Applied Number Theory

452 Pages·2015·2.578 MB·English

by  Harald Niederreiter, Arne Winterhof

Harald Niederreiter
Arne Winterhof
Applied
Number Theory
Applied Number Theory

Harald Niederreiter • Arne Winterhof
Applied Number Theory
123
HaraldNiederreiter ArneWinterhof
AustrianAcademyofSciences AustrianAcademyofSciences
JohannRadonInst.forComputational JohannRadonInst.forComputational
andAppliedMathematics(RICAM) andAppliedMathematics(RICAM)
Linz,Austria Linz,Austria
ISBN978-3-319-22320-9 ISBN978-3-319-22321-6 (eBook)
DOI10.1007/978-3-319-22321-6
LibraryofCongressControlNumber:2015949818
MathematicsSubjectClassification(2010):11-XX,65-XX,94-XX
SpringerChamHeidelbergNewYorkDordrechtLondon
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Preface
Etlequatrièmemystère,nondesmoindres,estceluidela
structuremathématiquedumonde:pourquoietquand
apparaît-elle,commentpeut-onlamodéliser,etcommentle
cerveauparvient-ilàl’élaborer,àpartirduchaosdanslequel
nousvivons?1
E.Abécassis,Lepalimpsested’Archimède
The Nobel laureate Eugene Wigner coined the phrase of the “unreasonableeffec-
tivenessofmathematics”inexplainingthephysicalworld,whilethenovelistEliette
Abécassis expressed this phenomenonin a more literary fashion, calling it one of
thefourgreatmysteriesoftheworld.Indeed,whetherviewedfromthescientist’sor
theartist’svantagepoint,thereisnodisputeabouttheapplicabilityofbroadpartsof
mathematics.Ontheotherhand,numbertheory—thepurestofpuremathematics—
haslongresistedthetemptationtobecomeapplicable,exceptfortrivialapplications
such as Pythagorean triples for the construction of right angles and very simple
cryptosystems. In 1940, the prominent number theorist G.H. Hardy confidently
asserted in his book A Mathematician’sApology that he had never done anything
useful and that no discovery of his was likely to make the least difference to the
amenityoftheworld.
Butthingschangeddramaticallyinthesecondhalfofthetwentiethcenturywhen,
drivenbytheimpetusofscienceandtechnology,entirelynewareasofmathematics
relyingheavilyonnumbertheorywerecreated.Today,numbertheoryisimplicitly
presentineverydaylife:insupermarketbarcodereaders,inourcars’GPSsystems,
intheerror-correctingcodesatworkinoursmartphones,andinonlinebanking,to
mentionbutafewexamples.
From our perspective, there are four major areas of application where number
theoryplaysafundamentalrole,namelycryptography,codingtheory,quasi-Monte
Carlo methods, and pseudorandom number generation. Excellent textbooks are
availableforeachoftheseareas.Thisbookpresentsthefirstunifiedaccountofall
theseapplications.Thisallowsustodelineatethemanifoldlinksandinterrelations
betweenthese areas.Chapters2–5coverthe fourmain areasofapplication,while
1Authors’ translation: And the fourth mystery, and not the least, is that of the mathematical
structureoftheworld:whyandwhendoesitarise,howcanonemodelit,andhowdoesthebrain
managetoworkitout,startingfromthechaosinwhichwelive?
v
vi Preface
the last chapter reviews variousadditionalapplicationsof number theory,ranging
from check-digit systems to quantum computation and the organizationof raster-
graphicsmemory.We hope that this panoramaof applications will inspire further
research in applied number theory. In order to enhance the accessibility of the
bookfor undergraduates,we haveincludeda briefintroductorycourseon number
theory in Chap. 1. The last section of each of Chaps. 2–5 offers a glimpse of
advancedresultsthatarestatedwithoutproofandrequireasomewhathigherlevel
ofmathematicalmaturity.
