Other eBook Options:

- 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

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.

**Introduction to Lattices**Euclidean space R

Geometry of numbers

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Two-dimensional lattices

Vallée's analysis of the Gaussian algorithm

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Complexity of the Gram-Schmidt process

Further results on the Gram-Schmidt process

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The original LLL algorithm

Analysis of the LLL algorithm

The closest vector problem

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Examples of deep insertion

Updating the GSO

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The modified LLL algorithm

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Knapsack cryptosystems

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Construction of the matrix

Determinant of the lattice

Application of the LLL algorithm

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Simultaneous Diophantine approximation

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Diagonalization of quadratic forms

The original Fincke-Pohst algorithm

The FP algorithm with LLL preprocessing

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Results from the geometry of numbers

Kannan’s algorithm

Complexity of Kannan’s algorithm

Improvements to Kannan’s algorithm

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A hierarchy of polynomial-time algorithms

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A brief introduction to NP-completeness

NP-completeness of SVP in the max norm

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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

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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

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**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.

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