Algorithmic Number Theory
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Algorithmic Number Theory
Algorithmic Number Theory provides a thorough introduction to the design and analysis of algorithms for problems from the theory of numbers. Although not an elementary textbook, it includes over 300 exercises with suggested solutions. Every theorem not proved in the text or left as an exercise has a reference in the notes section that appears at the end of each chapter. The bibliography contains over 1,750 citations to the literature. Finally, it successfully blends computational theory with practice by covering some of the practical aspects of algorithm implementations.The subject of algorithmic number theory represents the marriage of number theory with the theory of computational complexity. It may be briefly defined as finding integer solutions to equations, or proving their non-existence, making efficient use of resources such as time and space. Implicit in this definition is the question of how to efficiently represent the objects in question on a computer. The problems of algorithmic number theory are important both for their intrinsic mathematical interest and their application to random number generation, codes for reliable and secure information transmission, computer algebra, and other areas.Publisher's Note: Volume 2 was not written. Volume 1 is, therefore, a stand-alone publication.
A Course in Computational Algebraic Number Theory
Author: Henri Cohen
language: en
Publisher: Springer Science & Business Media
Release Date: 2013-04-17
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.