Johannes Buchmann & Ulrich Vollmer 
Binary Quadratic Forms 
An Algorithmic Approach

This book deals with algorithmic problems concerning binary quadratic forms 2 2 f(X, Y)= a X +b XY +c Y 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.

Table of Content

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.

About the author

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.