Finite Fields and Their Applications最新文献_第4页

Heffter spaces 赫夫特空间

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-08 DOI: 10.1016/j.ffa.2024.102464

M. Buratti , A. Pasotti

The notion of a Heffter array, which received much attention in the last decade, is equivalent to a pair of orthogonal Heffter systems. In this paper we study the existence problem of a set of r mutually orthogonal Heffter systems for any r. Such a set is equivalent to a resolvable partial linear space of degree r whose parallel classes are Heffter systems: this is a new combinatorial design that we call a Heffter space. We present a series of direct constructions of Heffter spaces with odd block size and arbitrarily large degree r obtained with the crucial use of finite fields. Among the applications we establish, in particular, that if $q = 2 k w + 1$ is a prime power with kw odd and $k \geq 3$ , then there are at least $⌈ \frac{w}{4 k^{4}} ⌉$ mutually orthogonal k-cycle systems of order q.

赫夫特阵列的概念在过去十年中备受关注，它等价于一对正交赫夫特系统。在本文中，我们研究了任意 r 的 r 个相互正交的赫夫特系统集合的存在性问题。这样的集合等价于一个度数为 r 的可解析偏线性空间，其并行类是赫夫特系统：这是一种新的组合设计，我们称之为赫夫特空间。我们介绍了一系列直接构造的赫夫特空间，这些空间具有奇数块大小和任意大的度数 r，是利用有限域获得的。在这些应用中，我们特别指出，如果 q=2kw+1 是 kw 为奇数且 k≥3 的素数幂，那么至少有 ⌈w4k4⌉ 个相互正交的 q 阶 k 循环系统。

引用次数: 0

Odd moments for the trace of Frobenius and the Sato–Tate conjecture in arithmetic progressions 算术级数中弗罗贝纽斯迹的奇矩和佐藤塔特猜想

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-07-08 DOI: 10.1016/j.ffa.2024.102465

Kathrin Bringmann , Ben Kane , Sudhir Pujahari

In this paper, we consider the moments of the trace of Frobenius of elliptic curves if the trace is restricted to a fixed arithmetic progression. We determine the asymptotic behavior for the ratio of the $(2 k + 1)$ -th moment to the zeroeth moment as the size of the finite field $F_{p^{r}}$ goes to infinity. These results follow from similar asymptotic formulas relating sums and moments of Hurwitz class numbers where the sums are restricted to certain arithmetic progressions. As an application, we prove that the distribution of the trace of Frobenius in arithmetic progressions is equidistributed with respect to the Sato–Tate measure.

在本文中，我们考虑了椭圆曲线 Frobenius 的迹的矩，如果迹被限制在一个固定的算术级数上。我们确定了当有限域 Fpr 的大小达到无穷大时，第 (2k+1)-th 矩与第零矩之比的渐近行为。这些结果源于赫维兹类数的和与矩的类似渐近公式，其中和被限制为某些算术级数。作为应用，我们证明了算术级数中弗罗贝尼斯迹的分布与萨托-塔特度量是等分布的。

引用次数: 0

On some congruences and exponential sums 关于一些全等和指数和

IF 1.2 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-06-27 DOI: 10.1016/j.ffa.2024.102451

Moubariz Z. Garaev , Igor E. Shparlinski

Let $ε > 0$ be a fixed small constant, $F_{p}$ be the finite field of p elements for prime p. We consider additive and multiplicative problems in $F_{p}$ that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let $M$ be an arbitrary subset of $F_{p}$ . If $# M > p^{1 / 3 + ε}$ and $H ⩾ p^{2 / 3}$ or if $# M > p^{3 / 5 + ε}$ and $H ⩾ p^{3 / 5 + ε}$ then all, but $O (p^{1 - δ})$ elements of $F_{p}$ can be represented in the form hm with $h \in [1, H]$ and $m \in M$ , where $δ > 0$ depends only on ε. Furthermore, let $X$ be an arbitrary interval of length H and s be a fixed positive integer. If $H > p^{17 / 35 + ε}, # M > p^{17 / 35 + ε},$ then the number $T_{6} (λ)$ of solutions to the congruence $\frac{m_{1}}{x_{1}^{s}} + \frac{m_{2}}{x_{2}^{s}} + \frac{m_{3}}{x_{3让 ε>0 是一个固定的小常数，Fp 是素数 p 的 p 元素有限域。我们考虑 Fp 中涉及区间和任意集合的加法和乘法问题。我们的代表性结果举例如下。设 M 是 Fp 的任意子集。如果 #M>p1/3+ε 和 H⩾p2/3，或者如果 #M>p3/5+ε 和 H⩾p3/5+ε，那么除了 O（p1-δ）个元素外，Fp 的所有元素都可以用 hm 的形式表示，其中 h∈[1,H]，m∈M，δ>0 只取决于 ε。此外，设 X 是长度为 H 的任意区间，s 是一个固定的正整数。若 H>p17/35+ε,#M>p17/35+ε，则全等m1x1s+m2x2s+m3x3s+m4x4s+m5x5s+m6x6s≡λmodp,mi∈M,xi∈X,i=1,...,6 的解的个数 T6(λ)满足T6(λ)=H6(#M)6p(1+O(p-δ))。下载PDF {"title":"On some congruences and exponential sums","authors":"Moubariz Z. Garaev , Igor E. Shparlinski","doi":"10.1016/j.ffa.2024.102451","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102451","url":null,"abstract":"<div>Let <math><mi>ε</mi><mo>></mo><mn>0</mn></math> be a fixed small constant, <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> be the finite field of p elements for prime p. We consider additive and multiplicative problems in <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> that involve intervals and arbitrary sets. Representative examples of our results are as follows. Let <math><mi>M</mi></math> be an arbitrary subset of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math>. If <math><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>3</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> and <math><mi>H</mi><mo>⩾</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msup></math> or if <math><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>5</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> and <math><mi>H</mi><mo>⩾</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>5</mn><mo>+</mo><mi>ε</mi></mrow></msup></math> then all, but <math><mi>O</mi><mo>(</mo><msup><mrow><mi>p</mi></mrow><mrow><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msup><mo>)</mo></math> elements of <math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math> can be represented in the form hm with <math><mi>h</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mi>H</mi><mo>]</mo></math> and <math><mi>m</mi><mo>∈</mo><mi>M</mi></math>, where <math><mi>δ</mi><mo>></mo><mn>0</mn></math> depends only on ε. Furthermore, let <math><mi>X</mi></math> be an arbitrary interval of length H and s be a fixed positive integer. If<math><mi>H</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>17</mn><mo>/</mo><mn>35</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>,</mo><mspace></mspace><mi>#</mi><mi>M</mi><mo>></mo><msup><mrow><mi>p</mi></mrow><mrow><mn>17</mn><mo>/</mo><mn>35</mn><mo>+</mo><mi>ε</mi></mrow></msup><mo>,</mo></math> then the number <math><msub><mrow><mi>T</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>(</mo><mi>λ</mi><mo>)</mo></math> of solutions to the congruence<math><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow><mrow><mi>s</mi></mrow></msubsup></mrow></mfrac><mo>+</mo><mfrac><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow><mrow><msubsup><mrow><mi>x</mi></mrow><mrow><mn>3","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S107157972400090X/pdfft?md5=73d751bad88083ca796c715f3b4d9bad&pid=1-s2.0-S107157972400090X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold 立方三折与非退化赫米提三折交点的最大点数 IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-06-25 DOI: 10.1016/j.ffa.2024.102462 Mrinmoy Datta , Subrata Manna It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in P^{4} (F_{q^{2}}) has at most d (q^{5} + q^{2}) + q^{3} + 1 points in common with a threefold of degree d defined over F_{q^{2}} . He proved the conjecture for d = 2 . In this paper, we show that the conjecture is true for d = 3 and q \geq 7 . 埃杜库（Edoukou）在 2008 年猜想，P4(Fq2) 中的非退化赫米提三重与定义在 Fq2 上的 d 度三重最多有 d(q5+q2)+q3+1 个共同点。他证明了 d=2 的猜想。在本文中，我们证明该猜想在 d=3 和 q≥7 时为真。