Algebraic Curves And Finite Fields

Author : J. W. P. Hirschfeld,G. Korchmáros,F. Torres
Publisher : Princeton University Press
Page : 744 pages
File Size : 42,6 Mb
Release : 2013-03-25
Category : Mathematics
ISBN : 9781400847419

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Algebraic Curves over a Finite Field by J. W. P. Hirschfeld,G. Korchmáros,F. Torres Pdf

This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Author : Harald Niederreiter,Alina Ostafe,Daniel Panario,Arne Winterhof
Publisher : Walter de Gruyter GmbH & Co KG
Page : 251 pages
File Size : 51,6 Mb
Release : 2014-08-20
Category : Mathematics
ISBN : 9783110317916

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Algebraic Curves and Finite Fields by Harald Niederreiter,Alina Ostafe,Daniel Panario,Arne Winterhof Pdf

Algebra and number theory have always been counted among the most beautiful and fundamental mathematical areas with deep proofs and elegant results. However, for a long time they were not considered of any substantial importance for real-life applications. This has dramatically changed with the appearance of new topics such as modern cryptography, coding theory, and wireless communication. Nowadays we find applications of algebra and number theory frequently in our daily life. We mention security and error detection for internet banking, check digit systems and the bar code, GPS and radar systems, pricing options at a stock market, and noise suppression on mobile phones as most common examples. This book collects the results of the workshops "Applications of algebraic curves" and "Applications of finite fields" of the RICAM Special Semester 2013. These workshops brought together the most prominent researchers in the area of finite fields and their applications around the world. They address old and new problems on curves and other aspects of finite fields, with emphasis on their diverse applications to many areas of pure and applied mathematics.

Author : Carlos Moreno
Publisher : Cambridge University Press
Page : 264 pages
File Size : 49,9 Mb
Release : 1993-10-14
Category : Mathematics
ISBN : 052145901X

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Algebraic Curves Over Finite Fields by Carlos Moreno Pdf

Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.

Author : Carlos Moreno
Publisher : Unknown
Page : 246 pages
File Size : 48,6 Mb
Release : 1993
Category : Electronic
ISBN : OCLC:1081580012

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Algebraic Curves Over Finite Fields by Carlos Moreno Pdf

Author : Harald Niederreiter,Chaoping Xing
Publisher : Cambridge University Press
Page : 260 pages
File Size : 51,5 Mb
Release : 2001-06-14
Category : Computers
ISBN : 0521665434

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Rational Points on Curves Over Finite Fields by Harald Niederreiter,Chaoping Xing Pdf

Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.

Author : Joint Summ Ams-Ims-Siam,Michael D. Fried,AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves over Finite Fields,Ams-Ims-Siam Joint Summer Research Conference on Applications of curve
Publisher : American Mathematical Soc.
Page : 226 pages
File Size : 41,5 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821809259

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Applications of Curves Over Finite Fields by Joint Summ Ams-Ims-Siam,Michael D. Fried,AMS-IMS-SIAM Joint Summer Research Conference on Applications of Curves over Finite Fields,Ams-Ims-Siam Joint Summer Research Conference on Applications of curve Pdf

This volume presents the results of the AMS-IMS-SIAM Joint Summer Research Conference held at the University of Washington (Seattle). The talks were devoted to various aspects of the theory of algebraic curves over finite fields and its numerous applications. The three basic themes are the following: Curves with many rational points. Several articles describe main approaches to the construction of such curves: the Drinfeld modules and fiber product methods, the moduli space approach, and the constructions using classical curves; Monodromy groups of characteristic $p$ covers. A number of authors presented the results and conjectures related to the study of the monodromy groups of curves over finite fields. In particular, they study the monodromy groups from genus $0$ covers, reductions of covers, and explicit computation of monodromy groups over finite fields; and, Zeta functions and trace formulas.To a large extent, papers devoted to this topic reflect the contributions of Professor Bernard Dwork and his students. This conference was the last attended by Professor Dwork before his death, and several papers inspired by his presence include commentaries about the applications of trace formulas and $L$-function. The volume also contains a detailed introduction paper by Professor Michael Fried, which helps the reader to navigate in the material presented in the book.

Author : Serguei A. Stepanov
Publisher : Springer Science & Business Media
Page : 352 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461547853

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Codes on Algebraic Curves by Serguei A. Stepanov Pdf

This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

Author : D. Kaledin,Y. Tschinkel
Publisher : IOS Press
Page : 356 pages
File Size : 46,7 Mb
Release : 2008-06-05
Category : Mathematics
ISBN : 9781607503255

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Higher-Dimensional Geometry Over Finite Fields by D. Kaledin,Y. Tschinkel Pdf

Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. The papers included in this publication introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension of at least 2.

