A Course in Computational Algebraic Number Theory

Transcription

1 Henri Cohen 2008 AGI-Information Management Consultants May be used for personal purporses only or by libraries associated to dandelon.com network. A Course in Computational Algebraic Number Theory Springer

2 Contents Chapter 1 Fundamental Number-Theoretic Algorithms Introduction Algorithms Multi-precision Base Fields and Rings Notations The Powering Algorithms Euclid's Algorithms Euclid's and Lehmer's Algorithms Euclid's Extended Algorithms The Chinese Remainder Theorem Continued Fraction Expansions of Real Numbers The Legendre Symbol The Groups (Z/nZ)* The Legendre-Jacobi-Kronecker Symbol Computing Square Roots Modulo p The Algorithm of Tonelli and Shanks The Algorithm of Cornacchia Solving Polynomial Equations Modulo p Power Detection Integer Square Roots Square Detection Prime Power Detection Exercises for Chapter 1. 42

3 XVI Contents Chapter 2 Algorithms for Linear Algebra and Lattices Introduction Linear Algebra Algorithms on Square Matrices Generalities on Linear Algebra"Algorithms Gaussian Elimination and Solving Linear Systems Computing Determinants Computing the Characteristic Polynomial Linear Algebra on General Matrices Kernel and Image Inverse Image and Supplement Operations on Subspaces Remarks on Modules Z-Modules and the Hermite and Smith Normal Forms Introduction to Z-Modules The Hermite Normal Form Applications of the Hermite Normal Form The Smith Normal Form and Applications Generalities on Lattices Lattices and Quadratic Forms The Gram-Schmidt Orthogonalization Procedure Lattice Reduction Algorithms The LLL Algorithm The LLL Algorithm with Deep Insertions The Integral LLL Algorithm LLL Algorithms for Linearly Dependent Vectors Applications of the LLL Algorithm Computing the Integer Kernel and Image of a Matrix Linear and Algebraic Dependence Using LLL Finding Small Vectors in Lattices Exercises for Chapter Chapter 3 Algorithms on Polynomials Basic Algorithms Representation of Polynomials 108 / Multiplication of Polynomials Division of Polynomials.. < Euclid's Algorithms for Polynomials Polynomials over a Field Unique Factorization Domains (UFD's) Polynomials over Unique Factorization Domains Euclid's Algorithm for Polynomials over a UFD 116

4 Contents XVII 3.3 The Sub-Resultant Algorithm Description of the Algorithm Resultants and Discriminants Resultants over a Non-Exact Domain Factorization of Polynomials Modulo p General Strategy Squarefree Factorization Distinct Degree Factorization Final Splitting Factorization of Polynomials over Z or Q Bounds on Polynomial Factors A First Approach to Factoring over Z Factorization Modulo p e : Hensel's Lemma Factorization of Polynomials over Z Discussion Additional Polynomial Algorithms Modular Methods for Computing GCD's in Z[X] Factorization of Polynomials over a Number Field A Root Finding Algorithm over C Exercises for Chapter Chapter 4 Algorithms for Algebraic Number Theory I Algebraic Numbers and Number Fields Basic Definitions and Properties of Algebraic Numbers Number Fields.... ' Representation_and Operations on Algebraic Numbers Algebraic Numbers as Roots of their Minimal Polynomial The Standard Representation of an Algebraic Number The Matrix (or Regular) Representation of an Algebraic Number The Conjugate Vector Representation of an Algebraic Number Trace, Norm and Characteristic Polynomial Discriminants, Integral Bases and Polynomial Reduction Discriminants and Integral Bases The Polynomial Reduction Algorithm The Subfield Problem and Applications The Subfield Problem Using the LLL Algorithm The Subfield Problem Using Linear Algebra over C The Subfield Problem Using Algebraic Algorithms Applications of the Solutions to the Subfield Problem 177

5 XVIII Contents 4.6 Orders and Ideals Basic Definitions., Ideals of Z K Representation of Modules and Ideals Modules and the Hermite Normal Form Representation of Ideals Decomposition of Prime Numbers I Definitions and Main Results A Simple Algorithm for the Decomposition of Primes Computing Valuations Ideal Inversion and the Different Units and Ideal Classes The Class Group Units and the Regulator Conclusion: the Main Computational Tasks of Algebraic Number Theory Exercises for Chapter Chapter 5 Algorithms for Quadratic Fields Discriminant, Integral Basis and Decomposition of Primes Ideals and Quadratic Forms Class Numbers of Imaginary Quadratic Fields 226 t Computing Class Numbers Using Reduced Forms Computing Class Numbers Using Modular Forms Computing Class Numbers Using Analytic Formulas Class Groups of Imaginary Quadratic Fields Shanks's Baby Step Giant Step Method Reduction and Composition of Quadratic Forms Class Groups Using Shanks's Method McCurley's Sub-exponential Algorithm Outline of the Algorithm Detailed Description of the Algorithm Atkin's Variant Class Groups of Real Quadratic Fields Computing Class Numbers Using Reduced Forms Computing Class Numbers Using Analytic Formulas A Heuristic Method of Shanks 263