求助PDF {"title":"Maximum number of points on an intersection of a cubic threefold and a non-degenerate Hermitian threefold","authors":"Mrinmoy Datta , Subrata Manna","doi":"10.1016/j.ffa.2024.102462","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102462","url":null,"abstract":"<div>It was conjectured by Edoukou in 2008 that a non-degenerate Hermitian threefold in <math><msup><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo></math> has at most <math><mi>d</mi><mo>(</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo><mo>+</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>+</mo><mn>1</mn></math> points in common with a threefold of degree d defined over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>. He proved the conjecture for <math><mi>d</mi><mo>=</mo><mn>2</mn></math>. In this paper, we show that the conjecture is true for <math><mi>d</mi><mo>=</mo><mn>3</mn></math> and <math><mi>q</mi><mo>≥</mo><mn>7</mn></math>.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用A further study on the Ness-Helleseth function 关于奈斯-赫勒塞斯函数的进一步研究 IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-06-25 DOI: 10.1016/j.ffa.2024.102453 Cheng Lyu, Xiaoqiang Wang, Dabin Zheng Let F_{p^{n}} be a finite field with p^{n} elements. Ness and Helleseth in [29] first studied a class of functions over F_{p^{n}} with the form f (x) = u x^{\frac{p^{n} - 3}{2}} + x^{p^{n} - 2}, u \in F_{p^{n}}^{⁎}, which is called the Ness-Helleseth function. The f (x) has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for p = 3 in [29] and by Zeng et al. for any odd prime p in [43] under the condition p^{n} \equiv 3 (\mod 4) and η (1 + u) = η (u - 1) . In this paper, we continue to study the Ness-Helleseth functions under the condition that p^{n} \equiv 3 (\mod 4) and η (1 + u) \neq η (u - 1) . Firstly, we prove that f (x) is a permutation polynomial with differential uniformity not more than 4 if η (1 + u) = η (1 - u) . Moreover, for some more special u, f is an involution with differential uniformity at most 3. Secondly, we show that f (x) is a locally-APN function for u = \pm 1 . In addition, the differential spectrum and boomerang spectrum of f (x)设 Fpn 是有 pn 个元素的有限域。Ness 和 Helleseth 在文献[29]中首次研究了 Fpn 上一类形式为 f(x)=uxpn-32+xpn-2,u\inFpn⁎ 的函数，称为 Ness-Helleseth 函数。在 pn\equiv3(mod4) 和 η(1+u)=η(u-1) 条件下，内斯和海勒塞斯在[29]中证明了 f(x) 是 p=3 的几乎完全非线性（APN）函数，曾等人在[43]中证明了 f(x) 是任意奇素数 p 的几乎完全非线性（APN）函数。本文继续研究 pn\equiv3(mod4)且 η(1+u)\neqη(u-1) 条件下的奈斯-赫勒塞特函数。首先，我们证明，如果 η(1+u)=η(1-u), f(x) 是微分均匀性不大于 4 的置换多项式。其次，我们证明了在 u=\pm1 时，f(x) 是局部 APN 函数。此外，通过判断一些特殊方程的解的个数，我们得到了 f(x) 的微分谱和回旋谱。我们得到了第一个其回旋均匀性可以达到 0 或 1 的非 APN 函数。求助PDF {"title":"A further study on the Ness-Helleseth function","authors":"Cheng Lyu, Xiaoqiang Wang, Dabin Zheng","doi":"10.1016/j.ffa.2024.102453","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102453","url":null,"abstract":"<div>Let <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> be a finite field with <math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></math> elements. Ness and Helleseth in [29] first studied a class of functions over <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></msub></math> with the form <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>u</mi><msup><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>-</mo><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>-</mo><mn>2</mn></mrow></msup><mo>,</mo><mspace></mspace><mi>u</mi><mo>\in</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>, which is called the Ness-Helleseth function. The <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for <math><mi>p</mi><mo>=</mo><mn>3</mn></math> in [29] and by Zeng et al. for any odd prime p in [43] under the condition <math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>\equiv</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math> and <math><mi>η</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>η</mi><mo>(</mo><mi>u</mi><mo>-</mo><mn>1</mn><mo>)</mo></math>. In this paper, we continue to study the Ness-Helleseth functions under the condition that <math><msup><mrow><mi>p</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>\equiv</mo><mn>3</mn><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><mn>4</mn><mo>)</mo></math> and <math><mi>η</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>u</mi><mo>)</mo><mo>\neq</mo><mi>η</mi><mo>(</mo><mi>u</mi><mo>-</mo><mn>1</mn><mo>)</mo></math>. Firstly, we prove that <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is a permutation polynomial with differential uniformity not more than 4 if <math><mi>η</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>u</mi><mo>)</mo><mo>=</mo><mi>η</mi><mo>(</mo><mn>1</mn><mo>-</mo><mi>u</mi><mo>)</mo></math>. Moreover, for some more special u, f is an involution with differential uniformity at most 3. Secondly, we show that <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> is a locally-APN function for <math><mi>u</mi><mo>=</mo><mo>\pm</mo><mn>1</mn></math>. In addition, the differential spectrum and boomerang spectrum of <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141482840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Cyclic 2-spreads in V(6,q) and flag-transitive linear spaces V(6,q)中的循环 2 展和旗帜传递线性空间 IF 1.2 3区数学 Q1 MATHEMATICS Finite Fields and Their Applications Pub Date : 2024-06-25 DOI: 10.1016/j.ffa.2024.102463 Cian Jameson, John Sheekey In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results. 