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On a recent extension of a family of biprojective APN functions

Lukas Kölsch

Abstract

APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension of a biprojective APN family by Göloğlu defined on $\mathbb{F}_{2^{2m}}$. We show that this generalization yields functions equivalent to Göloğlu's original family if $3\nmid m$. If $3|m$ we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.

On a recent extension of a family of biprojective APN functions

Abstract

APN functions play a big role as primitives in symmetric cryptography as building blocks that yield optimal resistance to differential attacks. In this note, we consider a recent extension of a biprojective APN family by Göloğlu defined on . We show that this generalization yields functions equivalent to Göloğlu's original family if . If we show exactly how many inequivalent APN functions this new family contains. We also show that the family has the minimal image set size for an APN function and determine its Walsh spectrum, hereby settling some open problems. In our proofs, we leverage a group theoretic technique recently developed by Göloğlu and the author in conjunction with a group action on the set of projective polynomials.
Paper Structure (6 sections, 15 theorems, 24 equations)

This paper contains 6 sections, 15 theorems, 24 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Determining equivalence
  3. A useful group action on (bi)projective polynomials
  4. Counting the number of inequivalent APN functions in the extended family
  5. The image sets and Walsh spectra of the APN functions
  6. Conclusion and open problems

Key Result

Theorem 1.4

Two quadratic APN functions $F,G \colon \mathbb F \rightarrow \mathbb F$ with $F(0)=G(0)=0$ are CCZ-equivalent if and only if they are EL-equivalent.

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • ...and 15 more