Modern Computer Algebra

Transcription

1 Modern Computer Algebra JOACHIM VON ZUR GATHEN and JURGEN GERHARD Universitat Paderborn CAMBRIDGE UNIVERSITY PRESS

2 Contents Introduction 1 1 Cyclohexane, cryptography, codes, and computer algebra Cyclohexane conformations The RSA cryptosystem Distributed data structures Computer algebra systems 17 1 Euclid 21 2 Fundamental algorithms Representation and addition of numbers Representation and addition of polynomials Multiplication Division with remainder 35 Notes 39 Exercises 39 3 The Euclidean Algorithm Euclidean domains The Extended Euclidean Algorithm Cost analysis for Z and F[x] 50 Notes 55 Exercises 57 4 Applications of the Euclidean Algorithm Modular arithmetic Modular inverses via Euclid Repeated squaring Modular inverses via Fermat Linear Diophantine equations 71 vii

3 viii Contents 4.6 Continued fractions and Diophantine approximation Calendars Musical scales 78 Notes 81 Exercises 84 5 Modular algorithms and interpolation Change of representation Evaluation and interpolation Application: Secret sharing The Chinese Remainder Algorithm Modular determinant computation Hermite interpolation Rational function reconstruction Cauchy interpolation Pade approximation Rational number reconstruction Partial fraction decomposition 119 Notes 122 Exercises The resultant and gcd computation Coefficient growth in the Euclidean Algorithm GauB' lemma The resultant Modular gcd algorithms Modular gcd algorithm in F[JC,V] Mignotte's factor bound and a modular gcd algorithm in Z[x] Small primes modular gcd algorithms Application: intersecting plane curves Nonzero preservation and the gcd of several polynomials Subresultants Modular Extended Euclidean Algorithms Pseudo-division and primitive Euclidean Algorithms Implementations 182 Notes 185 Exercises Application: Decoding BCH codes 197 Notes 203 Exercises 203

4 Contents ix II Newton Fast multiplication Karatsuba's multiplication algorithm The Discrete Fourier Transform and the Fast Fourier Transform Schonhage and Strassen's multiplication algorithm Multiplication in Z[x] and R[x,y] 233 Notes 234 Exercises Newton iteration Division with remainder using Newton iteration Generalized Taylor expansion and radix conversion Formal derivatives and Taylor expansion Solving polynomial equations via Newton iteration Computing integer roots Valuations, Newton iteration, and Julia sets Implementations of fast arithmetic 263 Notes 272 Exercises Fast polynomial evaluation and interpolation Fast multipoint evaluation Fast interpolation Fast Chinese remaindering 285 Notes 290 Exercises Fast Euclidean Algorithm A fast Euclidean Algorithm for polynomials Subresultants via Euclid's algorithm 306 Notes 310 Exercises Fast linear algebra Strassen's matrix multiplication Application: fast modular composition of polynomials Linearly recurrent sequences Wiedemann's algorithm and black box linear algebra 323 Notes 330 Exercises 331

5 x Contents 13 Fourier Transform and image compression The Continuous and the Discrete Fourier Transform Audio and video compression 339 Notes 344 Exercises 344 III GauB Factoring polynomials over finite fields Factorization of polynomials Distinct-degree factorization Equal-degree factorization: Cantor and Zassenhaus' algorithm A complete factoring algorithm Application: root finding Squarefree factorization The iterated Frobenius algorithm Algorithms based on linear algebra Testing irreducibility and constructing irreducible polynomials Cyclotomic polynomials and constructing BCH codes 387 Notes 393 Exercises Hensel lifting and factoring polynomials Factoring in X[x] andq[x]: the basic idea A factoring algorithm Frobenius'and Chebotarev's density theorems Hensel lifting Multifactor Hensel lifting Factoring using Hensel lifting: Zassenhaus'algorithm Implementations 435 Notes 440 Exercises Short vectors in lattices Lattices Lenstra, Lenstra and Lovasz'basis reduction algorithm Cost estimate for basis reduction From short vectors to factors A polynomial-time factoring algorithm for Q[x] Factoring multivariate polynomials 467 Notes 470 Exercises 472

6 Contents xi 17 Applications of basis reduction Breaking knapsack-type cryptosystems Pseudorandom numbers Simultaneous Diophantine approximation Disproof of Mertens' conjecture 482 Notes 483 Exercises 483 IV Fermat Primality testing Multiplicative order of integers The Fermat test The strong pseudoprimality test Finding primes The Solovay and Strassen test The complexity of primality testing 504 Notes 506 Exercises Factoring integers Factorization challenges Trial division Pollard's and Strassen's method Pollard's rho method Dixon's random squares method Pollard's p- 1 method Lenstra's elliptic curve method 531 Notes 541 Exercises Application: Public key cryptography Cryptosystems The RSA cryptosystem The Diffie-Hellman key exchange protocol The ElGamal cryptosystem Rabin's cryptosystem Elliptic curve systems Short vector cryptosystems 554 Notes 555 Exercises 555

7 xii Contents V Hilbert Grobner bases Polynomial ideals Monomial orders and multivariate division with remainder Monomial ideals and Hilbert's basis theorem Grobner bases and S-polynomials Buchberger's algorithm Geometric applications The complexity of computing Grobner bases 589 Notes 591 Exercises Symbolic integration Differential algebra ' Hermite's method The method of Rothstein and Trager 601 Notes 606 Exercises Symbolic summation Polynomial summation Harmonic numbers Greatest factorial factorization Hypergeometric summation: Gosper's algorithm 622 Notes 633 Exercises Applications Grobner proof systems Petrinets Proving identities and analysis of algorithms Cyclohexane revisited 649 Notes 661 Exercises 662 Appendix Fundamental concepts Groups Rings Polynomials and fields 672

8 Contents xiu 25.4 Finite fields Linear algebra Finite probability spaces "Big Oh" notation Complexity theory 685 Notes 688 Sources of illustrations 689 Sources of quotations 689 List of algorithms 694 List of figures and tables 696 References 698 List of notation 728 Index 729 Keeping up to date Addenda and corrigenda, comments, solutions to selected exercises, and ordering information can be found on the book's web page: