# Lattice Basis Reduction This book provides an introduction to the theory and applications of lattice basis reduction and the LLL algorithm.

Author: Murray R. Bremner

Publisher: CRC Press

ISBN: 9781439807026

Category: Computers

Page: 332

View: 392

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.
2011-08-12

# The LLL Algorithm 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.

Author: Phong Q. Nguyen

Publisher: Springer Science & Business Media

ISBN: 9783642022951

Category: Computers

Page: 496

View: 272

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.
2009-12-02

# An Application of the LLL Algorithm to Integer Factorization Solving the shortest vector problem algorithmically gained a boom with the publication of the LLL algorithm in 1982.

Author: Gerwin Pineda

Publisher:

ISBN: OCLC:1124768789

Category: Algorithms

Page: 56

View: 950

Solving the shortest vector problem algorithmically gained a boom with the publication of the LLL algorithm in 1982. Many problems can be reformulated as finding the shortest vector in a lattice, and the LLL can provide very good approximations to their true solutions. One of these problems is the factorization of a large integer given partial information about one of its factors. Coppersmith describes a novel method to do this in , which enables the factorization of a large integer N in polynomial time in log N, provided that (1/4) log2 N of the high order bits of one of the factors of N are given. However, in practice, this might require guessing some of the middle bits of the partially known factor and then apply the Coppersmith separately for each guess. In this thesis, we explore the LLL algorithm and how much information the Coppersmith method needs to factor N in one run. We also study the parameters of the Coppersmith algorithm with the hope of reducing the amount of information needed by it. We provide the Mathematica code for the various computations that we did, as well as instructive examples.
2018

# Modified LLL Algorithms Then we propose three modified algorithms to improve the computational efficiency, while the reduced matrices satisfy the LLL-reduced criteria. The first modified algorithm, to be referred to as MLLLPIVOT, uses a block pivoting strategy.

Author: Tianyang Zhou

Publisher:

ISBN: OCLC:253795976

Category:

Page: 83

View: 208

"Lattice basis reduction arises from many applications, such as cryptography, communications, GPS and so on. This thesis is concerned with the widely used LLL reduction. We cast it as a QRZ matrix factorization for real bases. Based on the matrix factorization, we first give the real version of the LLL algorithm (the original LLL algorithm is for integer bases). Then we propose three modified algorithms to improve the computational efficiency, while the reduced matrices satisfy the LLL-reduced criteria. The first modified algorithm, to be referred to as MLLLPIVOT, uses a block pivoting strategy. The second one, to be called MLLLINSERT, uses a greedy insertion strategy. The last one, to be called MLLLLAZY, uses a "lazy" size-reduction strategy. Extensive simulation results are given to show the improvements and different performance of the three algorithms. In addition, numerical stability of the LLL algorithm and the three modified algorithms is considered. The simulations indicate that on average the computational efficiency (measured by CPU time) of the four algorithms have the increasing order: LLL
2006

# Number Theoretic Algorithms in Cryptography A flowchart of the algorithm can be found in [ 57 , Ch . 2 , Algorithm 2.6.7 ] . The last modification of the LLL algorithm we mention here is Pohst's MLLL algorithm . It works with vectors bı , ... , bn E R spanning the set A = Zbı + .

Author: Oleg Nikolaevich Vasilenko

Publisher: American Mathematical Soc.

ISBN: 0821840908

Category: Mathematics

Page: 243

View: 472

Algorithmic number theory is a rapidly developing branch of number theory, which, in addition to its mathematical importance, has substantial applications in computer science and cryptography. Among the algorithms used in cryptography, the following are especially important: algorithms for primality testing; factorization algorithms for integers and for polynomials in one variable; applications of the theory of elliptic curves; algorithms for computation of discrete logarithms; algorithms for solving linear equations over finite fields; and, algorithms for performing arithmetic operations on large integers. The book describes the current state of these and some other algorithms. It also contains extensive bibliography. For this English translation, additional references were prepared and commented on by the author.
2007

# A Course in Computational Algebraic Number Theory Note that this algorithm is essentially a reformulation of the GramSchmidt orthogonalization procedure in the case where ... [LLL reduction] Apply the LLL algorithm to the n vectors formed by the rows of R", thus obtaining a unimodular ...

