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.

In Eurocrypt'18, Cid et al. proposed a new cryptanalysis tool called Boomerang Connectivity Table (BCT), to evaluate S-boxes of block ciphers. Later, Boura and Canteaut further investigated the new parameter Boomerang uniformity for cryptographic S-boxes. It is of great interest to find new S-boxes with low Boomerang uniformity for even dimensions. In this paper, we prove that a class of permutation quadrinomials over F2(2m)with m odd has Boomerang uniformity four, which gives the fifth class of such kind of permutation polynomials. Further, the occurrences of 0 and 4 in the BCTs of the investigated permutation polynomials are also completely determined.