Zeros of special polynomials and their impact on a class of APN functions

TL;DR

This paper investigates the zeros of specific polynomials related to APN functions over finite fields, establishing non-existence results for certain parameter ranges and thus constraining possible constructions of these cryptographic functions.

Contribution

It provides a non-existence proof for a class of APN functions based on polynomial zero conditions, especially for larger field sizes and certain parameter values.

Findings

Non-existence of the polynomial construction for m ≥ 8 when r=1.

Application of algebraic variety techniques over finite fields.

Constraints on the existence of APN functions in the studied family.

Abstract

In 2021, Calderini et al. introduced a construction for APN functions on $\mathbb{F}_{2^{2m}}$ in bivariate form $$ f(x,y)=\big(xy,\, x^{2^r+1} + x^{2^{r+m/2}} y^{2^{m/2}} + bxy^{2^r} + cy^{2^r+1}\big),\quad r < m/2,\quad \gcd(r, m) = 1. $$ They showed that this family exists provided the existence of a polynomial $$ P_{c,b}(X)=(cX^{2^r +1} + b X^{2^r}+1)^{2^{m/2}+1}+X^{2^{m/2}+1}, $$ with no zeros in $\mathbb{F}_{2^{2m}}$. For $m\le 6$ it was shown that we can have APN functions belonging to this family. However, up to now, no construction of such polynomials is known for $m\ge 8$. In this work we provide a non-existence result of such functions whenever $r<m/8-1$, by application of techniques from algebraic varieties over finite fields. In particular, for $r=1$ we have that the construction of Calderini et al. cannot provide an APN function for $m\ge 8$.