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.


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.


Number Theory In Progress

Author by : Kálmán Györy
Languange : en
Publisher by : Walter de Gruyter
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Description : Proceedings of the International Conference on Number Theory organized by the Stefan Banach International Mathematical Center in Honor of the 60th Birthday of Andrzej Schinzel, Zakopane, Poland, June 30-July 9, 1997.


Finite Fields And Applications

Author by : Dieter Jungnickel
Languange : en
Publisher by : Springer Science & Business Media
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Description : This volume represents the refereed proceedings of the Fifth International Conference on Finite Fields and Applications (F q5) held at the University of Augsburg (Germany) from August 2-6, 1999, and hosted by the Department of Mathematics. The conference continued a series of biennial international conferences on finite fields, following earlier conferences at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University ofGlasgow (Scotland) in July 1995, and the University ofWaterloo (Canada) in August 1997. The Organizing Committee of F q5 comprised Thomas Beth (


Number Theory

Author by : Kalman Gyoery
Languange : en
Publisher by : Walter de Gruyter
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Description : The series is aimed specifically at publishing peer reviewed reviews and contributions presented at workshops and conferences. Each volume is associated with a particular conference, symposium or workshop. These events cover various topics within pure and applied mathematics and provide up-to-date coverage of new developments, methods and applications.


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 title provides a self-contained introduction to the theory of algebraic curves over a finite field, whose origins can be traced back to the works of Gauss and Galois on algebraic equations in two variables with coefficients modulo a prime number.


Number Theory

Author by : R.P. Bambah
Languange : en
Publisher by : Birkhäuser
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Description : The Indian National Science Academy on the occasion ofthe Golden Jubilee Celebration (Fifty years of India's Independence) decided to publish a number of monographs on the selected fields. The editorial board of INS A invited us to prepare a special monograph in Number Theory. In reponse to this assignment, we invited several eminent Number Theorists to contribute expository/research articles for this monograph on Number Theory. Al though some ofthose invited, due to other preoccupations-could not respond positively to our invitation, we did receive fairly encouraging response from many eminent and creative number theorists throughout the world. These articles are presented herewith in a logical order. We are grateful to all those mathematicians who have sent us their articles. We hope that this monograph will have a significant impact on further development in this subject. R. P. Bambah v. C. Dumir R. J. Hans-Gill A Centennial History of the Prime Number Theorem Tom M. Apostol The Prime Number Theorem Among the thousands of discoveries made by mathematicians over the centuries, some stand out as significant landmarks. One of these is the prime number theorem, which describes the asymptotic distribution of prime numbers. It can be stated in various equivalent forms, two of which are: x (I) K(X) '" -I - as x --+ 00, ogx and Pn '" n log n as n --+ 00. (2) In (1), K(X) denotes the number of primes P ::s x for any x > O.


Applied Algebra And Number Theory

Author by : Gerhard Larcher
Languange : en
Publisher by : Cambridge University Press
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Description : Survey articles on modern topics related to the work of Harald Niederreiter, written by close colleagues and leading experts.


Monte Carlo And Quasi Monte Carlo Methods 1996

Author by : Harald Niederreiter
Languange : en
Publisher by : Springer Science & Business Media
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Description : Monte Carlo methods are numerical methods based on random sampling and quasi-Monte Carlo methods are their deterministic versions. This volume contains the refereed proceedings of the Second International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at the University of Salzburg (Austria) from July 9--12, 1996. The conference was a forum for recent progress in the theory and the applications of these methods. The topics covered in this volume range from theoretical issues in Monte Carlo and simulation methods, low-discrepancy point sets and sequences, lattice rules, and pseudorandom number generation to applications such as numerical integration, numerical linear algebra, integral equations, binary search, global optimization, computational physics, mathematical finance, and computer graphics. These proceedings will be of interest to graduate students and researchers in Monte Carlo and quasi-Monte Carlo methods, to numerical analysts, and to practitioners of simulation methods.


Finite Fields Theory Applications And Algorithms

Author by : Ronald Cleveland Mullin
Languange : en
Publisher by : American Mathematical Soc.
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Description : The Fourth International Conference on Finite Fields: Theory, Applications, and Algorithms was held at the University of Waterloo in August 1997. This volume presents the refereed proceedings. Because of its applications in so many diverse areas, finite fields continues to grow in importance in modern mathematics. Finite fields now play particularly important roles in number theory, algebra, and algebraic geometry. They also play a crucial role in computer science, statistics, and engineering. Areas of application include but are not limited to algebraic coding theory, cryptology, and combinatorial design theory. Computational and algorithmic aspects of finite field problems are also growing in significance. The conference drew workers in theoretical, applied, and algorithmic finite field theory. All papers were refereed. They are loosely classified as theoretical and applied and are listed under these general headings. The work contains up-to-date results from leading experts in the field.


Finite Fields And Applications

Author by : Stephen D. Cohen
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.


