On the exponents of APN power functions and Sidon sets, sum-free sets, and Dickson polynomials

We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents \begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document} , which are such that \begin{document}$ F(x) = x^d $\end{document} is an APN function over \begin{document}$ {\mathbb F}_{2^n} $\end{document} (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents \begin{document}$ d $\end{document} . We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to \begin{document}$ n = 48 $\end{document} , providing the number of exponents satisfying all the conditions for a function to be APN. We also show a new connection between APN exponents and Dickson polynomials: \begin{document}$ F(x) = x^d $\end{document} is APN if and only if the reciprocal polynomial of the Dickson polynomial of index \begin{document}$ d $\end{document} is an injective function from \begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document} to \begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document} . This also leads to a new and simple connection between Reversed Dickson polynomials and reciprocals of Dickson polynomials in characteristic 2 (which generalizes to every characteristic thanks to a small modification): the squared Reversed Dickson polynomial of some index and the reciprocal of the Dickson polynomial of the same index are equal.