P℘N functions, complete mappings and quasigroup difference sets

We investigate pairs of permutations of such that is a permutation for every . We show that, in that case, necessarily for some complete mapping of , and call the permutation a perfect nonlinear (PN) function. If , then is a PcN function, which have been considered in the literature, lately. With a binary operation on involving , we obtain a quasigroup, and show that the graph of a PN function is a difference set in the respective quasigroup. We further point to variants of symmetric designs obtained from such quasigroup difference sets. Finally, we analyze an equivalence (naturally defined via the automorphism group of the respective quasigroup) for PN functions, respectively, for the difference sets in the corresponding quasigroup.