Lattice basis reduction : an introduction to the LLL algorithm and its applications
 Author/Creator
 Bremner, Murray R.
 Language
 English.
 Imprint
 Boca Raton, FL : CRC Press, c2012.
 Physical description
 xvii, 316 p. : ill ; 25 cm.
 Series
 Monographs and textbooks in pure and applied mathematics ; .300.
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QA171.5 .B74 2012

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QA171.5 .B74 2012
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Contents/Summary
 Bibliography
 Includes bibliographical references (p. 299309) and index.
 Contents

 Introduction to Lattices Euclidean space Rn Lattices in Rn Geometry of numbers Projects Exercises TwoDimensional Lattices The Euclidean algorithm Twodimensional lattices Vallee's analysis of the Gaussian algorithm Projects Exercises GramSchmidt Orthogonalization The GramSchmidt theorem Complexity of the GramSchmidt process Further results on the GramSchmidt process Projects Exercises The LLL Algorithm Reduced lattice bases The original LLL algorithm Analysis of the LLL algorithm The closest vector problem Projects Exercises Deep Insertions Modifying the exchange condition Examples of deep insertion Updating the GSO Projects Exercises Linearly Dependent Vectors Embedding dependent vectors The modified LLL algorithm Projects Exercises The Knapsack Problem The subsetsum problem Knapsack cryptosystems Projects Exercises Coppersmith's Algorithm Introduction to the problem Construction of the matrix Determinant of the lattice Application of the LLL algorithm Projects Exercises Diophantine Approximation Continued fraction expansions Simultaneous Diophantine approximation Projects Exercises The FinckePohst Algorithm The rational Cholesky decomposition Diagonalization of quadratic forms The original FinckePohst algorithm The FP algorithm with LLL preprocessing Projects Exercises Kannan's Algorithm Basic definitions Results from the geometry of numbers Kannan's algorithm Complexity of Kannan's algorithm Improvements to Kannan's algorithm Projects Exercises Schnorr's Algorithm Basic definitions and theorems A hierarchy of polynomialtime algorithms Projects Exercises NPCompleteness Combinatorial problems for lattices A brief introduction to NPcompleteness NPcompleteness of SVP in the max norm Projects Exercises The Hermite Normal Form The row canonical form over a field The Hermite normal form over the integers The HNF with lattice basis reduction Systems of linear Diophantine equations Using linear algebra to compute the GCD The HMM algorithm for the GCD The HMM algorithm for the HNF Projects Exercises Polynomial Factorization The Euclidean algorithm for polynomials Structure theory of finite fields Distinctdegree decomposition of a polynomial Equaldegree decomposition of a polynomial Hensel lifting of polynomial factorizations Polynomials with integer coefficients Polynomial factorization using LLL Projects Exercises.
 (source: Nielsen Book Data)
 Publisher's Summary
 First developed in the early 1980s by Lenstra, Lenstra, and Lovasz, the LLL algorithm was originally used to provide a polynomialtime algorithm for factoring polynomials with rational coefficients. It very quickly became an essential tool in integer linear programming problems and was later adapted for use in cryptanalysis. This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm. With numerous examples and suggested exercises, the text discusses various applications of lattice basis reduction to cryptography, number theory, polynomial factorization, and matrix canonical forms.
(source: Nielsen Book Data)
Subjects
 Subject
 Lattice theory > Textbooks.
 Data reduction > Textbooks.
Bibliographic information
 Publication date
 2012
 Responsibility
 Murray R. Bremner.
 Series
 Pure and applied mathematics
 Monographs and textbooks in pure and applied mathematics ; .300
 ISBN
 9781439807026
 1439807027