In this paper, we consider two classes of permutation trinomials with Niho-type exponents over the finite field ${\mathbb{F}}_{2^{2m}}$, where m is a positive integer. We transform the problem into investigating on some quartic equations ($2^k$-th degree equations) over the subfield ${\mathbb{F}}_{2^m}$ in the first class (second class, respectively). We show that these equations have no solutions in ${\mathbb{F}}_{2^m}$. Some sufficient conditions are established to characterize the coefficients in the two classes of permutation polynomials. The numerical result suggests that the sufficient conditions on the coefficients for the case of m odd in the first class, and in the second class, are also necessary.

中文翻译：

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两类具有Niho指数的置换三项式

在本文中，我们考虑有限域上具有Niho型指数的两类置换三项式 ${\mathbb{F}}_{2^{2米}}$，其中m是一个正整数。我们将问题转化为研究一些四次方程（$2^{ķ}$子场的三阶方程） ${\mathbb{F}}_{2^{米}}$在头等舱（分别是第二类）中。我们证明这些方程没有解${\mathbb{F}}_{2^{米}}$。建立了一些足够的条件来表征两类置换多项式中的系数。数值结果表明，对于第一类和第二类中的m奇数的情况，系数的充分条件也是必要的。