On a conjecture on APN permutations

TL;DR

This paper investigates a specific family of APN permutations in finite fields, proving that no additional APN permutations exist in larger dimensions within this family using algebraic geometry tools.

Contribution

It extends previous work by analyzing a family of quadratic APN permutations and proving their non-existence in higher dimensions.

Findings

The family does not contain any other APN permutations in larger dimensions.

Algebraic geometry tools effectively prove non-existence results.

Supports the uniqueness of certain APN permutations in dimension nine.

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.