Pub Date : 2012-10-05DOI: 10.3233/978-1-60750-019-3-193
T. Shaska, G. Wijesiri
Let $X$ be an irreducible, smooth, projective curve of genus $g geq 2$ defined over the complex field $C.$ Then there is a covering $pi: X longrightarrow P^1,$ where $P^1$ denotes the projective line. The problem of expressing branch points of the covering $pi$ in terms of the transcendentals (period matrix, thetanulls, e.g.) is classical. It goes back to Riemann, Jacobi, Picard and Rosenhein. Many mathematicians, including Picard and Thomae, have offered partial treatments for this problem. In this work, we address the problem for cyclic curves of genus 2, 3, and 4 and find relations among theta functions for curves with automorphisms. We consider curves of genus $g > 1$ admitting an automorphism $sigma$ such that $X^sigma$ has genus zero and $sigma$ generates a normal subgroup of the automorphism group $Aut(X)$ of $X$. To characterize the locus of cyclic curves by analytic conditions on its Abelian coordinates, in other words, theta functions, we use some classical formulas, recent results of Hurwitz spaces, and symbolic computations, especially for genera 2 and 3. For hyperelliptic curves, we use Thomae's formula to invert the period map and discover relations among the classical thetanulls of cyclic curves. For non hyperelliptic curves, we write the equations in terms of thetanulls. Fast genus 2 curve arithmetic in the Jacobian of the curve is used in cryptography and is based on inverting the moduli map for genus 2 curves and on some other relations on theta functions. We determine similar formulas and relations for genus 3 hyperelliptic curves and offer an algorithm for how this can be done for higher genus curves. It is still to be determined whether our formulas for $g=3$ can be used in cryptographic applications as in $g=2.$
设$X$为复域上定义的属$g geq 2$的不可约光滑投影曲线$C.$，则有一个覆盖$pi: X longrightarrow P^1,$，其中$P^1$表示投影线。用超越量(周期矩阵，等)表示覆盖$pi$的分支点的问题是经典的。这可以追溯到黎曼，雅可比，皮卡德和罗森海因。许多数学家，包括皮卡德和托马斯，都对这个问题给出了部分的处理方法。在这项工作中，我们解决了2、3和4属循环曲线的问题，并找到了具有自同构曲线的函数之间的关系。我们考虑承认一个自同构$sigma$的属$g > 1$曲线，使得$X^sigma$的属为零，并且$sigma$产生$X$的自同构群$Aut(X)$的正规子群。为了在循环曲线的阿贝尔坐标(即函数)上用解析条件来表征循环曲线的轨迹，我们使用了一些经典公式、Hurwitz空间的最新结果以及符号计算，特别是对第2和第3类。对于超椭圆曲线，我们利用Thomae公式对周期映射进行了反演，并发现了经典循环曲线各环之间的关系。对于非超椭圆曲线，我们用tanulls表示方程。曲线雅可比矩阵中的快速格2曲线算法应用于密码学中，它基于对格2曲线的模映射的反演和函数上的其他关系。我们确定了类似的公式和关系的属3超椭圆曲线，并提供了一个算法，如何可以做到这一点，为更高的属曲线。我们的$g=3$公式是否可以在加密应用程序中使用还有待确定 $g=2.$
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Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-23
G. Nebe
A formal notion of a Typ T of a self-dual linear code over a nite left R- module V is introduced which allows to give explicit generators of a nite complex matrix group, the associated Clifford-Weil group C.T / • GLjVj.C/, such that the complete weight enumerators of self-dual isotropic codes of Type T span the ring of invariants of C.T /. This generalizes Gleason's 1970 theorem to a very wide class of rings and also includes multiple weight enumerators (see Section 2.7), as these are the complete weight enumerators cwem.C/ D cwe.Rm › C/ of Rm£m -linear self-dual codes Rm›C • .V m/N of Type T m with associated Clifford-Weil group Cm.T / D C.T m /. The nite Siegel 8-operator mapping cwem.C/ to cwemi1.C/ hence denes a ring epimorphism 8m V Inv.Cm.T // ! Inv.Cmi1.T // between invariant rings of complex matrix groups of different degrees. If R D V is a - nite eld, then the structure of Cm.T / allows to dene a commutative algebra of Cm.T / double cosets, called a Hecke algebra in analogy to the one in the theory of lattices and modular forms. This algebra consists of self-adjoint linear operators on Inv.Cm.T // commuting with 8m . The Hecke-eigenspaces yield explicit linear relations among the cwem of self-dual codes CV N.
