In this paper, we characterize the coefficients of $f(x)=x+a_1{x}^{s_1(2^m-1)+1}+a_2{x}^{s_2(2^m-1)+1}+a_3{x}^{s_3(2^m-1)+1}$ over ${\mathbb{F}}_{2^{2m}}$ that lead $f(x)$ to be a permutation of ${\mathbb{F}}_{2^{2m}}$ for the following three cases: 1) $(s_1,s_2,s_3)=(\frac{-1}{2^k-1},1,\frac{2^k}{2^k-1})$; 2) $(s_1,s_2,s_3)=(\frac{1}{2^k+1},1,\frac{2^k}{2^k+1})$, where m, k are two positive integers; 3) $(s_1,s_2,s_3)=(\frac{1}{4},1,\frac{3}{4})$. We transform the problems for the first two cases in the investigation of some $(2^k+1)$-th degree equations with variable in the unit circle. The numerical results suggest that the sufficient conditions that we find in this paper on the coefficients are also necessary. It turned out that some well-known results can be recovered (or covered) by our results.

在本文中，我们表征了 $F(X)=X+{\mathrm{一种}}_1{X}^{{秒}_1(2^{米}-1)+1}+{\mathrm{一种}}_2{X}^{{秒}_2(2^{米}-1)+1}+{\mathrm{一种}}_3{X}^{{秒}_3(2^{米}-1)+1}$ 超过 ${\mathbb{F}}_{2^{2米}}$ 那导致 $F(X)$ 是一个排列 ${\mathbb{F}}_{2^{2米}}$ 对于以下三种情况：1) $({秒}_1,{秒}_2,{秒}_3)=(\frac{-1}{2^{克}-1},1,\frac{2^{克}}{2^{克}-1})$; 2)$({秒}_1,{秒}_2,{秒}_3)=(\frac{1}{2^{克}+1},1,\frac{2^{克}}{2^{克}+1})$, 其中m , k是两个正整数；3)$({秒}_1,{秒}_2,{秒}_3)=(\frac{1}{4},1,\frac{3}{4})$. 我们在一些案件的调查中对前两个案件的问题进行了转化$(2^{克}+1)$-th 次方程在单位圆中具有变量。数值结果表明，我们在本文中找到的关于系数的充分条件也是必要的。事实证明，我们的结果可以恢复（或覆盖）一些众所周知的结果。