Motivated by the problem of constructing bijective maps with low differential uniformity, we introduce the notion of permutation resemblance of a function, which looks to measure the distance a given map is from being a permutation. We prove several results concerning permutation resemblance and show how it can be used to produce low differentially uniform bijections. We also study the permutation resemblance of planar functions, which over fields of odd characteristic are known not to be bijections and to have the optimal differential uniformity.

翻译：受构造低差分均匀性双射映射问题的启发，我们引入了函数的置换相似性概念，该概念用于衡量给定映射与置换之间的偏离程度。我们证明了关于置换相似性的若干结论，并展示了如何利用它来构造低差分均匀性的双射。我们还研究了平面函数的置换相似性——在奇特征域上，已知平面函数并非双射，但具有最优差分均匀性。