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

Dec 2022

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 thatnecessarily $G(x) = \wp(F(x))$ 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 havebeen considered in the literature, lately. With a binary operation on$\mathbb{F}_{p^n}\times\mathbb{F}_{p^n}$ involving $\wp$, we obtain aquasigroup, and show that the graph of a P$\wp$N function $F$ is a differenceset in the respective quasigroup. We further point to variants of symmetricdesigns obtained from such quasigroup difference sets. Finally, we analyze anequivalence (naturally defined via the automorphism group of the respectivequasigroup) for P$\wp$N functions, respectively, the difference sets in thecorresponding quasigroup.

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