Applications Of Curves Over Finite Fields

Author by : Joint Summ Ams-Ims-Siam
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 63
Total Download : 533
File Size : 40,9 Mb
GET BOOK

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.


Applications Of Finite Fields

Author by : Alfred J. Menezes
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 10
Total Download : 827
File Size : 53,7 Mb
GET BOOK

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.


Rational Points On Curves Over Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 27
Total Download : 955
File Size : 40,7 Mb
GET BOOK

Description : Discussion of theory and applications of algebraic curves over finite fields with many rational points.


Algebraic Curves Over A Finite Field

Author by : J. W. P. Hirschfeld
Languange : en
Publisher by : Princeton University Press
Format Available : PDF, ePub, Mobi
Total Read : 14
Total Download : 162
File Size : 54,6 Mb
GET BOOK

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.


Frobenius Action On Jacobians Of Curves Over Finite Fields

Author by : Wanlin Li
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 35
Total Download : 737
File Size : 49,6 Mb
GET BOOK

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.


Finite Fields And Applications

Author by : Dieter Jungnickel
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 46
Total Download : 693
File Size : 42,6 Mb
GET BOOK

Description : The Fifth International Conference on Finite Fields and Applications Fq5 held at the University of Augsburg, Germany, from August 2-6, 1999 continued a series of biennial international conferences on finite fields. The proceedings document the steadily increasing interest in this topic. Finite fields have an inherently fascinating structure and are important tools in discrete mathematics. Their applications range from combinatorial design theory, finite geometries, and algebraic geometry to coding theory, cryptology, and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields. This interplay has always been a dominant theme in Fq conferences and was very much in evidence at Fq5. The proceedings reflect this, and offer an up-to-date collection of surveys and original research articles by leading experts in the area.


Finite Fields And Applications

Author by : Gary L. Mullen
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 88
Total Download : 260
File Size : 40,6 Mb
GET BOOK

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.


Algebraic Curves And Finite Fields

Author by : Harald Niederreiter
Languange : en
Publisher by : Walter de Gruyter GmbH & Co KG
Format Available : PDF, ePub, Mobi
Total Read : 27
Total Download : 781
File Size : 45,9 Mb
GET BOOK

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 And Applications

Author by : London Mathematical Society. International Conference
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 32
Total Download : 591
File Size : 51,8 Mb
GET BOOK

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.


Finite Fields

Author by : Gary McGuire
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 26
Total Download : 179
File Size : 40,5 Mb
GET BOOK

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.


Algebraic Geometric Codes

Author by : Michael A. Tsfasman
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 35
Total Download : 602
File Size : 46,7 Mb
GET BOOK

Description : This book focuses on the theory of algebraic geometry codes, a subject that has emerged at the meeting point of several fields of mathematics. Unlike other texts, it consistently seeks interpretations that connect coding theory to algebraic geometry and number theory. This approach makes the book useful for both coding experts and experts in algebraic geometry.


Contemporary Developments In Finite Fields And Applications

Author by : Anne Canteaut
Languange : en
Publisher by : World Scientific
Format Available : PDF, ePub, Mobi
Total Read : 99
Total Download : 851
File Size : 40,5 Mb
GET BOOK

