Finite Fields and Their Applications最新文献

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Few-weight linear codes over Fp from t-to-one mappings 从 t 到一映射的 Fp 上的少权线性编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-11 DOI: 10.1016/j.ffa.2024.102510

René Rodríguez-Aldama

<div><div>For any prime number p, we provide two classes of linear codes with few weights over a p-ary alphabet. These codes are based on a well-known generic construction (the defining-set method), stemming on a class of monomials and a class of trinomials over finite fields. The considered monomials are Dembowski-Ostrom monomials <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math>, for a suitable choice of the exponent α, so that, when <math><mi>p</mi><mo>></mo><mn>2</mn></math> and <math><mi>n</mi><mo>≢</mo><mn>0</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math>, these monomials are planar. We study the properties of such monomials in detail for each integer <math><mi>n</mi><mo>></mo><mn>1</mn></math> and any prime number p. In particular, we show that they are t-to-one, where the parameter t depends on the field <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> and it takes the values <math><mn>1</mn><mo>,</mo><mn>2</mn></math> or <math><mi>p</mi><mo>+</mo><mn>1</mn></math>. Moreover, we give a simple proof of the fact that the functions are δ-uniform with <math><mi>δ</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mn>4</mn><mo>,</mo><mi>p</mi><mo>}</mo></math>. This result describes the differential behavior of these monomials for any p and n. For the second class of functions, we consider an affine equivalent trinomial to <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup></math>, namely, <math><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>λ</mi><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></msup><mo>+</mo><msup><mrow><mi>λ</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msup></mrow></msup><mi>x</mi></math> for <math><mi>λ</mi><mo>∈</mo><msubsup><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow><mrow><mo>⁎</mo></mrow></msubsup></math>. We prove that these trinomials satisfy certain regularity properties, which are useful for the specification of linear codes with three or four weights that are different than the monomial construction. These families of codes contain projective codes and optimal codes (with respect to the Griesmer bound). Remarkably, they contain infinite families of self-orthogonal and minimal p-ary linear codes for every prime number p. Our findings highlight the utility of st

对于任意素数 p，我们提供了两类 pary 字母表上权重较小的线性编码。这些编码基于一种著名的通用构造（定义集方法），源于有限域上的一类单项式和一类三项式。所考虑的单项式是登鲍斯基-奥斯特罗姆单项式 xpα+1，对于指数 α 的适当选择，当 p>2 和 n≢0(mod4)时，这些单项式是平面的。我们详细研究了每个整数 n>1 和任意素数 p 的此类单项式的性质。我们特别证明了它们是 t 对一的，其中参数 t 取决于场 Fpn，取值为 1、2 或 p+1。此外，我们还给出了函数δ∈{1,4,p}的δ均匀性的简单证明。对于第二类函数，我们考虑 xpα+1 的仿射等价三项式，即对于 λ∈Fpn⁎ 的 xpα+1+λxpα+λpαx 。我们证明了这些三项式满足某些正则特性，这对于规范与单项式构造不同的三重或四重线性编码非常有用。这些代码族包含投影代码和最优代码（关于格里斯梅尔约束）。值得注意的是，对于每个素数 p，它们都包含自正交和最小 pary 线性编码的无穷族。我们的发现突出了研究仿射等价函数的实用性，而这一点在这方面往往被忽视。

引用次数: 0

Drinfeld module and Weil pairing over Dedekind domain of class number two 二类 Dedekind 域上的 Drinfeld 模块和 Weil 配对

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102516

Chuangqiang Hu , Xiao-Min Huang

The primary objective of this paper is to derive explicit formulas for rank one and rank two Drinfeld modules over a specific domain denoted by

A

. This domain corresponds to the projective line associated with an infinite place of degree two. To achieve the goals, we construct a pair of standard rank one Drinfeld modules whose coefficients are in the Hilbert class field of

A

. We demonstrate that the period lattice of the exponential functions corresponding to both modules behaves similarly to the period lattice of the Carlitz module, the standard rank one Drinfeld module defined over rational function fields. Moreover, we employ Anderson's t-motive to obtain the complete family of rank two Drinfeld modules. This family is parameterized by the invariant

