Finite Fields and Their Applications最新文献

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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。

引用次数: 0

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

<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>∈</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>≡</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>≡</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>. 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>±</mo><mn>1</mn></math>. In addition, the differential spectrum and boomerang spectrum of <math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</

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∈Fpn⁎ 的函数，称为 Ness-Helleseth 函数。在 pn≡3(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≡3(mod4)且 η(1+u)≠η(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=±1 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。

引用次数: 0

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。

引用次数: 0

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。

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引用次数: 0

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。

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引用次数: 0

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∈Fq6 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。

引用次数: 0

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。

引用次数: 0

Saturating linear sets in PG(2,q4) PG(2,q4) 中的饱和线性集合

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-05-21 DOI: 10.1016/j.ffa.2024.102447

Ferdinando Zullo

Bonini, Borello and Byrne started the study of saturating linear sets in Desarguesian projective spaces, in connection with the covering problem in the rank metric. In this paper we study 1-saturating linear sets in PG $(2, q^{4})$ , that is $F_{q}$ -linear sets in PG $(2, q^{4})$ with the property that their secant lines cover the entire plane. By making use of a characterization of generalized Gabidulin codes, we prove that the rank of such a linear set is at least 5. This answers to a recent question posed by Bartoli, Borello and Marino.

Bonini、Borello 和 Byrne 结合秩度量中的覆盖问题，开始了对德萨格投影空间中饱和线性集的研究。在本文中，我们研究了 PG(2,q4) 中的 1 饱和线性集，即 PG(2,q4) 中的 Fq 线性集，它们的正割线覆盖整个平面。通过利用广义加比杜林码的特性，我们证明了这种线性集的秩至少为 5。这回答了巴托利、博雷洛和马里诺最近提出的一个问题。

引用次数: 0

On the generalized Fibonacci sequences of polynomials over finite fields 论有限域上多项式的广义斐波那契序列

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-05-15 DOI: 10.1016/j.ffa.2024.102446

Zekai Chen, Min Sha, Chen Wei

In this paper, as an analogue of the integer case, we study detailedly the period and the rank of the generalized Fibonacci sequences of polynomials over a finite field modulo an arbitrary polynomial. We establish some formulas to compute them, and we also obtain some properties about their quotient. We find that the polynomial case is much more complicated than the integer case.

在本文中，作为整数情况的类比，我们详细研究了有限域上任意多项式模的广义斐波那契序列的周期和秩。我们建立了一些公式来计算它们，还获得了它们商的一些性质。我们发现多项式情况比整数情况复杂得多。

引用次数: 0

On the Rosenhain forms of superspecial curves of genus two 论二属超特殊曲线的罗森海恩形式

IF 1 3区数学 Q1 MATHEMATICS

Finite Fields and Their Applications

Pub Date : 2024-05-13 DOI: 10.1016/j.ffa.2024.102445

Ryo Ohashi

In this paper, we examine superspecial genus-2 curves $C : y^{2} = x (x - 1) (x - λ) (x - μ) (x - ν)$ in odd characteristic p. As a main result, we show that the difference between any two elements in ${0, 1, λ, μ, ν}$ is a square in $F_{p^{2}}$ . Moreover, we show that C is maximal or minimal over $F_{p^{2}}$ without taking its $F_{p^{2}}$ -form (we give an explicit criterion in terms of p that tells whether C is maximal or minimal). As an application, we also study the maximality of superspecial hyperelliptic curves of genera 3 and 4 whose automorphism groups contain $Z / 2 Z \times Z / 2 Z$ .

本文研究了奇特征 p 中的超特殊属 2 曲线 C:y2=x(x-1)(x-λ)(x-μ)(x-ν)。作为主要结果，我们证明了{0,1,λ,μ,ν}中任意两个元素之差都是 Fp2 中的平方。此外，我们还证明了 C 在 Fp2 上是最大的或最小的，而无需考虑它的 Fp2 形式（我们给出了一个明确的 p 准则，告诉我们 C 是最大的还是最小的）。作为应用，我们还研究了属 3 和属 4 的超特殊超椭圆曲线的极大性，它们的自变群包含 Z/2Z×Z/2Z。

引用次数: 0

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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. Chem. Res. ACS Applied Bio Materials ACS Appl. Electron. Mater. ACS Appl. Energy Mater. ACS Appl. Mater. Interfaces ACS Appl. Nano Mater. ACS Appl. Polym. Mater. ACS BIOMATER-SCI ENG ACS Catal. ACS Cent. Sci. ACS Chem. Biol. ACS Chemical Health & Safety ACS Chem. Neurosci. ACS Comb. Sci. ACS Earth Space Chem. ACS Energy Lett. ACS Infect. Dis. ACS Macro Lett. ACS Mater. Lett. ACS Med. Chem. Lett. ACS Nano ACS Omega ACS Photonics ACS Sens. ACS Sustainable Chem. Eng. ACS Synth. Biol. Anal. Chem. BIOCHEMISTRY-US Bioconjugate Chem. BIOMACROMOLECULES Chem. Res. Toxicol. Chem. Rev. Chem. Mater. CRYST GROWTH DES ENERG FUEL Environ. Sci. Technol. Environ. Sci. Technol. Lett. Eur. J. Inorg. Chem. IND ENG CHEM RES Inorg. Chem. J. Agric. Food. Chem. J. Chem. Eng. Data J. Chem. Educ. J. Chem. Inf. Model. J. Chem. Theory Comput. J. Med. Chem. J. Nat. Prod. J PROTEOME RES J. Am. Chem. Soc. LANGMUIR MACROMOLECULES Mol. Pharmaceutics Nano Lett. Org. Lett. ORG PROCESS RES DEV ORGANOMETALLICS J. Org. Chem. J. Phys. Chem. J. Phys. Chem. A J. Phys. Chem. B J. Phys. Chem. C J. Phys. Chem. Lett. Analyst Anal. Methods Biomater. Sci. Catal. Sci. Technol. Chem. Commun. Chem. Soc. Rev. CHEM EDUC RES PRACT CRYSTENGCOMM Dalton Trans. Energy Environ. Sci. ENVIRON SCI-NANO ENVIRON SCI-PROC IMP ENVIRON SCI-WAT RES Faraday Discuss. Food Funct. Green Chem. Inorg. Chem. Front. Integr. Biol. J. Anal. At. Spectrom. J. Mater. Chem. A J. Mater. Chem. B J. Mater. Chem. C Lab Chip Mater. Chem. Front. Mater. Horiz. MEDCHEMCOMM Metallomics Mol. Biosyst. Mol. Syst. Des. Eng. Nanoscale Nanoscale Horiz. Nat. Prod. Rep. New J. Chem. Org. Biomol. Chem. Org. Chem. Front. PHOTOCH PHOTOBIO SCI PCCP Polym. Chem.

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