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.


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.


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.


Hypergeometric Functions Over Finite Fields And Their Relations To Algebraic Curves

Author by : Maria Valentina Vega Veglio
Languange : en
Publisher by : Unknown
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Description : Classical hypergeometric functions and their relations to counting points on curves over finite fields have been investigated by mathematicians since the beginnings of 1900. In the mid 1980s, John Greene developed the theory of hypergeometric functions over finite fi elds. He explored the properties of these functions and found that they satisfy many summation and transformation formulas analogous to those satisfi ed by the classical functions. These similarities generated interest in finding connections that hypergeometric functions over finite fields may have with other objects. In recent years, connections between these functions and elliptic curves and other Calabi-Yau varieties have been investigated by mathematicians such as Ahlgren, Frechette, Fuselier, Koike, Ono and Papanikolas. A survey of these results is given at the beginning of this dissertation. We then introduce hypergeometric functions over finite fi elds and some of their properties. Next, we focus our attention on a particular family of curves and give an explicit relationship between the number of points on this family over Fq and sums of values of certain hypergeometric functions over Fq. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over Fq in some particular cases. Based on numerical computations, we are able to state a conjecture relating these values in a more general setting, and advances toward the proof of this result are shown in the last chapter of this dissertation. We nish by giving various avenues for future study.


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.


Higher Dimensional Geometry Over Finite Fields

Author by : D. Kaledin
Languange : en
Publisher by : IOS Press
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Description : 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.


Algebraic Curves And Cryptography

Author by : Vijaya Kumar Murty
Languange : en
Publisher by : American Mathematical Soc.
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Description : It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered 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. This volume is based on seminars on algebraic curves and cryptography held at the GANITA Lab of the University of Toronto during 2001-2008. The articles are mostly suitable for independent study by graduate students who wish to enter the field, both in terms of introducing basic material as well as guiding them in the literature. The literature in cryptography seems to be growing at an exponential rate. For a new entrant into the subject, navigating through this ocean can seem quite daunting. In this volume, the reader is steered toward a discussion of a few key ideas of the subject, together with some brief guidance for further reading. It is hoped that this approach may render the subject more approachable. Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).|It is by now a well-known paradigm that public-key cryptosystems can be built using finite Abelian groups and that algebraic geometry provides a supply of such groups through Abelian varieties over finite fields. Of special interest are the Abelian varieties that are Jacobians of algebraic curves. All of the articles in this volume are centered on the theme of point counting and explicit arithmetic on the Jacobians of curves over finite fields. The topics covered 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. This volume is based on seminars on algebraic curves and cryptography held at the GANITA Lab of the University of Toronto during 2001-2008. The articles are mostly suitable for independent study by graduate students who wish to enter the field, both in terms of introducing basic material as well as guiding them in the literature. The literature in cryptography seems to be growing at an exponential rate. For a new entrant into the subject, navigating through this ocean can seem quite daunting. In this volume, the reader is steered toward a discussion of a few key ideas of the subject, together with some brief guidance for further reading. It is hoped that this approach may render the subject more approachable. Titles in this series are co-published with the Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).


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.


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.


Computational Aspects Of Algebraic Curves

Author by : Tanush Shaska
Languange : en
Publisher by : World Scientific Publishing Company Incorporated
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Description : 1. Preface -- 2. Foreword by the editor -- 3. A new proof for the non-degeneracy of the Frey-Rück pairing and a connection to isogenics over the base field / E. F. Schaefer -- 4. Elliptic curve torsion points and division polynomials / I. A. Burhanuddin and M. A. Huang -- 5. Detecting complex multiplication / J. D. Achter -- 6. Simple numerical uniformatization of elliptic curves / M. Seppälä -- 7. On the moduli space of Klein four covers of the projective line / D. Glass and R. Pries -- 8. Field of moduli and field of definition for curves of genus 2 / G. Cardona and J. Quer -- 9. Explicit computation of Hurwitz spectra / R. Vogeler -- 10. Non-normal Belyi p-gonal surfaces / A. Wootton -- 11. Hyperelliptic curves of genus 3 with prescribed automorphism group / J. Gutierrez, D. Sevilla, and T. Shaska -- 12. Curves over finite fields with many points : an introduction / J. Voight -- 13. Hyperelliptic curves of genus 3 and 4 in characteristic 2 / Y. Demirbas -- 14. Modular representations on some Riemann-Roch spaces of modular curves X(N) / D. Joyner and A. Ksir -- 15. Genus two curves covering elliptic curves : a computational approach / T. Shaska -- 16. A question about Pic(X) as a G-module / D. Goldstein, R. Guralnick, and D. Joyner -- 17. Galois groups of prime degree polynomials with nonreal roots / A. Bialostocki and T. Shaska -- 18. Counting generating systems of a finite group from given conjugacy classes / R. Staszewski, H. Völklein, and G. Wiesend -- 19. Group action on genus 3 curves and their Weierstrass points / H. Babu and P. Venkataraman


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.


