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

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.