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On the Walsh spectra of quadratic APN functions

Sophie Hannah Bénéteau, Nicolas Goluboff, Lukas Kölsch, Divyesh Vaghasiya

TL;DR

This paper addresses the Walsh spectra of quadratic APN functions on even $n$ by establishing two novel geometric connections: a vector space partition interpretation of amplitude distributions and a blocking-set interpretation of non-bent components. The authors prove a tight relation $|W_F(b,a)|=2^{(n+\dim(V_b))/2}$ linking component amplitudes to partition dimensions, derive a packing condition and bounds on large amplitudes, and show that at most one component can exceed $2^{3n/4}$. They also obtain the first nontrivial upper bound on the number of bent components for quadratic APN functions and derive CCZ-equivalence based constraints, including a stronger bound when a function is CCZ-equivalent to a permutation. The work further analyzes amplitude distributions in dimensions $n=6,8,10$, provides explicit vector-space-partition types, and outlines open problems guiding future study of non-classical Walsh spectra in APN functions with potential cryptographic implications.

Abstract

APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function $F$ operating on $n=2k$ bits is uniquely associated with a vector space partition of $\mathbb{F}_2^n$ and a specific blocking set in the corresponding projective space $PG(n-1,2)$. These connections allow us to prove a variety of results on the Walsh spectrum of $F$. We prove for instance that $F$ can have at most one component function of amplitude larger than $2^{3n/4}$. We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.

On the Walsh spectra of quadratic APN functions

TL;DR

This paper addresses the Walsh spectra of quadratic APN functions on even by establishing two novel geometric connections: a vector space partition interpretation of amplitude distributions and a blocking-set interpretation of non-bent components. The authors prove a tight relation linking component amplitudes to partition dimensions, derive a packing condition and bounds on large amplitudes, and show that at most one component can exceed . They also obtain the first nontrivial upper bound on the number of bent components for quadratic APN functions and derive CCZ-equivalence based constraints, including a stronger bound when a function is CCZ-equivalent to a permutation. The work further analyzes amplitude distributions in dimensions , provides explicit vector-space-partition types, and outlines open problems guiding future study of non-classical Walsh spectra in APN functions with potential cryptographic implications.

Abstract

APN functions play a central role as building blocks in the design of many block ciphers, serving as optimal functions to resist differential attacks. One of the most important properties of APN functions is their linearity, which is directly related to the Walsh spectrum of the function. In this paper, we establish two novel connections that allow us to derive strong conditions on the Walsh spectra of quadratic APN functions. We prove that the Walsh transform of a quadratic APN function operating on bits is uniquely associated with a vector space partition of and a specific blocking set in the corresponding projective space . These connections allow us to prove a variety of results on the Walsh spectrum of . We prove for instance that can have at most one component function of amplitude larger than . We also find the first nontrivial upper bound on the number of bent component functions of a quadratic APN function, and provide conditions for a function to be CCZ-equivalent to a permutation based on its number of bent components.

Paper Structure

This paper contains 20 sections, 33 theorems, 27 equations.

Table of Contents

  1. Introduction and Notations
  2. Our contributions and structure of the paper
  3. Vector Space Partitions in Relation to Amplitudes
  4. Conditions on Vector Space Partitions and their Consequences to Amplitude Distributions of APN Functions
  5. Explicit constructions of vector space partitions
  6. Analysis for small dimensions
  7. Dimension 6
  8. Dimension 8
  9. Dimension 10
  10. Blocking sets associated to quadratic functions
  11. General case
  12. The case of quadratic APN functions
  13. Combining vector space partitions and blocking sets
  14. Comparison of lower bounds on $|N_F|$
  15. The cases of small dimensions
  16. ...and 5 more sections

Key Result

Proposition 1.5

Let $F \colon \mathbb F_2^n \rightarrow \mathbb F_2^n$ be a quadratic APN function, and let $2^{\frac{n+l_b}{2}}$ be the amplitude of component $F_b$. Then

Theorems & Definitions (65)

  • Definition 1.1: Differential uniformity
  • Definition 1.2: Walsh transform and linearity
  • Definition 1.3: Amplitude distribution
  • Example 1.4
  • Proposition 1.5: carlet2015boolean
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Theorem 2.3
  • proof
  • ...and 55 more