Coding Theory And Number Theory

Author by : T. Hiramatsu
Languange : en
Publisher by : Unknown
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One Semester Of Elliptic Curves

Author by : Torsten Ekedahl
Languange : en
Publisher by : European Mathematical Society
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Description : These lecture notes grew out of a one semester introductory course on elliptic curves given to an audience of computer science and mathematics students, and assume only minimal background knowledge. After having covered basic analytic and algebraic aspects, putting special emphasis on explaining the interplay between algebraic and analytic formulas, they go on to some more specialized topics. These include the $j$-function from an algebraic and analytic perspective, a discussion of elliptic curves over finite fields, derivation of recursion formulas for the division polynomials, the algebraic structure of the torsion points of an elliptic curve, complex multiplication, and modular forms. In an effort to motivate basic problems the book starts very slowly but considers some aspects such as modular forms of higher level which are not usually treated. It presents more than 100 exercises and a Mathematica TM notebook that treats a number of calculations involving elliptic curves. The book is aimed at students of mathematics with a general interest in elliptic curves but also at students of computer science interested in their cryptographic aspects.


Coding Theory And Algebraic Geometry

Author by : Henning Stichtenoth
Languange : en
Publisher by : Springer
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Description : About ten years ago, V.D. Goppa found a surprising connection between the theory of algebraic curves over a finite field and error-correcting codes. The aim of the meeting "Algebraic Geometry and Coding Theory" was to give a survey on the present state of research in this field and related topics. The proceedings contain research papers on several aspects of the theory, among them: Codes constructed from special curves and from higher-dimensional varieties, Decoding of algebraic geometric codes, Trace codes, Exponen- tial sums, Fast multiplication in finite fields, Asymptotic number of points on algebraic curves, Sphere packings.


An Introduction To Algebraic Geometry

Author by : Kenji Ueno
Languange : en
Publisher by : American Mathematical Soc.
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Description : This introduction to algebraic geometry allows readers to grasp the fundamentals of the subject with only linear algebra and calculus as prerequisites. After a brief history of the subject, the book introduces projective spaces and projective varieties, and explains plane curves and resolution of their singularities. The volume further develops the geometry of algebraic curves and treats congruence zeta functions of algebraic curves over a finite field. It concludes with a complex analytical discussion of algebraic curves. The author emphasizes computation of concrete examples rather than proofs, and these examples are discussed from various viewpoints. This approach allows readers to develop a deeper understanding of the theorems.


The Derivative Operator On Weighted Bergman Spaces And Quantized Number Theory

Author by : Timothy Logan Cobler
Languange : en
Publisher by : Unknown
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Description : Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, relying on the Frobenius operator on the finite field. The zeta function is then expressed as a determinant allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this dissertation we study the properties of the derivative operator on a particular family of weighted Bergman spaces. This operator is meant to be the replacement for the Frobenius in the general case and is first used to give a method of quantizing elliptic curves and modular forms; then to construct an operator associated to any given meromorphic function. With this construction, we show that for a wide class of functions, the function can be recovered using a regularized Berezinian determinant involving the operator constructed from the meromorphic function. This is shown in some special cases: rational functions, zeta functions of algebraic curves over finite fields, geometric zeta functions of lattice self-similar strings, the gamma function, the Riemann zeta function and culminating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed.


The Large Sieve And Its Applications

Author by : E. Kowalski
Languange : en
Publisher by : Cambridge University Press
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Description : Among the modern methods used to study prime numbers, the 'sieve' has been one of the most efficient. Originally conceived by Linnik in 1941, the 'large sieve' has developed extensively since the 1960s, with a recent realisation that the underlying principles were capable of applications going well beyond prime number theory. This book develops a general form of sieve inequality, and describes its varied applications, including the study of families of zeta functions of algebraic curves over finite fields; arithmetic properties of characteristic polynomials of random unimodular matrices; homological properties of random 3-manifolds; and the average number of primes dividing the denominators of rational points on elliptic curves. Also covered in detail are the tools of harmonic analysis used to implement the forms of the large sieve inequality, including the Riemann Hypothesis over finite fields, and Property (T) or Property (tau) for discrete groups.


Applications Of Finite Fields

Author by : Alfred J. Menezes
Languange : en
Publisher by : Springer Science & Business Media
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Description : 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.


Elementary Number Theory Cryptography And Codes

Author by : M. Welleda Baldoni
Languange : en
Publisher by : Springer Science & Business Media
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Description : In this volume one finds basic techniques from algebra and number theory (e.g. congruences, unique factorization domains, finite fields, quadratic residues, primality tests, continued fractions, etc.) which in recent years have proven to be extremely useful for applications to cryptography and coding theory. Both cryptography and codes have crucial applications in our daily lives, and they are described here, while the complexity problems that arise in implementing the related numerical algorithms are also taken into due account. Cryptography has been developed in great detail, both in its classical and more recent aspects. In particular public key cryptography is extensively discussed, the use of algebraic geometry, specifically of elliptic curves over finite fields, is illustrated, and a final chapter is devoted to quantum cryptography, which is the new frontier of the field. Coding theory is not discussed in full; however a chapter, sufficient for a good introduction to the subject, has been devoted to linear codes. Each chapter ends with several complements and with an extensive list of exercises, the solutions to most of which are included in the last chapter. Though the book contains advanced material, such as cryptography on elliptic curves, Goppa codes using algebraic curves over finite fields, and the recent AKS polynomial primality test, the authors' objective has been to keep the exposition as self-contained and elementary as possible. Therefore the book will be useful to students and researchers, both in theoretical (e.g. mathematicians) and in applied sciences (e.g. physicists, engineers, computer scientists, etc.) seeking a friendly introduction to the important subjects treated here. The book will also be useful for teachers who intend to give courses on these topics.


Arithmetic Of Algebraic Curves

Author by : Serguei A. Stepanov
Languange : en
Publisher by : Springer Science & Business Media
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Total Read : 26
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Description : 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.