本文引入了一个关于nite左R模V上的自对偶线性码的T型的形式化概念，它允许给出一个nite复矩阵群的显式生成器，即相关的Clifford-Weil群C.T /•GLjVj.C/，使得T型自对偶各向同性码的完全权枚举子跨出C.T /的不变量环。这将Gleason 1970年的定理推广到一个非常广泛的环类，并且还包括多个权重枚举数(参见第2.7节)，因为这些是完整的权重枚举数。Rm›C / of Rm£m -线性自对偶代码Rm›C••v m/N of T型m与相关的Clifford-Weil组Cm。T / D / c / T / m /。新Siegel 8-算子映射cvem . c /到cvem . c /，因此得到了一个环上射。Inv.Cmi1。T //在不同程度的复矩阵群的不变环之间。如果rdvv是一个- nite场，则Cm的结构。T /允许求Cm的交换代数。T /双辅助集，称为赫克代数，类似于格和模形式理论中的赫克代数。这个代数由v. cm . t //与8m可交换上的自伴随线性算子组成。hecke -特征空间给出了自对偶码cvn的向量之间的显式线性关系。
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Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-100
A. Stein
{"title":"Real and imaginary hyperelliptic curve cryptography - Aspects of curve cryptography","authors":"A. Stein","doi":"10.3233/978-1-60750-019-3-100","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-100","url":null,"abstract":"","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"213 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132417173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-256
V. Ustimenko
{"title":"On the Cryptographical Properties of Extremal Algebraic Graphs","authors":"V. Ustimenko","doi":"10.3233/978-1-60750-019-3-256","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-256","url":null,"abstract":"","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"92 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125101460","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-238
A. Elezi
Inevitably, reading is one of the requirements to be undergone. To improve the performance and quality, someone needs to have something new every day. It will suggest you to have more inspirations, then. However, the needs of inspirations will make you searching for some sources. Even from the other people experience, internet, and many books. Books and internet are the recommended media to help you improving your quality and performance.
{"title":"Enumerative Geometry and String Theory","authors":"A. Elezi","doi":"10.3233/978-1-60750-019-3-238","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-238","url":null,"abstract":"Inevitably, reading is one of the requirements to be undergone. To improve the performance and quality, someone needs to have something new every day. It will suggest you to have more inspirations, then. However, the needs of inspirations will make you searching for some sources. Even from the other people experience, internet, and many books. Books and internet are the recommended media to help you improving your quality and performance.","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"18 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123725302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-81
V. Tonchev
Combinatorial Designs and Code Synchronization – p. 1/2
组合设计与代码同步- p. 1/2
{"title":"Combinatorial Designs and Code Synchronization","authors":"V. Tonchev","doi":"10.3233/978-1-60750-019-3-81","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-81","url":null,"abstract":"Combinatorial Designs and Code Synchronization – p. 1/2","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122184956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-42
E. Previato
{"title":"Vector Bundles in Error-Correcting for Geometric Goppa Codes","authors":"E. Previato","doi":"10.3233/978-1-60750-019-3-42","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-42","url":null,"abstract":"","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130972585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-1
W. C. Huffman
{"title":"Additive Codes over F4 with Automorphisms","authors":"W. C. Huffman","doi":"10.3233/978-1-60750-019-3-1","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-1","url":null,"abstract":"","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"229 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133848078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.3233/978-1-60750-019-3-174
K. Magaard, S. Shpectorov
{"title":"A variant of the Reidemeister-Schreier algorithm for the fundamental groups of Riemann surfaces","authors":"K. Magaard, S. Shpectorov","doi":"10.3233/978-1-60750-019-3-174","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-174","url":null,"abstract":"","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"37 3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116733799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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