Description : The volume is a collection of 20 refereed articles written in connection with lectures presented at the 12th International Conference on Finite Fields and Their Applications ("Fq12") at Skidmore College in Saratoga Springs, NY in July 2015. Finite fields are central to modern cryptography and secure digital communication, and hence must evolve rapidly to keep pace with new technologies. Topics in this volume include cryptography, coding theory, structure of finite fields, algorithms, curves over finite fields, and further applications. Contributors will include: Antoine Joux (Fondation Partenariale de l'UPMC, France); Gary Mullen (Penn State University, USA); Gohar Kyureghyan (Otto-von-Guericke Universität, Germany); Gary McGuire (University College Dublin, Ireland); Michel Lavrauw (Università degli Studi di Padova, Italy); Kirsten Eisentraeger (Penn State University, USA); Renate Scheidler (University of Calgary, Canada); Michael Zieve (University of Michigan, USA). Contents:Divisibility of L-Polynomials for a Family of Curves (I Blanco-Chacón, R Chapman, S Fordham and G McGuire)Divisibility of Exponential Sums Associated to Binomials Over 𝔽p (F Castro, R Figueroa, P Guan and J Ortiz-Ubarri)Dickson Polynomials that are Involutions (P Charpin, S Mesnager and S Sarkar)Constructing Elliptic Curves and Curves of Genus 2 over Finite Fields (K Eisenträger)A Family of Plane Curves with Two or More Galois Points in Positive Characteristic (S Fukasawa)Permutation Polynomials of 𝔽q2 of the Form αX + Xr(q-1)+1 (X-D Hou)Character Sums and Generating Sets (M-D A Huang and L Liu)Nearly Sparse Linear Algebra and Application to Discrete Logarithms Computations (A Joux and C Pierrot)Full Degree Two del Pezzo Surfaces over Small Finite Fields (A Knecht and K Reyes)Diameter of Some Monomial Digraphs (A Kodess, F Lazebnik, S Smith and J Sporre)Permutation Polynomials of the Form X + γTr(Xk) (G Kyureghyan and M Zieve)Scattered Spaces in Galois Geometry (M Lavrauw)On the Value Set of Small Families of Polynomials over a Finite Field, III (G Matera, M Pérez and Melina Privitelli)The Density of Unimodular Matrices over Integrally Closed Subrings of Function Fields (G Micheli and R Schnyder)Some Open Problems Arising from My Recent Finite Field Research (G L Mullen)On Coefficients of Powers of Polynomials and Their Compositions over Finite Fields (G L Mullen, A Muratović-Ribić and Q Wang)On the Structure of Certain Reduced Linear Modular Systems (E Orozco)Finding a Gröbner Basis for the Ideal of Recurrence Relations on m-Dimensional Periodic Arrays (I M Rubio, M Sweedler and C Heegard)An Introduction to Hyperelliptic Curve Arithmetic (R Scheidler)On the Existence of Aperiodic Complementary Hexagonal Lattice Arrays (Y Tan and G Gong) Readership: Researchers in combinatorics and graph theory, numerical analysis and computational mathematics, and coding theory.


Finite Fields And Applications

Author by : Gary L. Mullen
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 57
Total Download : 888
File Size : 47,7 Mb
GET BOOK

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.


Applications Of Finite Fields

Author by : Alfred J. Menezes
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 95
Total Download : 822
File Size : 40,6 Mb
GET BOOK

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 of mathematics, in recent years there has been a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. Applications of Finite Fields introduces some of these recent developments. This book focuses attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, Applications of Finite Fields does not attempt 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. This book is developed from a seminar held at the University of Waterloo. The purpose of the seminar was to bridge the knowledge of the participants whose expertise and interests ranged from the purely theoretical to the applied. As a result, this book will be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics. Applications of Finite Fields is an excellent reference and may be used as a text for a course on the subject.


Applications Of Algebraic Geometry To Coding Theory Physics And Computation

Author by : Ciro Ciliberto
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 86
Total Download : 849
File Size : 50,7 Mb
GET BOOK

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.


Applied Algebra And Number Theory

Author by : Gerhard Larcher
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 95
Total Download : 782
File Size : 53,8 Mb
GET BOOK

Description : Survey articles on modern topics related to the work of Harald Niederreiter, written by close colleagues and leading experts.


Finite Fields

Author by : Gary L. Mullen
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 61
Total Download : 305
File Size : 46,9 Mb
GET BOOK

Description : Because of their applications in so many diverse areas, finite fields continue to play increasingly important roles in various branches of modern mathematics, including number theory, algebra, and algebraic geometry, as well as in computer science, information theory, statistics, and engineering. Computational and algorithmic aspects of finite field problems also continue to grow in importance. This volume contains the refereed proceedings of a conference entitled Finite Fields: Theory, Applications and Algorithms, held in August 1993 at the University of Nevada at Las Vegas. Among the topics treated are theoretical aspects of finite fields, coding theory, cryptology, combinatorial design theory, and algorithms related to finite fields. Also included is a list of open problems and conjectures. This volume is an excellent reference for applied and research mathematicians as well as specialists and graduate students in information theory, computer science, and electrical engineering.


Finite Fields Theory And Computation

Author by : Igor Shparlinski
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 31
Total Download : 382
File Size : 54,7 Mb
GET BOOK

Description : 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.


Progress In Cryptology Indocrypt 2003

Author by : Thomas Johansson
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 35
Total Download : 371
File Size : 45,5 Mb
GET BOOK

Description : This book constitutes the refereed proceedings of the 4th International Conference on Cryptology in India, INDOCRYPT 2003, held in New Delhi, India in December 2003. The 29 revised full papers presented together with 2 invited papers were carefully reviewed and selected from 101 submissions. The papers are organized in topical sections on stream ciphers, block ciphers, Boolean functions, secret sharing, bilinear pairings, public key cryptography, signature schemes, protocols, elliptic curve cryptography and algebraic geometry, implementation and digital watermarking, and authentication.


