On the Conjecture on APN Functions

An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field F is called exceptional APN, if it is also APN on infinitely many extensions of F. In this article we consider the most studied case of F = F2n. 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, where n = 2 + 1 or n = 2 − 2 + 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 +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.