IntroductionOutlineThe OrganizationPART I: Computational Number TheoryComputational Number TheoryGroups, Rings, and FieldsElements of Number TheoryInteger Rings and FieldsChinese Remainder Theorem for IntegersNumber Theory for Finite Integer RingsPolynomial AlgebraAlgebra of Polynomials over a FieldRoots of a PolynomialPolynomial Fields and RingsThe Chinese Remainder Theorem for PolynomialsCRT-P in Matrix FormLagrange InterpolationPolynomial Algebra over GF(p)Order of an ElementTheoretical Aspects of Discrete Fourier Transform and ConvolutionThe Discrete Fourier TransformBasic Formulation of ConvolutionBounds on the Multiplicative ComplexityBasic Formulation of Convolution AlgorithmsMatrix Exchange PropertyCyclotomic Polynomial Factorization and Associated FieldsCyclotomic Polynomial Factorization over Complex and Real NumbersCyclotomic Polynomial Factorization over Rational NumbersCyclotomic Fields and Cyclotomic Polynomial FactorizationsExtension Fields of Cyclotomic Fields and Cyclotomic Polynomial FactorizationA Preview of Applications to Digital Signal ProcessingCyclotomic Polynomial Factorization in Finite FieldsCyclotomic Polynomial FactorizationFactorization of (un - 1) over GF (p)Primitive Polynomials over GF (p)Complex Finite Fields and Cyclotomic Polynomial FactorizationFinite Integer Rings: Polynomial Algebra and Cyclotomic FactorizationPolynomial Algebra over a RingLagrange InterpolationNumber Theoretic TransformsMonic Polynomial FactorizationExtension of CRT-P over Finite Integer RingsPolynomial Algebra and CRT-PR: The Complex CaseNumber Theoretic Transforms: The Complex CasePseudo Number Theoretic TransformsPolynomial Algebra and Direct Sum Properties in Integer Polynomial RingsPART II: Convolution AlgorithmsThoughts on Part IIFast Algorithms for Acyclic ConvolutionCRT-P Based Fast Algorithms for One-Dimensional Acyclic ConvolutionCasting the Algorithm in Bilinear FormulationMultidimensional Approaches to One-Dimensional Acyclic ConvolutionMultidimensional Acyclic Convolution AlgorithmsNesting and Split Nesting Algorithms for Multidimensional ConvolutionAcyclic Convolution Algorithms over Finite Fields and RingsFast One-Dimensional Cyclic Convolution AlgorithmsBilinear Forms and Cyclic ConvolutionCyclotomic Polynomials and Related Algorithms over Re and CCyclotomic Polynomials and Related Algorithms over ZOther ConsiderationsComplex Cyclotomic Polynomials and Related Algorithms over CZThe Agarwal-Cooley AlgorithmCyclic Convolution Algorithms over Finite Fields and RingsTwo- and Higher Dimensional Cyclic Convolution AlgorithmsPolynomial Formulation and an AlgorithmImprovements and Related AlgorithmsDiscrete Fourier Transform Based AlgorithmsAlgorithms Based on Extension FieldsAlgorithms for Multidimensional Cyclic ConvolutionAlgorithms for Two-Dimensional Cyclic Convolution in Finite Integer RingsValidity of Fast Algorithms over Different Number SystemsIntroductionMathematical PreliminariesChinese Remainder Theorem over Finite Integer RingsInterrelationships among Algorithms over Different Number SystemsAnalysis of Two-Dimensional Cyclic Convolution AlgorithmsFault Tolerance for Integer SequencesA Framework for Fault ToleranceMathematical Structure of C over Z(M)Coding Techniques over Z(q)Examples and SFC-DFD CodesPART III: Fast Fourier Transform (FFT) AlgorithmsThoughts on Part IIIFast Fourier Transform: One-Dimensional Data SequencesThe DFT: Definitions and PropertiesRader's FFT Algorithm, n=p, p an Odd PrimeRader's FFT Algorithm, n=pc, p an Odd PrimeCooley-Tukey FFT Algorithm, n=a . bFFT Algorithms for n a Power of 2The Prime Factor FFT n=a . b, (a,b) =1The Winograd FFT AlgorithmFast Fourier Transform: Multidimensional Data SequencesThe Multidimensional DFT: Definition and PropertiesFFT for n=p, p an Odd PrimeMultidimensional FFT Algorithms for n a Power of 2Matrix Formulation of Multidimensional DFT and Related AlgorithmsPolynomial Version of Rader's AlgorithmPolynomial Transform Based FFT AlgorithmsPART IV: Recent Results on Algorithms in Finite Integer RingsThoughts on Part IVPaper One: A Number Theoretic Approach to Fast Algorithms for Two-Dimensional Digital Signal Processing in Finite Integer RingsPaper Two: On Fast Algorithms for One-Dimensional Digital Signal Processing in Finite Integer and Complex Integer RingsPaper Three: Cyclotomic Polynomial Factorization in Finite Integer Rings with Applications to Digital Signal ProcessingPaper Four: Error Control Techniques for Data Sequences Defined in Finite Integer RingsA. Small Length Acyclic Convolution AlgorithmsB. Classification of Cyclotomic PolynomialsIndex