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The LLL Algorithmby Phong Q. Nguyen
Synopses & Reviews
The LLL algorithm is a polynomial-time lattice reduction algorithm, named after its inventors, Arjen Lenstra, Hendrik Lenstra and László Lovász. The algorithm has revolutionized computational aspects of the geometry of numbers since its introduction in 1982, leading to breakthroughs in fields as diverse as computer algebra, cryptology and algorithmic number theory. This book consists of 15 survey chapters on computational aspects of Euclidean lattices and their main applications. Topics covered include polynomial factorization, lattice reduction algorithms, applications in number theory, integer programming, provable security, lattice-based cryptography and complexity. The authors include many detailed motivations, explanations and examples, and the contributions are largely self-contained. The book will be of value to a wide range of researchers and graduate students working in related fields of theoretical computer science and mathematics.
The first book to offer a comprehensive view of the LLL algorithm, this text surveys computational aspects of Euclidean lattices and their main applications. It includes many detailed motivations, explanations and examples.
Table of Contents
A Tale of Two Papers.- Polynomial Factorization and Lattices in the Very Early 1980s.- Floating-Point LLL: Theoretical and Practical Aspects.- Progress on LLL and Lattice Reduction.- Probabilistic Analyses of Lattice Reduction Algorithms.- LLL: A Tool for Effective Diophantine Approximation.- Selected Applications of LLL in Number Theory.- The van Hoeij Algorithm to Factor Polynomials.- The LLL-Algorithm and Integer Programming.- The Geometry of Provable Security: Some Proofs of Security in Which Lattices Make a Surprise Appearance.- Practical Lattice-Based Cryptography: NTRUEncrypt and NTRUSign.- Using LLL-Reduction for Solving RSA and Factorization Problems: A Survey.- Lattice-Based Cryptanalysis.- Inapproximability Results for Computational Problems on Lattices.- On the Complexity of Lattice Problems with Polynomial Approximation Factors.- Cryptographic Functions from Worst-Case Complexity Assumptions.
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