Handbook of Finite Fields

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.

IntroductionHistory of Finite Fields, Roderick GowFinite fields in the 18th and 19th centuries Introduction to Finite FieldsBasic properties of finite fields, Gary L. Mullen and Daniel PanarioTables, David Thomson Theoretical PropertiesIrreducible Polynomials Counting irreducible polynomials, Joseph L. YucasConstruction 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 JungnickelNormal bases, Shuhong Gao and Qunying LiaoComplexity of normal bases, Shuhong Gao and David ThomsonCompletely normal bases, Dirk Hachenberger Exponential and Character SumsGauss, Jacobi, and Kloosterman sums, Ronald J. EvansMore general exponential and character sums, Antonio Rojas-León Some applications of character sums, Alina Ostafe and Arne WinterhofSum-product theorems and applications, Moubariz Z. Garaev Equations over Finite FieldsGeneral forms, Daqing WanQuadratic forms, Robert FitzgeraldDiagonal equations, Francis Castro and Ivelisse Rubio Permutation PolynomialsOne variable, Gary L. Mullen and Qiang WangSeveral variables, Rudolf Lidl and Gary L. Mullen Value sets of polynomials, Gary L. Mullen and Michael E. ZieveExceptional polynomials, Michael E. Zieve Special Functions over Finite FieldsBoolean functions, Claude CarletPN and APN functions, Pascale CharpinBent and related functions, Alexander Kholosha and Alexander Pottk-polynomials and related algebraic objects, Robert Coulter Planar functions and commutative semifields, Robert CoulterDickson polynomials, Qiang Wang and Joseph L. YucasSchur’s conjecture and exceptional covers, Michael D. Fried Sequences over Finite FieldsFinite field transforms, Gary McGuireLFSR sequences and maximal period sequences, Harald Niederreiter Correlation and autocorrelation of sequences, Tor Helleseth Linear complexity of sequences and multisequences, Wilfried Meidl and Arne WinterhofAlgebraic dynamical systems over finite fields, Igor Shparlinski AlgorithmsComputational techniques, Christophe DocheUnivariate polynomial counting and algorithms, Daniel PanarioAlgorithms for irreducibility testing and for constructing irreducible polynomials, Mark GiesbrechtFactorization of univariate polynomials, Joachim von zur GathenFactorization of multivariate polynomials, Erich Kaltofen and Grégoire LecerfDiscrete logarithms over finite fields, Andrew OdlyzkoStandard models for finite fields, Bart de Smit and Hendrik Lenstra Curves over Finite FieldsIntroduction to function fields and curves, Arnaldo Garcia and Henning StichtenothElliptic curves, Joseph SilvermanAddition formulas for elliptic curves, Daniel J. Bernstein and Tanja LangeHyperelliptic curves, Michael John Jacobson, Jr. and Renate ScheidlerRational 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 TopicsRelations between integers and polynomials over finite fields, Gove EffingerMatrices over finite fields, Dieter JungnickelClassical groups over finite fields, Zhe-Xian WanComputational linear algebra over finite fields, Jean-Guillaume Dumas and Clément PernetCarlitz and Drinfeld modules, David Goss ApplicationsCombinatorialLatin squares, Gary L. MullenLacunary polynomials over finite fields, Simeon Ball and Aart BlokhuisAffine and projective planes, Gary Ebert and Leo StormeProjective spaces, James W.P. Hirschfeld and Joseph A. ThasBlock designs, Charles J. Colbourn and Jeffrey H. DinitzDifference sets, Alexander PottOther combinatorial structures, Jeffrey H. Dinitz and Charles J. Colbourn(t, m, s)-nets and (t, s)-sequences, Harald NiederreiterApplications and weights of multiples of primitive and other polynomials, Brett StevensRamanujan and expander graphs, M. Ram Murty and Sebastian M. Cioaba Algebraic Coding TheoryBasic coding properties and bounds, Ian Blake and W. Cary HuffmanAlgebraic-geometry codes, Harald Niederreiter LDPC and Gallager codes over finite fields, Ian Blake and W. Cary Huffman Turbo codes over finite fields, Oscar TakeshitaRaptor codes, Ian Blake and W. Cary HuffmanPolar codes, Simon Litsyn CryptographyIntroduction to cryptography, Alfred MenezesStream and block ciphers, Guang Gong and Kishan Chand Gupta Multivariate cryptographic systems, Jintai DingElliptic curve cryptographic systems, Andreas EngeHyperelliptic curve cryptographic systems, Nicolas ThériaultCryptosystems arising from Abelian varieties, Kumar MurtyBinary extension field arithmetic for hardware implementations, M. Anwarul Hasan and Haining Fan Miscellaneous ApplicationsFinite fields in biology, Franziska Hinkelmann and Reinhard LaubenbacherFinite fields in quantum information theory, Martin Roetteler and Arne WinterhofFinite 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.