Lattice Basis Reduction

08/18/2011 · No Responses

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.

Includes numerous algorithms in structured form (without goto statements) in both pseudocode and Maple
Presents the essential concepts that should be familiar to all users of lattice algorithms
Based on fundamental research papers on lattice basis reduction and its applications
Designed as a complete introduction for non-specialists: the only prerequisites are basic linear algebra and elementary number theory
Includes two applications to cryptography: knapsack cryptosystems, and Coppersmith’s algorithm
Includes two applications to computer algebra: polynomial factorization, and the Hermite normal form of an integer matrix

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.