On the Conjecture on APN Functions

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field $\mathbb{F}$ is called exceptional APN, if it is also APN on infinitely many extensions of $\mathbb{F}$. In this article we consider the most studied case of $\mathbb{F}=\mathbb{F}_{2^n}$. A conjecture of Janwa-Wilson and McGuire-Janwa-Wilson (1993/1996), settled in 2011, was that the only exceptional monomial APN functions are the monomials $x^n$, where $n=2^i+1$ or $n={2^{2i}-2i+1}$ (the Gold or the Kasami exponents respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our result is that all functions of the form $f(x)=x^{2k+1}+h(x)$ (for any odd degree $h(x)$, with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture.