Theory And Applications Of Finite Fields

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Applications of Finite Fields

Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian
Applications of Finite Fields Book PDF

Author : Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian
Publisher : Springer Science & Business Media
Page : 229 pages
File Size : 46,6 Mb
Release : 2013-04-17
Category : Technology & Engineering
ISBN : 9781475722260

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Applications of Finite Fields by Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian Pdf

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.

Finite Fields and Applications

Gary L. Mullen,Carl Mummert
Finite Fields and Applications Book PDF

Author : Gary L. Mullen,Carl Mummert
Publisher : American Mathematical Soc.
Page : 175 pages
File Size : 44,5 Mb
Release : 2007
Category : Mathematics
ISBN : 9780821844182

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Finite Fields and Applications by Gary L. Mullen,Carl Mummert Pdf

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.

Finite Fields

Rudolf Lidl,Harald Niederreiter
Finite Fields Book PDF

Author : Rudolf Lidl,Harald Niederreiter
Publisher : Cambridge University Press
Page : 784 pages
File Size : 46,7 Mb
Release : 1997
Category : Mathematics
ISBN : 0521392314

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Finite Fields by Rudolf Lidl,Harald Niederreiter Pdf

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

Finite Fields: Theory and Computation

Igor Shparlinski
Finite Fields Theory and Computation Book PDF

Author : Igor Shparlinski
Publisher : Springer Science & Business Media
Page : 532 pages
File Size : 42,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401592390

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Finite Fields: Theory and Computation by Igor Shparlinski Pdf

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.

Handbook of Finite Fields

Gary L. Mullen,Daniel Panario
Handbook of Finite Fields Book PDF

Author : Gary L. Mullen,Daniel Panario
Publisher : CRC Press
Page : 1048 pages
File Size : 50,6 Mb
Release : 2013-06-17
Category : Computers
ISBN : 9781439873823

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Handbook of Finite Fields by Gary L. Mullen,Daniel Panario Pdf

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

Lectures on Finite Fields

Xiang-dong Hou
Lectures on Finite Fields Book PDF

Author : Xiang-dong Hou
Publisher : American Mathematical Soc.
Page : 240 pages
File Size : 44,7 Mb
Release : 2018-06-07
Category : Finite fields (Algebra)
ISBN : 9781470442897

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Lectures on Finite Fields by Xiang-dong Hou Pdf

The theory of finite fields encompasses algebra, combinatorics, and number theory and has furnished widespread applications in other areas of mathematics and computer science. This book is a collection of selected topics in the theory of finite fields and related areas. The topics include basic facts about finite fields, polynomials over finite fields, Gauss sums, algebraic number theory and cyclotomic fields, zeros of polynomials over finite fields, and classical groups over finite fields. The book is mostly self-contained, and the material covered is accessible to readers with the knowledge of graduate algebra; the only exception is a section on function fields. Each chapter is supplied with a set of exercises. The book can be adopted as a text for a second year graduate course or used as a reference by researchers.

Finite Fields and Their Applications

Pascale Charpin,Alexander Pott,Arne Winterhof
Finite Fields and Their Applications Book PDF

Author : Pascale Charpin,Alexander Pott,Arne Winterhof
Publisher : Walter de Gruyter
Page : 288 pages
File Size : 54,9 Mb
Release : 2013-05-28
Category : Mathematics
ISBN : 9783110283600

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Finite Fields and Their Applications by Pascale Charpin,Alexander Pott,Arne Winterhof Pdf

This book is based on the invited talks of the "RICAM-Workshop on Finite Fields and Their Applications: Character Sums and Polynomials" held at the Federal Institute for Adult Education (BIfEB) in Strobl, Austria, from September 2-7, 2012. Finite fields play important roles in many application areas such as coding theory, cryptography, Monte Carlo and quasi-Monte Carlo methods, pseudorandom number generation, quantum computing, and wireless communication. In this book we will focus on sequences, character sums, and polynomials over finite fields in view of the above mentioned application areas: Chapters 1 and 2 deal with sequences mainly constructed via characters and analyzed using bounds on character sums. Chapters 3, 5, and 6 deal with polynomials over finite fields. Chapters 4 and 9 consider problems related to coding theory studied via finite geometry and additive combinatorics, respectively. Chapter 7 deals with quasirandom points in view of applications to numerical integration using quasi-Monte Carlo methods and simulation. Chapter 8 studies aspects of iterations of rational functions from which pseudorandom numbers for Monte Carlo methods can be derived. The goal of this book is giving an overview of several recent research directions as well as stimulating research in sequences and polynomials under the unified framework of character theory.

Introduction to Finite Fields and Their Applications

Rudolf Lidl,Harald Niederreiter
Introduction to Finite Fields and Their Applications Book PDF

Author : Rudolf Lidl,Harald Niederreiter
Publisher : Cambridge University Press
Page : 446 pages
File Size : 54,5 Mb
Release : 1994-07-21
Category : Mathematics
ISBN : 0521460948

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Introduction to Finite Fields and Their Applications by Rudolf Lidl,Harald Niederreiter Pdf

Presents an introduction to the theory of finite fields and some of its most important applications.

Finite Fields, with Applications to Combinatorics

Kannan Soundararajan
Finite Fields with Applications to Combinatorics Book PDF

Author : Kannan Soundararajan
Publisher : American Mathematical Society
Page : 100 pages
File Size : 52,7 Mb
Release : 2022-11-09
Category : Mathematics
ISBN : 9781470469306

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Finite Fields, with Applications to Combinatorics by Kannan Soundararajan Pdf

This book uses finite field theory as a hook to introduce the reader to a range of ideas from algebra and number theory. It constructs all finite fields from scratch and shows that they are unique up to isomorphism. As a payoff, several combinatorial applications of finite fields are given: Sidon sets and perfect difference sets, de Bruijn sequences and a magic trick of Persi Diaconis, and the polynomial time algorithm for primality testing due to Agrawal, Kayal and Saxena. The book forms the basis for a one term intensive course with students meeting weekly for multiple lectures and a discussion session. Readers can expect to develop familiarity with ideas in algebra (groups, rings and fields), and elementary number theory, which would help with later classes where these are developed in greater detail. And they will enjoy seeing the AKS primality test application tying together the many disparate topics from the book. The pre-requisites for reading this book are minimal: familiarity with proof writing, some linear algebra, and one variable calculus is assumed. This book is aimed at incoming undergraduate students with a strong interest in mathematics or computer science.

Combinatorics and Finite Fields

Kai-Uwe Schmidt,Arne Winterhof
Combinatorics and Finite Fields Book PDF

Author : Kai-Uwe Schmidt,Arne Winterhof
Publisher : Walter de Gruyter GmbH & Co KG
Page : 506 pages
File Size : 52,6 Mb
Release : 2019-07-08
Category : Mathematics
ISBN : 9783110641967

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Combinatorics and Finite Fields by Kai-Uwe Schmidt,Arne Winterhof Pdf

The series is devoted to the publication of high-level monographs, surveys and proceedings which cover the whole spectrum of computational and applied mathematics. The books of this series are addressed to both specialists and advanced students. Interested authors may submit book proposals to the Managing Editor or to any member of the Editorial Board. Managing EditorUlrich Langer, Johannes Kepler University Linz, Austria Editorial BoardHansj rg Albrecher, University of Lausanne, SwitzerlandRonald H. W. Hoppe, University of Houston, USAKarl Kunisch, RICAM, Linz, Austria; University of Graz, AustriaHarald Niederreiter, RICAM, Linz, AustriaChristian Schmeiser, University of Vienna, Austria

Finite Fields

Janet Simmons
Finite Fields Book PDF

Author : Janet Simmons
Publisher : Nova Publishers
Page : 125 pages
File Size : 48,9 Mb
Release : 2017
Category : Mathematics
ISBN : 1536104000

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Finite Fields by Janet Simmons Pdf

This book provides new research in finite fields. Chapter One presents some techniques that rely on a combination of results from graph theory, finite fields, matrix theory, and finite geometry to researchers working in the area of preserver problems. It also gives a brief presentation of this research field to other mathematicians. Chapter Two contains a basic and self-contained introduction to classical coherent state transforms, namely classical wavelet and classical wave-packet transforms, on finite fields. Chapter Three proposes an intrinsic representation of finite m? extension as this is a tradition for finite extension fields. Chapter Four reviews m? cyclic codes on a m? field. Chapter Five discusses two problems of Carlitz and their generalizations.

Finite Fields for Computer Scientists and Engineers

Robert J. McEliece
Finite Fields for Computer Scientists and Engineers Book PDF

Author : Robert J. McEliece
Publisher : Springer Science & Business Media
Page : 212 pages
File Size : 47,7 Mb
Release : 2012-12-06
Category : Technology & Engineering
ISBN : 9781461319832

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Finite Fields for Computer Scientists and Engineers by Robert J. McEliece Pdf

This book developed from a course on finite fields I gave at the University of Illinois at Urbana-Champaign in the Spring semester of 1979. The course was taught at the request of an exceptional group of graduate students (includ ing Anselm Blumer, Fred Garber, Evaggelos Geraniotis, Jim Lehnert, Wayne Stark, and Mark Wallace) who had just taken a course on coding theory from me. The theory of finite fields is the mathematical foundation of algebraic coding theory, but in coding theory courses there is never much time to give more than a "Volkswagen" treatment of them. But my 1979 students wanted a "Cadillac" treatment, and this book differs very little from the course I gave in response. Since 1979 I have used a subset of my course notes (correspond ing roughly to Chapters 1-6) as the text for my "Volkswagen" treatment of finite fields whenever I teach coding theory. There is, ironically, no coding theory anywhere in the book! If this book had a longer title it would be "Finite fields, mostly of char acteristic 2, for engineering and computer science applications. " It certainly does not pretend to cover the general theory of finite fields in the profound depth that the recent book of Lidl and Neidereitter (see the Bibliography) does.

Finite Fields and their Applications

James A. Davis
Finite Fields and their Applications Book PDF

Author : James A. Davis
Publisher : Walter de Gruyter GmbH & Co KG
Page : 214 pages
File Size : 53,8 Mb
Release : 2020-10-26
Category : Mathematics
ISBN : 9783110621730

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Finite Fields and their Applications by James A. Davis Pdf

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.

Algebraic Curves and Finite Fields

Harald Niederreiter,Alina Ostafe,Daniel Panario,Arne Winterhof
Algebraic Curves and Finite Fields Book PDF

Author : Harald Niederreiter,Alina Ostafe,Daniel Panario,Arne Winterhof
Publisher : Walter de Gruyter GmbH & Co KG
Page : 251 pages
File Size : 48,8 Mb
Release : 2014-08-20
Category : Mathematics
ISBN : 9783110317916

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Algebraic Curves and Finite Fields by Harald Niederreiter,Alina Ostafe,Daniel Panario,Arne Winterhof Pdf

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 and Algorithmic Problems in Finite Fields

Igor Shparlinski
Computational and Algorithmic Problems in Finite Fields Book PDF

Author : Igor Shparlinski
Publisher : Springer Science & Business Media
Page : 253 pages
File Size : 41,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401118064

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Computational and Algorithmic Problems in Finite Fields by Igor Shparlinski Pdf

This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include polynomial factorization, finding irreducible and primitive polynomials, 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 araes of mathematics. For completeness, also included are two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number generators, modular arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.