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.
The History of the LLL-Algorithm.- Hermite#x2019;s Constant and Lattice Algorithms.- Probabilistic Analyses of Lattice Reduction Algorithms.- Progress on LLL and Lattice Reduction.- Floating-Point LLL: Theoretical and Practical Aspects.- LLL: A Tool for Effective Diophantine Approximation.- Selected Applications of LLL in Number Theory.- The van Hoeij Algorithm for Factoring Polynomials.- The LLL Algorithm and Integer Programming.- Using LLL-Reduction for Solving RSA and Factorization Problems.- Practical Lattice-Based Cryptography: NTRUEncrypt and NTRUSign.- The Geometry of Provable Security: Some Proofs of Security in Which Lattices Make a Surprise Appearance.- Cryptographic Functions from Worst-Case Complexity Assumptions.- Inapproximability Results for Computational Problems on Lattices.- On the Complexity of Lattice Problems with Polynomial Approximation Factors.
From the reviews:
"Tells the history of the LLL algorithm and paper. ? this helpful and useful volume is a welcome reference book that covers nearly all applications of lattice reduction."
[Samuel S. Wagstaff, Jr., Mathematical Reviews, Issue 2011 m]
"This book is a compilation of survey-cum-expository articles contributed by leading experts ... The LLL algorithm embodies the power of lattice reduction on a wide range of problems in pure and applied fields [... and] the success of LLL attests to the triumph of theory in computer science. This book provides a broad survey of the developments in various fields of mathematics and computer science emanating from the LLL algorithm. As well-known researchers in their areas, the authors present an invaluable perspective on the topics by sharing their insights and understanding. The book is an exemplar of the unity of computer science in bringing a broad array of concepts, tools and techniques to the study of lattice problems. The many open problems and questions stated in every chapter of the book will inspire researchers to explore the LLL algorithm and its variants further. Graduate students in computer science and mathematics and researchers in theoretical computer science will find this book very useful. Finally, it is simply a pleasure to read this lovely book."
[Krishnan Narayanan, SIGACT News Book Review Column 45(4) 2014]
