论文标题
$ \ mathbb {f} _ {q^2} $的置换多项式
Permutation Polynomials of $\mathbb{F}_{q^2}$ : A Linear Algebraic Approach
论文作者
论文摘要
在本文中，我们提出了一种线性代数方法，用于研究置换多项式的研究，该方法是由有限字段$ \ mathbb {f} _ {q^2} $的线性图引起的。我们在$ \ mathbb {f} _ {q^2} $上研究特定类别的置换多项式，在等级缺陷和完整的等级线性映射上的上下文上，$ \ mathbb {f} _ {q^2} $。我们得出了必要和充分的条件，在给定类别的多项式是置换多项式的情况下。我们进一步表明，此类置换多项式的数量很容易被列举。早期文献中仅报道了这些置换多项式的子集。事实证明，这类置换多项式具有相同类型的组成逆，我们提供了算法来评估大多数这些置换多项式的组成逆。
In this paper, we present a linear algebraic approach to the study of permutation polynomials that arise from linear maps over a finite field $\mathbb{F}_{q^2}$. We study a particular class of permutation polynomials over $\mathbb{F}_{q^2}$, in the context of rank deficient and full rank linear maps over $\mathbb{F}_{q^2}$. We derive necessary and sufficient conditions under which the given class of polynomials are permutation polynomials. We further show that the number of such permutation polynomials can be easily enumerated. Only a subset of these permutation polynomials have been reported in literature earlier. It turns out that this class of permutation polynomials have compositional inverses of the same kind and we provide algorithms to evaluate the compositional inverses of most of these permutation polynomials.
