Lattice Basis Reduction：An Introduction to the LLL Algorithm and Its Applications

First developed in the early 1980s by Lenstra, Lenstra, and Lovász, the LLL algorithm was originally used to provide a polynomial-time 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.

Murray R. Bremner received a Bachelor of Science from the University of Saskatchewan in 1981, a Master of Computer Science from Concordia University in Montreal in 1984, and a Doctorate in Mathematics from Yale University in 1989. He spent one year as a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, and three years as an Assistant Professor in the Department of Mathematics at the University of Toronto. He returned to the Department of Mathematics and Statistics at the University of Saskatchewan in 1993 and was promoted to Professor in 2002. His research interests focus on the application of computational methods to problems in the theory of linear nonassociative algebras, and he has had more than 50 papers published or accepted by refereed journals in this area.

Introduction to LatticesEuclidean space RnLattices in Rn Geometry of numbers Projects Exercises Two-Dimensional LatticesThe Euclidean algorithm Two-dimensional lattices Vallée's analysis of the Gaussian algorithm Projects Exercises Gram-Schmidt OrthogonalizationThe Gram-Schmidt theorem Complexity of the Gram-Schmidt process Further results on the Gram-Schmidt process Projects Exercises The LLL AlgorithmReduced lattice bases The original LLL algorithm Analysis of the LLL algorithm The closest vector problem Projects Exercises Deep InsertionsModifying the exchange condition Examples of deep insertion Updating the GSO Projects Exercises Linearly Dependent VectorsEmbedding dependent vectors The modified LLL algorithm Projects Exercises The Knapsack ProblemThe subset-sum problem Knapsack cryptosystems Projects Exercises Coppersmith’s AlgorithmIntroduction to the problem Construction of the matrix Determinant of the lattice Application of the LLL algorithm Projects Exercises Diophantine ApproximationContinued fraction expansions Simultaneous Diophantine approximation Projects Exercises The Fincke-Pohst AlgorithmThe rational Cholesky decomposition Diagonalization of quadratic forms The original Fincke-Pohst algorithm The FP algorithm with LLL preprocessing Projects Exercises Kannan’s AlgorithmBasic definitions Results from the geometry of numbers Kannan’s algorithm Complexity of Kannan’s algorithm Improvements to Kannan’s algorithm Projects Exercises Schnorr’s AlgorithmBasic definitions and theorems A hierarchy of polynomial-time algorithms Projects Exercises NP-CompletenessCombinatorial problems for lattices A brief introduction to NP-completeness NP-completeness of SVP in the max norm Projects Exercises The Hermite Normal FormThe 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 FactorizationThe Euclidean algorithm for polynomials Structure theory of finite fields Distinct-degree decomposition of a polynomial Equal-degree decomposition of a polynomial Hensel lifting of polynomial factorizations Polynomials with integer coefficients Polynomial factorization using LLL Projects Exercises