Finite Fields and Their Applications

289 Pages·2013·1.881 MB·English

by  Pascale Charpin, Alexander Pott, Arne Winterhof

PascaleCharpin,AlexanderPott,ArneWinterhof(Eds.)
FiniteFieldsandTheirApplications
Radon Series on Computational
and Applied Mathematics
Managing Editor
Heinz W. Engl,Linz/Vienna, Austria
Editorial Board
Hansjörg Albrecher, Lausanne, Switzerland
Ronald H. W. Hoppe, Houston, USA
Karl Kunisch, Linz/Graz, Austria
Ulrich Langer, Linz, Austria
Harald Niederreiter, Linz, Austria
Christian Schmeiser, Vienna, Austria
Volume 11
Finite Fields and
Their Applications
Character Sumsand Polynomials
Edited by
Pascale Charpin
Alexander Pott
Arne Winterhof
2010MathematicsSubjectClassification
11BXX,11CXX,11KXX,11LXX,11TXX,12CXX,12YXX,37PXX,51EXX,94AXX
Editors
PascaleCharpin
ResearchDirectorSECRET
Inria
Rocquencourt,France
[email protected]
AlexanderPott
ProfessorforDiscreteMathematics
InstituteforAlgebraandGeometry(IAG)
FacultyofMathematics
Magdeburg,Germany
[email protected]
ArneWinterhof
ProjectLeaderAppliedDiscreteMathematicsandCryptography
JohannRadonInstituteforComputationalandAppliedMathematics(RICAM)
AustrianAcademyofSciences
Linz,Austria
[email protected]
ISBN978-3-11-028240-5
e-ISBN978-3-11-028360-0
Set-ISBN978-3-11-028361-7
ISSN1865-3707
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ACIPcatalogrecordforthisbookhasbeenappliedforattheLibraryofCongress.
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;detailed
bibliographicdataareavailableintheInternetathttp://dnb.dnb.de.
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Typesetting:le-texpublishingservicesGmbH,Leipzig
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Preface
ThisbookisbasedontheinvitedtalksoftheRICAM-WorkshoponFiniteFieldsand
TheirApplications:CharacterSumsandPolynomialsheldattheFederalInstitutefor
AdultEducation(BIfEB)inStrobl,Austria,fromSeptember2–7,2012.
The topic of the book is the theory of finite fields. Finite fields play important
rolesinmanyapplicationareassuchascodingtheory,cryptography,MonteCarloand
quasi-MonteCarlomethods,pseudorandomnumbergeneration, quantumcomputing,
andwirelesscommunication.Inthisbookwewillfocusonsequences,charactersums,
andpolynomialsoverfinitefieldsinviewoftheabovementionedapplicationareas.
Thegoalofthisbookistogiveanoverviewofseveralrecentresearchdirectionsas
wellastostimulateresearchinsequencesandpolynomialsundertheunifiedframe-
workofcharactertheory.
Chapters1and2dealwithsequencesmainlyconstructedviacharactersandana-
lyzedusingboundsoncharactersums.InChapter1measuresofpseudorandomness
inviewofapplicationstowirelesscommunicationarementioned,whereasChapter2
containsasurveyonmeasuresofpseudorandomnessfromamoretheoreticalpoint
ofviewwherecryptographymaybethemostimportantapplicationarea.
Chapters3,5,and6dealwithpolynomialsoverfinitefields. Chapter3givesan
overviewaboutresults onpolynomialswith someproperties described. Chapters5
and6discusspolynomialswhicharesuitableforcryptographicapplications. Chap-
ters 4and9considerproblemsrelated tocodingtheory studiedviafinitegeometry
andadditivecombinatorics,respectively.Chapter7dealswithquasirandompointsin
viewofapplicationstonumericalintegrationusingquasi-MonteCarlomethodsand
simulation. Chapter8studiesaspectsofiterationsofrationalfunctionsfromwhich
pseudorandom numbers for Monte Carlo methods can be derived. For Monte Car-
loandquasi-Monte Carlomethods uniformlydistributed sequences areneeded. In
manycasesameasurefortheuniformdistribution,thediscrepancy,canbeestimat-
edintermsofadditivecharactersums.
