Differential uniformity properties of some classes of permutation polynomials

The notion of c-differential uniformity has received a lot of attention since its proposal [5], and recently a characterization of perfect c-nonlinear functions in terms of difference sets in some quasigroups was obtained in [1]. Moreover, in a very recent manuscript by Pal and Stănică [19], an intriguing connection was discovered showing that in fact, the boomerang uniformity for an odd APN function (odd characteristic) equals its c-differential uniformity when $c=-1$, if the function is a permutation, otherwise it is the maximum of the $(-1)$-DDT entries disregarding the first row/column. The construction of functions, especially permutations, with low c-differential uniformity is an interesting and difficult mathematical problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low c-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest, from a mathematical perspective.