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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, inZentralblatt 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, inSIGACT News