J = λ^{q^{2} + 1}

which effectively serves as the counterpart of the j-invariant for elliptic curves. Building upon the concepts introduced by van der Heiden, particularly with regard to rank two Drinfeld modules, we are able to reformulate the Weil pairing of Drinfeld modules of any rank using a specialized polynomial in multiple variables known as the Weil operator. As an illustrative example, we provide a detailed examination of a more explicit formula for the Weil pairing and the Weil operator of rank two Drinfeld modules over the domain

A

本文的主要目的是推导出一个特定域（用 A 表示）上的秩一和秩二 Drinfeld 模块的明确公式。为了实现这些目标，我们构建了一对系数在 A 的希尔伯特类域中的标准一级德林菲尔德模块。我们证明，这两个模块对应的指数函数的周期网格与卡利茨模块（定义在有理函数域上的标准一级德林菲尔德模块）的周期网格表现类似。此外，我们利用安德森 t 动力得到了完整的二阶德林菲尔德模块族。这个族的参数是不变式 J=λq2+1，它实际上是椭圆曲线 j 不变式的对应变量。基于范德尔海登提出的概念，特别是关于秩二的 Drinfeld 模块的概念，我们能够使用称为 Weil 算子的专门多变量多项式来重新表述任意秩的 Drinfeld 模块的 Weil 配对。举例说明，我们将详细研究域 A 上的二阶 Drinfeld 模块的 Weil 配对和 Weil 算子的更明确公式。

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引用次数: 0

A spherical extension theorem and applications in positive characteristic 正特征球面扩展定理及其应用

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102515

Doowon Koh , Thang Pham

In this paper, we prove an extension theorem for spheres of square radii in

F_{q}^{d}

, which improves a result obtained by Iosevich and Koh [9] (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem due to the authors and Lee (2022). Applications on the distance problems will be also discussed.

在本文中，我们证明了 Fqd 中方形半径球面的扩展定理，它改进了 Iosevich 和 Koh [9] (2010) 所获得的结果。我们的主要工具是一个新的点-超平面入射界限，它将通过作者和 Lee (2022) 提出的圆锥限制定理得到。我们还将讨论它在距离问题上的应用。

引用次数: 0

Quadratic residue patterns, algebraic curves and a K3 surface 二次残差模式、代数曲线和 K3 曲面

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-10 DOI: 10.1016/j.ffa.2024.102517

Valentina Kiritchenko , Michael Tsfasman , Serge Vlăduţ , Ilya Zakharevich

Quadratic residue patterns modulo a prime are studied since 19th century. In the first part we extend existing results on the number of consecutive ℓ-tuples of quadratic residues, studying corresponding algebraic curves and their Jacobians, which happen to be products of Jacobians of hyperelliptic curves. In the second part we state the last unpublished result of Lydia Goncharova on squares such that their differences are also squares, reformulate it in terms of algebraic geometry of a K3 surface, and prove it. The core of this theorem is an unexpected relation between the number of points on the K3 surface and that on a CM elliptic curve.

自 19 世纪以来，人们一直在研究调制素数的二次残差模式。在第一部分中，我们扩展了关于二次残差的连续 ℓ-tuples 数的现有结果，研究了相应的代数曲线及其雅各比，它们恰好是超椭圆曲线雅各比的乘积。在第二部分中，我们陈述了莉迪亚-冈察洛娃（Lydia Goncharova）最后一个未发表的关于正方形的结果，即它们的差也是正方形，用 K3 曲面的代数几何重新表述并证明了这一结果。该定理的核心是 K3 曲面上的点数与 CM 椭圆曲线上的点数之间的意外关系。

引用次数: 0

An approach to the moments subset sum problem through systems of diagonal equations over finite fields 通过有限域对角方程组解决矩子集和问题的方法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-07 DOI: 10.1016/j.ffa.2024.102511

Juan Francisco Gottig , Mariana Pérez , Melina Privitelli

Let

F_{q}

be the finite field of q elements. Let m and k be positive integers,

b \in F_{q}^{m}

and

D \subset F_{q}

with

k \leq | D |

. Our aim is to determine the existence of a subset

S \subset D

of cardinality k such that

\sum_{a \in S} a^{i} = b_{i}

for

i = 1, \dots, m

. This problem is known as the moments subset sum problem. We take a novel approach to this problem by establishing a connection between the existence of such a subset S with the problem of determining if a certain system of diagonal equations has at least one rational solution with distinct coordinates. To achieve this, we study some relevant geometric properties of the set of solutions of the mentioned system. This analysis allows us to provide estimates on the number of rational solutions of systems of diagonal equations and we subsequently apply these results to address the moments subset sum problem.