在这篇论文中，我们完整地分类了在特征非二或三的有限域上的 6 维向量空间的 2 维子空间的扩散，在这些扩散上有一个循环群起横向作用。这解决了旗跃线性空间分类中的一个未决问题。我们利用保利和班贝格创新的多项式方法来获得我们的结果。下载PDF {"title":"Cyclic 2-spreads in V(6,q) and flag-transitive linear spaces","authors":"Cian Jameson, John Sheekey","doi":"10.1016/j.ffa.2024.102463","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102463","url":null,"abstract":"<div>In this paper we completely classify spreads of 2-dimensional subspaces of a 6-dimensional vector space over a finite field of characteristic not two or three upon which a cyclic group acts transitively. This addresses one of the remaining open cases in the classification of flag-transitive linear spaces. We utilise the polynomial approach innovated by Pauley and Bamberg to obtain our results.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724001023/pdfft?md5=cd6ef2ef8226a10487b87f668b7b3d4e&pid=1-s2.0-S1071579724001023-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141485873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2 关于 p2 阶子群的 p 特性赫米蒂曲线的伽罗瓦子覆盖率 IF 1.2 3区数学 Q1 Mathematics Finite Fields and Their Applications Pub Date : 2024-06-21 DOI: 10.1016/j.ffa.2024.102450 Barbara Gatti , Gábor Korchmáros A (projective, geometrically irreducible, non-singular) curve X defined over a finite field F_{q^{2}} is maximal if the number N_{q^{2}} of its F_{q^{2}} -rational points attains the Hasse-Weil upper bound, that is N_{q^{2}} = q^{2} + 2 g q + 1 where g is the genus of X . An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order p^{2} where p is the characteristic of F_{q^{2}} . Doing so we also determine the F_{q^{2}} -isomorphism classes of such curves and describe their full F_{q^{2}} -automorphism groups. 如果在有限域 Fq2 上定义的（投影的、几何上不可还原的、非奇异的）曲线 X 的 Fq2 有理点数 Nq2 达到哈塞-韦尔（Hasse-Weil）上限，即 Nq2=q2+2gq+1 其中 g 是 X 的属。迄今为止，对于赫尔墨斯曲线的伽罗瓦覆盖的一些曲线，特别是在伽罗瓦群有素数阶的情况下，人们已经临时完成了这一工作。在本文中，我们得到了赫尔墨斯曲线的所有伽罗华盖的明确方程，这些曲线的伽罗华群为 p2 阶，其中 p 是 Fq2 的特征。为此，我们还确定了这些曲线的 Fq2-同构类，并描述了它们的全 Fq2-同构群。下载PDF {"title":"Galois subcovers of the Hermitian curve in characteristic p with respect to subgroups of order p2","authors":"Barbara Gatti , Gábor Korchmáros","doi":"10.1016/j.ffa.2024.102450","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102450","url":null,"abstract":"<div>A (projective, geometrically irreducible, non-singular) curve <math><mi>X</mi></math> defined over a finite field <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> is maximal if the number <math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math> of its <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>-rational points attains the Hasse-Weil upper bound, that is <math><msub><mrow><mi>N</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>=</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>2</mn><mi>g</mi><mi>q</mi><mo>+</mo><mn>1</mn></math> where <math><mi>g</mi></math> is the genus of <math><mi>X</mi></math>. An important question, also motivated by applications to algebraic-geometry codes, is to find explicit equations for maximal curves. For a few curves which are Galois covered of the Hermitian curve, this has been done so far ad hoc, in particular in the cases where the Galois group has prime order. In this paper we obtain explicit equations of all Galois covers of the Hermitian curve with Galois group of order <math><msup><mrow><mi>p</mi></mrow><mrow><mn>2</mn></mrow></msup></math> where p is the characteristic of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>. Doing so we also determine the <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>-isomorphism classes of such curves and describe their full <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub></math>-automorphism groups.