計(jì)算代數(shù)數(shù)值論教程

內(nèi)容概要

With the advent of powerful computing tools and numerous advances in mathematics, computer science and cryptography, algorithmic number theory has become an important subject in its own right. Both external and internal pressures gave a powerful impetus to the development of more powerful algorithms. These in turn led to a large number of spectacular breakthroughs. To mention but a few, the LLL algorithm which has a wide range of applications, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc ...

書(shū)籍目錄

Chapter 1 Fundamental Number-Theoretic Algorithms 　1.1 Introduction 　　1.1.1 Algorithms 　　1.1.2 Multi-precision 　　1.1.3 Base Fields and Rings 　　1.1.4 Notations 　1.2 The Powering Algorithms 　1.3 Euclid's Algorithms 　　1.3.1 Euclid's and Lehmer's Algorithms 　　1.3.2 Euclid's Extended Algorithms 　　1.3.3 The Chinese Remainder Theorem 　　1.3.4 Continued Fraction Expansions of Real Numbers 　1.4 The Legendre Symbol 　　1.4.1 The Groups (Z/nZ)* 　　1.4.2 The Legendre-Jacobi-Kronecker Symbol 　1.5 Computing Square Roots Modulo p 　　1.5.1 The Algorithm of Tonelli and Shanks 　　1.5.2 The Algorithm of Cornacchia 　1.6 Solving Polynomial Equations Modulo p 　1.7 Power Detection   ……  1.8 Exercises for Chapter 1Chapter 2 Algorithms for Linear AQlgebra and Lattices  2.1 Introducion  2.2 Linear Algebra Algorithms on Square Matrices  2.3 Linear Algebra on General Matrices  2.4 Z-Modules and the Hermite and Smith Normal Forms  2.5 Generalities on Lattices  2.6 Lattice Reducion Algorithms  2.7 Applications of the LLL Algorithm  2.8 Exercises for Chapter 2Chapter 3 Algorithms on Polynomials  3.1 Basic Algorithms  3.2 Euclid's Algorithms for Polynomials  3.3 The Sub-Resultant Algorithm  3.4 Factorization of Polynomials Modulo p  3.5 Factoriztion of Polynomials over Z or Q  3.6 Additiional Polynomial Algoritms  3.7 Exercises for Chapter 3Cahpter 4 Algorithms for Algebraic Number Theory ICahpter 5 Algorithms for Quadratic FieldsCahpter 6 Algorithms for Algebraic Number Theory IICahpter 7 Introducion to Elliptic CurvesCahpter 8 Factoring in the Dark AgesCahpter 9 Modern Primality TestsCahpter 10 Modern Factoring MethodsAppendix A Packages for Number TheoryAppendix B Some Useful TablesBibliographyIndex

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