Finite Fields and Their Applications最新文献

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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。

引用次数: 0

On the duality of cyclic codes of length ps over Fpm[u]〈u3〉论Fpm[u]〈u3〉上长度为ps的循环码的对偶性

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-09-04 DOI: 10.1016/j.ffa.2024.102500

Ahmad Erfanian , Roghaye Mohammadi Hesari

In this paper, we determine the dual codes of cyclic codes of length $p^{s}$ over $R_{3} = \frac{F_{p^{m}} [u]}{〈 u^{3} 〉}$ , where p is a prime number and $u^{3} = 0$ . Also, we improve and give correction of the results stated by B. Kim and J. Lee (2020) in [11]. Finally, we provide some examples of optimal and near-MDS cyclic codes of length $p^{s}$ over $R_{3}$ and compute dual of them.

在本文中，我们确定了 R3 上长度为 ps 的循环码的对偶码=Fpm[u]〈u3〉，其中 p 是素数且 u3=0。同时，我们改进并修正了 B. Kim 和 J. Lee (2020) 在 [11] 中所述的结果。最后，我们举例说明了 R3 上长度为 ps 的最优和近 MDS 循环码，并计算了它们的对偶性。

引用次数: 0

Denniston partial difference sets exist in the odd prime case 奇素数情况下存在丹尼斯顿偏差集

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-09-03 DOI: 10.1016/j.ffa.2024.102499

James A. Davis , Sophie Huczynska , Laura Johnson , John Polhill

<div>Denniston constructed partial difference sets (PDSs) with the parameters <math><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></math> in elementary abelian groups of order <math><msup><mrow><mn>2</mn></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup></math> for all <math><mi>m</mi><mo>≥</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo>≤</mo><mi>r</mi><mo><</mo><mi>m</mi></math>. These correspond to maximal arcs in Desarguesian projective planes of even order. In this paper, we show that - although maximal arcs do not exist in Desarguesian projective planes of odd order - PDSs with the Denniston parameters <math><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mi>m</mi></mrow></msup><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><mn>1</mn><mo>)</mo><mo>,</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>+</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup><mo>−</mo><mn>2</mn><mo>)</mo><mo>,</mo><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi><mo>+</mo><mi>r</mi></mrow></msup><mo>−</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><msup><mrow><mi>p</mi></mrow><mrow

丹尼斯顿构造了参数为(23m,(2m+r-2m+2r)(2m-1),2m-2r+(2m+r-2m+2r)(2r-2),(2m+r-2m+2r)(2r-1))的23m阶基本阿贝尔群中的局部差集（PDSs），对于所有m≥2,1≤r<m。这些弧对应于偶数阶笛卡尔投影面中的最大弧。在本文中，我们将证明--尽管奇阶笛卡尔投影面中不存在最大弧--但具有丹尼斯顿参数 (p3m,(pm+r-pm+pr)(pm-1)、pm-pr+(pm+r-pm+pr)(pr-2),(pm+r-pm+pr)(pr-1))的 PDS 存在于所有 m≥2,r∈{1,m-1} 的 p3m 阶基本阿贝尔群中，其中 p 是奇素数。我们的方法使用的 PDS 是环类的联合。

引用次数: 0

The resultant method in higher dimensions 高维度结果法

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-08-30 DOI: 10.1016/j.ffa.2024.102493

N. Harrach , L. Storme , P. Sziklai , M. Takáts

Stability results play an important role in Galois geometries. The famous resultant method, developed by Szőnyi and Weiner [12], [11], became very fruitful and resulted in many stability theorems in the last two decades. This method is based on some bivariate polynomials associated to point sets. In this paper we generalize the method for the multidimensional case and show some new applications. We build up the multivariate polynomial machinery and apply it for Rédei polynomials. We can prove a high dimensional analogue of the result of Szőnyi-Weiner [9], concerning the number of hyperplanes being skew to a point set of the space. We prove general results on “partial blocking sets”, using the tools we have developed.

稳定性结果在伽罗瓦几何中发挥着重要作用。由 Szőnyi 和 Weiner [12], [11] 提出的著名的结果法在过去二十年中取得了丰硕成果，并产生了许多稳定性定理。该方法基于与点集相关的一些双变量多项式。在本文中，我们将该方法推广到多维情况，并展示了一些新的应用。我们建立了多变量多项式机制，并将其应用于雷代多项式。我们可以证明 Szőnyi-Weiner [9] 结果的高维类比，涉及向空间点集倾斜的超平面数量。我们利用所开发的工具证明了 "部分阻塞集 "的一般结果。

引用次数: 0

A note on (2,2)-isogenies via theta coordinates 通过 Theta 坐标的 (2,2)-isogenies 注释

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-08-30 DOI: 10.1016/j.ffa.2024.102496

Jianming Lin , Saiyu Wang , Chang-An Zhao

In this paper, we revisit the algorithm for computing chains of $(2, 2)$ -isogenies between products of elliptic curves via theta coordinates proposed by Dartois et al. For each fundamental block of this algorithm, we provide an explicit inversion-free version. Besides, we exploit the technique of x-only ladder to speed up the computation of gluing isogeny. Finally, we present a mixed optimal strategy, which combines the inversion-elimination tool with the original methods together to execute a chain of $(2, 2)$ -isogenies.

We make a cost analysis and present a concrete comparison between ours and the previously known methods for inversion elimination. Furthermore, we implement the mixed optimal strategy for benchmark. The results show that when computing $(2, 2)$ -isogeny chains with lengths of 126, 208 and 632, compared to Dartois, Maino, Pope and Robert's latest implementation, utilizing our techniques can reduce 9.7%, 9.5% and 9.6% multiplications over the base field $F_{p}$ , respectively. Therefore, even for the updated version that employs their inversion-free algorithms, our tools still possess an advantage.

本文重温了 Dartois 等人提出的通过 Theta 坐标计算椭圆曲线乘积间 (2,2)-isogeny 链的算法。此外，我们还利用仅 x 梯形技术加快了胶合同源性的计算速度。最后，我们提出了一种混合最优策略，它将反转消除工具和原始方法结合在一起，以执行 (2,2)-isogenies 链。我们进行了成本分析，并具体比较了我们的方法和之前已知的反转消除方法。此外，我们还实施了混合最优策略作为基准。结果表明，在计算长度为 126、208 和 632 的 (2,2)-isogeny 链时，与 Dartois、Maino、Pope 和 Robert 的最新实现相比，利用我们的技术可以在基域 Fp 上分别减少 9.7%、9.5% 和 9.6% 的乘法运算。因此，即使是采用他们的无反转算法的更新版本，我们的工具仍然具有优势。

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引用次数: 0

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