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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 this paper, a family of non-monomial permutations over the finite field $${F_{{2^n}}}$$F2nwith differential uniformity at most 6 is proposed, where nis a positive integer. The algebraic degree of these functions is also determined.