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Changing almost perfect nonlinear functions on affine subspaces of small codimensions

Hiroaki Taniguchi, Alexandr Polujan, Alexander Pott, Razi Arshad

TL;DR

This work advances secondary constructions of APN functions by analyzing modifications on affine subspaces with small codimensions. It develops precise APN-ness criteria for hyperplane and codimension-two subspace modifications, introduces and leverages $H$-equivalence to unify quadratic APN classifications (notably showing that in dimension $6$ every quadratic APN can be represented as $x^3+\mathrm{Tr}(x)L(x)$ up to $H$-equivalence), and provides an exponential-sum condition for the APN-ness of such forms. The results yield numerous EA-inequivalent APN functions from a single representative and demonstrate concrete constructions (including a codimension-two subspace example in dimension $8$) with explicit univariate representations. Overall, the paper deepens understanding of the APN landscape, offering practical criteria for generating new APN families and enriching the taxonomy of equivalent and inequivalent classes in cryptographic settings.

Abstract

In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function yield new ones. Furthermore, we show that some of the newly constructed functions are extended-affine inequivalent to the original ones.

Changing almost perfect nonlinear functions on affine subspaces of small codimensions

TL;DR

This work advances secondary constructions of APN functions by analyzing modifications on affine subspaces with small codimensions. It develops precise APN-ness criteria for hyperplane and codimension-two subspace modifications, introduces and leverages -equivalence to unify quadratic APN classifications (notably showing that in dimension every quadratic APN can be represented as up to -equivalence), and provides an exponential-sum condition for the APN-ness of such forms. The results yield numerous EA-inequivalent APN functions from a single representative and demonstrate concrete constructions (including a codimension-two subspace example in dimension ) with explicit univariate representations. Overall, the paper deepens understanding of the APN landscape, offering practical criteria for generating new APN families and enriching the taxonomy of equivalent and inequivalent classes in cryptographic settings.

Abstract

In this article, we study algebraic decompositions and secondary constructions of almost perfect nonlinear (APN) functions. In many cases, we establish precise criteria which characterize when certain modifications of a given APN function yield new ones. Furthermore, we show that some of the newly constructed functions are extended-affine inequivalent to the original ones.
Paper Structure (11 sections, 12 theorems, 45 equations, 1 table)

This paper contains 11 sections, 12 theorems, 45 equations, 1 table.

Table of Contents

  1. Introduction
  2. Our contribution and the structure of the paper
  3. Preliminaries
  4. Constructing $(n,m)$-APN functions
  5. Revisiting the switching construction for $(n,m)$-APN functions
  6. Constructing $(n,m)$-APN functions from $(n-1,m)$-APN functions
  7. Modifying quadratic APN functions on hyperplanes
  8. A characterization for the quadratic case and the notion of H-equivalence
  9. A necessary and sufficient condition for the APN-ness of $G(x)=x^3+\operatorname{Tr}(x)L(x)$
  10. Modifying APN functions on affine subspaces of codimension two
  11. Conclusion and open problems

Key Result

Lemma 2.2

Let $f\colon {\Bbb F}_2^n\rightarrow {\Bbb F}_2^m$ be an APN function. Let $l<m$ be a positive integer and let $\pi\colon{\Bbb F}_2^m\rightarrow {\Bbb F}_2^{l}$ be an ${\Bbb F}_2$-linear surjection. Then, $\pi\circ f$ is an APN function if and only if $D_f^*({\Bbb F}_2^n)\cap \ker(\pi)=\emptyset$.

Theorems & Definitions (29)

  • Definition 2.1: Definition 1 of taniguchi
  • Lemma 2.2: Lemma 2 of taniguchi
  • Lemma 2.3: Lemma 3 of taniguchi
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Lemma 2.6: See nyberg
  • Example 2.7
  • Proposition 2.8
  • proof
  • ...and 19 more