Handbook of Finite Fields: 1st Edition (Hardback) book cover

Handbook of Finite Fields

1st Edition

By Gary L. Mullen, Daniel Panario

Chapman and Hall/CRC

1,068 pages | 12 B/W Illus.

Purchasing Options:$ = USD
Hardback: 9781439873786
pub: 2013-06-17
SAVE ~$33.00
eBook (VitalSource) : 9780429105197
pub: 2013-06-17
from $28.98

FREE Standard Shipping!


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.


"… 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

Table of Contents


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


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


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



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


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



About the Authors

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

Daniel Panario is a professor of mathematics at Carleton University.

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Security / Cryptography
MATHEMATICS / Number Theory
MATHEMATICS / Combinatorics