Some Theorems on Planar Mappings

Abstract

A mapping $f:{\mathbb{F}}_p^n\to {\mathbb{F}}_p^n$ is called planar if for every nonzero $a \in {\mathbb{F}}_p^n$ the difference mapping D _f,a: x ↦f(x + a) − f(x) is a permutation of ${\mathbb{F}}_p^n$. In this note we prove that two planar functions are CCZ-equivalent exactly when they are EA-equivalent. We give a sharp lower bound on the size of the image set of a planar function. Further we observe that all currently known main examples of planar functions have image sets of that minimal size.