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 \(n={2^{2i}-2^i+1}\) (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 \(f(x)=x^{2^k+1}+h(x)\) (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.