More Differential Properties of the Ness-Helleseth Function

Let
$n\geq 3$
be an odd integer,
$d_{1} = {\frac {3^{n}-1}{2}} -1 $
,
$d_{2} =3^{n}-2$
and u be an element of the finite field
$\mathbb {F}_{3^{n}}$
. This paper shows that
$f_{u}(x)=ux^{d_{1}}+x^{d_{2}}$
is an almost perfect nonlinear (APN) function on
$\mathbb {F}_{3^{n}}$
if and only if
$\chi (u+1)=\chi (u-1)=\chi (u)$
, where
$\chi (\cdot)$
denotes the quadratic character of
$\mathbb {F}_{3^{n}}$
. This settles the open problem raised by Ness and Helleseth in IEEE Trans. Inf. Theory 53(7): 2581-2586, 2007, where only the sufficiency part of the result was proved. Furthermore, we investigate the differential spectra of
$f_{u}(x)$
for elements u satisfying
$\chi (u+1)=\chi (u-1)$
and express them in terms of several quadratic character sums of cubic polynomials.