Revisiting some results on APN and algebraic immune functions

<p style='text-indent:20px;'>We push a little further the study of two recent characterizations of almost perfect nonlinear (APN) functions. We state open problems about them, and we revisit in their perspective a well-known result from Dobbertin on APN exponents. This leads us to a new result about APN power functions and more general APN polynomials with coefficients in a subfield <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{F}_{2^k} $\end{document}</tex-math></inline-formula>, which eases the research of such functions. It also allows to construct automatically many differentially uniform functions from them (this avoids calculations for proving their differential uniformity as done in a recent paper, which are tedious and specific to each APN function). In a second part, we give simple proofs of two important results on Boolean functions, one of which deserves to be better known but needed clarification, while the other needed correction.</p>

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

<p style='text-indent:20px;'>We derive necessary conditions related to the notions, in additive combinatorics, of Sidon sets and sum-free sets, on those exponents <inline-formula><tex-math id="M1">\begin{document}$ d\in {\mathbb Z}/(2^n-1){\mathbb Z} $\end{document}</tex-math></inline-formula>, which are such that <inline-formula><tex-math id="M2">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is an APN function over <inline-formula><tex-math id="M3">\begin{document}$ {\mathbb F}_{2^n} $\end{document}</tex-math></inline-formula> (which is an important cryptographic property). We study to what extent these new conditions may speed up the search for new APN exponents <inline-formula><tex-math id="M4">\begin{document}$ d $\end{document}</tex-math></inline-formula>. We summarize all the necessary conditions that an exponent must satisfy for having a chance of being an APN, including the new conditions presented in this work. Next, we give results up to <inline-formula><tex-math id="M5">\begin{document}$ n = 48 $\end{document}</tex-math></inline-formula>, providing the number of exponents satisfying all the conditions for a function to be APN.</p><p style='text-indent:20px;'>We also show a new connection between APN exponents and Dickson polynomials: <inline-formula><tex-math id="M6">\begin{document}$ F(x) = x^d $\end{document}</tex-math></inline-formula> is APN if and only if the reciprocal polynomial of the Dickson polynomial of index <inline-formula><tex-math id="M7">\begin{document}$ d $\end{document}</tex-math></inline-formula> is an injective function from <inline-formula><tex-math id="M8">\begin{document}$ \{y\in {\Bbb F}_{2^n}^*; tr_n(y) = 0\} $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M9">\begin{document}$ {\Bbb F}_{2^n}\setminus \{1\} $\end{document}</tex-math></inline-formula>. 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.</p>