设 Fq 是有 q 个元素的有限域。设 m 和 k 为正整数，b∈Fqm，D⊂Fq，k≤|D|。我们的目标是确定是否存在一个心数为 k 的子集 S⊂D，使得 i=1,...m 时，∑a∈Sai=bi。这个问题被称为矩子集和问题。我们对这一问题采取了一种新的方法，即在这样一个子集 S 的存在与确定某个对角方程组是否至少有一个具有不同坐标的有理解这一问题之间建立联系。为此，我们研究了上述系统解集的一些相关几何性质。通过分析，我们可以对对角方程组的有理解的数量进行估计，随后我们将应用这些结果来解决矩子集和问题。

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引用次数: 0

On hyperelliptic Jacobians with complex multiplication by Q(−2+2) 关于 Q(-2+2) 复乘法的超椭圆雅各比

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-07 DOI: 10.1016/j.ffa.2024.102512

Tomasz Jędrzejak

<div><div>Consider a one-parameter family of hyperelliptic curves <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>:</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><mn>3</mn><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>−</mo><mn>2</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>−</mo><mn>6</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msup><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>3</mn><msup><mrow><mi>a</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>x</mi><mo>+</mo><msup><mrow><mi>a</mi></mrow><mrow><mn>5</mn></mrow></msup></math> defined over <math><mi>Q</mi></math>, and their Jacobians <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> where without loss of generality a is a non-zero squarefree integer. Clearly, the curve <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> is a quadratic twist by a of <math><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></math>. Note that <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> has complex multiplication by the quartic field <math><mi>Q</mi><mrow><mo>(</mo><msqrt><mrow><mo>−</mo><mn>2</mn><mo>+</mo><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></msqrt><mo>)</mo></mrow></math>. For a prime <math><mi>p</mi><mo>∤</mo><mn>2</mn><mi>a</mi></math> we obtain types of reduction (supersingular, superspecial, ordinary) of <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub></math> at p in terms of congruences modulo 16 and the exact formulas for the zeta function of <math><msub><mrow><mi>C</mi></mrow><mrow><mi>a</mi></mrow></msub></math> over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> for <math><mi>p</mi><mo>≢</mo><mn>1</mn><mo>,</mo><mn>7</mn><mrow><mo>(</mo><mi>mod</mi><mspace></mspace><mn>16</mn><mo>)</mo></mrow></math>. We deduce as conclusions the complete characterization of torsion subgroups of <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow></math>, namely <math><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mi>tors</mi></mrow></msub><mo>=</mo><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><mo>)</mo></mrow><mrow><mo>[</mo><mn>2</mn><mo>]</mo></mrow><mo>≃</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi></math>, and some information about <math><mi>rank</mi><mspace></mspace><msub><mrow><mi>J</mi></mrow><mrow><mi>a</mi></mrow></msub><mrow><mo>(</mo><mi>Q</mi><m

考虑定义在 Q 上的超椭圆曲线 Ca:y2=x5+3ax4-2a2x3-6a3x2+3a4x+a5 的单参数族，以及它们的 Jacobian Ja（在不失一般性的前提下，a 为非零的无平方整数）。显然，曲线 Ca 是 C1 的 a 二次扭曲。请注意，Ja 与四元数域 Q(-2+2) 有复乘法关系。对于素数 p∤2a，我们根据同余式 modulo 16 得到了 Ja 在 p 处的还原类型（超同余式、超特异式、普通式），以及对于 p≢1,7(mod16)，Ca 在 Fp 上的 zeta 函数的精确公式。作为结论，我们推导出 Ja(Q) 扭转子群的完整表征，即 Ja(Q)tors=Ja(Q)[2]≃Z/2Z 以及关于秩 Ja(Q) 的一些信息。

引用次数: 0

Erasure list-decodable codes and Turán hypercube problems 可擦除列表解码和图兰超立方问题

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-10-02 DOI: 10.1016/j.ffa.2024.102513

Noga Alon

We observe that several vertex Turán type problems for the hypercube that received a considerable amount of attention in the combinatorial community are equivalent to questions about erasure list-decodable codes. Analyzing a recent construction of Ellis, Ivan and Leader, and determining the Turán density of certain hypergraph augmentations we obtain improved bounds for some of these problems.