Arithmetic Of Finite Fields

Author by : Çetin Kaya Koç
Languange : en
Publisher by : Springer
Format Available : PDF, ePub, Mobi
Total Read : 28
Total Download : 451
File Size : 40,7 Mb
GET BOOK

Description : This book constitutes the refereed proceedings of the 5th International Workshop on the Arithmetic of Finite Field, WAIFI 2014, held in Gebze, Turkey, in September 2014. The 9 revised full papers and 43 invited talks presented were carefully reviewed and selected from 27 submissions. This workshop is a forum of mathematicians, computer scientists, engineers and physicists performing research on finite field arithmetic, interested in communicating the advances in the theory, applications, and implementations of finite fields. The workshop will help to bridge the gap between the mathematical theory of finite fields and their hardware/software implementations and technical applications.


Elliptic Curves

Author by : Susanne Schmitt
Languange : en
Publisher by : Walter de Gruyter
Format Available : PDF, ePub, Mobi
Total Read : 40
Total Download : 464
File Size : 45,9 Mb
GET BOOK

Description : The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.


Number Theory

Author by : R.P. Bambah
Languange : en
Publisher by : Birkhäuser
Format Available : PDF, ePub, Mobi
Total Read : 84
Total Download : 591
File Size : 54,6 Mb
GET BOOK

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.


Finite Fields And Their Applications

Author by : James A. Davis
Languange : en
Publisher by : Walter de Gruyter GmbH & Co KG
Format Available : PDF, ePub, Mobi
Total Read : 48
Total Download : 186
File Size : 54,5 Mb
GET BOOK

Description : The volume covers wide-ranging topics from Theory: structure of finite fields, normal bases, polynomials, function fields, APN functions. Computation: algorithms and complexity, polynomial factorization, decomposition and irreducibility testing, sequences and functions. Applications: algebraic coding theory, cryptography, algebraic geometry over finite fields, finite incidence geometry, designs, combinatorics, quantum information science.


Handbook Of Finite Fields

Author by : Gary L. Mullen
Languange : en
Publisher by : CRC Press
Format Available : PDF, ePub, Mobi
Total Read : 93
Total Download : 522
File Size : 40,8 Mb
GET BOOK

Description : Poised to become the leading reference in the field, the Handbook of Finite Fields is exclusively devoted to the theory and applications of finite fields. More than 80 international contributors compile state-of-the-art research in this definitive handbook. Edited by two renowned researchers, the book uses a uniform style and format throughout and


Topics In Geometry Coding Theory And Cryptography

Author by : Arnaldo Garcia
Languange : en
Publisher by : Springer Science & Business Media
Format Available : PDF, ePub, Mobi
Total Read : 33
Total Download : 956
File Size : 45,8 Mb
GET BOOK

Description : The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.


The Riemann Hypothesis For Elliptic Curves Over Finite Fields

Author by : Connor Cassady
Languange : en
Publisher by : Unknown
Format Available : PDF, ePub, Mobi
Total Read : 13
Total Download : 553
File Size : 48,8 Mb
GET BOOK

Description : The Riemann Hypothesis has been a result eluding mathematicians for nearly 200 years. Analogs of this result have been found for elliptic curves over finite fields, which is the subject of this thesis. We begin by establishing algebraic foundations that will be used to prove larger results regarding the Riemann Hypothesis. Next, we explore an elementary number theoretic approach to this problem, and deal with a very particular type of elliptic curve. The crux of this paper is found in the next chapter, where we state and prove the Riemann Hypothesis for elliptic curves over finite fields. Finally, we investigate some examples of specific elliptic curves to see the applications of the theorems proved earlier.


Algebraic Curves Over Finite Fields

Author by : Carlos Moreno
Languange : en
Publisher by : Cambridge University Press
Format Available : PDF, ePub, Mobi
Total Read : 89
Total Download : 778
File Size : 47,5 Mb
GET BOOK

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


Arithmetic Geometry Computation And Applications

Author by : Yves Aubry
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 21
Total Download : 266
File Size : 42,6 Mb
GET BOOK

Description : For thirty years, the biennial international conference AGC T (Arithmetic, Geometry, Cryptography, and Coding Theory) has brought researchers to Marseille to build connections between arithmetic geometry and its applications, originally highlighting coding theory but more recently including cryptography and other areas as well. This volume contains the proceedings of the 16th international conference, held from June 19–23, 2017. The papers are original research articles covering a large range of topics, including weight enumerators for codes, function field analogs of the Brauer–Siegel theorem, the computation of cohomological invariants of curves, the trace distributions of algebraic groups, and applications of the computation of zeta functions of curves. Despite the varied topics, the papers share a common thread: the beautiful interplay between abstract theory and explicit results.