On the Differential Spectrum and the APcN Property of a Class of Power Functions Over Finite Fields

In this paper, we investigate the power function
$F(x)=x^{d}$
over the finite field
$\mathbb {F}_{2^{4n}}$
, where
$n$
is a positive integer and
$d=2^{3n}+2^{2n}+2^{n}-1$
. We prove that this power function is AP
$c\text{N}$
with respect to all
$c\in \mathbb {F}_{2^{4n}}\setminus \{1\}$
satisfying
$c^{2^{2n}+1}=1$
, and we determine its
$c$
-differential spectrum. To the best of our knowledge, this is the second class of AP
$c\text{N}$
power functions over finite fields of even characteristic. By the same proof ideas, we completely determine the differential spectrum of this function, and give an affirmative answer to a recent conjecture proposed by Budaghyan, Calderini, Carlet, Davidova and Kaleyski.