Preface
Figures
Tables
Algorithms
Acronyms
1IntroductionandSomeAlgebraPreliminaries
1.1Notatio
1.1.1SetNotation
1.1.2MatrixNotation
1.1.3AsymptoticNotation
1.1.4GeneralNotation
1.2AbstractAlgebraicBackground
1.2.1Group
1.2.2Ring
1.2.3Field
1.2.4IrreducibleandPrimitivePolynomials
1.2.5Co tructionofExte ionFields
1.2.6Module
1.2.7Algebra
1.3LinearAlgebraBackground
1.3.1Involution
1.3.2SesquilinearForm
1.3.3UnitaryMatrix
1.3.4ParaunitaryMatrix
IFinite-FieldWavelets
2BackgroundReviewandMotivation
2.1WaveletsforDiscrete-TimeSignals
2.2CyclicWaveletTra forms
2.3ReviewofTra formsoverFiniteFields
2.3.1DiscreteFourierTra formoverFiniteFields
2.3.2Base-FieldTra formsoverFiniteFields
2.3.3RelatedWorkonFinite-FieldWavelets
3Finite-FieldWaveletBasisFunctio
3.1Finite-FieldDiscrete-TimeBasis
3.1.1Non-DegenerateBilinearForm
3.1.2OrthonormalWaveletBasisoverFiniteFields
3.1.3CompletenessoftheOrthonormalSet
3.2Co tructionofMotherWaveletandScalingFunction
3.3Summary
4TheoryofParaunitaryFilterBanksoverFieldsofCharacteristic2
4.1BackgroundReview
4.1.1Degree-1ParaunitaryBuildingBlockoverGF(2)
4.1.2Degree-2ParaunitaryBuildingBlocksoverGF(2)
4.1.3LappedOrthogonalTra formsoverGF(2)
4.2UnitaryMatricesoverGF(2r)
4.3ParaunitaryMatricesoverFieldsofCharacteristic2
4.3.1Propertiesof2x2ParaunitaryMatricesoverGF(2r)
4.4FactorizationofParaunitaryMatricesoverGF(2r)
4.4.1Degree-1ParaunitaryBuildingBlockoverGF(2r)
4.4.2Degree-2ParaunitaryBuildingBlockoverGF(2r)
4.4.3Degree-2rParaunitaryBuildingBlockoverGF(2r)
4.4.4Factorizationof2x2ParaunitaryMatricesoverGF(2r)
4.4.5Degree-MrParaunitaryBuildingBlockoverGF(2r)
4.4.6FactorizationofMxMParaunitaryMatricesoverGF(2r)
4.5Summary
IIMultivariateCryptography
5Introduction
5.1HistoricalBackgroundandMotivation
5.2RSA
5.3EllipticCurveCryptography
5.4MultivariateCryptography
6WaveletSelf-SynchronizingStreamCipher
6.1BackgroundReview
6.1.1ClassificationofStreamCiphe
6.2WaveletSelf-SynchronizingStreamCipher(WSSC)
6.2.1ModifiedWaveletTra form
6.2.2BasicRoundoftheWSSC
6.2.3MultipleRoundsoftheWSSC
6.2.4KeySetup
6.3CryptanalysisoftheWSSC
6.3.1InterpolationAttack
6.3.2AlgebraicAttacks
6.3.3DeltaAttack
6.3.4Time-MemoryTradeoffAttack
6.3.5Divide-and-ConquerAttack
6.3.6CorrelationandDistinguishingAttacks
6.4PerformanceEvaluation
6.5Summary
7WaveletBlockCipher
7.1BackgroundReview
7.1.1FeistelCipherandDataEncryptionStandard(DES)
7.1.2AdvancedEncryptionStandard(AES)
7.2WaveletBlockCipher(WBC)
7.2.1LinearComponentsoftheWBC
7.2.2NonlinearComponentsoftheWBC
7.3Two-RoundWaveletBlockCipher
7.3.1KeySetup
7.4CryptanalysisoftheWBC
7.4.1DifferentialandLinearAttacks
7.4.2Divide-and-ConquerAttack
7.4.3InterpolationAttack
7.4.4DeltaAttack
7.5PerformanceEvaluation
7.6Summary
8ParaunitaryPublic-KeyCryptography
8.1BackgroundReview
8.1.1SignatureBasedonBirationalPermutatio
8.1.2TameTra formationMethods
8.1.3TractableRationalMapCryptosystem
8.1.4C*AlgorithmanditsVariants
8.2ParaunitaryAsymmetricCryptosystem(PAC)
8.2.1BijectiveMappings
8.2.2PolynomialVector
8.2.3SetupAlgorithms
8.3ProbabilisticPAC
8.4OntheComputationalSecurityofthePAC
8.5APracticalI tanceofthePAC
8.5.1Co tructingthePolynomialVector
8.5.2ComplexityofthePAC
8.6CryptanalysisoftheI tanceofthePAC
8.6.1GrobnerBasis
8.6.2UnivariatePolynomialRepresentationofthePublicPolynomials
8.6.3XLandFXLAlgorithms
8.6.4AnAttackforSmallr
8.7ParaunitaryDigitalSignatureScheme(PDSS)
8.7.1PolynomialVector
8.7.2SetupAlgorithm
8.7.3APracticalI tanceofthePDSS
8.8Summary
IIIError-ControlCoding
9SomeBasicConceptsofError-ControlCoding
10Double-CirculantWaveletBlockCodes
10.1StructureofDouble-CirculantWaveletCoding
10.1.1WaveletStructuresforEncodingandDecoding
10.2Maximum-DistanceSeparableCodes
10.3Double-CirculantSelf-DualCodes
10.3.1FundamentalStructureofSelf-DualWaveletCodes
10.3.2Maximum-DistanceSeparableSelf-DualCodes
10.4DecodingWaveletCodes
10.4.1Bounded-DistanceDecodingof(20,10,6)Double-CirculantWaveletCode
10.4.2Bounded-DistanceDecodingoftheWavelet-GolayCode
10.5Summary
11Arbitrary-RateWaveletBlockCodes
11.1StructureofWaveletCoding
11.1.1WaveletStructureforEncodingandDecoding
11.2Rate-1/LMaximum-DistanceSeparableCodes
11.3Arbitrary-RateWaveletBlockCodes
11.4Arbitrary-RateMaximum-DistanceSeparableCodes
11.5DecodingArbitrary-RateWaveletBlockCodes
11.5.1Bounded-DistanceDecodingofthe(12,4,6)WaveletCode
11.5.2SymbolErrorCorrectioninthe(7,3,5)MDSCode
11.5.3Tail-BitingTrellisesforWaveletBlockCodes
11.6Summary
12WaveletConvolutionalCodes
12.1StructureofWaveletConvolutionalCodes
12.2AlgebraicPropertiesofWaveletConvolutionalEncode
12.3SyndromeGenerato andDualEncode
12.4Self-DualandSelf-OrthogonalConvolutionalCodes
12.5Time-VaryingWaveletConvolutionalCodesandBipartiteTrellises
12.6Summary
Appendices
AProofsofChap.4inPartI
BEfficientGenerationofPUMatrices
CToyExamplesofthePACandthePDSS
DProofsofChap.10inPartIII
EBriefReviewofTrellisStructures
Bibliography
Index