有限域上n-to-1映射的刻画与构造
Characterizations and constructions of n-to-1 mappings over finite fields
摘要：
$n$ -to- $1$映射在密码学、有限几何、编码理论和组合设计等领域有着广泛的应用。本文研究了有限域上的许多类$n$ -to- $1$映射。首先，我们利用Walsh变换给出了$\mathbb{F}_{p^m}$上一般$n$到$1$映射的一个特征.然后，我们完全确定了次数不超过$\mathbb{F}_{p^{m}}$上$4$的$3$到$1$多项式.此外，我们还得到了一个类AGW准则，用来刻画有限集$A$上的一个映射的$n$ -to- $1$性质与$A$子集上的另一个映射的性质之间的等价关系.最后，我们将AGW准则应用于几种多项式形式，得到了一些显式的$n$ -to- $1$映射.特别是，从分圆的角度提供了形式$x^rh\left( x^s \right) $的三个显式构造，以及形式$ g\left( x^{q^k} -x +δ\right) +cx$的几类$n$到$1$映射。
英文摘要：
$n$-to-$1$ mappings have wide applications in many areas, especially in cryptography, finite geometry, coding theory and combinatorial design. In this paper, many classes of $n$-to-$1$ mappings over finite fields are studied. First, we provide a characterization of general $n$-to-$1$ mappings over $\mathbb{F}_{p^m}$ by means of the Walsh transform. Then, we completely determine $3$-to-$1$ polynomials with degree no more than $4$ over $\mathbb{F}_{p^{m}}$. Furthermore, we obtain an AGW-like criterion for characterizing an equivalent relationship between the $n$-to-$1$ property of a mapping over finite set $A$ and that of another mapping over a subset of $A$. Finally, we apply the AGW-like criterion into several forms of polynomials and obtain some explicit $n$-to-$1$ mappings. Especially, three explicit constructions of the form $x^rh\left( x^s \right) $ from the cyclotomic perspective, and several classes of $n$-to-$1$ mappings of the form $ g\left( x^{q^k} -x +δ\right) +cx$ are provided.
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