Binary Quadratic Forms

An Algorithmic Approach

  • Johannes Buchmann
  • Ulrich Vollmer

Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 20)

Table of contents

  1. Front Matter
    Pages I-XIV
  2. Johannes Buchmann, Ulrich Vollmer
    Pages 1-7
  3. Johannes Buchmann, Ulrich Vollmer
    Pages 9-20
  4. Johannes Buchmann, Ulrich Vollmer
    Pages 21-34
  5. Johannes Buchmann, Ulrich Vollmer
    Pages 35-56
  6. Johannes Buchmann, Ulrich Vollmer
    Pages 57-84
  7. Johannes Buchmann, Ulrich Vollmer
    Pages 85-105
  8. Johannes Buchmann, Ulrich Vollmer
    Pages 107-142
  9. Johannes Buchmann, Ulrich Vollmer
    Pages 143-156
  10. Johannes Buchmann, Ulrich Vollmer
    Pages 157-176
  11. Johannes Buchmann, Ulrich Vollmer
    Pages 177-216
  12. Johannes Buchmann, Ulrich Vollmer
    Pages 217-232
  13. Johannes Buchmann, Ulrich Vollmer
    Pages 233-271
  14. Johannes Buchmann, Ulrich Vollmer
    Pages 273-287
  15. Back Matter
    Pages 289-318

About this book

Introduction

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.

Keywords

Number theory algebra algebraic number theory algorithmic number theory algorithms cryptography quadratic forms

Authors and affiliations

  • Johannes Buchmann
    • 1
  • Ulrich Vollmer
    • 1
  1. 1.Department of Computer ScienceTechnical UniversityDarmstadtGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-540-46368-9
  • Copyright Information Springer 2007
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-540-46367-2
  • Online ISBN 978-3-540-46368-9
  • Series Print ISSN 1431-1550
  • About this book