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 \(\mathbb {F}_{2^n}\). 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 \(\frac{2^n+1}{3}+1 \le |{\text {Im}}(f)| \le 2^n-2^{(n-1)/2}\).

我们考虑有限域的差分d均匀映射的图像集。我们提出了此类地图图像大小的下限，并研究了它们的原像分布。此外，我们关注二进制字段\(\mathbb {F}_{2^n}\)上的一个特别有趣的 APN 映射案例。我们展示了具有最小图像大小的 APN 映射非常接近于 3 比 1。我们证明，对于n ，即使是几个重要的 APN 图族的图像集也是最小的，因此它们具有经典的 Walsh 谱。最后，我们提出了 APN 地图图像大小的上限。对于非双射几乎弯曲的映射f，这些结果意味着\(\frac{2^n+1}{3}+1 \le |{\text {Im}}(f)| \le 2^n-2 ^{(n-1)/2}\)。