Permutation polynomials over finite fields constitute an active research area. Permutation trinomials attract researchers' interest due to their simple algebraic form and some additional extraordinary properties. In this paper, we present six new classes of permutation trinomials over $\mathbb{F}_{2^{n}}$ which have explicit forms by determining the solutions of some equations.

Based on a recent work of Ding, Wang, and Xiong, we present some cyclotomic constructions of zero-difference balanced (ZDB) functions, which interpret the previous construction proposed by Cai, Zeng, Helleseth, Tang, and Yang. The resulted ZDB functions can be used to generate optimal constant composition codes and perfect difference systems of sets with new parameters.

In this paper, two new families of functions in odd characteristic are constructed, and they are proved to be almost perfect nonlinear (APN) functions. Some of the “open cases” which were introduced by Helleseth are answered by these new APN functions.