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On lower bounds for the distances between APN functions

Maria Mihaila, Darrion Thornburgh

TL;DR

This work addresses whether two APN functions can be distance 1 apart by leveraging the CCZ-invariant $\\Pi_F$ and recasting it through exclude multiplicities of the APN graph $\\mathcal{G}_F$ viewed as a Sidon set. It derives new, strong lower bounds on the distance $d(F,G)$ for several families, including plateaued APN (even $n$), all plateaued 3-to-1 functions, and the APN inverse when $n$ is odd, by linking $\\Pi_F$ to the minimal exclude multiplicity $e_{\\min}(\\mathcal{G}_F)$. The analysis connects these multiplicities to linear structures of the Boolean function $\\gamma_F$ and to the ortho-derivative in the quadratic case, yielding structural consequences such as non-CCZ-equivalence between the Brinkmann-Leander-Edel-Pott function and plateaued functions. Overall, the results deepen the understanding of APN distance problems, provide practical criteria for ruling out CCZ-equivalences, and illuminate the Sidon-set structure underlying APN graphs.

Abstract

Whether two distinct APN functions can have a Hamming distance of $1$ remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant $Π_F$ which can be used to provide lower bounds on the Hamming distance between a given APN function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ and other APN functions. Lower bounds on the distance from an APN function $F$ to any other are known for almost bent (AB) functions and when $F$ is a 3-to-1 quadratic function with $n$ even. In this paper, we reinterpret $Π_F$ in terms of the exclude multiplicities of the graph $\mathcal{G}_F=\{(x, F(x)) : x \in \mathbb{F}_2^n\}$ of $F$ as a Sidon set. We establish lower bounds on the distance for even $n$ when $F$ is plateaued APN, generalize the known lower bounds for quadratic $3$-to-$1$ function to all 3-to-1 plateaued functions (e.g. Kasami functions), and derive new lower bounds for when $F$ is the APN inverse function over $\mathbb{F}_{2^n}$ for $n$ odd. We also study how the exclude multiplicities of $\mathcal{G}_F$ are directly connected to the existence of linear structures of $γ_F$ when $F$ is plateaued and APN and the ortho-derivative when $F$ is a quadratic APN function. We also use the CCZ-invariance of exclude multiplicities to prove that the Brinkmann-Leander-Edel-Pott function is not CCZ-equivalent to a plateaued function.

On lower bounds for the distances between APN functions

TL;DR

This work addresses whether two APN functions can be distance 1 apart by leveraging the CCZ-invariant and recasting it through exclude multiplicities of the APN graph viewed as a Sidon set. It derives new, strong lower bounds on the distance for several families, including plateaued APN (even ), all plateaued 3-to-1 functions, and the APN inverse when is odd, by linking to the minimal exclude multiplicity . The analysis connects these multiplicities to linear structures of the Boolean function and to the ortho-derivative in the quadratic case, yielding structural consequences such as non-CCZ-equivalence between the Brinkmann-Leander-Edel-Pott function and plateaued functions. Overall, the results deepen the understanding of APN distance problems, provide practical criteria for ruling out CCZ-equivalences, and illuminate the Sidon-set structure underlying APN graphs.

Abstract

Whether two distinct APN functions can have a Hamming distance of remains an open problem. In 2020, L. Budaghyan et al. introduced a new CCZ-invariant which can be used to provide lower bounds on the Hamming distance between a given APN function and other APN functions. Lower bounds on the distance from an APN function to any other are known for almost bent (AB) functions and when is a 3-to-1 quadratic function with even. In this paper, we reinterpret in terms of the exclude multiplicities of the graph of as a Sidon set. We establish lower bounds on the distance for even when is plateaued APN, generalize the known lower bounds for quadratic -to- function to all 3-to-1 plateaued functions (e.g. Kasami functions), and derive new lower bounds for when is the APN inverse function over for odd. We also study how the exclude multiplicities of are directly connected to the existence of linear structures of when is plateaued and APN and the ortho-derivative when is a quadratic APN function. We also use the CCZ-invariance of exclude multiplicities to prove that the Brinkmann-Leander-Edel-Pott function is not CCZ-equivalent to a plateaued function.

Paper Structure

This paper contains 12 sections, 39 theorems, 79 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Distance between APN functions
  4. Plateaued APN functions
  5. AB functions
  6. Properties of the exclude multiplicities of $\mathcal{G}_{F}$ when $F$ is plateaued APN
  7. 3-to-1 plateaued functions
  8. Quadratic APN functions
  9. Power functions
  10. Lower bounds on distances from the APN inverse function
  11. On the existence of linear structures of $\gamma_F$
  12. Open problems

Key Result

Theorem 1.1

Suppose $n\geq 4$ is even, let $F,G \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ be distinct APN functions such that $F$ is plateaued. Then $d(F,G) \geq 2^{\frac{n}{2}-1}$.

Theorems & Definitions (73)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Conjecture 2.1: budaghyanCarletHellesetUpperBoundsDegree
  • Conjecture 2.2: carlet_apnGraphMaximal
  • Lemma 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 63 more