On APN functions in odd characteristic, the disproof of a conjecture and related problems

TL;DR

This paper disproves a conjecture about APN permutations in odd characteristic fields, showing they do not exist for large fields and analyzing related functions, thereby clarifying the landscape of APN functions.

Contribution

It disproves a conjecture on APN permutations for large fields and analyzes related functions, identifying limitations for their APN properties.

Findings

APN permutations do not exist for fields larger than 9587.

Certain functions are not suitable as APN candidates in large fields.

Small field APN behavior does not extend to larger fields.

Abstract

In this paper disprove a conjecture by Pal and Budaghyan (DCC, 2024) on the existence of a family of APN permutations, but showing that if the field's cardinality $q$ is larger than~$9587$, then those functions will never be APN. Moreover, we discuss other connected families of functions, for potential APN functions, but we show that they are not good candidates for APNess if the underlying field is large, in spite of the fact that they though they are APN for small environments.