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

摘要

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 $ctext{N}$ with respect to all $cin mathbb {F}_{2^{4n}}setminus {1}$ 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 $ctext{N}$ 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.