On the infiniteness of a family of APN functions

TL;DR

This paper investigates whether a specific family of quadratic APN functions exists for infinitely many dimensions, using algebraic and combinatorial techniques to establish their existence for all dimensions m ≥ 3.

Contribution

The paper proves the existence of a family of APN functions for all dimensions m ≥ 3, employing novel algebraic and combinatorial methods.

Findings

Existence of the APN family for all m ≥ 3 established.

Uses algebraic varieties and vector space partitions in the proof.

Advances understanding of APN functions' infinite families.

Abstract

APN functions play a fundamental role in cryptography against attacks on block ciphers. Several families of quadratic APN functions have been proposed in the recent years, whose construction relies on the existence of specific families of polynomials. A key question connected with such constructions is to determine whether such APN functions exist for infinitely many dimensions or not. In this paper we consider a family of functions recently introduced by Li et al. in 2021 showing that for any dimension $m\geq 3$ there exists an APN function belonging to such a family. Our main result is proved by a combination of different techniques arising from both algebraic varieties over finite fields connected with linearized permutation rational functions and {partial vector space partitions}, together with investigations on the kernels of linearized polynomials.