Algebraic Aspects of Digital Communications最新文献

Theta functions and algebraic curves with automorphisms 函数与自同构代数曲线

Algebraic Aspects of Digital Communications Pub Date : 2012-10-05 DOI: 10.3233/978-1-60750-019-3-193

T. Shaska, G. Wijesiri

{"title":"Theta functions and algebraic curves with automorphisms","authors":"T. Shaska, G. Wijesiri","doi":"10.3233/978-1-60750-019-3-193","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-193","url":null,"abstract":"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 $X$. \u0000To 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. \u0000Fast 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 $g=2.$","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2012-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126159177","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}

引用次数: 10

Self-dual codes and invariant theory 自对偶码与不变理论

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-23

G. Nebe

{"title":"Self-dual codes and invariant theory","authors":"G. Nebe","doi":"10.3233/978-1-60750-019-3-23","DOIUrl":"https://doi.org/10.3233/978-1-60750-019-3-23","url":null,"abstract":"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.","PeriodicalId":185285,"journal":{"name":"Algebraic Aspects of Digital Communications","volume":"34 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":"123775657","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}

引用次数: 177

Real and imaginary hyperelliptic curve cryptography - Aspects of curve cryptography 实和虚超椭圆曲线密码学。曲线密码学的几个方面

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-100

A. Stein

引用次数: 0

On the Cryptographical Properties of Extremal Algebraic Graphs 极值代数图的密码学性质

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-256

V. Ustimenko

引用次数: 15

Enumerative Geometry and String Theory 枚举几何与弦理论

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-238

A. Elezi

引用次数: 28

Combinatorial Designs and Code Synchronization 组合设计和代码同步

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-81

V. Tonchev

引用次数: 0

Vector Bundles in Error-Correcting for Geometric Goppa Codes 几何Goppa码的矢量束纠错

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-42

E. Previato

引用次数: 0

Additive Codes over F4 with Automorphisms 具有自同构的F4上的加性码

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-1

W. C. Huffman

引用次数: 0

A variant of the Reidemeister-Schreier algorithm for the fundamental groups of Riemann surfaces Riemann曲面基本群的Reidemeister-Schreier算法的一个变体

Algebraic Aspects of Digital Communications Pub Date : 1900-01-01 DOI: 10.3233/978-1-60750-019-3-174

K. Magaard, S. Shpectorov

引用次数: 0