The study of permutation polynomials over finite fields has attracted many scholars' attentions due to their wide applications and there are several different forms of permutations over finite fields. However, there is little literature on the relation between different forms of permutations.

In this paper, we find an equivalent relation between permutation polynomials of the form $f(x)=xh\left(x^{2^n-1}\right)$ over ${\mathbb{F}}_{2^{2n}}$ and permutations of the form $F(x)=\frac{H_1(\mathrm{\Delta })}{H_1(\mathrm{\Delta })+H_2(\mathrm{\Delta })}+x$ on ${\mathbb{F}}_{2^n}$, where $\mathrm{\Delta }=\frac{1}{x^2+x+\delta }$, $\delta \in {\mathbb{F}}_{2^n}$ with ${\mathrm{Tr}}_{2^n}(\delta )=1$ and $H_1,H_2$ are the x-coordinate and 1-coordinate of h (see Definition 2.4), respectively. A special case with $H_1+H_2=1$, which means $F(x)=H_1\left(\frac{1}{x^2+x+\delta }\right)+x$, has been already studied [8], [16], [21], [22]. We summarize the known permutations and then construct 8 new classes of permutations of such form, from which one can derive some permutation polynomials of the form $xh\left(x^{2^n-1}\right)$ over ${\mathbb{F}}_{2^{2n}}$ directly. Finally, we give an asymptotic result about ${\left(\frac{1}{x^2+x+\delta }\right)}^{2^k}+x$ to be a permutation over ${\mathbb{F}}_{2^n}$ using the Hasse-Weil bound.

有限域置换多项式的研究由于其广泛的应用而引起了许多学者的关注，有限域上的置换有几种不同形式。但是，关于不同排列形式之间关系的文献很少。

在本文中，我们发现形式的置换多项式之间的等价关系 $F（X）=XH（X^{2^{ñ}-\mathrm{1个}}）$ 过度 ${\mathbb{F}}_{2^{2ñ}}$ 和形式的排列 $F（X）=\frac{H_{\mathrm{1个}}（\mathrm{\Delta }）}{H_{\mathrm{1个}}（\mathrm{\Delta }）+H_2（\mathrm{\Delta }）}+X$ 上 ${\mathbb{F}}_{2^{ñ}}$，在哪里 $\mathrm{\Delta }=\frac{\mathrm{1个}}{X^2+X+\delta }$， $\delta \in {\mathbb{F}}_{2^{ñ}}$ 与 ${\mathrm{Tr}}_{2^{ñ}}（\delta ）=\mathrm{1个}$ 和 $H_{\mathrm{1个}}，H_2$分别是h的x坐标和1坐标（请参见定义2.4）。特例$H_{\mathrm{1个}}+H_2=\mathrm{1个}$， 意思是 $F（X）=H_{\mathrm{1个}}（\frac{\mathrm{1个}}{X^2+X+\delta }）+X$，已经被研究过[8]，[16]，[21]，[22]。我们总结了已知的置换，然后构造了8种此类形式的置换，从中可以得出某种形式的置换多项式$XH（X^{2^{ñ}-\mathrm{1个}}）$ 过度 ${\mathbb{F}}_{2^{2ñ}}$直。最后，我们给出关于的渐近结果${（\frac{\mathrm{1个}}{X^2+X+\delta }）}^{2^{ķ}}+X$ 作为一个排列 ${\mathbb{F}}_{2^{ñ}}$ 使用Hasse-Weil界线。