A Complete Characterization of the APN Property of a Class of Quadrinomials

In this paper, by the Hasse-Weil bound, we determine the necessary and sufficient condition on coefficients
$a_{1},a_{2},a_{3}\in {\mathbb F} _{2^{n}}$
with
$n=2m$
such that
$f(x) = {x}^{3\cdot 2^{m}} + a_{1}x^{2^{m+1}+1} + a_{2} x^{2^{m}+2} + a_{3}x^{3}$
is an APN function over
${\mathbb F}_{2^{n}}$
. Our work together with the follow-up work by Chase and Lisoněk indicates that all such APN quadrinomials
$f(x)$
are affine equivalent to two instances of Gold functions, which resolves the first half of an open problem by Carlet at the International Workshop on the Arithmetic of Finite Fields, 83-107, 2014.