1070 Pages 12 B/W Illustrations
    by Chapman & Hall

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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 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.

    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

    Biography

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

    Daniel Panario is a professor of mathematics at Carleton University.

    "... 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