我们观察到，在组合学界受到极大关注的超立方体的几个顶点图兰类型问题等同于可擦除列表解码问题。通过分析 Ellis、Ivan 和 Leader 最近的一个构造，并确定某些超图扩增的图兰密度，我们得到了其中一些问题的改进边界。

引用次数: 0

Linear codes with few weights over finite fields 有限域上权重较少的线性编码

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-09-27 DOI: 10.1016/j.ffa.2024.102509

Yan Wang , Jiayi Fan , Nian Li , Fangyuan Liu

Linear codes with a few weights have wide applications in digital signatures, authentication codes, secret sharing protocols and some other fields. Using definition sets to construct linear codes is an effective method. In this paper, we investigate a new defining set and obtain linear codes with four weights, five weights and six weights over

F_{p}

, where p is an odd prime number. The parameters and weight distribution of the constructed linear code are completely determined by accurately calculating the exponential sum over the finite field.

只有几个权重的线性编码在数字签名、验证码、秘密共享协议和其他一些领域有着广泛的应用。利用定义集来构造线性编码是一种有效的方法。本文研究了一种新的定义集，并在 Fp（其中 p 是奇素数）上获得了四重、五重和六重的线性编码。通过精确计算有限域上的指数和，可以完全确定所构建线性码的参数和权重分布。

引用次数: 0

The explicit values of the UBCT, the LBCT and the DBCT of the inverse function 反函数的 UBCT、LBCT 和 DBCT 的显式值

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-09-17 DOI: 10.1016/j.ffa.2024.102508

Yuying Man , Nian Li , Zhen Liu , Xiangyong Zeng

Substitution boxes (S-boxes) play a significant role in ensuring the resistance of block ciphers against various attacks. The Upper Boomerang Connectivity Table (UBCT), the Lower Boomerang Connectivity Table (LBCT) and the Double Boomerang Connectivity Table (DBCT) of a given S-box are crucial tools to analyze its security concerning specific attacks. However, there are currently no research results on determining these tables of a function. The inverse function is crucial for constructing S-boxes of block ciphers with good cryptographic properties in symmetric cryptography. Therefore, extensive research has been conducted on the inverse function, exploring various properties related to standard attacks. Thanks to the recent advances in boomerang cryptanalysis, particularly the introduction of concepts such as UBCT, LBCT, and DBCT, this paper aims to further investigate the properties of the inverse function $F (x) = x^{2^{n} - 2}$ over $F_{2^{n}}$ for arbitrary n. As a consequence, by carrying out certain finer manipulations of solving specific equations over $F_{2^{n}}$ , we give all entries of the UBCT, LBCT of $F (x)$ over $F_{2^{n}}$ for arbitrary n. Besides, based on the results of the UBCT and LBCT for the inverse function, we determine that $F (x)$ is hard when n is odd. Furthermore, we completely compute all entries of the DBCT of $F (x)$ over $F_{2^{n}}$ for arbitrary n. Additionally, we provide the precise number of elements with a given entry by means of the values of some Kloosterman sums. Further, we determine the double boomerang uniformity of $F (x)$ over $F_{2^{n}}$ for arbitrary n. Our in-depth analysis of the DBCT of $F (x)$ contributes to a better evaluation of the S-box's resistance against boomerang attacks.

