# On Some Properties of Quadratic APN Functions of a Special Form

## TL;DR

This paper investigates specific quadratic functions formed by linear combinations of $x^3$ and $x^9$, providing necessary and sufficient conditions for these functions to be Almost Perfect Nonlinear (APN), which are valuable in cryptography.

## Contribution

It offers a detailed characterization of when quadratic functions of a special form are APN, advancing understanding of their cryptographic properties.

## Key findings

- Derived necessary and sufficient conditions for APN property
- Identified new classes of APN functions of the given form
- Enhanced methods for constructing APN functions

## Abstract

In a recent paper, it is shown that functions of the form $L_1(x^3)+L_2(x^9)$, where $L_1$ and $L_2$ are linear, are a good source for construction of new infinite families of APN functions. In the present work we study necessary and sufficient conditions for such functions to be APN.