Image sets of perfectly nonlinear maps

We consider image sets of differentially d -uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a particularly interesting case of APN maps on binary fields 𝔽_2^n . We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for n even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map f , these results imply 2^n+1/3+1 ≤ |Im(f)| ≤ 2^n-2^(n-1)/2 .