Author: Henri Cohen

Publisher: Springer Science & Business Media

ISBN: 9783662029459

Category: Mathematics

Page: 536

View: 819

A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
2013-04-17

# China Satellite Navigation Conference CSNC 2014 Proceedings Table 13.1 Comparison of indicators between LLL and improved LLL algorithm Decorrelation algorithms Spectral condition number (Log10) Average correlation coefficient Reduction time (s) Original 10.82 0.60 LLL algorithm 5.71 0.22 0.04 ...

ISBN: 9783642547409

Category: Technology & Engineering

Page: 733

View: 962

China Satellite Navigation Conference (CSNC) 2014 Proceedings presents selected research papers from CSNC2014, held on 21-23 May in Nanjing, China. The theme of CSNC2014 is 'BDS Application: Innovation, Integration and Sharing'. These papers discuss the technologies and applications of the Global Navigation Satellite System (GNSS) and the latest progress made in the China BeiDou System (BDS) especially. They are divided into 9 topics to match the corresponding sessions in CSNC2014, which broadly covered key topics in GNSS. Readers can learn about the BDS and keep abreast of the latest advances in GNSS techniques and applications. SUN Jiadong is the Chief Designer of the Compass/ BDS, and the Academician of Chinese Academy of Sciences (CAS); JIAO Wenhai is a researcher at China Satellite Navigation Office; WU Haitao is a professor at Navigation Headquarters, CAS; LU Mingquan is a professor at Department of Electronic Engineering of Tsinghua University.
2014-04-22

# Algorithms ESA 99 There is already a wide number of variations around the LLL algorithm (due for instance to Kannan or Schnorr [6,13]) whose goal is to find lattice bases with sharper Euclidean properties than the original LLL algorithm.

Author: Esa 9

Publisher: Springer Science & Business Media

ISBN: 9783540662518

Category: Computers

Page: 552

View: 768

This book constitutes the refereed proceedings of the 7th Annual European Symposium on Algorithms, ESA '99, held in Prague, Czech Republic, in July 1999. The 44 revised papers presented were carefully reviewed and selected from a total of 122 submissions. All areas of algorithmic research are covered, in particular approximation algorithms, combinatorial optimization, computational mathematics, computational science, databases and information retrieval, graph computations, network algorithms, online algorithms, pattern matching, data compression, parallel algorithms, distributed algorithms, and sequential algorithms.
1999-07-07 By Esa 9

# Basis Reduction Algorithms and Subset Sum Problems Seysen's technique, used in combination with the LLL algorithm, and other heuristics, enables us to solve a much larger class of subset sum problems than was previously possible.

Author: Brian Andrew LaMacchia

Publisher:

ISBN: OCLC:25053550

Category: Cryptography

Page: 98

View: 265

This thesis investigates a new approach to lattice basis reduction suggested by M. Seysen. Seysen's algorithm attempts to globally reduce a lattice basis, whereas the Lenstra, Lenstra, Lov\'asz (LLL) family of reduction algorithms concentrates on local reductions. We show that Seysen's algorithm is well suited for reducing certain classes of lattice bases, and often requires much less time in practice than the LLL algorithm. We also demonstrate how Seysen's algorithm for basis reduction may be applied to subset sum problems. Seysen's technique, used in combination with the LLL algorithm, and other heuristics, enables us to solve a much larger class of subset sum problems than was previously possible.
1991

# LATIN 2010 Theoretical Informatics Furthermore, none of these studies is dedicated to the fine understanding of the internal structure of the algorithm. The LLL algorithm is a multidimensional extension, in dimension n, of the Euclid algorithm (obtained for n = 1) or the ...

Author: Alejandro López-Ortiz

Publisher: Springer

ISBN: 9783642122002

Category: Computers

Page: 706

View: 459

This book constitutes the refereed proceedings of the 9th International Latin American Symposium on Theoretical Informatics, LATIN 2010, held in Oaxaca, Mexico; in April 2010. The 56 revised full papers presented together with the abstracts of 4 invited plenary talks were carefully reviewed and selected from 155 submissions. The papers address a variety of topics in theoretical computer science with a certain focus on algorithms, automata theory and formal languages, coding theory and data compression, algorithmic graph theory and combinatorics, complexity theory, computational algebra, computational biology, computational geometry, computational number theory, cryptography, theoretical aspects of databases and information retrieval, data structures, networks, logic in computer science, machine learning, mathematical programming, parallel and distributed computing, pattern matching, quantum computing and random structures.
2010-04-22

# An Introduction to Mathematical Cryptography Before describing the technicalities of the LLL algorithm, we make some brief remarks indicating the general underlying idea. Given a basis {v1 ,v 2 ,...,v n}, it is easy to form a new basis that satisfies the Size Condition.

