Boolean and affine functions

TL;DR

This work advances understanding of how far vectorial Boolean functions are from affine mappings by precisely determining the distance d_H to affine functions for three infinite classes of -bent functions derived from pre-quasifields, and by deriving tight bounds for particular monomial APN functions such as Nyberg's inverse and a Gold-type monomial. It leverages Sidon-set structures, gerbera configurations, and detailed Walsh-spectrum analysis to connect nonlinearity, differential uniformity, and distance to affine functions, providing evidence for the Liu-Mesnager-Chen conjecture in specific bent contexts and delivering exact distances in several small-dimension APN families. The results have cryptographic relevance in S-box design due to refined characterizations of nonlinearity and differential resistance, and they open avenues to extend these exact-distance analyses to broader classes. Overall, the paper integrates algebraic, combinatorial, and computational techniques to sharpen bounds and exact values for d_H(f,\mathcal{A}) across bent and APN function families.

Abstract

In this paper, we study the Hamming distance between vectorial Boolean functions and affine functions. This parameter is known to be related to the non-linearity and differential uniformity of vectorial functions, while the calculation of it is in general difficult. In 2017, Liu, Mesnager and Chen conjectured an upper bound for this metric. We prove this bound for two classes of vectorial bent functions, obtained from finite quasigroups in characteristic two, and we improve the known bounds for two classes of monomial functions of differential uniformity two or four. For many of the known APN functions of dimension at most nine, we compute the exact distance to affine functions.

Paper Structure

This paper contains 14 sections, 21 theorems, 48 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The distance of $(2m,t)$-bent functions to affine functions
  4. Maiorana-McFarland type $(n,m)$-bent functions
  5. Dillon type $(2m,t)$-bent functions
  6. Quasifield type $(2m,t)$-bent functions
  7. Proof of Theorem \ref{['thm:main-bent']}
  8. The distance of monomial functions to affine functions
  9. Distance results in small dimensions
  10. APN functions in small dimension
  11. Sidon sets in small dimension
  12. Graphs of APN functions and gerbera configurations
  13. Conclusion
  14. Algorithms for Sidon sets in $\mathsf{GF}(2)^n$

Key Result

Theorem 1

Let $t\leq m$ be positive integers and $(\mathsf{GF}(2^m),+,\star)$ be a left pre-quasifield. Let the maps $\gamma, \tau, h:\mathsf{GF}(2^m)\to \mathsf{GF}(2^t)$ and $\sigma:\mathsf{GF}(2^m)\to \mathsf{GF}(2^m)$ be such that $\gamma$ is balanced, $\tau$ is surjective linear, $\sigma$ is invertible, Then

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Definition 4
  • Theorem 5: Theorem 3.1 of Wu2013 and Corollary 17 of Wan2023
  • Proposition 6
  • proof
  • Lemma 7
  • proof
  • ...and 29 more