Author : Vijaya Kumar Murty
Publisher : American Mathematical Soc.
Page : 142 pages
File Size : 47,8 Mb
Release : 2010
Category : Coding theory
ISBN : 9780821843116

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Algebraic Curves and Cryptography by Vijaya Kumar Murty Pdf

Focusing on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields the topics covered in this volume include Schoof's $\ell$-adic point counting algorithm, the $p$-adic algorithms of Kedlaya and Denef-Vercauteren, explicit arithmetic on the Jacobians of $C_{ab}$ curves and zeta functions.

Author : Oliver Pretzel
Publisher : Clarendon Press
Page : 209 pages
File Size : 51,5 Mb
Release : 1998-01-08
Category : Mathematics
ISBN : 9780191589041

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Codes and Algebraic Curves by Oliver Pretzel Pdf

The geometry of curves has fascinated mathematicians for 2500 years, and the theory has become highly abstract. Recently links have been made with the subject of error correction, leading to the creation of geometric Goppa codes, a new and important area of coding theory. This book is an updated and extended version of the last part of the successful book Error-Correcting Codes and Finite Fields. It provides an elementary introduction to Goppa codes, and includes many examples, calculations, and applications. The book is in two parts with an emphasis on motivation, and applications of the theory take precedence over proofs of theorems. The formal theory is, however, provided in the second part of the book, and several of the concepts and proofs have been simplified without sacrificing rigour.

Author : Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian
Publisher : Springer Science & Business Media
Page : 229 pages
File Size : 52,7 Mb
Release : 2013-04-17
Category : Technology & Engineering
ISBN : 9781475722260

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Applications of Finite Fields by Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian Pdf

The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches in mathematics. Inrecent years we have witnessed a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. The purpose of this book is to introduce the reader to some of these recent developments. It should be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. We shall focus our attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, we have not attempted to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes. To limit the size of the volume we have been forced to omit some important applications of finite fields. Some of these missing applications are briefly mentioned in the Appendix along with some key references.

Author : Igor Shparlinski
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 51,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401118064

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Computational and Algorithmic Problems in Finite Fields by Igor Shparlinski Pdf

This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include polynomial factorization, finding irreducible and primitive polynomials, distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types, and new applications of finite fields to other araes of mathematics. For completeness, also included are two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number generators, modular arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.

Author : Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin
Publisher : American Mathematical Soc.
Page : 453 pages
File Size : 47,9 Mb
Release : 2019-07-02
Category : Coding theory
ISBN : 9781470448653

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Algebraic Geometry Codes: Advanced Chapters by Michael Tsfasman,Serge Vlǎduţ,Dmitry Nogin Pdf

Algebraic Geometry Codes: Advanced Chapters is devoted to the theory of algebraic geometry codes, a subject related to local_libraryBook Catalogseveral domains of mathematics. On one hand, it involves such classical areas as algebraic geometry and number theory; on the other, it is connected to information transmission theory, combinatorics, finite geometries, dense packings, and so on. The book gives a unique perspective on the subject. Whereas most books on coding theory start with elementary concepts and then develop them in the framework of coding theory itself within, this book systematically presents meaningful and important connections of coding theory with algebraic geometry and number theory. Among many topics treated in the book, the following should be mentioned: curves with many points over finite fields, class field theory, asymptotic theory of global fields, decoding, sphere packing, codes from multi-dimensional varieties, and applications of algebraic geometry codes. The book is the natural continuation of Algebraic Geometric Codes: Basic Notions by the same authors. The concise exposition of the first volume is included as an appendix.

Author : Serguei A. Stepanov
Publisher : Springer Science & Business Media
Page : 444 pages
File Size : 50,6 Mb
Release : 1994-12-31
Category : Mathematics
ISBN : 0306110369

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Arithmetic of Algebraic Curves by Serguei A. Stepanov Pdf

Author S.A. Stepanov thoroughly investigates the current state of the theory of Diophantine equations and its related methods. Discussions focus on arithmetic, algebraic-geometric, and logical aspects of the problem. Designed for students as well as researchers, the book includes over 250 excercises accompanied by hints, instructions, and references. Written in a clear manner, this text does not require readers to have special knowledge of modern methods of algebraic geometry.

Author : Igor Shparlinski
Publisher : Springer Science & Business Media
Page : 532 pages
File Size : 40,6 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401592390

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Finite Fields: Theory and Computation by Igor Shparlinski Pdf

This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.