Allthesechapterswerereviewedandwewishtothanktheanonymousreferees
fortheirprecioushelp.
Wealsothanktheotherparticipantsoftheworkshoplistedbelowwhocontribut-
ed withexcellent talksandmadethe workshopagreatsuccess: Jürgen Bierbrauer,
HeriveltoBorges,NinaBrandstätter,ClaudeCarlet,FrancisCastro,AycaCesmelioglu,
StephenD.Cohen,DomingoGomez-Perez,CemGüneri,JingHe,PeterHellekalek,Tor
Helleseth,RoswithaHofer,LeylaIsik,JonathanJedwab,GiorgosKapetanakis,Daniel
Katz,AlexanderKholosha,PeterKritzer,MichelLavrauw,VsevolodLev,PetrLisonek,
Florian Luca, Christian Mauduit, Wilfried Meidl, Sihem Mesnager, Sylvia Morris,
GaryMullen, FerruhÖzbudak, BuketÖzkaya, DanielPanario,Gottlieb Pirsic,Clau-
dioQureshi,AndrasSarközy,Kai-UweSchmidt,JohnSheekey,HenningStichtenoth,
vi Preface
ValentinSuder,DavidThomson,AlevTopuzoglu,SimoneUgolini,Christiaanvande
Woestijne,JoachimvonzurGathen,QiWang,andQiangWang.
Moredetailsonthisworkshopcanbefoundonthewebpagehttp://www.ricam.
oeaw.ac.at/events/workshops/ffta2012/
We also thank the Radon Institute for Computational and Applied Mathemat-
ics (RICAM) of the Austrian Academy of Sciences for financial support, BIfEB for
theirhospitality,thepublisherforthepleasantcooperation,ourco-organizersGary
Mullen,HaraldNiederreiter,andDanielPanario,andlastbutnotleastAnnetteWeihs
andWolfgangForsthuberfortheirgreatsupportduringthepreparationofthework-
shop.
Paris,Magdeburg,andLinz,December2012 PascaleCharpin
AlexanderPott
ArneWinterhof
Contents
Preface v
GuangGong
CharacterSumsandPolyphaseSequenceFamilieswithLowCorrelation,
DiscreteFourierTransform(DFT),andAmbiguity 1
1 Introduction 1
2 BasicDefinitionsandConcepts 2
2.1 Notations 2
2.2 PolynomialFunctionsoverF 3
q
2.3 CharactersofFiniteFields 4
2.4 TheWeilBoundsonCharacterSums 4
3 Correlation,DFT,andAmbiguityFunctions 5
3.1 OperatorsonSequences 5
3.2 CorrelationFunctions 6
3.3 AmbiguityFunctions 8
3.4 ConvolutionandCorrelation 10
3.5 OptimalCorrelation,DFT,andAmbiguity 10
4 PolyphaseSequencesforThreeMetrics 11
4.1 SequencesfromtheAdditiveGroupofZ and
N
theAdditiveGroupofZ 11
p
4.1.1 Frank–Zadoff–Chu(FZC)Sequences 11
4.1.2 AnotherClassforZ 13
N
4.1.3 SequencesfromF AdditiveCharacters 13
p
4.2 SequencesfromF MultiplicativeCharacters 13
p
4.3 SequencesfromF AdditiveCharacters 15
q
4.4 SequencesfromF MultiplicativeCharacters 17
q
4.5 SequencesDefinedbyIndexingFieldElementsAlternatively 20
5 SequenceswithLowDegreePolynomials 22
5.1 MethodsforGeneratingSignalSetsfromaSingleSequence 22
5.2 SequenceswithLowOddDegreePolynomials 23
5.2.1 F AdditiveSequenceswithLowOddDegreePolynomials 23
q
5.2.2 F MultiplicativeSequenceswithLowOddDegreePolynomials 25
q
5.3 SequencesfromPowerResidueandSidel’nikovSequences 26
5.3.1 InterleavedStructureofSidel’nikovSequences 26
