The LLL Algorithm: Survey and Applications

ISBN-13: 9783642261640 / Angielski / Miękka / 2012 / 496 str.

Nguyen, Phong Q.

Opis

Computational aspects of geometry of numbers have been revolutionized by the Lenstra Lenstra Lovasz lattice reduction algorithm (LLL), which has led to bre- throughs in elds as diverse as computer algebra, cryptology, and algorithmic number theory. After its publication in 1982, LLL was immediately recognized as one of the most important algorithmic achievements of the twentieth century, because of its broad applicability and apparent simplicity. Its popularity has kept growing since, as testi ed by the hundreds of citations of the original article, and the ever more frequent use of LLL as a synonym to lattice reduction. As an unfortunate consequence of the pervasiveness of the LLL algorithm, researchers studying and applying it belong to diverse scienti c communities, and seldom meet. While discussing that particular issue with Damien Stehle at the 7th Algorithmic Number Theory Symposium (ANTS VII) held in Berlin in July 2006, John Cremona accuratelyremarkedthat 2007would be the 25th anniversaryof LLL and this deserveda meetingto celebrate that event. The year 2007was also involved in another arithmetical story. In 2003 and 2005, Ali Akhavi, Fabien Laguillaumie, and Brigitte Vallee with other colleagues organized two workshops on cryptology and algorithms with a strong emphasis on lattice reduction: CAEN 03 and CAEN 05, CAEN denoting both the location and the content (Cryptologie et Algori- miqueEn Normandie). Veryquicklyafterthe ANTSconference, AliAkhavi, Fabien Laguillaumie, and Brigitte Vallee were thus readily contacted and reacted very enthusiastically about organizing the LLL birthday conference. The organization committee was formed."

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.

Recenzje

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]

Spis treści

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.