Six New Classes of Permutation Trinomials over $\mathbb{F}_{2^{n}}$

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.