Binary Quadratic Forms
An Algorithmic Approach
Buchmann, Johannes.
creator
author.
Vollmer, Ulrich.
author.
SpringerLink (Online service)
text
gw
2007
monographic
eng
access
XIV, 318 p. online resource.
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.
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.
by Johannes Buchmann, Ulrich Vollmer.
Mathematics
Data encryption (Computer science)
Computer science
Mathematics
Algebra
Number theory
Mathematics
Algebra
Number Theory
Mathematics of Computing
Data Encryption
QA150-272
512
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