Buchmann J., Vollmer U. Binary Quadratic Forms. An Algorithmic Approach

Springer, 2007. — 326 p. — (Algorithms and Computation in Mathematics, Volume 20)This book Binary Quadratic Forms - An Algorithmic Approach has been published as Volume 20 in the series ”Algorithms and Computation in Mathematics” of Springer. The main author of the book Johannes Buchmann is a well-known computational number theory expert. The book consists of twelve chapters and an appendix. The appendix includes information about vectors, matrices, a lemma of Gauss, lattices, and linear algebra. The book explains the theory, algorithms, and applications of binary quadratic forms. The coverage is good and the description is exhaustive. The authors also try to focus on recent developments such as cryptographic applications of binary quadratic forms. Exercises have been
included at the end of chapters along with references and suggestions for further reading. This will be very useful for pedagogy and self-study. The authors have to be appreciated for considering an algorithmic approach to binary quadratic forms something which has not been done in books published earlier than this one. It is stated in the book that only basic mathematical knowledge is required. However, I wish to state that this is adequate for getting a feel of the subject, however, advanced concepts demand expertise in the subject.The book will be useful for mathematicians especially number theorists, theoretical computer science specialists, students, faculty members and researchers in mathematics as well as computer science. I recommend this book as a useful reference and textbook on binary quadratic forms.This book deals with algorithmic problems concerning binary quadratic forms f(X,Y)=aX²+bXY+cY² with integer coefficients a, b, c, the mathematical 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 research. 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.
Thus, number theory has become a research area not only in mathematics but also in computer science. This book tries to combine both worlds. It talks about mathematical theory, algorithms, and complexity.
The material presented is suitable as an introduction to many areas of (algorithmic) number theory for which the theory of binary quadratic forms is a starting point. We illustrate this for the areas Diophantine equations, geometry of numbers, and algebraic number theory.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