Cyclotomic polynomial

Abstract: In this paper we present a polynomial-time algorithm to solve the following problem: given a non-zero polynomial fe Q(X) in one variable with rational coefficients, find the decomposition of f into irreducible factors in Q(X). It is well known that this is equivalent to factoring primitive polynomials feZ(X) into irreducible factors in Z(X). Here we call f~ Z(X) primitive if the greatest common divisor of its coefficients (the content of f) is 1. Our algorithm performs well in practice, cf. (8). Its running time, measured in bit operations, is O(nl2+n9(log(fD3).

Abstract: 1 Introduction.- 1.1 Introductory Remarks.- 1.2 Notations.- 1.3 The Structure of the Book.- 2 Elements of Number Theory and Polynomial Algebra.- 2.1 Elementary Number Theory.- 2.1.1 Divisibility of Integers.- 2.1.2 Congruences and Residues.- 2.1.3 Primitive Roots.- 2.1.4 Quadratic Residues.- 2.1.5 Mersenne and Fermat Numbers.- 2.2 Polynomial Algebra.- 2.2.1 Groups.- 2.2.2 Rings and Fields.- 2.2.3 Residue Polynomials.- 2.2.4 Convolution and Polynomial Product Algorithms in Polynomial Algebra.- 3 Fast Convolution Algorithms.- 3.1 Digital Filtering Using Cyclic Convolutions.- 3.1.1 Overlap-Add Algorithm.- 3.1.2 Overlap-Save Algorithm.- 3.2 Computation of Short Convolutions and Polynomial Products.- 3.2.1 Computation of Short Convolutions by the Chinese Remainder Theorem.- 3.2.2 Multiplications Modulo Cyclotomic Polynomials.- 3.2.3 Matrix Exchange Algorithm.- 3.3 Computation of Large Convolutions by Nesting of Small Convolutions.- 3.3.1 The Agarwal-Cooley Algorithm.- 3.3.2 The Split Nesting Algorithm.- 3.3.3 Complex Convolutions.- 3.3.4 Optimum Block Length for Digital Filters.- 3.4 Digital Filtering by Multidimensional Techniques.- 3.5 Computation of Convolutions by Recursive Nesting of Polynomials.- 3.6 Distributed Arithmetic.- 3.7 Short Convolution and Polynomial Product Algorithms.- 3.7.1 Short Circular Convolution Algorithms.- 3.7.2 Short Polynomial Product Algorithms.- 3.7.3 Short Aperiodic Convolution Algorithms.- 4 The Fast Fourier Transform.- 4.1 The Discrete Fourier Transform.- 4.1.1 Properties of the DFT.- 4.1.2 DFTs of Real Sequences.- 4.1.3 DFTs of Odd and Even Sequences.- 4.2 The Fast Fourier Transform Algorithm.- 4.2.1 The Radix-2 FFT Algorithm.- 4.2.2 The Radix-4 FFT Algorithm.- 4.2.3 Implementation of FFT Algorithms.- 4.2.4 Quantization Effects in the FFT.- 4.3 The Rader-Brenner FFT.- 4.4 Multidimensional FFTs.- 4.5 The Bruun Algorithm.- 4.6 FFT Computation of Convolutions.- 5 Linear Filtering Computation of Discrete Fourier Transforms.- 5.1 The Chirp z-Transform Algorithm.- 5.1.1 Real Time Computation of Convolutions and DFTs Using the Chirp z-Transform.- 5.1.2 Recursive Computation of the Chirp z-Transform.- 5.1.3 Factorizations in the Chirp Filter.- 5.2 Rader's Algorithm.- 5.2.1 Composite Algorithms.- 5.2.2 Polynomial Formulation of Rader's Algorithm.- 5.2.3 Short DFT Algorithms.- 5.3 The Prime Factor FFT.- 5.3.1 Multidimensional Mapping of One-Dimensional DFTs.- 5.3.2 The Prime Factor Algorithm.- 5.3.3 The Split Prime Factor Algorithm.- 5.4 The Winograd Fourier Transform Algorithm (WFTA).- 5.4.1 Derivation of the Algorithm.- 5.4.2 Hybrid Algorithms.- 5.4.3 Split Nesting Algorithms.- 5.4.4 Multidimensional DFTs.