论文标题
在有限字段上的n-to-1映射的特征和构造
Characterizations and constructions of n-to-1 mappings over finite fields
论文作者
论文摘要
$ n $ -to- $ 1 $映射在许多领域都有广泛的应用，尤其是在密码学，有限的几何，编码理论和组合设计中。在本文中，研究了许多类别的$ n $ - 至$ 1 $映射的有限字段。首先，我们通过Walsh Transform提供了$ \ Mathbb {f} _ {p^m} $的一般$ n $ -to-to- $ 1 $ 1 $映射的特征。然后，我们完全确定$ 3 $ -to- $ 1 $多项式，学位不超过$ \ \ \ \ \玛理布{f} _ {p^{m}} $。此外，我们获得了一个类似AGW的标准，用于表征$ n $ -to- $ 1 $ 1 $的属性之间的等效关系，该映射属于有限的套件$ a $，以及$ a $ a $的另一个映射的另一个映射。最后，我们将类似AGW的标准应用于多种形式的多项式，并获得一些明确的$ n $ -to-to- $ 1 $映射。尤其是，从环体观点的角度来看，$ x^rh \ left（x^s \ right）$的三个明确构造，以及提供$ n $ -to- $ 1 $ 1 $映射的几类$ g \ left（x^{q^k} -x +δ\ uight） +cx $。
$n$-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.
