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Equivalences of biprojective almost perfect nonlinear functions

Faruk Göloğlu, Lukas Kölsch

TL;DR

This work develops a robust group-theoretic framework for biprojective APN functions that exploits a large cyclic automorphism subgroup to resolve equivalence both within and between known biprojective families. It introduces a new infinite APN family, $ F_4$, with explicit parameter constraints, and proves its APN property via differential analysis on the biprojective structure. The central contribution is a theorem that reduces EL-equivalence testing to structured, monomial-block linear-algebra under a centralizer condition, enabling comprehensive enumeration of inequivalent APN functions across families and establishing the exponential growth of inequivalents in $ F_4$. The results not only recover and unify prior findings for Taniguchi and Zhou-Pott, but also provide a generic, scalable method applicable to a broad super-class of biprojective APN functions, significantly advancing understanding of APN equivalence and diversity. The paper also analyzes the Walsh spectrum for the new family, showing a classical pattern consistent with other known biprojective APN families.

Abstract

Two important problems on almost perfect nonlinear (APN) functions are the enumeration and equivalence problems. In this paper, we solve these two problems for any biprojective APN function family by introducing a strong group theoretic method for those functions. Roughly half of the known APN families of functions on even dimensions are biprojective. By our method, we settle the equivalence problem for all known biprojective APN functions. Furthermore, we give a new family of biprojective APN functions. Using our method, we count the number of inequivalent APN functions in all known biprojective APN families and show that the new family found in this paper gives exponentially many new inequivalent APN functions. Quite recently, the Taniguchi family of APN functions was shown to contain an exponential number of inequivalent APN functions by Kaspers and Zhou (J. Cryptol. 34 (1), 2021) which improved their previous count (J. Comb. Th. A 186, 2022) for the Zhou-Pott family. Our group theoretic method substantially simplifies the work required for proving those results and provides a generic natural method for every family in the large super-class of biprojective APN functions that contains these two family along with many others.

Equivalences of biprojective almost perfect nonlinear functions

TL;DR

This work develops a robust group-theoretic framework for biprojective APN functions that exploits a large cyclic automorphism subgroup to resolve equivalence both within and between known biprojective families. It introduces a new infinite APN family, , with explicit parameter constraints, and proves its APN property via differential analysis on the biprojective structure. The central contribution is a theorem that reduces EL-equivalence testing to structured, monomial-block linear-algebra under a centralizer condition, enabling comprehensive enumeration of inequivalent APN functions across families and establishing the exponential growth of inequivalents in . The results not only recover and unify prior findings for Taniguchi and Zhou-Pott, but also provide a generic, scalable method applicable to a broad super-class of biprojective APN functions, significantly advancing understanding of APN equivalence and diversity. The paper also analyzes the Walsh spectrum for the new family, showing a classical pattern consistent with other known biprojective APN families.

Abstract

Two important problems on almost perfect nonlinear (APN) functions are the enumeration and equivalence problems. In this paper, we solve these two problems for any biprojective APN function family by introducing a strong group theoretic method for those functions. Roughly half of the known APN families of functions on even dimensions are biprojective. By our method, we settle the equivalence problem for all known biprojective APN functions. Furthermore, we give a new family of biprojective APN functions. Using our method, we count the number of inequivalent APN functions in all known biprojective APN families and show that the new family found in this paper gives exponentially many new inequivalent APN functions. Quite recently, the Taniguchi family of APN functions was shown to contain an exponential number of inequivalent APN functions by Kaspers and Zhou (J. Cryptol. 34 (1), 2021) which improved their previous count (J. Comb. Th. A 186, 2022) for the Zhou-Pott family. Our group theoretic method substantially simplifies the work required for proving those results and provides a generic natural method for every family in the large super-class of biprojective APN functions that contains these two family along with many others.

Paper Structure

This paper contains 13 sections, 18 theorems, 77 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Projective and biprojective polynomials
  4. The new APN family
  5. Family $\mathcal{F}_{4}$
  6. A technique to determine equivalence
  7. The group theoretic framework
  8. A theorem to determine equivalence of projective polynomial pairs
  9. Inequivalences within biprojective families
  10. Inequivalences between biprojective families
  11. The centralizer condition
  12. The Carlet family $\mathcal{C}$
  13. Walsh spectra of biprojective APN functions

Key Result

Lemma 2.1

Let $F(x,y) = (f(x,y),g(x,y))$ be a $(q,r)$-biprojective polynomial pair. Then $F$ is APN on $\mathbb M \times \mathbb M$ if and only if ${\mathsf{D}_{f}^{{u}}}(x,y) = 0 = {\mathsf{D}_{g}^{{u}}}(x,y)$ has exactly two solutions for each $u \in {\mathbb{P}^1}(\mathbb M)$.

Theorems & Definitions (34)

  • Lemma 2.1
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Definition 4.1
  • Theorem 4.2
  • Proposition 4.3
  • proof
  • Lemma 4.5
  • ...and 24 more