Coding Theory And Cryptology

Author by : Harald Niederreiter
Languange : en
Publisher by : World Scientific
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Description : The inaugural research program of the Institute for Mathematical Sciences at the National University of Singapore took place from July to December 2001 and was devoted to coding theory and cryptology. As part of the program, tutorials for graduate students and junior researchers were given by world-renowned scholars. These tutorials covered fundamental aspects of coding theory and cryptology and were designed to prepare for original research in these areas. The present volume collects the expanded lecture notes of these tutorials. The topics range from mathematical areas such as computational number theory, exponential sums and algebraic function fields through coding-theory subjects such as extremal problems, quantum error-correcting codes and algebraic-geometry codes to cryptologic subjects such as stream ciphers, public-key infrastructures, key management, authentication schemes and distributed system security. Contents:Extremal Problems of Coding Theory (A Barg)Analysis and Design Issues for Synchronous Stream Ciphers (E Dawson & L Simpson)Quantum Error-Correcting Codes (K Feng)Public Key Infrastructures (D Gollmann)Computational Methods in Public Key Cryptology (A K Lenstra)Detecting and Revoking Compromised Keys (T Matsumoto)Algebraic Function Fields Over Finite Fields (H Niederreiter)Authentication Schemes (D Y Pei)Exponential Sums in Coding Theory, Cryptology and Algorithms (I E Shparlinski)Distributed Authorization: Principles and Practice (V Varadharajan)Introduction to Algebraic Geometry Codes (C P Xing) Readership: Graduate students and researchers in number theory, discrete mathematics, coding theory, cryptology and IT security. Keywords:Coding Theory;Cryptology;Number Theory;Algebraic-Geometry Codes;Public-Key Infrastructures;Error-Correcting Codes


Finite Fields And Applications

Author by : International Conference on Finite Fields and Applications
Languange : en
Publisher by : Springer Science & Business Media
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Description : This book constitutes the thoroughly refereed post-proceedings of the 7th International Conference on Finite Fields and Applications, Fq7, held in Toulouse, France, in May 2004. The 19 revised full papers presented were carefully selected from around 60 presentations at the conference during two rounds of reviewing and revision. Among the topics addressed are Weierstrass semigroups, Galois rings, hyperelliptic curves, polynomial irreducibility, pseudorandom number sequences, permutation polynomials, random polynomials, matrices, function fields, ramified towers, BCH codes, cyclic codes, primitive polynomials, covering sequences, cyclic decompositions.


Algebraic Geometric Codes

Author by : M. Tsfasman
Languange : en
Publisher by : Springer Science & Business Media
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Description : 'Et moi ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series.


Algebraic Curves Over Finite Fields

Author by : Carlos Moreno
Languange : en
Publisher by : Cambridge University Press
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Description : Develops the theory of algebraic curves over finite fields, their zeta and L-functions and the theory of algebraic geometric Goppa codes.


Random And Quasi Random Point Sets

Author by : Peter Hellekalek
Languange : en
Publisher by : Springer Science & Business Media
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Description : This volume is a collection of survey papers on recent developments in the fields of quasi-Monte Carlo methods and uniform random number generation. We will cover a broad spectrum of questions, from advanced metric number theory to pricing financial derivatives. The Monte Carlo method is one of the most important tools of system modeling. Deterministic algorithms, so-called uniform random number gen erators, are used to produce the input for the model systems on computers. Such generators are assessed by theoretical ("a priori") and by empirical tests. In the a priori analysis, we study figures of merit that measure the uniformity of certain high-dimensional "random" point sets. The degree of uniformity is strongly related to the degree of correlations within the random numbers. The quasi-Monte Carlo approach aims at improving the rate of conver gence in the Monte Carlo method by number-theoretic techniques. It yields deterministic bounds for the approximation error. The main mathematical tool here are so-called low-discrepancy sequences. These "quasi-random" points are produced by deterministic algorithms and should be as "super" uniformly distributed as possible. Hence, both in uniform random number generation and in quasi-Monte Carlo methods, we study the uniformity of deterministically generated point sets in high dimensions. By a (common) abuse oflanguage, one speaks of random and quasi-random point sets. The central questions treated in this book are (i) how to generate, (ii) how to analyze, and (iii) how to apply such high-dimensional point sets.


Applications Of Algebraic Geometry To Coding Theory Physics And Computation

Author by : Ciro Ciliberto
Languange : en
Publisher by : Springer Science & Business Media
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Description : An up-to-date report on the current status of important research topics in algebraic geometry and its applications, such as computational algebra and geometry, singularity theory algorithms, numerical solutions of polynomial systems, coding theory, communication networks, and computer vision. Contributions on more fundamental aspects of algebraic geometry include expositions related to counting points on varieties over finite fields, Mori theory, linear systems, Abelian varieties, vector bundles on singular curves, degenerations of surfaces, and mirror symmetry of Calabi-Yau manifolds.


