Algebraic Curves over a Finite Field
Algebraic Curves over a Finite Field

Author: J. W. P. Hirschfeld

Publisher: Princeton University Press

Published: 2013-03-25

Total Pages: 717

ISBN-13: 1400847419

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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
Algebraic Curves Over Finite Fields

Author: Carlos Moreno

Publisher: Cambridge University Press

Published: 1993-10-14

Total Pages: 264

ISBN-13: 9780521459013

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

Handbook of Finite Fields
Handbook of Finite Fields

Author: Gary L. Mullen

Publisher: CRC Press

Published: 2013-06-17

Total Pages: 1048

ISBN-13: 1439873828

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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

Finite Fields
Finite Fields

Author: Rudolf Lidl

Publisher: Cambridge University Press

Published: 1997

Total Pages: 784

ISBN-13: 9780521392310

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This book is devoted entirely to the theory of finite fields.

Lectures on Finite Fields and Galois Rings
Lectures on Finite Fields and Galois Rings

Author: Zhe-Xian Wan

Publisher: World Scientific

Published: 2003

Total Pages: 360

ISBN-13: 9789812385703

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This is a textbook for graduate and upper level undergraduate students in mathematics, computer science, communication engineering and other fields. The explicit construction of finite fields and the computation in finite fields are emphasised. In particular, the construction of irreducible polynomials and the normal basis of finite fields are included. The essentials of Galois rings are also presented. This invaluable book has been written in a friendly style, so that lecturers can easily use it as a text and students can use it for self-study. A great number of exercises have been incorporated.

Advances in Cryptology - CRYPTO '90
Advances in Cryptology - CRYPTO '90

Author: Alfred J. Menezes

Publisher: Springer

Published: 2003-06-30

Total Pages: 630

ISBN-13: 3540384243

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Crypto '90 marked the tenth anniversary of the Crypto conferences held at the University of California at Santa Barbara. The conference was held from August 11 to August 15, 1990 and was sponsored by the International Association for Cryptologic Research, in cooperation with the IEEE Computer Society Technical Committee on Security and Privacy and the Department of Computer Science of the University of California at Santa Barbara. 227 participants from twenty countries around the world. Crypto '90 attracted Roughly 35% of attendees were from academia, 45% from industry and 20% from government. The program was intended to provide a balance between the purely theoretical and the purely practical aspects of cryptography to meet the needs and diversified interests of these various groups. The overall organization of the conference was superbly handled by the general chairperson Sherry McMahan. All of the outstanding features of Crypto, which we have come to expect over the years, were again present and, in addition to all of this, she did a magnificent job in the preparation of the book of abstracts. This is a crucial part of the program and we owe her a great deal of thanks.

The Finite Field Distance Problem
The Finite Field Distance Problem

Author: David J. Covert

Publisher: American Mathematical Soc.

Published: 2021-06-21

Total Pages: 181

ISBN-13: 1470460319

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Erdős asked how many distinct distances must there be in a set of n n points in the plane. Falconer asked a continuous analogue, essentially asking what is the minimal Hausdorff dimension required of a compact set in order to guarantee that the set of distinct distances has positive Lebesgue measure in R R. The finite field distance problem poses the analogous question in a vector space over a finite field. The problem is relatively new but remains tantalizingly out of reach. This book provides an accessible, exciting summary of known results. The tools used range over combinatorics, number theory, analysis, and algebra. The intended audience is graduate students and advanced undergraduates interested in investigating the unknown dimensions of the problem. Results available until now only in the research literature are clearly explained and beautifully motivated. A concluding chapter opens up connections to related topics in combinatorics and number theory: incidence theory, sum-product phenomena, Waring's problem, and the Kakeya conjecture.