Author: Jeffrey Hoffstein

Publisher: Springer

ISBN: 9781493917112

Category: Mathematics

Page: 538

View: 895

This self-contained introduction to modern cryptography emphasizes the mathematics behind the theory of public key cryptosystems and digital signature schemes. The book focuses on these key topics while developing the mathematical tools needed for the construction and security analysis of diverse cryptosystems. Only basic linear algebra is required of the reader; techniques from algebra, number theory, and probability are introduced and developed as required. This text provides an ideal introduction for mathematics and computer science students to the mathematical foundations of modern cryptography. The book includes an extensive bibliography and index; supplementary materials are available online. The book covers a variety of topics that are considered central to mathematical cryptography. Key topics include: classical cryptographic constructions, such as Diffie–Hellmann key exchange, discrete logarithm-based cryptosystems, the RSA cryptosystem, and digital signatures; fundamental mathematical tools for cryptography, including primality testing, factorization algorithms, probability theory, information theory, and collision algorithms; an in-depth treatment of important cryptographic innovations, such as elliptic curves, elliptic curve and pairing-based cryptography, lattices, lattice-based cryptography, and the NTRU cryptosystem. The second edition of An Introduction to Mathematical Cryptography includes a significant revision of the material on digital signatures, including an earlier introduction to RSA, Elgamal, and DSA signatures, and new material on lattice-based signatures and rejection sampling. Many sections have been rewritten or expanded for clarity, especially in the chapters on information theory, elliptic curves, and lattices, and the chapter of additional topics has been expanded to include sections on digital cash and homomorphic encryption. Numerous new exercises have been included.
2014-09-11

# Complexity of Lattice Problems Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid.

Author: Daniele Micciancio

Publisher: Springer Science & Business Media

ISBN: 9781461508977

Category: Computers

Page: 220

View: 810

Lattices are geometric objects that can be pictorially described as the set of intersection points of an infinite, regular n-dimensional grid. De spite their apparent simplicity, lattices hide a rich combinatorial struc ture, which has attracted the attention of great mathematicians over the last two centuries. Not surprisingly, lattices have found numerous ap plications in mathematics and computer science, ranging from number theory and Diophantine approximation, to combinatorial optimization and cryptography. The study of lattices, specifically from a computational point of view, was marked by two major breakthroughs: the development of the LLL lattice reduction algorithm by Lenstra, Lenstra and Lovasz in the early 80's, and Ajtai's discovery of a connection between the worst-case and average-case hardness of certain lattice problems in the late 90's. The LLL algorithm, despite the relatively poor quality of the solution it gives in the worst case, allowed to devise polynomial time solutions to many classical problems in computer science. These include, solving integer programs in a fixed number of variables, factoring polynomials over the rationals, breaking knapsack based cryptosystems, and finding solutions to many other Diophantine and cryptanalysis problems.
2012-12-06

# Algorithms ESA 2003 This implies that the LLL-algorithm has polynomial complexity. Theorem 3 (Lenstra, Lenstra and Lovász). Let A ∈ Zn×n be a lattice basis and let A0 be the number A0 = max{aj | j = 1,... ,n}. The LLL-algorithm performs O(n4 logA 0) ...

Author: Giuseppe Di Battista

Publisher: Springer Science & Business Media

ISBN: 9783540200642

Category: Computers

Page: 790

View: 324

This book constitutes the refereed proceedings of the 11th Annual European Symposium on Algorithms, ESA 2003, held in Budapest, Hungary, in September 2003. The 66 revised full papers presented were carefully reviewed and selected from 165 submissions. The scope of the papers spans the entire range of algorithmics from design and mathematical analysis issues to real-world applications, engineering, and experimental analysis of algorithms.
2003-09-15

# Optimization and Control for Systems in the Big Data Era We denote the checks whether some fixed value column-wise any and column bQnC1;j of LLL D BQLO reduced has the form matrix of bQ of i;j BLO by BQLO. The algorithm 2 f0; g, i D 1;2;:::;n, for repeats bQi;j with b replaced by 2 f0; g, ...

