Bounds on the degree of APN polynomials: the case of x −1 + g(x)

In this paper we consider APN functions $${f:\mathcal{F}_{2^m}\to \mathcal{F}_{2^m}}$$ of the form f(x) = x −1 + g(x) where g is any non $${\mathcal{F}_{2}}$$-affine polynomial. We prove a lower bound on the degree of the polynomial g. This bound in particular implies that such a function f is APN on at most a finite number of fields $${\mathcal{F}_{2^m}}$$. Furthermore we prove that when the degree of g is less than 7 such functions are APN only if m ≤ 3 where these functions are equivalent to x 3.