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- Gives a complete account of state-of-the-art theoretical and applied topics in finite fields
- Describes numerous applications from the fields of computer science and engineering
- Presents the history of finite fields and a brief summary of basic results
- Discusses theoretical properties of finite fields
- Covers applications in cryptography, coding theory, and combinatorics
- Includes many remarks to further explain the various results
- Contains more than 3,000 references, including citations to proofs of important results
- Offers extensive tables of polynomials useful for computational issues, with even larger tables available on the book’s CRC Press web page

Watch Gary L. Mullen discuss the book.

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*

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,

Construction of irreducible,

Conditions for reducible polynomials,

Weights of irreducible polynomials,

Prescribed coefficients,

Multivariate polynomials,

**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,

**Equations over Finite Fields**General forms,

Quadratic forms,

Diagonal equations,

**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,

**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,

Other combinatorial structures,

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**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,

Polar codes,

**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,

Hyperelliptic curve cryptographic systems,

Cryptosystems arising from Abelian varieties,

Binary extension field arithmetic for hardware implementations,

**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

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

Daniel Panario is a professor of mathematics at Carleton University.

Resource | OS Platform | Updated | Description | Instructions |
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Cross Platform | August 20, 2013 | Web Links | click on http://people.math.carleton.ca/~daniel/hff/ |