- 5.4.5 Programming and Quantization Noise Issues.- 5.5 Short DFT Algorithms.- 5.5.1 2-Point DFT.- 5.5.2 3-Point DFT.- 5.5.3 4-Point DFT.- 5.5.4 5-Point DFT.- 5.5.5 7-Point DFT.- 5.5.6 8-Point DFT.- 5.5.7 9-Point DFT.- 5.5.8 16-Point DFT.- 6 Polynomial Transforms.- 6.1 Introduction to Polynomial Transforms.- 6.2 General Definition of Polynomial Transforms.- 6.2.1 Polynomial Transforms with Roots in a Field of Polynomials.- 6.2.2 Polynomial Transforms with Composite Roots.- 6.3 Computation of Polynomial Transforms and Reductions.- 6.4 Two-Dimensional Filtering Using Polynomial Transforms.- 6.4.1 Two-Dimensional Convolutions Evaluated by Polynomial Transforms and Polynomial Product Algorithms.- 6.4.2 Example of a Two-Dimensional Convolution Computed by Polynomial Transforms.- 6.4.3 Nesting Algorithms.- 6.4.4 Comparison with Conventional Convolution Algorithms.- 6.5 Polynomial Transforms Defined in Modified Rings.- 6.6 Complex Convolutions.- 6.7 Multidimensional Polynomial Transforms.- 7 Computation of Discrete Fourier Transforms by Polynomial Transforms.- 7.1 Computation of Multidimensional DFTs by Polynomial Transforms.- 7.1.1 The Reduced DFT Algorithm.- 7.1.2 General Definition of the Algorithm.- 7.1.3 Multidimensional DFTs.- 7.1.4 Nesting and Prime Factor Algorithms.- 7.1.5 DFT Computation Using Polynomial Transforms Defined in Modified Rings of Polynomials.- 7.2 DFTs Evaluated by Multidimensional Correlations and Polynomial Transforms.- 7.2.1 Derivation of the Algorithm.- 7.2.2 Combination of the Two Polynomial Transform Methods.- 7.3 Comparison with the Conventional FFT.- 7.4 Odd DFT Algorithms.- 7.4.1 Reduced DFT Algorithm. N = 4.- 7.4.2 Reduced DFT Algorithm. N = 8.- 7.4.3 Reduced DFT Algorithm. N = 9.- 7.4.4 Reduced DFT Algorithm. N = 16.- 8 Number Theoretic Transforms.- 8.1 Definition of the Number Theoretic Transforms.- 8.1.1 General Properties of NTTs.- 8.2 Mersenne Transforms.- 8.2.1 Definition of Mersenne Transforms.- 8.2.2 Arithmetic Modulo Mersenne Numbers.- 8.2.3 Illustrative Example.- 8.3 Fermat Number Transforms.- 8.3.1 Definition of Fermat Number Transforms.- 8.3.2 Arithmetic Modulo Fermat Numbers.- 8.3.3 Computation of Complex Convolutions by FNTs.- 8.4 Word Length and Transform Length Limitations.- 8.5 Pseudo Transforms.- 8.5.1 Pseudo Mersenne Transforms.- 8.5.2 Pseudo Fermat Number Transforms.- 8.6 Complex NTTs.- 8.7 Comparison with the FFT.- Appendix A Relationship Between DFT and Conyolution Polynomial Transform Algorithms.- A.1 Computation of Multidimensional DFT's by the Inverse Polynomial Transform Algorithm.- A.1.1 The Inverse Polynomial Transform Algorithm.- A.1.2 Complex Polynomial Transform Algorithms.- A.1.3 Round-off Error Analysis.- A.2 Computation of Multidimensional Convolutions by a Combination of the Direct and Inverse Polynomial Transform Methods.- A.2.1 Computation of Convolutions by DFT Polynomial Transform Algorithms.- A.2.2 Convolution Algorithms Based on Polynomial Transforms and Permutations.- A.3 Computation of Multidimensional Discrete Cosine Transforms by Polynomial Transforms.- A.3.1 Computation of Direct Multidimensional DCT's.- A.3.2 Computation of Inverse Multidimensional DCT's.- Appendix B Short Polynomial Product Algorithms.- Problems.- References.

