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 <inline-formula> <tex-math notation="LaTeX">$F(x)=x^{d}$ </tex-math></inline-formula> over the finite field <inline-formula> <tex-math notation="LaTeX">$\mathbb {F}_{2^{4n}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> is a positive integer and <inline-formula> <tex-math notation="LaTeX">$d=2^{3n}+2^{2n}+2^{n}-1$ </tex-math></inline-formula>. We prove that this power function is AP<inline-formula> <tex-math notation="LaTeX">$c\text{N}$ </tex-math></inline-formula> with respect to all <inline-formula> <tex-math notation="LaTeX">$c\in \mathbb {F}_{2^{4n}}\setminus \{1\}$ </tex-math></inline-formula> satisfying <inline-formula> <tex-math notation="LaTeX">$c^{2^{2n}+1}=1$ </tex-math></inline-formula>, and we determine its <inline-formula> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula>-differential spectrum. To the best of our knowledge, this is the second class of AP<inline-formula> <tex-math notation="LaTeX">$c\text{N}$ </tex-math></inline-formula> 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.