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

by Murray R. Bremner

Overview

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.

Product Details

ISBN-13:	9781032921822
Publisher:	CRC Press
Publication date:	10/14/2024
Series:	Chapman & Hall Pure and Applied Mathematics
Pages:	334
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

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.

Table of Contents

Introduction to Lattices. Two-Dimensional Lattices. Gram-Schmidt Orthogonalization. The LLL Algorithm. Deep Insertions. Linearly Dependent Vectors. The Knapsack Problem. Coppersmith’s Algorithm. Diophantine Approximation. The Fincke-Pohst Algorithm. Kannan’s Algorithm. Schnorr’s Algorithm. NP-Completeness. The Hermite Normal Form. Polynomial Factorization.

What People are Saying About This

From the Publisher

the book succeeds in making accessible to nonspecialists the area of lattice algorithms, which is remarkable because some of the most important results in the field are fairly recent.
—M. Zimand, Computing Reviews, March 2012

Editorial Reviews

the book succeeds in making accessible to nonspecialists the area of lattice algorithms, which is remarkable because some of the most important results in the field are fairly recent.
—M. Zimand, Computing Reviews, March 2012

This text is meant as a survey of lattice basis reduction at a level suitable for students and interested researchers with a solid background in undergraduate linear algebra. … The writing is clear and quite concise.
—Zentralblatt MATH 1237

From the Publisher