We have sought to minimize the prerequisites for the book. A background in
number theory is not necessary, although it is certainly helpful. Elementary facts
from calculus are used as a matter of course. Linear algebra appears only in a
limited context, and the important special case of linear algebra over finite fields
is developedfrom scratch. The chapters on coding theory and quasi-Monte Carlo
methods are quite extensive, so that they could be used to teach separate courses
on each of these topics. But we believe that a single course stressing the unity of
appliednumbertheoryisinbetterconformitywiththephilosophyofthebook.
Writing a book is not possible without the help of many. We are particularly
indebted to Professor Friedrich Pillichshammer of the University of Linz for his
assistance with the figures, to Professors Sheldon Axler and Ken Ribet for their
comments on a preliminary version of the book, to our institutions for providing
excellent research facilities, and to Edward Lear for developing limericks into a
veritableartform.The limericksat the beginningof eachchapterare notcredited
since they were written by the first author (our apologies if you find them silly).
We also wish to extend our special gratitude to Ruth Allewelt and Martin Peters
atSpringer-Verlagfortheirunfailingsupportofourprojectandtoourfamiliesfor
theirpatienceandindulgence.
Linz,Austria HaraldNiederreiter
March2015 ArneWinterhof
Contents
1 AReviewofNumberTheoryandAlgebra................................ 1
1.1 IntegerArithmetic....................................................... 1
1.2 Congruences............................................................. 5
1.3 GroupsandCharacters.................................................. 12
1.3.1 AbelianGroups................................................. 12
1.3.2 Characters....................................................... 19
1.4 FiniteFields............................................................. 23
1.4.1 FundamentalProperties ........................................ 23
1.4.2 Polynomials..................................................... 27
1.4.3 ConstructionsofFiniteFields.................................. 33
1.4.4 TraceMapandCharacters...................................... 40
Exercises....................................................................... 43
2 Cryptography ................................................................ 47
2.1 ClassicalCryptosystems................................................ 47
2.1.1 BasicPrinciples................................................. 47
2.1.2 SubstitutionCiphers............................................ 50
2.2 SymmetricBlockCiphers .............................................. 52
2.2.1 DataEncryptionStandard(DES) .............................. 52
2.2.2 AdvancedEncryptionStandard(AES) ........................ 54
2.3 Public-KeyCryptosystems ............................................. 56
2.3.1 BackgroundandBasics......................................... 56
2.3.2 TheRSACryptosystem ........................................ 59
2.3.3 FactorizationMethods.......................................... 62
2.4 CryptosystemsBasedonDiscreteLogarithms ........................ 67
2.4.1 TheCryptosystems ............................................. 67
2.4.2 ComputingDiscreteLogarithms............................... 69
2.5 DigitalSignatures....................................................... 73
2.5.1 DigitalSignaturesfromPublic-KeyCryptosystems .......... 73
2.5.2 DSSandRelatedSchemes ..................................... 75
2.6 ThresholdSchemes ..................................................... 77
vii
viii Contents
2.7 PrimalityTests .......................................................... 80
2.7.1 FermatTestandCarmichaelNumbers......................... 80
2.7.2 Solovay-StrassenTest .......................................... 83
2.7.3 PrimalityTestsforSpecialNumbers........................... 86
2.8 AGlimpseofAdvancedTopics........................................ 89
Exercises....................................................................... 94
3 CodingTheory ............................................................... 99
3.1 IntroductiontoError-CorrectingCodes................................ 99
3.1.1 BasicDefinitions................................................ 99
3.1.2 ErrorCorrection ................................................ 102
3.2 LinearCodes ............................................................ 106
3.2.1 VectorSpacesOverFiniteFields .............................. 106
3.2.2 FundamentalPropertiesofLinearCodes...................... 109
3.2.3 MatricesOverFiniteFields .................................... 112
3.2.4 GeneratorMatrix................................................ 114
3.2.5 TheDualCode.................................................. 117
3.2.6 Parity-CheckMatrix............................................ 118
3.2.7 TheSyndromeDecodingAlgorithm........................... 121
