An infinite family of 0-APN monomials with two parameters

TL;DR

This paper introduces an infinite family of exponents characterized by two parameters that generate 0-APN monomials over finite fields, expanding understanding of APN functions and their classifications.

Contribution

The paper defines a new two-parameter family of exponents, provides conditions for 0-APN property, and characterizes their relation to known APN exponent families.

Findings

Generated infinite 0-APN monomials for various parameters.

Identified relationships between the new family and known exponent classes.

Found no new APN monomials outside known classes in computational tests.

Abstract

We consider an infinite family of exponents $e(l,k)$ with two parameters, $l$ and $k$, and derive sufficient conditions for $e(l,k)$ to be 0-APN over $\mathbb{F}_{2^n}$. These conditions allow us to generate, for each choice of $l$ and $k$, an infinite list of dimensions $n$ where $x^{e(l,k)}$ is 0-APN much more efficiently than in general. We observe that the Gold and Inverse exponents, as well as the inverses of the Gold exponents can be expressed in the form $e(l,k)$ for suitable $l$ and $k$. We characterize all cases in which $e(l,k)$ can be cyclotomic equivalent to a representative from the Gold, Kasami, Welch, Niho, and Inverse families of exponents. We characterize when $e(l,k)$ can lie in the same cyclotomic coset as the Dobbertin exponent (without considering inverses) and provide computational data showing that the Dobbertin inverse is never equivalent to $e(l,k)$. We computationally test the APN-ness of $e(l,k)$ for small values of $l$ and $k$ over $\mathbb{F}_{2^n}$ for $n \le 100$, and sketch the limits to which such tests can be performed using currently available technology. We conclude that there are no APN monomials among the tested functions, outside of known classes.