P℘N functions, complete mappings and quasigroup difference sets

Abstract

We investigate pairs of permutations $F,G$ of $F_{p^n}$ such that $F\left(x+a\right)-G\left(x\right)$ is a permutation for every $a\in F_{p^n}$. We show that, in that case, necessarily $G\left(x\right)=\wp \left(F\left(x\right)\right)$ for some complete mapping $-\wp$ of $F_{p^n}$, and call the permutation $F$ a perfect $\wp$ nonlinear (P$\wp$N) function. If $\wp \left(x\right)=cx$, then $F$ is a PcN function, which have been considered in the literature, lately. With a binary operation on $F_{p^n}\times F_{p^n}$ involving $\wp$, we obtain a quasigroup, and show that the graph of a P$\wp$N function $F$ 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 P$\wp$N functions, respectively, for the difference sets in the corresponding quasigroup.