New results of $0$-APN power functions over $\mathbb{F}_{2^n}$

TL;DR

This paper introduces new infinite classes of $0$-APN power functions over finite fields, providing complete characterizations and explaining previously observed examples, advancing the understanding of APN functions.

Contribution

It proposes several new infinite classes of $0$-APN power functions over $F_{2^n}$ using multivariate and resultant elimination methods, with complete characterizations.

Findings

Several new infinite classes of $0$-APN power functions are proposed.

Complete characterization of certain $0$-APN power functions based on modular conditions.

The new classes explain some previously known exponents in the literature.

Abstract

Partially APN functions attract researchers' particular interest recently. It plays an important role in studying APN functions. In this paper, based on the multivariate method and resultant elimination, we propose several new infinite classes of $0$-APN power functions over $\mathbb{F}_{2^n}$. Furthermore, two infinite classes of $0$-APN power functions $x^d$ over $\mathbb{F}_{2^n}$ are characterized completely where $(2^k-1)d\equiv 2^m-1~({\rm mod}\ 2^n-1)$ or $(2^k+1)d\equiv 2^m+1~({\rm mod}\ 2^n-1)$ for some positive integers $n, m, k$. These infinite classes of $0$-APN power functions can explain some examples of exponents of Table $1$ in \cite{BKRS2020}.