置换盒（S-boxes）在确保块密码免受各种攻击方面发挥着重要作用。给定 S 盒的上回旋镖连接表（UBCT）、下回旋镖连接表（LBCT）和双回旋镖连接表（DBCT）是分析其针对特定攻击的安全性的重要工具。然而，目前还没有关于确定这些函数表的研究成果。在对称密码学中，反函数对于构建具有良好密码特性的块密码 S 盒至关重要。因此，人们对反函数进行了广泛的研究，探索与标准攻击相关的各种特性。得益于最近在回旋镖密码分析领域取得的进展，特别是 UBCT、LBCT 和 DBCT 等概念的引入，本文旨在进一步研究任意 n 的反函数 F(x)=x2n-2 over F2n 的性质。因此，通过对 F2n 上的特定方程进行某些更精细的求解操作，我们给出了任意 n 时 F2n 上 F(x) 的 UBCT、LBCT 的所有项。此外，对于任意 n，我们完全计算了 F(x) 在 F2n 上的 DBCT 的所有条目。我们对 F(x) DBCT 的深入分析有助于更好地评估 S 盒对回旋镖攻击的抵抗力。

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引用次数: 0

Oriented supersingular elliptic curves and Eichler orders of prime level 有向超星椭圆曲线和素级艾希勒阶

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-09-17 DOI: 10.1016/j.ffa.2024.102501

Guanju Xiao , Zijian Zhou , Longjiang Qu

<div>Let <math><mi>p</mi><mo>></mo><mn>3</mn></math> be a prime and E be a supersingular elliptic curve defined over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>. Let c be a prime with <math><mi>c</mi><mo><</mo><mn>3</mn><mi>p</mi><mo>/</mo><mn>16</mn></math> and G be a subgroup of <math><mi>E</mi><mo>[</mo><mi>c</mi><mo>]</mo></math> of order c. The pair <math><mo>(</mo><mi>E</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> is called a supersingular elliptic curve with level-c structure, and the endomorphism ring <math><mtext>End</mtext><mo>(</mo><mi>E</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> is isomorphic to an Eichler order with level c. We construct two kinds of Eichler orders <math><msub><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></math> and <math><msubsup><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>q</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math> with level c. Interestingly, we prove that each <math><msub><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></math> or <math><msubsup><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>q</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math> can represent a primitive reduced binary quadratic form with discriminant <math><mo>−</mo><mn>16</mn><mi>c</mi><mi>p</mi></math> or <math><mo>−</mo><mi>c</mi><mi>p</mi></math> respectively. If a curve E is <math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mi>c</mi><mi>p</mi></mrow></msqrt><mo>]</mo></math>-oriented or <math><mi>Z</mi><mo>[</mo><mfrac><mrow><mn>1</mn><mo>+</mo><msqrt><mrow><mo>−</mo><mi>c</mi><mi>p</mi></mrow></msqrt></mrow><mrow><mn>2</mn></mrow></mfrac><mo>]</mo></math>-oriented, then we prove that <math><mtext>End</mtext><mo>(</mo><mi>E</mi><mo>,</mo><mi>G</mi><mo>)</mo></math> is isomorphic to <math><msub><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>,</mo><mi>r</mi><mo>)</mo></math> or <math><msubsup><mrow><mi>O</mi></mrow><mrow><mi>c</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>q</mi><mo>,</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math> respectively. Due to the fact that <math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mi>c</mi><mi>p</mi></mrow></msqrt><mo>]</mo></math>-oriented isogenies between <math><mi>Z</mi><mo>[</mo><msqrt><mrow><mo>−</mo><mi>c</mi><mi>p</mi></mrow></msqrt><mo>]</mo></math></

设 p>3 是素数，E 是定义在 Fp2 上的超椭圆曲线。让 c 是 c<3p/16 的素数，G 是 E[c] 的一个阶为 c 的子群。这对 (E,G) 称为具有 c 级结构的超椭圆曲线，其内定环 End(E,G) 与具有 c 级结构的艾希勒阶同构。有趣的是，我们证明了每个 Oc(q,r)或 Oc′(q,r′)可以分别表示一个判别式为 -16cp 或 -cp 的原始还原二元二次型。如果曲线 E 是 Z[-cp]-oriented 或 Z[1+-cp2]-oriented 的，那么我们证明 End(E,G) 分别与 Oc(q,r) 或 Oc′(q,r′)同构。由于 Z[-cp]-oriented 椭圆曲线之间的 Z[-cp]-oriented 同素异形可以用二次型来表示，我们证明了这些同素异形通过相应二次型的组成法则反映在相应的艾希勒阶中。

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