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000893/pdfft?md5=4d7dc430a08ae7fb7ea1e9c89490e861&pid=1-s2.0-S1071579724000893-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141438233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用A multistep strategy for polynomial system solving over finite fields and a new algebraic attack on the stream cipher Trivium 有限域上多项式系统求解的多步策略和对流密码 Trivium 的新代数攻击 IF 1 3区数学 Q1 Mathematics Finite Fields and Their Applications Pub Date : 2024-06-19 DOI: 10.1016/j.ffa.2024.102452 Roberto La Scala , Federico Pintore , Sharwan K. Tiwari , Andrea Visconti In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a subset of variables stepwise, that is, by incrementing the size of such subset each time that an evaluation leads to a polynomial system which is possibly unfeasible to solve. The decision about which evaluation to extend is based on a preprocessing consisting in computing an incomplete Gröbner basis after the current evaluation, which possibly generates linear polynomials that are used to eliminate further variables. If the number of remaining variables in the system is deemed still too high, the evaluation is extended and the preprocessing is iterated. Otherwise, we solve the system by a complete Gröbner basis computation. Having in mind cryptanalytic applications, we present an implementation of this strategy in an algorithm called MultiSolve which is designed for polynomial systems having at most one solution. We prove explicit formulas for its complexity which are based on probability distributions that can be easily estimated by performing the proposed preprocessing on a testset of evaluations for different subsets of variables. We prove that an optimal complexity of MultiSolve is achieved by using a full multistep strategy with a maximum number of steps and in turn the standard guess-and-determine strategy, which essentially is a strategy consisting of a single step, is the worst choice. Finally, we extensively study the behaviour of MultiSolve when performing an algebraic attack on the well-known stream cipher Trivium . 在本文中，我们介绍了一种用于求解有限域上多元多项式方程组的 "猜测-确定 "或 "混合 "策略的多步骤推广方法。具体而言，我们建议逐步对变量子集进行穷举求解，即每次求解导致多项式方程组可能无法求解时，就增加子集的大小。在当前评估结束后，通过计算不完整的格罗布纳基础（可能会生成线性多项式，用于消除更多变量）进行预处理，然后决定扩大评估范围。如果认为系统中剩余变量的数量仍然过多，则会延长评估时间，并反复进行预处理。考虑到密码分析的应用，我们在名为 MultiSolve 的算法中介绍了这一策略的实现方法，该算法专为最多只有一个解的多项式系统而设计。我们证明了其复杂度的明确公式，这些公式基于概率分布，通过对不同变量子集的评估测试集执行建议的预处理，可以轻松估算出这些概率分布。我们证明，使用具有最大步数的完整多步策略可以实现 MultiSolve 的最佳复杂度，而标准的 "猜测-确定 "策略则是最差的选择。最后，我们广泛研究了对著名的流密码Trivium进行代数攻击时MultiSolve的表现。求助PDF {"title":"A multistep strategy for polynomial system solving over finite fields and a new algebraic attack on the stream cipher Trivium","authors":"Roberto La Scala , Federico Pintore , Sharwan K. Tiwari , Andrea Visconti","doi":"10.1016/j.ffa.2024.102452","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102452","url":null,"abstract":"<div>In this paper we introduce a multistep generalization of the guess-and-determine or hybrid strategy for solving a system of multivariate polynomial equations over a finite field. In particular, we propose performing the exhaustive evaluation of a subset of variables stepwise, that is, by incrementing the size of such subset each time that an evaluation leads to a polynomial system which is possibly unfeasible to solve. The decision about which evaluation to extend is based on a preprocessing consisting in computing an incomplete Gröbner basis after the current evaluation, which possibly generates linear polynomials that are used to eliminate further variables. If the number of remaining variables in the system is deemed still too high, the evaluation is extended and the preprocessing is iterated. Otherwise, we solve the system by a complete Gröbner basis computation.Having in mind cryptanalytic applications, we present an implementation of this strategy in an algorithm called MultiSolve which is designed for polynomial systems having at most one solution. We prove explicit formulas for its complexity which are based on probability distributions that can be easily estimated by performing the proposed preprocessing on a testset of evaluations for different subsets of variables. We prove that an optimal complexity of MultiSolve is achieved by using a full multistep strategy with a maximum number of steps and in turn the standard guess-and-determine strategy, which essentially is a strategy consisting of a single step, is the worst choice. Finally, we extensively study the behaviour of MultiSolve when performing an algebraic attack on the well-known stream cipher Trivium.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141429874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用Scattered trinomials of Fq6[X] in even characteristic 偶数特征中 Fq6[X] 的散点三项式 IF 1 3区数学 Q1 Mathematics Finite Fields and Their Applications Pub Date : 2024-06-07 DOI: 10.1016/j.ffa.2024.102449 Daniele Bartoli , Giovanni Longobardi , Giuseppe Marino , Marco Timpanella In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [9], [24], the authors proved that the trinomial f_{c} (X) = X^{q} + X^{q^{3}} + c X^{q^{5}} of F_{q^{6}} [X] is scattered under the assumptions that q is odd and c^{2} + c = 1 . They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q . Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly q^{3} elements c \in F_{q^{6}} such that f_{c} (X) is scattered. Also, we prove that the corresponding MRD-codes and F_{q} -linear sets of PG (1, q^{6}) are not equivalent to the previously known ones. 近年来，文献中研究了多个散点多项式族。然而，它们大多只存在于奇数特征中。在 [9], [24] 中，作者证明了 Fq6[X] 的三项式 fc(X)=Xq+Xq3+cXq5 在 q 为奇数且 c2+c=1 的假设条件下是分散的。他们还明确地指出，当 q 为偶数时，这种情况是错误的。本文利用正特征代数几何工具，证明当 q 为偶数且足够大时，大致有 q3 个元素 c\inFq6 使得 fc(X) 是分散的。此外，我们还证明了 PG(1,q6) 的相应 MRD 代码和 Fq 线性集合与之前已知的并不等价。下载PDF {"title":"Scattered trinomials of Fq6[X] in even characteristic","authors":"Daniele Bartoli , Giovanni Longobardi , Giuseppe Marino , Marco Timpanella","doi":"10.1016/j.ffa.2024.102449","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102449","url":null,"abstract":"<div>In recent years, several families of scattered polynomials have been investigated in the literature. However, most of them only exist in odd characteristic. In [9], [24], the authors proved that the trinomial <math><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>+</mo><msup><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></mrow></msup><mo>+</mo><mi>c</mi><msup><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>5</mn></mrow></msup></mrow></msup></math> of <math><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup></mrow></msub><mo>[</mo><mi>X</mi><mo>]</mo></math> is scattered under the assumptions that q is odd and <math><msup><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>c</mi><mo>=</mo><mn>1</mn></math>. They also explicitly observed that this is false when q is even. In this paper, we provide a different set of conditions on c for which this trinomial is scattered in the case of even q. Using tools of algebraic geometry in positive characteristic, we show that when q is even and sufficiently large, there are roughly <math><msup><mrow><mi>q</mi></mrow><mrow><mn>3</mn></mrow></msup></math> elements <math><mi>c</mi><mo>\in</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup></mrow></msub></math> such that <math><msub><mrow><mi>f</mi></mrow><mrow><mi>c</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math> is scattered. Also, we prove that the corresponding MRD-codes and <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>-linear sets of <math><mrow><mi>PG</mi></mrow><mo>(</mo><mn>1</mn><mo>,</mo><msup><mrow><mi>q</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math> are not equivalent to the previously known ones.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1071579724000881/pdfft?md5=8a841f52ac38f09246cba1d4c920247c&pid=1-s2.0-S1071579724000881-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141286330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用On the P-construction of algebraic-geometry codes 论代数几何代码的 P 构造 IF 1 3区数学 Q1 Mathematics Finite Fields and Their Applications Pub Date : 2024-06-05 DOI: 10.1016/j.ffa.2024.102448 R. Toledano , M. Vides We construct algebraic-geometry codes by using projective systems from projective curves over a finite field and the global sections of invertible sheaves on these curves. We also prove a formula for the Hilbert function of a finite set of points in a projective space in terms of the rank of a matrix constructed with the Veronese embedding and we use it to estimate the minimum distance of the dual codes. 我们利用有限域上的投影曲线的投影系统以及这些曲线上的可逆剪切的全局截面来构造代数几何代码。我们还证明了一个投影空间中有限点集的希尔伯特函数公式，该公式与用维罗内嵌入构建的矩阵的秩有关，我们用它来估计对偶码的最小距离。求助PDF {"title":"On the P-construction of algebraic-geometry codes","authors":"R. Toledano , M. Vides","doi":"10.1016/j.ffa.2024.102448","DOIUrl":"https://doi.org/10.1016/j.ffa.2024.102448","url":null,"abstract":"<div>We construct algebraic-geometry codes by using projective systems from projective curves over a finite field and the global sections of invertible sheaves on these curves. We also prove a formula for the Hilbert function of a finite set of points in a projective space in terms of the rank of a matrix constructed with the Veronese embedding and we use it to estimate the minimum distance of the dual codes.</div>","PeriodicalId":50446,"journal":{"name":"Finite Fields and Their Applications","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141264015","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0} 引用次数: 0 引用批量引用首页上一页 1234567 ... 10 下一页尾页类型全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程数据库全部 ACS Publications Elsevier ieeexplore Springer The Royal Society of Chemistry Wiley 期刊 Finite Fields and Their Applications 全部 Acc. 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window.open("/literature/literaturelist?keyword=" + encodeURIComponent($("#fkeyword").val()));
}
else {
window.location.href = "/literature/literaturelist?keyword=" + encodeURIComponent($("#fkeyword").val());
}
});