6 Contents XIX 5.7 Computation of the Fundamental Unit and of the Regulator Description of the Algorithms Analysis of the Continued Fraction Algorithm Computation of the Regulator The Infrastructure Method of Shanks The Distance Function Description of the Algorithm Compact Representation of the Fundamental Unit Other Application and Generalization of the Distance Function Buchmann's Sub-exponential Algorithm Outline of the Algorithm Detailed Description of Buchmann's Sub-exponential Algorithm The Cohen-Lenstra Heuristics Results and Heuristics for Imaginary Quadratic Fields Results and Heuristics for Real Quadratic Fields Exercises for Chapter Chapter 6 Algorithms for Algebraic Number Theory II Computing the Maximal Order The Pohst-Zassenhaus Theorem The Dedekind Criterion Outline of the Round 2 Algorithm Detailed Description of the Round 2 Algorithm Decomposition of Prime Numbers II Newton Polygons ' 3g? Theoretical Description of the Buchmann-Lenstra Method Multiplying and Dividing Ideals Modulo p Splitting of Separable Algebras over W p Detailed Description of the Algorithm for Prime Decomposition Computing Galois Groups The Resolvent Method -, Degree 3, Degree Degree Degree Degree A List of Test Polynomials 327

7 XX Contents 6.4 Examples of Families of Number Fields Making Tables of Number Fields Cyclic Cubic Fields Pure Cubic Fields._ Decomposition of Primes in Pure Cubic Fields General Cubic Fields Computing the Class Group, Regulator and Fundamental Units Ideal Reduction Computing the Relation Matrix Computing the Regulator and a System of Fundamental Units The General Class Group and Unit Algorithm The Principal Ideal Problem Exercises for Chapter Chapter 7 Introduction to Elliptic Curves Basic Definitions Introduction Elliptic Integrals and Elliptic Functions Elliptic Curves over a Field Points on Elliptic Curves Complex Multiplication and Class Numbers Maps Between Complex Elliptic Curves Isogenies Complex Multiplication Complex Multiplication and Hilbert Class Fields Modular Equations Rank and L-functions :3.1 The Zeta Function of a Variety L-functions of Elliptic Curves The Taniyama-Weil Conjecture The Birch and Swinnerton-Dyer Conjecture Algorithms for Elliptic Curves Algorithms for Elliptic Curves over C Algorithm for Reducing a General Cubic Algorithms for Elliptic Curves over-f p Algorithms for Elliptic Curves over Q Tate's algorithm Computing rational points Algorithms for computing the L-function 405

8 Contents XXI 7.6 Algorithms for Elliptic Curves with pomplex Multiplication Computing the Complex Values of j(r) Computing the Hilbert Class Polynomials Computing Weber Class Polynomials Exercises for Chapter Chapter 8 Factoring in the Dark Ages Factoring and Primality Testing Compositeness Tests Primality Tests The Pocklington-Lehmer N- 1 Test Briefly, Other Tests Lehman's Method Pollard's p Method Outline of the Method Methods for Detecting Periodicity Brent's Modified Algorithm Analysis of the Algorithm Shanks's Class Group Method Shanks's SQUFOF The p - 1-method The First Stage The Second Stage Other Algorithms of the Same Type Exercises for Chapter Chapter 9 Modern Primality Tests The Jacobi Sum Test Group Rings of Cyclotomic Extensions Characters, Gauss Sums and Jacobi Sums The Basic Test Checking Condition C p The Use of Jacobi Sums Detailed Description of the Algorithm Discussion ' ^ The Elliptic Curve Test The Goldwasser-Kilian Test Atkin's Test, Exercises for Chapter 9 467

9 XXII x. Contents Chapter 10 Modern Factoring Methods The Continued Fraction Method The Class Group Method Sketch of the Method The Schnorr-Lenstra Factoring Method The Elliptic Curve Method Sketch of the Method Elliptic Curves Modulo N The ECM Factoring Method of Lenstra Practical Considerations The Multiple Polynomial Quadratic Sieve The Basic Quadratic Sieve Algorithm The Multiple Polynomial Quadratic Sieve Improvements to the MPQS Algorithm The Number Field Sieve Introduction Description of the Special NFS when h(k) = Description of the Special NFS when h(k) > Description of the General NFS Miscellaneous Improvements to the Number Field Sieve Exercises for Chapter Appendix A Packages for Number Theory 498 Appendix B Some Useful Tables 503 B.I Table of Class Numbers of Complex Quadratic Fields B.2 Table of Class Numbers and Units of Real Quadratic Fields 505 B.3 Table of Class Numbers and Units of Complex Cubic Fields 509 B.4 Table of Class Numbers and Units of Totally Real Cubic Fields 511 B.5 Table of Elliptic Curves ' 514 Bibliography 517 Index i 529