Abstract: 1. The Number of Primes Below a Given Limit.- What Is a Prime Number?.- The Fundamental Theorem of Arithmetic.- Which Numbers Are Primes? The Sieve of Eratosthenes.- General Remarks Concerning Computer Programs.- A Sieve Program.- Compact Prime Tables.- Hexadecimal Compact Prime Tables.- Difference Between Consecutive Primes.- The Number of Primes Below x.- Meissel's Formula.- Evaluation of Pk(x, a).- Lehmer's Formula.- Computations.- A Computation Using Meissel's Formula.- A Computation Using Lehmer's Formula.- A Computer Program Using Lehmer's Formula.- Mapes' Method.- Deduction of Formulas.- A Worked Example.- Mapes' Algorithm.- Programming Mapes' Algorithm.- Recent Developments.- Results.- Computational Complexity.- Comparison Between the Methods Discussed.- 2. The Primes Viewed at Large.- No Polynomial Can Produce Only Primes.- Formulas Yielding All Primes.- The Distribution of Primes Viewed at Large. Euclid's Theorem.- The Formulas of Gauss and Legendre for ?(x). The Prime Number Theorem.- The Chebyshev Function ?(x).- The Riemann Zeta-function.- The Zeros of the Zeta-function.- Conversion From f(x) Back to ?(x).- The Riemann Prime Number Formula.- The Sign of li x ? ?(x).- The Influence of the Complex Zeros of ?(s) on ?(x).- The Remainder Term in the Prime Number Theorem.- Effective Inequalities for ?(x), pn, and ?(x).- The Number of Primes in Arithmetic Progressions.- 3. Subtleties in the Distribution of Primes.- The Distribution of Primes in Short Intervals.- Twins and Some Other Constellations of Primes.- Admissible Constellations of Primes.- The Hardy-Littlewood Constants.- The Prime k-Tuples Conjecture.- Theoretical Evidence in Favour of the Prime k-Tuples Conjecture.- Numerical Evidence in Favour of the Prime k-Tuples Conjecture.- The Second Hardy-Littlewood Conjecture.- The Midpoint Sieve.- Modification of the Midpoint Sieve.- Construction of Superdense Admissible Constellations.- Some Dense Clusters of Primes.- The Distribution of Primes Between the Two Series 4n + 1 and 4n + 3.- Graph of the Function ?4,3(x) ? ?4,1(x).- The Negative Regions.- The Negative Blocks.- Large Gaps Between Consecutive Primes.- The Cramer Conjecture.- 4. The Recognition of Primes.- Tests of Primality and of Compositeness.- Factorization Methods as Tests of Compositeness.- Fermat's Theorem as Compositeness Test.- Fermat's Theorem as Primality Test.- Pseudoprimes and Probable Primes.- A Computer Program for Fermat's Test.- The Labor Involved in a Fermat Test.- Carmichael Numbers.- Euler Pseudoprimes.- Strong Pseudoprimes and a Primality Test.- A Computer Program for Strong Pseudoprime Tests.- Counts of Pseudoprimes and Carmichael Numbers.- Rigorous Primality Proofs.- Lehmer's Converse of Fermat's Theorem.- Formal Proof of Theorem 4.3.- Ad Hoc Search for a Primitive Root.- The Use of Several Bases.- Fermat Numbers and Pepin's Theorem.- Cofactors of Fermat Numbers.- Generalized Fermat Numbers.- A Relaxed Converse of Fermat's Theorem.- Proth's Theorem.- Tests of Compositeness for Numbers of the form N = h * 2n +- k.- An Alternative Approach.- Certificates of Primality.- Primality Tests of Lucasian Type.- Lucas Sequences.- The Fibonacci Numbers.- Large Subscripts.- An Alternative Deduction.- Divisibility Properties of the Numbers Un.- Primality Proofs by Aid of Lucas Sequences.- Lucas Tests for Mersenne Numbers.- A Relaxation of Theorem 4.8.- Pocklington's Theorem.- Lehmer-Pocklington's Theorem.- Pocklington-Type Theorems for Lucas Sequences.- Primality Tests for Integers of the form N = h * 2n ? 1, when 3?h.- Primality Tests for N = h * 2n ? 1, when 3?h.- The Combined N ? 1 and N + 1 Test.- Lucas Pseudoprimes.- Modern Primality Proofs.- The Jacobi Sum Primality Test.- Three Lemmas.- Lenstra's Theorem.- The Sets P and Q.- Running Time for the APRCL Test.- Elliptic Curve Primality Proving, ECPP.- The Goldwasser-Kilian Test.- Atkin's Test.- 5. Classical Methods of Factorization.- When Do We Attempt Factorization?.- Trial Division.- A Computer Implementation of Trial Division.- Euclid's Algorithm as an Aid to Factorization.- Fermat's Factoring Method.- Legendre's Congruence.- Euler's Factoring Method.- Gauss' Factoring Method.- Legendre's Factoring Method.- The Number of Prime Factors of Large Numbers.- How Does a Typical Factorization Look?.- The Erd?s-Kac Theorem.- The Distribution of Prime Factors of Various Sizes.- Dickman's Version of Theorem 5.4.- A More Detailed Theory.- The Size of the kth Largest Prime Factor of N.- Smooth Integers.- Searching for Factors of Certain Forms.- Legendre's Theorem for the Factors of N = an +- bn.- Adaptation to Search for Factors of the Form p = 2kn + 1.- Adaptation of Trial Division.- Adaptation of Fermat's Factoring Method.- Adaptation of Euclid's Algorithm as an Aid to Factorization.- 6. Modem Factorization Methods.- Choice of Method.- Pollard's (p ? 1)-Method.- Phase 2 of the (p ? 1)-Method.- The (p + 1)-Method.- Pollard's rho Method.- A Computer Program for Pollard's rho Method.- An Algebraic Description of Pollard's rho Method.- Brent's Modification of Pollard's rho Method.- The Pollard-Brent Method for p = 2kn + 1.- Shanks' Factoring Method SQUFOF.- A Computer Program for SQUFOF.- Comparison Between Pollard's rho Method and SQUFOF.- Morrison and Brillhart's Continued Fraction Method CFRAC.- The Factor Base.- An Example of a Factorization with CFRAC.- Further Details of CFRAC.- The Early Abort Strategy.- Results Achieved with CFRAC.- Running Time Analysis of CFRAC.- The Quadratic Sieve, QS.- Smallest Solutions to Q(x) ? 0 mod p.- Special Factors.- Results Achieved with QS.- The Multiple Polynomial Quadratic Sieve, MPQS.- Results Achieved with MPQS.- Running Time Analysis of QS and MPQS.- The Schnorr-Lenstra Method.- Two Categories of Factorization Methods.- Lenstra's Elliptic Curve Method, ECM.- Phase 2 of ECM.- The Choice of A, B, and P1.- Running Times of ECM.- Recent Results Achieved with ECM.- The Number Field Sieve, NFS.- Factoring Both in Z and in Z(z).- A Numerical Example.- The General Number Field Sieve, GNFS.- Running Times of NFS and GNFS.- Results Achieved with NFS. Factorization of F9.- Strategies in Factoring.- How Fast Can a Factorization Algorithm Be?.- 7. Prime Numbers and Cryptography.- Practical Secrecy.- Keys in Cryptography.- Arithmetical Formulation.- RSA Cryptosystems.- How to Find the Recovery Exponent.- A Worked Example.- Selecting Keys.- Finding Suitable Primes.- The Fixed Points of an RSA System.- How Safe is an RSA Cryptosystem?.- Superior Factorization.- Appendix 1. Basic Concepts in Higher Algebra.- Modules.- Euclid's Algorithm.- The Labor Involved in Euclid's Algorithm.- A Definition Taken from the Theory of Algorithms.- A Computer Program for Euclid's Algorithm.- Reducing the Labor.- Binary Form of Euclid's Algorithm.- Groups.- Lagrange's Theorem. Cosets.- Abstract Groups. Isomorphic Groups.- The Direct Product of Two Given Groups.- Cyclic Groups.- Rings.- Zero Divisors.- Fields.- Mappings. Isomorphisms and Homomorphisms.- Group Characters.- The Conjugate or Inverse Character.- Homomorphisms and Group Characters.- Appendix 2. Basic Concepts in Higher Arithmetic.- Divisors. Common Divisors.- The Fundamental Theorem of Arithmetic.- Congruences.- Linear Congruences.- Linear Congruences and Euclid's Algorithm.- Systems of Linear Congruences.- Carmichael's Function.- Carmichael's Theorem.- Appendix 3. Quadratic Residues.- Legendre's Symbol.- Arithmetic Rules for Residues and Non-Residues.- The Law of Quadratic Reciprocity.- Jacobi's Symbol.- Appendix 4. The Arithmetic of Quadratic Fields.- Appendix 5. Higher Algebraic Number Fields.- Algebraic Numbers.- Appendix 6. Algebraic Factors.- Factorization of Polynomials.- The Cyclotomic Polynomials.- Aurifeuillian Factorizations.- Factorization Formulas.- The Algebraic Structure of Aurifeuillian Numbers.- Appendix 7. Elliptic Curves.- Cubics.- Rational Points on Rational Cubics.- Homogeneous Coordinates.- Elliptic Curves.- Rational Points on Elliptic Curves.- Appendix 8. Continued Fractions.- What Is a Continued Fraction?.- Regular Continued Fractions. Expansions.- Evaluating a Continued Fraction.- Continued Fractions as Approximations.- Euclid's Algorithm and Continued Fractions.- Linear Diophantine Equations and Continued Fractions.- A Computer Program.- Continued Fraction Expansions of Square Roots.- Proof of Periodicity.- The Maximal Period-Length.- Short Periods.- Continued Fractions and Quadratic Residues.- Appendix 9. Multiple-Precision Arithmetic.- Various Objectives for a Multiple-Precision Package.- How to Store Multi-Precise Integers.- Addition and Subtraction of Multi-Precise Integers.- Reduction in Length of Multi-Precise Integers.