1st Edition

Handbook of Finite Fields




ISBN 9781439873786
Published June 17, 2013 by Chapman and Hall/CRC
1068 Pages 12 B/W Illustrations

USD $175.00

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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 each chapter is self contained and peer reviewed.

The first part of the book traces the history of finite fields through the eighteenth and nineteenth centuries. The second part presents theoretical properties of finite fields, covering polynomials, special functions, sequences, algorithms, curves, and related computational aspects. The final part describes various mathematical and practical applications of finite fields in combinatorics, algebraic coding theory, cryptographic systems, biology, quantum information theory, engineering, and other areas. The book provides a comprehensive index and easy access to over 3,000 references, enabling you to quickly locate up-to-date facts and results regarding finite fields.

Table of Contents

Introduction
History of Finite Fields, Roderick Gow
Finite fields in the 18th and 19th centuries

Introduction to Finite Fields
Basic properties of finite fields, Gary L. Mullen and Daniel Panario
Tables, David Thomson

Theoretical Properties
Irreducible Polynomials

Counting irreducible polynomials, Joseph L. Yucas
Construction of irreducible, Melsik Kyuregyan
Conditions for reducible polynomials, Daniel Panario
Weights of irreducible polynomials, Omran Ahmadi
Prescribed coefficients, Stephen D. Cohen
Multivariate polynomials, Xiang-dong Hou

Primitive Polynomials
Introduction to primitive polynomials, Gary L. Mullen and Daniel Panario
Prescribed coefficients, Stephen D. Cohen
Weights of primitive polynomials, Stephen D. Cohen
Elements of high order, José Felipe Voloch

Bases
Duality theory of bases, Dieter Jungnickel
Normal bases, Shuhong Gao and Qunying Liao
Complexity of normal bases, Shuhong Gao and David Thomson
Completely normal bases, Dirk Hachenberger

Exponential and Character Sums
Gauss, Jacobi, and Kloosterman sums, Ronald J. Evans
More general exponential and character sums, Antonio Rojas-León
Some applications of character sums, Alina Ostafe and Arne Winterhof
Sum-product theorems and applications, Moubariz Z. Garaev

Equations over Finite Fields
General forms, Daqing Wan
Quadratic forms, Robert Fitzgerald
Diagonal equations, Francis Castro and Ivelisse Rubio

Permutation Polynomials
One variable, Gary L. Mullen and Qiang Wang
Several variables, Rudolf Lidl and Gary L. Mullen
Value sets of polynomials, Gary L. Mullen and Michael E. Zieve
Exceptional polynomials, Michael E. Zieve

Special Functions over Finite Fields
Boolean functions, Claude Carlet
PN and APN functions, Pascale Charpin
Bent and related functions, Alexander Kholosha and Alexander Pott
k-polynomials and related algebraic objects, Robert Coulter
Planar functions and commutative semifields, Robert Coulter
Dickson polynomials, Qiang Wang and Joseph L. Yucas
Schur’s conjecture and exceptional covers, Michael D. Fried

Sequences over Finite Fields
Finite field transforms, Gary McGuire
LFSR sequences and maximal period sequences, Harald Niederreiter
Correlation and autocorrelation of sequences, Tor Helleseth
Linear complexity of sequences and multisequences, Wilfried Meidl and Arne Winterhof
Algebraic dynamical systems over finite fields, Igor Shparlinski

Algorithms
Computational techniques, Christophe Doche
Univariate polynomial counting and algorithms, Daniel Panario
Algorithms for irreducibility testing and for constructing irreducible polynomials, Mark Giesbrecht
Factorization of univariate polynomials, Joachim von zur Gathen
Factorization of multivariate polynomials, Erich Kaltofen and Grégoire Lecerf
Discrete logarithms over finite fields, Andrew Odlyzko
Standard models for finite fields, Bart de Smit and Hendrik Lenstra

Curves over Finite Fields
Introduction to function fields and curves, Arnaldo Garcia and Henning Stichtenoth
Elliptic curves, Joseph Silverman
Addition formulas for elliptic curves, Daniel J. Bernstein and Tanja Lange
Hyperelliptic curves, Michael John Jacobson, Jr. and Renate Scheidler
Rational points on curves, Arnaldo Garcia and Henning Stichtenoth
Towers, Arnaldo Garcia and Henning Stichtenoth
Zeta functions and L-functions, Lei Fu
p-adic estimates of zeta functions and L-functions, Régis Blache
Computing the number of rational points and zeta functions, Daqing Wan

Miscellaneous Theoretical Topics
Relations between integers and polynomials over finite fields, Gove Effinger
Matrices over finite fields, Dieter Jungnickel
Classical groups over finite fields, Zhe-Xian Wan
Computational linear algebra over finite fields, Jean-Guillaume Dumas and Clément Pernet
Carlitz and Drinfeld modules, David Goss

Applications
Combinatorial
Latin squares, Gary L. Mullen
Lacunary polynomials over finite fields, Simeon Ball and Aart Blokhuis
Affine and projective planes, Gary Ebert and Leo Storme
Projective spaces, James W.P. Hirschfeld and Joseph A. Thas
Block designs, Charles J. Colbourn and Jeffrey H. Dinitz
Difference sets, Alexander Pott
Other combinatorial structures, Jeffrey H. Dinitz and Charles J. Colbourn
(t, m, s)-nets and (t, s)-sequences, Harald Niederreiter
Applications and weights of multiples of primitive and other polynomials, Brett Stevens
Ramanujan and expander graphs, M. Ram Murty and Sebastian M. Cioaba

Algebraic Coding Theory
Basic coding properties and bounds, Ian Blake and W. Cary Huffman
Algebraic-geometry codes, Harald Niederreiter
LDPC and Gallager codes over finite fields, Ian Blake and W. Cary Huffman
Turbo codes over finite fields, Oscar Takeshita
Raptor codes, Ian Blake and W. Cary Huffman
Polar codes, Simon Litsyn

Cryptography
Introduction to cryptography, Alfred Menezes
Stream and block ciphers, Guang Gong and Kishan Chand Gupta
Multivariate cryptographic systems, Jintai Ding
Elliptic curve cryptographic systems, Andreas Enge
Hyperelliptic curve cryptographic systems, Nicolas Thériault
Cryptosystems arising from Abelian varieties, Kumar Murty
Binary extension field arithmetic for hardware implementations, M. Anwarul Hasan and Haining Fan

Miscellaneous Applications
Finite fields in biology, Franziska Hinkelmann and Reinhard Laubenbacher
Finite fields in quantum information theory, Martin Roetteler and Arne Winterhof
Finite fields in engineering, Jonathan Jedwab and Kai-Uwe Schmidt

Bibliography

Index

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Author(s)

Biography

Gary L. Mullen is a professor of mathematics at The Pennsylvania State University.

Daniel Panario is a professor of mathematics at Carleton University.

Reviews

"... a brilliant, monumental work on the state of the art in theory and applications of finite fields. It's a must for everyone doing research in finite fields and their related areas. ... It presents such a huge amount of information readily available and masterly presented. Editors, contributors, and the publisher are equally congratulated for providing such a beautiful result not only for the finite field community, but also for those from the applied areas, especially cryptographers and coding theorists."
—Olaf Ninnemann, Berlin, in Zentralblatt MATH 1319

"The handbook will be very useful for senior-level students, teachers, and researchers in mathematics and computer science. It will also be useful for scientists, engineers, and practitioners. The handbook is well organized. ... It is likely to become a standard reference book for the theory and applications of finite fields. ... It will be a very useful addition to the libraries of academic and research institutions."
—S.V.Nagaraj, Chennai, India, in SIGACT News