On a Class of Quadratic Polynomials with no Zeros and its Application to APN Functions

TL;DR

This paper constructs an infinite family of APN functions over finite fields by analyzing quadratic polynomials with no zeros, extending previous computational findings to a theoretical framework for specific field degrees.

Contribution

It provides a theoretical proof for the existence of certain APN functions over fields with even degree and specific conditions, expanding on prior computational results.

Findings

Existence of APN functions for even k with 3 not dividing k

Construction of an infinite family of APN functions

Extension of computational results to theoretical proof

Abstract

We show that the there exists an infinite family of APN functions of the form $F(x)=x^{2^{s}+1} + x^{2^{k+s}+2^k} + cx^{2^{k+s}+1} + c^{2^k}x^{2^k + 2^s} + \delta x^{2^{k}+1}$, over $\gf_{2^{2k}}$, where $k$ is an even integer and $\gcd(2k,s)=1, 3\nmid k$. This is actually a proposed APN family of Lilya Budaghyan and Claude Carlet who show in \cite{carlet-1} that the function is APN when there exists $c$ such that the polynomial $y^{2^s+1}+cy^{2^s}+c^{2^k}y+1=0$ has no solutions in the field $\gf_{2^{2k}}$. In \cite{carlet-1} they demonstrate by computer that such elements $c$ can be found over many fields, particularly when the degree of the field is not divisible by 3. We show that such $c$ exists when $k$ is even and $3\nmid k$ (and demonstrate why the $k$ odd case only re-describes an existing family of APN functions). The form of these coefficients is given so that we may write the infinite family of APN functions.