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Lattice Basis Reduction

An Introduction to the LLL Algorithm and Its Applications

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## Book Description

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.

## Table of Contents

**Introduction to Lattices**Euclidean space R

^{n}Lattices in R

^{n}

Geometry of numbers

Projects

Exercises

**Two-Dimensional Lattices**

The Euclidean algorithm

Two-dimensional lattices

Vallée's analysis of the Gaussian algorithm

Projects

Exercises

**Gram-Schmidt Orthogonalization**

The Gram-Schmidt theorem

Complexity of the Gram-Schmidt process

Further results on the Gram-Schmidt 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 subset-sum 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 Fincke-Pohst Algorithm**

The rational Cholesky decomposition

Diagonalization of quadratic forms

The original Fincke-Pohst 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 polynomial-time algorithms

Projects

Exercises

**NP-Completeness**

Combinatorial problems for lattices

A brief introduction to NP-completeness

NP-completeness 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

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

## Author(s)

### Biography

**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.

## 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 2012This 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