In this paper, we report some interesting results on permutations on \({\mathbb {Z}}_{n}\), the ring of integers modulo n, having full differential uniformity. By full differential uniformity of a permutation f on \({\mathbb {Z}}_{n}\), we mean that the cardinality of the set \(\{x\in {\mathbb {Z}}_{n}: f(x+a)-f(x)=b\}\) is exactly n for some \(a,b\in {\mathbb {Z}}_{n}\setminus \{0\}\). We give a sufficient condition for an arbitrary map on \({\mathbb {Z}}_{n}\) to have full differential uniformity. A necessary and sufficient condition for a permutation to have full differential uniformity over the ring of integers modulo n is also given. Further, we propose an upper bound and two lower bounds on permutations with full differential uniformity on \({\mathbb {Z}}_{n}\). We prove that these bounds are non-trivial bounds and give the exact number of permutations with full differential uniformity for a certain class of moduli.

在本文中，我们报告了关于\（{\ mathbb {Z}} _ {n} \）上的置换的一些有趣结果，该整数环的模数为n，具有完全微分均匀性。通过\（{\ mathbb {Z}} _ {n} \）上的置换f的完全微分均匀性，我们表示集{（mathbb {Z}} _ {n }：对于某些\（a，b \ in {\ mathbb {Z}} _ {n} \ setminus \ {0 \} \，f（x + a）-f（x）= b \} \）的正好是n ）。我们为\（{\ mathbb {Z}} _ {n} \）上的任意映射提供充分的差分均匀性的充分条件。置换在整数模环上具有完全微分均匀性的必要和充分条件n也给定。此外，我们在\（{\ mathbb {Z}} _ {n} \）上具有完全差分均匀性的置换中提出了一个上限和两个下限。我们证明了这些界限是非平凡的界限，并给出了特定类别模数具有完全微分均匀性的排列的精确数量。