5.3.2 SequencesfromLinearand/orQuadratic/InversePolynomials 27
5.4 SequencesfromHybridCharacters 29
5.4.1 SequencesUsingWeilRepresentationandTheirGeneralizations 29
5.4.2 GeneralizationtoF HybridSequences 30
q
5.5 ANewConstruction 32
viii Contents
6 Two-LevelAutocorrelationSequencesandDoubleExponential
Sums 33
6.1 PrimeTwo-LevelAutocorrelationSequences 33
6.2 HadamardTransform,Second-OrderDecimation-HadamardTransform,
andHadamardEquivalence 34
6.3 ConjecturesonTernary2-LevelAutocorrelationSequences 35
7 SomeOpenProblems 37
7.1 CurrentStatusoftheConjecturesonTernary2-Level
Autocorrelation 37
7.2 PossibilityofMultiplicativeSequenceswithLowAutocorrelation 38
7.3 ProblemsinFourAlternativeClassesofSequencesandtheGeneral
HybridConstruction 38
8 Conclusions 38
KatalinGyarmati
MeasuresofPseudorandomness 43
1 Introduction 43
2 DefinitionofthePseudorandomMeasures 44
3 TypicalValuesofPseudorandomMeasures 46
4 MinimumValuesofPseudorandomMeasures 47
5 ConnectionbetweenPseudorandomMeasures 49
6 Constructions 50
7 FamilyMeasures 52
8 LinearComplexity 54
9 MultidimensionalTheory 56
10 Extensions 57
SophieHuczynska
ExistenceResultsforFiniteFieldPolynomialswithSpecifiedProperties 65
1 Introduction 65
2 ASurveyofKnownResults 66
2.1 NormalBases 67
2.2 PrimitiveNormalBases 68
2.3 PrescribedCoefficients 70
2.4 PrimitivePolynomials:PrescribedCoefficients 70
2.5 PrimitiveNormalPolynomials:PrescribedCoefficients 73
3 ASurveyofMethodologyandTechniques 75
3.1 BasicApproach 76
3.2 Ap-adicApproachtoCoefficientConstraints 78
3.3 TheSievingTechnique 81
4 Conclusion 83
Contents ix
DieterJungnickel
IncidenceStructures,Codes,andGaloisGeometries 89
1 Introduction 89
2 GaloisClosedCodes 91
3 ExtensionCodesofSimplexandFirst-OrderReed–MullerCodes 93
4 SimpleIncidenceStructuresandTheirCodes 97
5 EmbeddingTheorems 99
6 DesignswithClassicalParameters 105
7 Two-WeightCodes 108
8 SteinerSystems 109
9 Configurations 111
10 ConclusionandOpenProblems 113
GoharM.Kyureghyan
SpecialMappingsofFiniteFields 117
1 Introduction 117
2 DifferentNotionsforOptimalNon-linearity 120
2.1 AlmostPerfectNonlinear(APN)Mappings 121
2.2 BentandAlmostBent(AB)Mappings 124
3 FunctionswithaLinearStructure 125
4 CrookedMappings 128
5 PlanarMappings 130
6 SwitchingConstruction 133
7 ProductsofLinearizedPolynomials 137
FernandoHernandoandGaryMcGuire
OnTheClassificationofPerfectNonlinear(PN)andAlmostPerfectNonlinear(APN)
MonomialFunctions 145
1 Introduction 145
2 BackgroundandMotivation 146
2.1 PNandPlanarFunctions 146
2.2 APNFunctions 148
3 OutlineofAPNFunctionsClassificationProof 149
3.1 SingularitiesinAPNcase 151
3.2 AWarm-UpCase 153
4 PNFunctionsClassificationProof:AnalysisofSingularities 154
4.1 SingularPointsinCase(b.1) 156
4.2 SingularPointsatInfinity 157
4.3 TheMultiplicities 159
4.4 FurtherAnalysis 159
4.5 Type(i) 161