A Course in Computational Algebraic Number Theory

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Springer Science & Business Media, 17.04.2013 - 536 Seiten
With the advent of powerful computing tools and numerous advances in math ematics, 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 al gorithms. 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 appli cations, including real world applications to integer programming, primality testing and factoring algorithms, sub-exponential class group and regulator algorithms, etc ... Several books exist which treat parts of this subject. (It is essentially impossible for an author to keep up with the rapid pace of progress in all areas of this subject.) Each book emphasizes a different area, corresponding to the author's tastes and interests. The most famous, but unfortunately the oldest, is Knuth's Art of Computer Programming, especially Chapter 4. The present book has two goals. First, to give a reasonably comprehensive introductory course in computational number theory. In particular, although we study some subjects in great detail, others are only mentioned, but with suitable pointers to the literature. Hence, we hope that this book can serve as a first course on the subject. A natural sequel would be to study more specialized subjects in the existing literature.
 

Inhalt

Fundamental NumberTheoretic Algorithms
1
Algorithms for Quadratic Fields
5
Algorithms for Linear Algebra and Lattices
45
Algorithms on Polynomials
108
Algorithms for Algebraic Number Theory I
151
Number Theory
214
Algorithms for Algebraic Number Theory II
297
Introduction to Elliptic Curves
360
Factoring in the Dark Ages
396
Modern Primality Tests
437
Modern Factoring Methods
469
Appendix A Packages for Number Theory
498
Table of Class Numbers and Units of Totally Real Cubic Fields
511
Index
529
214
534
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