P$\wp$N functions, complete mappings and quasigroup difference sets

TL;DR

This paper introduces P$ ext{ extbackslash}wp$N functions, explores their connection to complete mappings and quasigroup difference sets, and analyzes their structural properties and equivalences in finite fields.

Contribution

It characterizes P$ ext{ extbackslash}wp$N functions, links them to quasigroup difference sets, and studies their automorphism-based equivalences, advancing understanding of their algebraic and combinatorial structures.

Findings

P$ ext{ extbackslash}wp$N functions are characterized by $G(x)= ext{ extbackslash}wp(F(x))$ for complete mappings.

Graph of P$ ext{ extbackslash}wp$N functions forms difference sets in associated quasigroups.

Variants of symmetric designs can be derived from these quasigroup difference sets.

Abstract

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