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On a conjecture on APN permutations

Daniele Bartoli, Marco Timpanella

TL;DR

Using tools from algebraic geometry over finite fields, it is proved that such a family of sporadic quadratic APN permutations found by Beierle and Leander does not contain any other APN permits for larger dimensions.

Abstract

The single trivariate representation proposed in [C. Beierle, C. Carlet, G. Leander, L. Perrin, A Further Study of Quadratic APN Permutations in Dimension Nine, arXiv:2104.08008] of the two sporadic quadratic APN permutations in dimension 9 found by Beierle and Leander \cite{Beierle} is further investigated. In particular, using tools from algebraic geometry over finite fields, we prove that such a family does not contain any other APN permutation for larger dimensions.

On a conjecture on APN permutations

TL;DR

Using tools from algebraic geometry over finite fields, it is proved that such a family of sporadic quadratic APN permutations found by Beierle and Leander does not contain any other APN permits for larger dimensions.

Abstract

The single trivariate representation proposed in [C. Beierle, C. Carlet, G. Leander, L. Perrin, A Further Study of Quadratic APN Permutations in Dimension Nine, arXiv:2104.08008] of the two sporadic quadratic APN permutations in dimension 9 found by Beierle and Leander \cite{Beierle} is further investigated. In particular, using tools from algebraic geometry over finite fields, we prove that such a family does not contain any other APN permutation for larger dimensions.

Paper Structure

This paper contains 2 sections, 4 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Connection with algebraic surfaces

Key Result

Proposition 2.2

Let $H$ be a plane of $\mathrm{PG}(3,\overline{\mathbb{F}}_2)$ such that $V\cap H$ contains a non-repeated absolutely irreducible component defined over $\mathbb{F}_q$. Then $V$ possesses a non-repeated absolutely irreducible component defined over $\mathbb{F}_q$.

Theorems & Definitions (7)

  • Definition 1.1
  • Remark 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof