# Introduction To Finite Fields And Their Applications

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## Introduction to Finite Fields and Their Applications

Author | : Rudolf Lidl,Harald Niederreiter |

Publsiher | : Cambridge University Press |

Total Pages | : 416 |

Release | : 1994-07-21 |

ISBN | : 9780521460941 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Finite Fields and Their Applications Book Excerpt:**

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

## Introduction to Finite Fields and Their Applications

Author | : Rudolf Lidl,Harald Niederreiter |

Publsiher | : Unknown |

Total Pages | : 407 |

Release | : 1986 |

ISBN | : 9780521307062 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Finite Fields and Their Applications Book Excerpt:**

The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

## Finite Fields and their Applications

Author | : James A. Davis |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 214 |

Release | : 2020-10-26 |

ISBN | : 3110622173 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields and their Applications Book Excerpt:**

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.

## Finite Fields and Their Applications

Author | : Pascale Charpin,Alexander Pott,Arne Winterhof |

Publsiher | : Walter de Gruyter |

Total Pages | : 285 |

Release | : 2013-05-28 |

ISBN | : 3110283603 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields and Their Applications Book Excerpt:**

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

Author | : R. Lidl |

Publsiher | : Unknown |

Total Pages | : 135 |

Release | : 1986 |

ISBN | : 1928374650XXX |

Category | : Electronic Book |

Language | : EN, FR, DE, ES & NL |

**INTRODUCTION TO FINITE FIELDS AND THEIR APPLICATIONS Book Excerpt:**

## Applications of Finite Fields

Author | : Alfred J. Menezes,Ian F. Blake,XuHong Gao,Ronald C. Mullin,Scott A. Vanstone,Tomik Yaghoobian |

Publsiher | : Springer Science & Business Media |

Total Pages | : 218 |

Release | : 2013-04-17 |

ISBN | : 1475722265 |

Category | : Technology & Engineering |

Language | : EN, FR, DE, ES & NL |

**Applications of Finite Fields Book Excerpt:**

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.

## Sequences and Their Applications SETA 2010

Author | : Claude Carlet,Alexander Pott |

Publsiher | : Springer |

Total Pages | : 465 |

Release | : 2010-09-07 |

ISBN | : 3642158749 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Sequences and Their Applications SETA 2010 Book Excerpt:**

This book constitutes the proceedings of the 6th International Conference on Sequences and Their Applications held in Paris, France, in September 2010.

## Finite Fields and Applications

Author | : Gary L. Mullen,Carl Mummert |

Publsiher | : American Mathematical Soc. |

Total Pages | : 175 |

Release | : 2007 |

ISBN | : 0821844180 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields and Applications Book Excerpt:**

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 Theory Applications and Algorithms

Author | : Ronald Cleveland Mullin,Applications International Conference on Finite Fields: Theory, and Algorithms (4th : 1997 : University of Waterloo),Gary L. Mullen,International Conference on Finite Fields: Theory, Applications, and Algorithms (4th : 1997 : University of Waterloo) |

Publsiher | : American Mathematical Soc. |

Total Pages | : 243 |

Release | : 1999 |

ISBN | : 0821808176 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields Theory Applications and Algorithms Book Excerpt:**

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 with Applications to Coding Theory Cryptography and Related Areas

Author | : Gary L. Mullen,Henning Stichtenoth,Horacio Tapia-Recillas |

Publsiher | : Springer Science & Business Media |

Total Pages | : 335 |

Release | : 2012-12-06 |

ISBN | : 3642594352 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields with Applications to Coding Theory Cryptography and Related Areas Book Excerpt:**

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.

## Finite Fields Theory and Computation

Author | : Igor Shparlinski |

Publsiher | : Springer Science & Business Media |

Total Pages | : 528 |

Release | : 2013-03-09 |

ISBN | : 940159239X |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Finite Fields Theory and Computation Book Excerpt:**

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.

## Reed Solomon Codes and Their Applications

Author | : Stephen B. Wicker,Vijay K. Bhargava |

Publsiher | : John Wiley & Sons |

Total Pages | : 336 |

Release | : 1999-10-12 |

ISBN | : 9780780353916 |

Category | : Technology & Engineering |

Language | : EN, FR, DE, ES & NL |

**Reed Solomon Codes and Their Applications Book Excerpt:**

Electrical Engineering/Communications/Information Theory "The Berlekamp article alone will make this book worth having." --David Forney, Vice President, Motorola Codex Reed-Solomon Codes and Their Applications Edited by Stephen B. Wicker, Georgia Institute of Technology and Vijay K. Bhargava, University of Victoria On the Voyager spacecraft, they were responsible for sending clear pictures of the planets back to earth. They have also played a key role in the digital audio revolution. They are Reed-Solomon error codes: the extremely powerful codes that provide critical error control for many different types of digital communications systems. This outstanding collection of thirteen original articles written by leading researchers in the field provides a uniquely comprehensive overview of the history and practical applications--some never before published--of these important codes. Key features include: * Thirteen original articles from leading researchers in the field, with a historical overview by Reed and Solomon * An explanation of how Reed-Solomon codes were used in the Voyager spacecraft and how they are currently used in the compact disc player * Specific applications for digital audio, data transfer over mobile radio, satellite communications, spread spectrum systems, and more * New techniques for improving the performance of your own communications systems This book will be of interest to design and research engineers in the telecommunications field, particularly those in the aerospace/satellite and mobile radio industries. It is also well-suited for use as an advanced-level textbook on the subject of error control coding. Books of Related Interest from IEEE Press Clauide Elwood Shannon: Collected Papers Edited by N. J. A. Sloane and A. D. Wyner. AT&T Bell Labs The first published collection of papers by Claude E. Shannon, including his seminal article "The Mathematical Theory of Communication." 1993 Hardcover 968 pp IEEE Order Number PC0331-9 ISBN 0-7803-0434-9 Multiple Access Communications: Foundations for Emerging Technologies Edited by Norman Abramson, University of Hawaii at Manoa The first book to explain the connection between spread spectrum and ALOHA channels, providing a collection of key developments in the theory and practice of multiple user communications channels. 1993 Hardcover 528pp IEEE Order Number PC0287-3 ISBN 0-87942-292-0

## Infinite Algebraic Extensions of Finite Fields

Author | : Joel V. Brawley,George E. Schnibben |

Publsiher | : American Mathematical Soc. |

Total Pages | : 104 |

Release | : 1989 |

ISBN | : 0821851012 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Infinite Algebraic Extensions of Finite Fields Book Excerpt:**

Over the last several decades there has been a renewed interest in finite field theory, partly as a result of important applications in a number of diverse areas such as electronic communications, coding theory, combinatorics, designs, finite geometries, cryptography, and other portions of discrete mathematics. In addition, a number of recent books have been devoted to the subject. Despite the resurgence in interest, it is not widely known that many results concerning finite fields have natural generalizations to abritrary algebraic extensions of finite fields. The purpose of this book is to describe these generalizations. After an introductory chapter surveying pertinent results about finite fields, the book describes the lattice structure of fields between the finite field $GF(q)$ and its algebraic closure $\Gamma (q)$. The authors introduce a notion, due to Steinitz, of an extended positive integer $N$ which includes each ordinary positive integer $n$ as a special case. With the aid of these Steinitz numbers, the algebraic extensions of $GF(q)$ are represented by symbols of the form $GF(q^N)$. When $N$ is an ordinary integer $n$, this notation agrees with the usual notation $GF(q^n)$ for a dimension $n$ extension of $GF(q)$. The authors then show that many of the finite field results concerning $GF(q^n)$ are also true for $GF(q^N)$. One chapter is devoted to giving explicit algorithms for computing in several of the infinite fields $GF(q^N)$ using the notion of an explicit basis for $GF(q^N)$ over $GF(q)$. Another chapter considers polynomials and polynomial-like functions on $GF(q^N)$ and contains a description of several classes of permutation polynomials, including the $q$-polynomials and the Dickson polynomials. Also included is a brief chapter describing two of many potential applications. Aimed at the level of a beginning graduate student or advanced undergraduate, this book could serve well as a supplementary text for a course in finite field theory.

## Introduction to Modern Algebra and Its Applications

Author | : Nadiya Gubareni |

Publsiher | : CRC Press |

Total Pages | : 394 |

Release | : 2021-06-23 |

ISBN | : 1000209539 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Modern Algebra and Its Applications Book Excerpt:**

The book provides an introduction to modern abstract algebra and its applications. It covers all major topics of classical theory of numbers, groups, rings, fields and finite dimensional algebras. The book also provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. In particular, it considers algorithm RSA, secret sharing algorithms, Diffie-Hellman Scheme and ElGamal cryptosystem based on discrete logarithm problem. It also presents Buchberger’s algorithm which is one of the important algorithms for constructing Gröbner basis. Key Features: Covers all major topics of classical theory of modern abstract algebra such as groups, rings and fields and their applications. In addition it provides the introduction to the number theory, theory of finite fields, finite dimensional algebras and their applications. Provides interesting and important modern applications in such subjects as Cryptography, Coding Theory, Computer Science and Physics. Presents numerous examples illustrating the theory and applications. It is also filled with a number of exercises of various difficulty. Describes in detail the construction of the Cayley-Dickson construction for finite dimensional algebras, in particular, algebras of quaternions and octonions and gives their applications in the number theory and computer graphics.

## Computational and Algorithmic Problems in Finite Fields

Author | : Igor Shparlinski |

Publsiher | : Springer Science & Business Media |

Total Pages | : 240 |

Release | : 2012-12-06 |

ISBN | : 940111806X |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Computational and Algorithmic Problems in Finite Fields Book Excerpt:**

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.

## Arithmetic and Algebraic Circuits

Author | : Antonio Lloris Ruiz,Encarnación Castillo Morales,Luis Parrilla Roure,Antonio García Ríos,María José Lloris Meseguer |

Publsiher | : Springer Nature |

Total Pages | : 682 |

Release | : 2021-03-27 |

ISBN | : 3030672662 |

Category | : Technology & Engineering |

Language | : EN, FR, DE, ES & NL |

**Arithmetic and Algebraic Circuits Book Excerpt:**

This book presents a complete and accurate study of arithmetic and algebraic circuits. The first part offers a review of all important basic concepts: it describes simple circuits for the implementation of some basic arithmetic operations; it introduces theoretical basis for residue number systems; and describes some fundamental circuits for implementing the main modular operations that will be used in the text. Moreover, the book discusses floating-point representation of real numbers and the IEEE 754 standard. The second and core part of the book offers a deep study of arithmetic circuits and specific algorithms for their implementation. It covers the CORDIC algorithm, and optimized arithmetic circuits recently developed by the authors for adders and subtractors, as well as multipliers, dividers and special functions. It describes the implementation of basic algebraic circuits, such as LFSRs and cellular automata. Finally, it offers a complete study of Galois fields, showing some exemplary applications and discussing the advantages in comparison to other methods. This dense, self-contained text provides students, researchers and engineers, with extensive knowledge on and a deep understanding of arithmetic and algebraic circuits and their implementation.

## Arithmetic of Finite Fields

Author | : Claude Carlet,Berk Sunar |

Publsiher | : Springer |

Total Pages | : 360 |

Release | : 2007-09-21 |

ISBN | : 3540730745 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Arithmetic of Finite Fields Book Excerpt:**

This book constitutes the refereed proceedings of the First International Workshop on the Arithmetic of Finite Fields, WAIFI 2007, held in Madrid, Spain in June 2007. It covers structures in finite fields, efficient implementation and architectures, efficient finite field arithmetic, classification and construction of mappings over finite fields, curve algebra, cryptography, codes, and discrete structures.

## Arithmetic of Finite Fields

Author | : Charles Small |

Publsiher | : CRC Press |

Total Pages | : 240 |

Release | : 1991-04-24 |

ISBN | : 9780824785260 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Arithmetic of Finite Fields Book Excerpt:**

Text for a one-semester course at the advanced undergraduate/beginning graduate level, or reference for algebraists and mathematicians interested in algebra, algebraic geometry, and number theory, examines counting or estimating numbers of solutions of equations in finite fields concentrating on top

## Sequences and Their Applications SETA 2014

Author | : Kai-Uwe Schmidt,Arne Winterhof |

Publsiher | : Springer |

Total Pages | : 315 |

Release | : 2014-11-17 |

ISBN | : 3319123254 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Sequences and Their Applications SETA 2014 Book Excerpt:**

This book constitutes the refereed proceedings of the 8th International Conference on Sequences and Their Applications, SETA 2014, held in Melbourne, VIC, Australia, in November 2014. The 24 full papers presented together with 2 invited papers were carefully reviewed and selected from 36 submissions. The papers have been organized in topical sections on Boolean functions, perfect sequences, correlation of arrays, relative difference sets, aperiodic correlation, pseudorandom sequences and stream ciphers, crosscorrelation of sequences, prime numbers in sequences, OFDM and CDMA, and frequency-hopping sequences.

## Arithmetic of Finite Fields

Author | : Lilya Budaghyan,Francisco Rodríguez-Henríquez |

Publsiher | : Springer |

Total Pages | : 309 |

Release | : 2018-12-29 |

ISBN | : 3030051536 |

Category | : Computers |

Language | : EN, FR, DE, ES & NL |

**Arithmetic of Finite Fields Book Excerpt:**

This book constitutes the thoroughly refereed post-workshop proceedings of the 7th International Workshop on the Arithmetic of Finite Field, WAIFI 2018, held in Bergen, Norway, in June 2018. The 14 revised full papers and six invited talks presented were carefully reviewed and selected from 26 submissions. The papers are organized in topical sections on invited talks; elliptic curves; hardware implementations; arithmetic and applications of finite fields and cryptography.