$(window).scroll(function () {
var scrollTop = $(this).scrollTop();

if (scrollTop > $("#menufloat").offset().top) {
$("#fkeyword").val($("#tkeyword").val());
$("#menufloat2").show();
$("#menufloat").css("visibility", "hidden");
}
else if (scrollTop <= $("#menufloat").offset().top) {
$("#menufloat2").hide();
$("#menufloat").css("visibility", "");
}
})
})

function isPaperGo(s) {
let pos = 0;
let digitCount = 0;
let flag = false;
const searchCleanArr = s.split(/[., ]/).map(item => item.trim()).filter(item => item !== '');
for (const item of searchCleanArr) {
if (!flag) {
if (!isNaN(parseInt(item[0]))) {
flag = !flag;
} else {
pos += item.length;
}
} else {
for (const i of item) {
if (!isNaN(i)) {
digitCount++;
}
}
}
}
const sLength = (searchCleanArr.join('')).substring(pos).length;
if (sLength === 0 || digitCount / sLength < 0.5) {
return false;
}
return true;
}var clearShow = false;
$(function () {
if ($("#tkeyword").val().trim().length > 0) {
$("#clear").show();
clearShow = true;
}
$("#tkeyword").keydown(function (e) {
var curKey = e.which;
if (curKey == 13) {

