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Cover: 9783642079719 | Binary Quadratic Forms | An Algorithmic Approach | Vollmer (u. a.)
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Binary Quadratic Forms
An Algorithmic Approach
Taschenbuch von Ulrich Vollmer (u. a.)
Sprache: Englisch

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Beschreibung
This book deals with algorithmic problems concerning binary quadratic forms 2 2 f(X,Y)= aX +bXY +cY with integer coe?cients a, b, c, the mathem- ical theories that permit the solution of these problems, and applications to cryptography. A considerable part of the theory is developed for forms with real coe?cients and it is shown that forms with integer coe?cients 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 ofmanyof theseproblemspossible.Algorithmicsolutionsandtheirproperties became an object of study in their own right. Thisbookintertwinestheexpositionofoneveryclassicalstrandofnumber 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 e?ciency 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 pr- lems such as the solution of Pell equations with large coe?cients. On the other, the application of number theory in public-key cryptography increased the urgency for establishing the complexity of several computational pr- lems: many a computer system stays only secure as long as these problems remain intractable.
This book deals with algorithmic problems concerning binary quadratic forms 2 2 f(X,Y)= aX +bXY +cY with integer coe?cients a, b, c, the mathem- ical theories that permit the solution of these problems, and applications to cryptography. A considerable part of the theory is developed for forms with real coe?cients and it is shown that forms with integer coe?cients 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 ofmanyof theseproblemspossible.Algorithmicsolutionsandtheirproperties became an object of study in their own right. Thisbookintertwinestheexpositionofoneveryclassicalstrandofnumber 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 e?ciency 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 pr- lems such as the solution of Pell equations with large coe?cients. On the other, the application of number theory in public-key cryptography increased the urgency for establishing the complexity of several computational pr- lems: many a computer system stays only secure as long as these problems remain intractable.
Über den Autor

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.

Zusammenfassung
The book deals with algorithmic problems related to binary quadratic forms. Written by a world leader in number theory, it is the only book focusing on the algorithmic aspects of the theory. It deals with problems such as finding the representations of an integer by a form with integer coefficients, finding the minimum of a form with real coefficients and deciding equivalence of two forms. In order to solve those problems, 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. It requires only basic mathematical knowledge.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Algorithms and Computation in Mathematics
Inhalt: xiv
318 S.
ISBN-13: 9783642079719
ISBN-10: 3642079717
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Vollmer, Ulrich
Buchmann, Johannes
Auflage: Softcover reprint of hardcover 1st ed. 2007
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Algorithms and Computation in Mathematics
Maße: 235 x 155 x 19 mm
Von/Mit: Ulrich Vollmer (u. a.)
Erscheinungsdatum: 25.11.2010
Gewicht: 0,511 kg
Artikel-ID: 107176039
Über den Autor

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.

Zusammenfassung
The book deals with algorithmic problems related to binary quadratic forms. Written by a world leader in number theory, it is the only book focusing on the algorithmic aspects of the theory. It deals with problems such as finding the representations of an integer by a form with integer coefficients, finding the minimum of a form with real coefficients and deciding equivalence of two forms. In order to solve those problems, 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. It requires only basic mathematical knowledge.
Inhaltsverzeichnis
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.
Details
Erscheinungsjahr: 2010
Fachbereich: Arithmetik & Algebra
Genre: Mathematik
Rubrik: Naturwissenschaften & Technik
Medium: Taschenbuch
Reihe: Algorithms and Computation in Mathematics
Inhalt: xiv
318 S.
ISBN-13: 9783642079719
ISBN-10: 3642079717
Sprache: Englisch
Ausstattung / Beilage: Paperback
Einband: Kartoniert / Broschiert
Autor: Vollmer, Ulrich
Buchmann, Johannes
Auflage: Softcover reprint of hardcover 1st ed. 2007
Hersteller: Springer-Verlag GmbH
Springer Berlin Heidelberg
Algorithms and Computation in Mathematics
Maße: 235 x 155 x 19 mm
Von/Mit: Ulrich Vollmer (u. a.)
Erscheinungsdatum: 25.11.2010
Gewicht: 0,511 kg
Artikel-ID: 107176039
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