3.2.8 TheMacWilliamsIdentity...................................... 124
3.2.9 Self-OrthogonalandSelf-DualCodes......................... 127
3.3 CyclicCodes ............................................................ 128
3.3.1 CyclicCodesandIdeals........................................ 128
3.3.2 TheGeneratorPolynomial ..................................... 133
3.3.3 GeneratorMatrix................................................ 135
3.3.4 DualCodeandParity-CheckMatrix........................... 138
3.3.5 CyclicCodesfromRoots....................................... 140
3.3.6 IrreducibleCyclicCodes....................................... 143
3.3.7 DecodingAlgorithmsforCyclicCodes ....................... 146
3.4 BoundsinCodingTheory .............................................. 151
3.4.1 ExistenceTheoremsforGoodCodes.......................... 151
3.4.2 LimitationsontheParametersofCodes....................... 153
3.5 SomeSpecialLinearCodes ............................................ 157
3.5.1 HammingCodes................................................ 157
3.5.2 GolayCodes .................................................... 165
3.5.3 Reed-SolomonCodesandBCHCodes........................ 168
3.6 AGlimpseofAdvancedTopics........................................ 173
Exercises....................................................................... 180
4 Quasi-MonteCarloMethods ............................................... 185
4.1 NumericalIntegrationandUniformDistribution...................... 185
4.1.1 TheOne-DimensionalCase.................................... 185
4.1.2 TheMultidimensionalCase.................................... 204
4.2 ClassicalLow-DiscrepancySequences ................................ 216
4.2.1 KroneckerSequencesandContinuedFractions............... 216
4.2.2 HaltonSequences............................................... 223
Contents ix
4.3 LatticeRules ............................................................ 227
4.3.1 GoodLatticePoints............................................. 227
4.3.2 GeneralLatticeRules........................................... 244
4.4 Netsand.t;s/-Sequences............................................... 251
4.4.1 BasicFactsAboutNets......................................... 251
4.4.2 DigitalNetsandDualityTheory............................... 258
4.4.3 ConstructionsofDigitalNets .................................. 268
4.4.4 .t;s/-Sequences................................................. 287
4.4.5 AConstructionof.t;s/-Sequences ............................ 294
4.5 AGlimpseofAdvancedTopics........................................ 299
Exercises....................................................................... 303
5 PseudorandomNumbers.................................................... 307
5.1 GeneralPrinciples....................................................... 307
5.1.1 RandomNumberGeneration................................... 307
5.1.2 TestingPseudorandomNumbers............................... 312
5.2 TheLinearCongruentialMethod ...................................... 316
5.2.1 BasicProperties................................................. 316
5.2.2 ConnectionswithGoodLatticePoints ........................ 324
5.3 NonlinearMethods...................................................... 330
5.3.1 TheGeneralNonlinearMethod................................ 330
5.3.2 InversiveMethods .............................................. 340
5.4 PseudorandomBits ..................................................... 350
5.5 AGlimpseofAdvancedTopics........................................ 359
Exercises....................................................................... 362
6 FurtherApplications ........................................................ 367
6.1 Check-DigitSystems ................................................... 367
6.1.1 DefinitionandExamples ....................................... 367
6.1.2 NeighborTranspositionsandOrthomorphisms............... 369
6.1.3 PermutationsforDetectingOtherFrequentErrors............ 372
6.2 CoveringSetsandPackingSets........................................ 377
6.2.1 CoveringSetsandRewritingSchemes ........................ 377
6.2.2 PackingSetsandLimited-MagnitudeErrorCorrection ...... 379
6.3 Waring’sProblemforFiniteFields..................................... 381
6.3.1 Waring’sProblem............................................... 381
6.3.2 AdditionTheorems............................................. 383
6.3.3 Sum-ProductTheorems ........................................ 387
6.3.4 CoveringCodes................................................. 391
6.4 HadamardMatricesandApplications.................................. 394
6.4.1 BasicConstructions............................................. 394
6.4.2 HadamardCodes................................................ 398
6.4.3 SignalCorrelation .............................................. 400
6.4.4 HadamardTransformandBentFunctions..................... 402