- Multiplication of Multi-Precise Integers.- Division of Multi-Precise Integers.- Input and Output of Multi-Precise Integers.- A Complete Package for Multiple-Precision Arithmetic.- A Computer Program for Pollard's rho Method.- Appendix 10. Fast Multiplication of Large Integers.- The Ordinary Multiplication Algorithm.- Double Length Multiplication.- Recursive Use of Double Length Multiplication Formula.- A Recursive Procedure for Squaring Large Integers.- Fractal Structure of Recursive Squaring.- Large Mersenne Primes.- Appendix 11. The Stieltjes Integral.- Functions With Jump Discontinuities.- The Riemann Integral.- Definition of the Stieltjes Integral.- Rules of Integration for Stieltjes Integrals.- Integration by Parts of Stieltjes Integrals.- The Mean Value Theorem.- Applications.- Tables. For Contents.- List of Textbooks.

Abstract: Analysis and synthesis techniques for a class of sequential discrete-state networks are discussed. These networks, made up of arbitrary interconnections of unit-delay elements (or of trigger flip-flops), modulo-p adders, and scalar multipliers (modulo \alpha , prime p ), are of importance in unconventional radar and communication systems, in automatic error-correction circuits, and in the control circuits of digital computers. In addition, these networks are of theoretical significance to the study of more general sequential networks. The basic problem with which this paper is concerned is that of finding economical realizations of such networks for prescribed autonomous (excitation-free) behavior. To this end, an analytical-algebraic model is described which permits the investigation of the relation between network logical structure and state-sequential behavior. This relation is studied in detail for nonsingular networks (those with purely cyclic behavior). Among the results of this investigation is the establishment of relations between the state diagram of the network and a characteristic polynomial derived from its logical structure, An operation of multiplication of state diagrams is shown to correspond to multiplication of the corresponding polynomials. A criterion is established for the realizability of prescribed cyclic behavior by means of linear autonomous sequential networks. An effective procedure for the economical realization of such networks is described, and it is shown that linear feedback shift registers constitute a canonical class of realizations. Examples are given of the realization procedure. The problem of synthesis with only one-cycle length specified is also discussed. A partial solution is obtained to this "don't care" problem. Some special families of feedback shift registers are investigated in detail, and the state-diagram structures are obtained for an arbitrary number of stages and an arbitrary (prime) modulus. Mathematical appendixes are included which summarize the pertinent results in Galois field theory and in the factorization of cyclotomic polynomials into irreducible factors over a modular field. The relation of the theory developed in this paper to Huffman's description of linear sequence transducers in terms of the D operator is discussed, as well as unsolved problems and directions for further generalization.

Abstract: The computation of Feynman integrals in massive higher order perturbative calculations in renormalizable Quantum Field Theories requires extensions of multiply nested harmonic sums, which can be generated as real representations by Mellin transforms of Poincare- iterated integrals including denominators of higher cyclotomic polynomials. We derive the cyclotomic harmonic polylogarithms and harmonic sums and study their algebraic and structural relations. The analytic continuation of cyclotomic harmonic sums to complex values of N is performed using analytic representations. We also consider special values of the cyclotomic harmonic polylogarithms at argument x = 1, resp., for the cyclotomic harmonic sums at N → ∞, which are related to colored multiple zeta values, deriving various of their relations, based on the stuffle and shuffle alge bras and three multiple ar- gument relations. We also consider infinite generalized nested harmonic sums at roots of unity which are related to the infinite cyclotomic harmonic sums. Basis representations are derived for weight w = 1,2 sums up to cyclotomy l = 20.