Rational Points On Curves Over Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Cambridge University Press
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Description : Discussion of theory and applications of algebraic curves over finite fields with many rational points.


Lecture Notes Series

Author by : Anonim
Languange : en
Publisher by : Unknown
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Rational Points On Curves Over Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Unknown
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Description : Discussion of theory and applications of algebraic curves over finite fields with many rational points.


Applications Of Curves Over Finite Fields

Author by : Joint Summ Ams-Ims-Siam
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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.


Codes On Algebraic Curves

Author by : Serguei A. Stepanov
Languange : en
Publisher by : Springer Science & Business Media
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Description : 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.


Many Rational Points

Author by : N.E. Hurt
Languange : en
Publisher by : Springer Science & Business Media
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Description : This volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators." "The book will be of use to mathematicians, physicists and engineers interested in the mathematical methods of algebraic geometry as they apply to coding theory and cryptography."--Jacket.


Algebraic Curves Over A Finite Field

Author by : J. W. P. Hirschfeld
Languange : en
Publisher by : Princeton University Press
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Description : 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.


Finite Fields And Applications

Author by : Gary L. Mullen
Languange : en
Publisher by : American Mathematical Soc.
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Description : Introduction to the theory of finite fields and to some of their many applications. The first chapter is devoted to the theory of finite fields. After covering their construction and elementary properties, the authors discuss the trace and norm functions, bases for finite fields, and properties of polynomials over finite fields. Chapter 2 deals with combinatorial topics such as the construction of sets of orthogonal Latin squares, affine and projective planes, block designs, and Hadamard matrices. Chapters 3 and 4 provide a number of constructions and basic properties of error-correcting codes and cryptographic systems using finite fields. Appendix A provides a brief review of the basic number theory and abstract algebra used in the text. Appendix B provides hints and partial solutions for many of the exercises in each chapter.--From publisher description.


Frobenius Action On Jacobians Of Curves Over Finite Fields

Author by : Wanlin Li
Languange : en
Publisher by : Unknown
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Description : This thesis focuses on studying the eigenvalues of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Some of the results have applications to answering questions in analytic number theory over function fields. The study of zeros of L-functions associated to Dirichlet characters has been a topic of interest in analytic number theory. Questions and conjectures arising there could also be studied in the function field setting. With the field of rational numbers replaced by the field of rational functions over a finite field, those questions are closely related to the study of the Frobenius action on the l-adic Tate modules of Jacobians of curves over finite fields. Chowla conjectured that the L-function of any quadratic Dirichlet character does not vanish at the central point s=1/2. Soundararajan showed that Chowla's conjecture holds for a positive proportion of quadratic characters ordered by conductor. Over the function field F_q(t), the analogous statement can be phrased but the situation can be very different. Quadratic characters correspond to hyperelliptic curves over F_q and their L-functions are closely related to the Hasse-Weil zeta functions of the curves. To construct quadratic characters whose L-functions vanish at the central point s=1/2 is equivalent to constructing hyperelliptic curves whose Jacobians admit sqrt(q) as an eigenvalue of the Frobenius action on its l-adic Tate module. Over any given finite field F_q, I use the Honda-Tate theory and other previous results to show the existence of such hyperelliptic curves which then give quadratic characters over the function field F_q(t) whose L-functions vanish at the central point s=1/2. This is in contrast with the situation over the rational numbers. Moreover, using a counting result of Poonen on the number of squarefree values of squarefree polynomials over the function field, I give a lower bound on the number of such characters which grows to infinity when the conductor is allowed to be arbitrarily large. Although the analogous statement of Chowla's conjecture does not hold over the function field, it is still believed that 100% of the quadratic characters satisfy the condition that their L-functions do not vanish at the central point s=1/2. So in order to approach this conjecture, joint with J. Ellenberg and M. Shusterman, we use the idea of reduction to give an upper bound on the number of quadratic characters whose L-functions vanish at a given point of the critical line. This upper bound gets better when the size of the constant field is large and the density of such characters goes to 0 when the size of the constant field grows to infinity. Geometrically, we realize the number of hyperelliptic curves whose Jacobians admit some fixed number as an eigenvalue of the Frobenius action on its l-torsion subgroup can be counted by the number of rational points of a twisted Hurwitz scheme over finite fields. Using an earlier result of Ellenberg--Venkatesh--Westerland on the homological stability for Hurwitz spaces, we give an upper bound on the number of rational points of the twisted Hurwitz scheme to get the result. The previous work are all related to studying Weil integers realized as Frobenius eigenvalues for curves over finite fields. From Honda-Tate theory, it is known that every Weil integer appears as a Frobenius eigenvalue for some abelian variety over finite fields. To show the same holds for Jacobian varieties, it suffices to show that every abelian variety over the finite field is covered by a Jacobian variety. This result can be deduced from Poonen's work on the Bertini theorem over finite fields. But there was not an effective bound on the dimension of the Jacobian variety with respect to the degree and dimension of the abelian variety and this is the topic of the last part of my thesis. Given an abelian variety in a projective space over a finite field, joint with J. Bruce, we show the existence of a smooth curve whose Jacobian admits a dominant map to the given abelian variety with an explicit upper bound on its genus. Applying this to simple abelian varieties combined with the theory of Honda-Tate, one can deduce the existence of smooth curves whose Jacobians admit some fixed Weil integer as an eigenvalue with an upper bound on its genus.


