# On Dillon's property of $(n,m)$-functions

## TL;DR

This paper investigates Dillon's property for $(n,m)$-functions, providing bounds, characterizations, and simplified checks, thereby extending the understanding of APN functions and plateaued functions with this property.

## Contribution

It extends Dillon's property to $(n,m)$-functions with $m
eq n$, offering new bounds, characterizations via Walsh transform and ANF, and simplified criteria for quadratic functions.

## Key findings

- Derived combinatorial bounds on the dimension $m$ for Dillon's property.
- Characterized the property using Walsh transform and ANF for quadratic functions.
- Extended families of APN functions satisfying Dillon's property.

## Abstract

Dillon observed that an APN function $F$ over $\mathbb{F}_2^{n}$ with $n$ greater than $2$ must satisfy the condition $\{F(x) + F(y) + F(z) + F(x + y + z) \,:\, x,y,z \in\mathbb{F}_2^n\}= \mathbb{F}_2^n$. Recently, Taniguchi (2023) generalized this condition to functions defined from $\mathbb{F}_2^n$ to $\mathbb{F}_2^m$, with $m>n$, calling it the D-property. Taniguchi gave some characterizations of APN functions satisfying the D-property and provided some families of APN functions from $\mathbb{F}_2^n$ to $\mathbb{F}_2^{n+1}$ satisfying this property. In this work, we further study the D-property for $(n,m)$-functions with $m\ge n$. We give some combinatorial bounds on the dimension $m$ for the existence of such functions. Then, we characterize the D-property in terms of the Walsh transform and for quadratic functions we give a characterization of this property in terms of the ANF. We also give a simplification on checking the D-property for quadratic functions, which permits to extend some of the APN families provided by Taniguchi. We further focus on the class of the plateaued functions, providing conditions for the D-property.