}
});
$("#SearchBtn").click(function () {
var keyWordHome = $("#tkeyword").val();
var fullWidthRegex = /[\u3000-\u303F\u3040-\u309F\u30A0-\u30FF\u3100-\u312F\u3130-\u318F\u3190-\u319F\u31A0-\u31BF\u31C0-\u31EF\u31F0-\u31FF\u3200-\u32FF\u3300-\u33FF\uFF00-\uFFEF\u4E00-\u9FFF]/;

if (keyWordHome.trim().length == 0) {
alert("请输入搜索关键字");
$("#tkeyword").focus();
return;
}
if (/[\u4e00-\u9fa5]/.test($("#tkeyword").val())) {
if ($("#tkeyword").val().trim().length > 200) {
alert("关键词长度不能大于200个字符");
return;
}
}
else {
if ($("#tkeyword").val().trim().length > 500) {
alert("关键词长度不能大于500个字符");
return;
}
}
if (fullWidthRegex.test(keyWordHome)) {
keyWordHome = toHalfWidth(keyWordHome);
}

var regex = /\b(10[.][0-9]{4,}(?:[.][0-9]+)*\/(?:(?![\|"&\'])\S)+)(?:\+?|\b)/g;
const optionarr = keyWordHome.trim().match(regex);
if ((optionarr != undefined && optionarr.length > 0) || isPaperGo(keyWordHome.trim())) {
window.open('/literature/literaturelist?keyword=' + encodeURIComponent(keyWordHome));
}
else {
window.location.href = '/literature/literaturelist?keyword=' + encodeURIComponent(keyWordHome);
}
});
$(":text.ssi").unbind().keyup(function (event) {
var which = event.which;
if (which != 13) {
if ($("[data-module='scisearch']").length > 0) {
$("[data-module='scisearch']").attr("data-search-value", $(this).val());
}
}
else {
$("#SearchBtn").click();
}
});
$(":text.ssi").on('input', function () {
if ($("[data-module='scisearch']").length > 0) {
$("[data-module='scisearch']").attr("data-search-value", $(this).val());
}
});

$("#clear").click(function () {
$("#tkeyword").val("");
$("#clear").hide();
clearShow = false;
$("#tkeyword").focus();
});

$("#tkeyword").keyup(function () {
if ($(this).val().trim().length > 0) {
if (!clearShow) {
$("#clear").show();
clearShow = true;
}
}
else {
if (clearShow) {
$("#clear").hide();
clearShow = false;
}
}
});

$("#tkeyword").change(function () {
if ($(this).val().trim().length > 0) {
if (!clearShow) {
$("#clear").show();
clearShow = true;
}
}
else {
if (clearShow) {
$("#clear").hide();
clearShow = false;
}
}
})

if (getCookie("userName") == null) {
if (getCookie("_sci_uid") == null) {
var uuidstr = getuuid();
setCookie("_sci_uid", "sci_" + uuidstr, 9999);
}
}
});

function isPaperGo(s) {
let pos = 0;
let digitCount = 0;
let flag = false;
const searchCleanArr = s.split(/[., ]/).map(item => item.trim()).filter(item => item !== '');
for (const item of searchCleanArr) {
if (!flag) {
if (!isNaN(parseInt(item[0]))) {
flag = !flag;
} else {
pos += item.length;
}
} else {
for (const i of item) {
if (!isNaN(i)) {
digitCount++;
}
}
}
}
const sLength = (searchCleanArr.join('')).substring(pos).length;
if (sLength === 0 || digitCount / sLength < 0.5) {
return false;
}
return true;
}}}$