APN monomials over GF(2n) for infinitely many n

I present some results towards a complete classification of monomials that are Almost Perfect Nonlinear (APN), or equivalently differentially 2-uniform, over for infinitely many positive integers n. APN functions are useful in constructing S-boxes in AES-like cryptosystems. An application of a theorem by Weil [A. Weil, Sur les courbes algébriques et les variétés qui sʹen déduisent, in: Actualités Sci. Ind., vol. 1041, Hermann, Paris, 1948] on absolutely irreducible curves shows that a monomial xm is not APN over for all sufficiently large n if a related two variable polynomial has an absolutely irreducible factor defined over . I will show that the latter polynomialʹs singularities imply that except in three specific, narrowly defined cases, all monomials have such a factor over a finite field of characteristic 2. Two of these cases, those with exponents of the form 2k+1 or 4k−2k+1 for any integer k, are already known to be APN for infinitely many fields. The last, relatively rare case when a certain gcd is maximal is still unproven; my method fails. Some specific, special cases of power functions have already been known to be APN over only finitely many fields, but they also follow from the results below.