Algebraic Geometry Codes Advanced Chapters

Author by : Michael Tsfasman
Languange : en
Publisher by : American Mathematical Soc.
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Description : 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.


Encyclopaedia Of Mathematics

Author by : Michiel Hazewinkel
Languange : en
Publisher by : Springer Science & Business Media
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Description : This is the first Supplementary volume to Kluwer's highly acclaimed Encyclopaedia of Mathematics. This additional volume contains nearly 600 new entries written by experts and covers developments and topics not included in the already published 10-volume set. These entries have been arranged alphabetically throughout. A detailed index is included in the book. This Supplementary volume enhances the existing 10-volume set. Together, these eleven volumes represent the most authoritative, comprehensive up-to-date Encyclopaedia of Mathematics available.


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.


Surveys In Combinatorics 2015

Author by : Artur Czumaj
Languange : en
Publisher by : Cambridge University Press
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Description : Surveys of recent important developments in combinatorics covering a wide range of areas in the field.


Arithmetic Of Finite Fields

Author by : M. Anwar Hasan
Languange : en
Publisher by : Springer Science & Business Media
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Description : This book constitutes the refereed proceedings of the Third International Workshop on the Arithmetic of Finite Fields, WAIFI 2010, held in Istanbul, Turkey, in June 2010. The 15 revised full papers presented were carefully reviewed and selected from 33 submissions. The papers are organized in topical sections on efficient finite field arithmetic, pseudo-random numbers and sequences, Boolean functions, functions, Equations and modular multiplication, finite field arithmetic for pairing based cryptography, and finite field, cryptography and coding.


Monte Carlo And Quasi Monte Carlo Methods 2002

Author by : Harald Niederreiter
Languange : en
Publisher by : Springer Science & Business Media
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Description : This book represents the refereed proceedings of the Fifth International Conference on Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing which was held at the National University of Singapore in the year 2002. An important feature are invited surveys of the state of the art in key areas such as multidimensional numerical integration, low-discrepancy point sets, computational complexity, finance, and other applications of Monte Carlo and quasi-Monte Carlo methods. These proceedings also include carefully selected contributed papers on all aspects of Monte Carlo and quasi-Monte Carlo methods. The reader will be informed about current research in this very active area.


Characterization Of Multi Frobenius Non Classical Plane Curves And Construction Of Complete Plane N D Arcs

Author by : Herivelto Martins Borges Filho
Languange : en
Publisher by : Unknown
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Description : This work is composed of two independent parts, both addressing problems related to algebraic curves over finite fields. In the first part, we characterize all irreducible plane curves defined over Fq which are Frobenius non-classical for different powers of q. Such characterization gives rise to many previously unknown curves which turn out to have some interesting properties. For instance, for n [greater-than or equal to] 3 a curve which is both q- and qn-Frobenius non-classical will have its number of Fqn-rational points attaining the Stöhr-Voloch bound. In the second part, we study the arc property of several plane curves and present new complete (N, d)-arcs in PG(2, q). Some of these arcs (viewed as linear (N, 3,N - d)-codes) are just a small constant away from the Griesmer bound and for some small values of q the bound is achieved. In addition, this part also answers a question of Voloch about the arc property of a certain family of curves with many rational points, and another question of Giulietti et al about the arc property of q-Frobenius non-classical plane curves.


Codes And Curves

Author by : Judy L. Walker
Languange : en
Publisher by : American Mathematical Soc.
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Description : This book is based on a series of lectures the author gave as part of the IAS/Park City Mathematics Institute (Utah) program on arithmetic algebraic geometry. It introduces the reader to the exciting field of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. This book is published in cooperation with IAS/Park City Mathematics Institute.


Arithmetic Geometry Cryptography And Coding Theory

Author by : Alp Bassa
Languange : en
Publisher by : American Mathematical Soc.
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Description : This volume contains the proceedings of the 15th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held at the Centre International de Rencontres Mathématiques in Marseille, France, from May 18–22, 2015. Since the first meeting almost 30 years ago, the biennial AGCT meetings have been one of the main events bringing together researchers interested in explicit aspects of arithmetic geometry and applications to coding theory and cryptography. This volume contains original research articles reflecting recent developments in the field.


Finite Fields With Applications To Coding Theory Cryptography And Related Areas

Author by : Gary L. Mullen
Languange : en
Publisher by : Springer Science & Business Media
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Description : The Sixth International Conference on Finite Fields and Applications, Fq6, held in the city of Oaxaca, Mexico, from May 21-25, 2001, continued a series of biennial international conferences on finite fields. This volume documents the steadily increasing interest in this topic. Finite fields are an important tool in discrete mathematics and its applications cover algebraic geometry, coding theory, cryptology, design theory, finite geometries, and scientific computation, among others. An important feature is the interplay between theory and applications which has led to many new perspectives in research on finite fields and other areas. This interplay has been emphasized in this series of conferences and certainly was reflected in Fq6. This volume offers up-to-date original research papers by leading experts in the area.