Some new results on the conjecture on exceptional APN functions and absolutely irreducible polynomials: The gold case

An almost perfect nonlinear (APN) function \begin{document}$f:\mathbb{F}_{2^n}\rightarrow\mathbb{F}_{2^n}$\end{document} (necessarily polynomial) is called exceptional APN if it is APN on infinitely many extensions of \begin{document}$\mathbb{F}_{2^n}$\end{document} . Aubry, McGuire and Rodier conjectured that the only exceptional APN functions are the Gold and the Kasami-Welch monomial functions. They established that a polynomial function of odd degree is not exceptional APN provided the degree is not a Gold number \begin{document}$(2^k+1)$\end{document} or a Kasami-Welch number \begin{document}$(2^{2k}-2^k+1)$\end{document} . When the degree of the polynomial function is a Gold number or a Kasami-Welch number, several partial results have been obtained by several authors including us. In this article we address these exceptions. We almost prove the exceptional APN conjecture in the Gold degree case when \begin{document}$\deg{(h(x))}$\end{document} is odd. We also show exactly when the corresponding multivariate polynomial \begin{document}$φ(x, y, z)$\end{document} is absolutely irreducible. Also, there is only one result known when \begin{document}$f(x)=x^{2^{k}+1} + h(x)$\end{document} , and \begin{document}$\deg(h(x))$\end{document} is even. Here, we extend this result as well, thus making progress in this case that seems more difficult.