Quadratic Binomial APN Functions and Absolutely Irreducible Polynomials

TL;DR

This paper investigates quadratic binomial functions over finite fields of characteristic 2, demonstrating that many are not APN infinitely often, using algebraic geometry techniques like the Weil bound.

Contribution

It provides new insights into the APN properties of quadratic binomial functions and employs algebraic geometry to analyze their irreducibility and APN behavior.

Findings

Many quadratic binomial functions are not APN infinitely often

The proof utilizes the Weil bound from algebraic geometry

Advances understanding of APN properties in finite fields

Abstract

We show that many quadratic binomial functions on a finite field of characteristic 2 are not APN infinitely often. This is of interest in the light of recent discoveries of new families of quadratic binomial APN functions. The proof uses the Weil bound from algebraic geometry.