Binary Quadratic Forms
An Algorithmic Approach
(Sprache: Englisch)
The book deals with algorithmic problems related to binary quadratic forms. It uniquely focuses on the algorithmic aspects of the theory. The book introduces the reader to important areas of number theory such as diophantine equations, reduction theory of...
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The book deals with algorithmic problems related to binary quadratic forms. It uniquely focuses on the algorithmic aspects of the theory. The book introduces the reader to important areas of number theory such as diophantine equations, reduction theory of quadratic forms, geometry of numbers and algebraic number theory. The book explains applications to cryptography and requires only basic mathematical knowledge. The author is a world leader in number theory.
This book deals with algorithmic problems concerning binary quadratic forms 2 2 f(X,Y)= aX +bXY +cY with integer coefficients a, b, c, the mathemetical theories that permit the solution of these problems, and applications to cryptography. A considerable part of the theory is developed for forms with real coefficients and it is shown that forms with integer coefficients appear in a natural way. Much of the progress of number theory has been stimulated by the study of concrete computational problems. Deep theories were developed from the classic time of Euler and Gauss onwards to this day that made the solutions of many of these problems possible. Algorithmic solutions and their properties became an object of study in their own right.
This book intertwines the exposition of one very classical strand of number theory with the presentation and analysis of algorithms both classical and modern which solve its motivating problems. This algorithmic approach will lead the reader, we hope, not only to an understanding of theory and solution methods, but also to an appreciation of the efficiency with which solutions can be reached. The computer age has led to a marked advancement of algorithmic - search. On the one hand, computers make it feasible to solve very hard problems such as the solution of Pell equations with large coefficients. On the other, the application of number theory in public-key cryptography increased the urgency for establishing the complexity of several computational problems: many a computer system stays only secure as long as these problems remain intractable.
This book intertwines the exposition of one very classical strand of number theory with the presentation and analysis of algorithms both classical and modern which solve its motivating problems. This algorithmic approach will lead the reader, we hope, not only to an understanding of theory and solution methods, but also to an appreciation of the efficiency with which solutions can be reached. The computer age has led to a marked advancement of algorithmic - search. On the one hand, computers make it feasible to solve very hard problems such as the solution of Pell equations with large coefficients. On the other, the application of number theory in public-key cryptography increased the urgency for establishing the complexity of several computational problems: many a computer system stays only secure as long as these problems remain intractable.
Inhaltsverzeichnis zu „Binary Quadratic Forms “
Binary Quadratic Forms.- Equivalence of Forms.- Constructing Forms.- Forms, Bases, Points, and Lattices.- Reduction of Positive Definite Forms.- Reduction of Indefinite Forms.- Multiplicative Lattices.- Quadratic Number Fields.- Class Groups.- Infrastructure.- Subexponential Algorithms.- Cryptographic Applications.
Autoren-Porträt von Johannes Buchmann, Ulrich Vollmer
Buchmann: Professor of Computer Science and Mathematics
special areas number theory, computer algebra, cryptography
associate editor Journal of Cryptology
Leibniz Award of the Deutsche Forschungsgemeinschaft
Author of "Introduction to cryptography" UTM, translated into seven languages
Member of Berlin-Brandenburg Academy of Sciences
Member of Academy of Sciences and Literature, Mainz
Vollmer: Thesis and several articles on algorithms for Class Group and Regulator computation in quadratic fields.
Bibliographische Angaben
- Autoren: Johannes Buchmann , Ulrich Vollmer
- 2007, 2007, 318 Seiten, Maße: 16 x 24,1 cm, Gebunden, Englisch
- Verlag: Springer
- ISBN-10: 3540463674
- ISBN-13: 9783540463672
- Erscheinungsdatum: 28.02.2007
Sprache:
Englisch
Rezension zu „Binary Quadratic Forms “
From the reviews:"Quadratic Field Theory is the best platform for the development of a computer viewpoint. Such an idea is not dominant in earlier texts on quadratic forms ... . this book reads like a continuous program with major topics occurring as subroutines. The theory appears as 'program comments,' accompanied by numerical examples. ... An appendix explaining linear algebra (bases and matrices) helps make this work ideal as a self-contained well-motivated textbook for computer-oriented students at any level and as a reference book." (Harvey Cohn, Zentralblatt MATH, Vol. 1125 (2), 2008)
"The book under discussion contains the classical Gauß -Dirichlet representation theory of integral binary quadric forms. ... Many of the algorithms presented in this book are described in full detail. The whole text is very carefully written. It is therefore also well suited for beginners as in addition no special knowledge on Number Theory is necessary to understand the text. It can also be recommended to teachers who give courses in Number Theory or Computational Algebra." (J. Schoissengeier, Monatshefte für Mathematik, Vol. 156 (3), March, 2009)
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