Curves Over Finite Fields Of Characteristic Two With Many Rational Points

Author by : Gerard van der Geer
Languange : en
Publisher by : Unknown
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Description : Abstract: "In this note we construct curves over finite fields in characteristic 2 with many rational points. The methods of construction are inspired by considerations from coding theory."


Coding And Cryptology

Author by : Yeow Meng Chee
Languange : en
Publisher by : Springer Science & Business Media
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Description : The biennial International Workshop on Coding and Cryptology (IWCC) aims to bring together many of the world's greatest minds in coding and crypt- ogy to share ideas and exchange knowledge related to advancements in c- ing and cryptology, amidst an informal setting conducive for interaction and collaboration. It is well known that fascinating connections exist between coding and cr- tology. Therefore this workshop series was organized to facilitate a fruitful - teraction and stimulating discourse among experts from these two areas. The inaugural IWCC was held at Wuyi Mountain, Fujian Province, China, during June 11-15, 2007 and attracted over 80 participants. Following this s- cess, the second IWCC was held June 1-5, 2009 at Zhangjiajie, Hunan Province, China. Zhangjiajie is one of the most scenic areas in China. The proceedings of this workshop consist of 21 technical papers, covering a wide range of topics in coding and cryptology, as well as related ?elds such as combinatorics. All papers, except one, are contributed by the invited speakers of the workshop and each paper has been carefully reviewed. We are grateful to the external reviewers for their help, which has greatly strengthened the quality of the proceedings. IWCC 2009 was co-organizedby the National University of Defense Techn- ogy (NUDT), China and Nanyang Technological University (NTU), Singapore. We acknowledge with gratitude the ?nancial support from NUDT. We wouldliketo expressourthanks to Springer formaking it possible forthe proceedings to be published in the Lecture Notes in Computer Science series.


Algebraic Geometry In Coding Theory And Cryptography

Author by : Harald Niederreiter
Languange : en
Publisher by : Princeton University Press
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Description : This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books


Advances In Algebraic Geometry Codes

Author by : Edgar Martinez-Moro
Languange : en
Publisher by : World Scientific
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Description : Advances in Algebraic Geometry Codes presents the most successful applications of algebraic geometry to the field of error-correcting codes, which are used in the industry when one sends information through a noisy channel. The noise in a channel is the corruption of a part of the information due to either interferences in the telecommunications or degradation of the information-storing support (for instance, compact disc). An error-correcting code thus adds extra information to the message to be transmitted with the aim of recovering the sent information. With contributions from renowned researchers, this pioneering book will be of value to mathematicians, computer scientists, and engineers in information theory.


Applied Algebra Algebraic Algorithms And Error Correcting Codes

Author by : Marc Fossorier
Languange : en
Publisher by : Springer
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Description : This book constitutes the refereed proceedings of the 19th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-13, held in Honolulu, Hawaii, USA in November 1999. The 42 revised full papers presented together with six invited survey papers were carefully reviewed and selected from a total of 86 submissions. The papers are organized in sections on codes and iterative decoding, arithmetic, graphs and matrices, block codes, rings and fields, decoding methods, code construction, algebraic curves, cryptography, codes and decoding, convolutional codes, designs, decoding of block codes, modulation and codes, Gröbner bases and AG codes, and polynomials.


Algebraic Curves And Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Walter de Gruyter GmbH & Co KG
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Description : 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.


Finite Fields

Author by : Gary McGuire
Languange : en
Publisher by : American Mathematical Soc.
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Description : This volume contains the proceedings of the Ninth International Conference on Finite Fields and Applications, held in Ireland, July 13-17, 2009. It includes survey papers by all invited speakers as well as selected contributed papers. Finite fields continue to grow in mathematical importance due to applications in many diverse areas. This volume contains a variety of results advancing the theory of finite fields and connections with, as well as impact on, various directions in number theory, algebra, and algebraic geometry. Areas of application include algebric coding theory, cryptology, and combinatorial design theory.


Finite Fields And Applications

Author by : London Mathematical Society. International Conference
Languange : en
Publisher by : Cambridge University Press
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Description : Finite fields are algebraic structures in which there is much research interest. This book gives a state-of-the-art account of finite fields and their applications in communications (coding theory, cryptology), combinatorics, design theory, quasirandom points, algorithms and their complexity. Typically, theory and application are tightly interwoven in the survey articles and original research papers included here. The book also demonstrates interconnections with other branches of pure mathematics such as number theory, group theory and algebraic geometry. This volume is an invaluable resource for any researcher in finite fields or related areas.