Author: Tsan-Ming Choi

Publisher: Springer

ISBN: 9783319535180

Page: 280

View: 904

This book focuses on optimal control and systems engineering in the big data era. It examines the scientific innovations in optimization, control and resilience management that can be applied to further success. In both business operations and engineering applications, there are huge amounts of data that can overwhelm computing resources of large-scale systems. This “big data” provides new opportunities to improve decision making and addresses risk for individuals as well in organizations. While utilizing data smartly can enhance decision making, how to use and incorporate data into the decision making framework remains a challenging topic. Ultimately the chapters in this book present new models and frameworks to help overcome this obstacle. Optimization and Control for Systems in the Big-Data Era: Theory and Applications is divided into five parts. Part I offers reviews on optimization and control theories, and Part II examines the optimization and control applications. Part III provides novel insights and new findings in the area of financial optimization analysis. The chapters in Part IV deal with operations analysis, covering flow-shop operations and quick response systems. The book concludes with final remarks and a look to the future of big data related optimization and control problems.
2017-05-04

# Geodesy the Challenge of the 3rd Millennium of the GPS world, which will simply be called the LLL algorithm in the rest of this paper. The basic idea of the LLL method is to reach the goal of decorrelation by integer Gram-Schmidt orthogonalization. Because the LLL algorithm was ...

Author: Erik Grafarend

Publisher: Springer Science & Business Media

ISBN: 9783662052969

Category: Science

Page: 474

View: 637

Geodesy as the science which determines the figure of the earth, its orientation in space and its gravity field as well as its temporal changes, produces key elements in describing the kinematics and the dynamics of the deformable body "earth". It contributes in particular to geodynamics and opens the door to decode the complex interactions between components of "the system earth". In the breathtaking development recently a whole arsenal of new terrestrial, airborne as well as satelliteborne measurement techniques for earth sciences have been made available and have broadened the spectrum of measurable earth parameters with an unforeseen accuracy and precision, in particular to resolve the factor time. The book focusses on these topics and gives a state of the art of modern geodesy.
2013-12-11

# Interactive Theorem Proving A Formalization of the LLL Basis Reduction Algorithm Jose Divasón1, Sebastiaan Joosten2, René Thiemann3(B), and Akihisa Yamada4 1 University of La Rioja, Logro ̃no, Spain 2 University of Twente, Enschede, The Netherlands 3 University of ...

Publisher: Springer

ISBN: 9783319948218

Category: Mathematics

Page: 642

View: 826

This book constitutes the refereed proceedings of the 9th International Conference on Interactive Theorem Proving, ITP 2018, held in Oxford, UK, in July 2018. The 32 full papers and 5 short papers presented were carefully reviewed and selected from 65 submissions. The papers feature research in the area of logical frameworks and interactive proof assistants. The topics include theoretical foundations and implementation aspects of the technology, as well as applications to verifying hardware and software systems to ensure their safety and security, and applications to the formal verication of mathematical results. Chapters 2, 10, 26, 29, 30 and 37 are available open access under a Creative Commons Attribution 4.0 International License via link.springer.com.
2018-07-03

# Handbook of Floating Point Arithmetic the LLL algorithm is relatively simple to prove, but the complexity analysis is significantly more involved. We refer the interested reader to . The LLL algorithm has been extensively studied since its invention [324, 544, 576, ...

Author: Jean-Michel Muller

Publisher: Birkhäuser

ISBN: 9783319765266

Category: Mathematics

Page: 627

View: 683

Floating-point arithmetic is the most widely used way of implementing real-number arithmetic on modern computers. However, making such an arithmetic reliable and portable, yet fast, is a very difficult task. As a result, floating-point arithmetic is far from being exploited to its full potential. This handbook aims to provide a complete overview of modern floating-point arithmetic. So that the techniques presented can be put directly into practice in actual coding or design, they are illustrated, whenever possible, by a corresponding program. The handbook is designed for programmers of numerical applications, compiler designers, programmers of floating-point algorithms, designers of arithmetic operators, and more generally, students and researchers in numerical analysis who wish to better understand a tool used in their daily work and research.
2018-05-02

# Mathematics of Public Key Cryptography 17.6 Variants of the LLL algorithm There are many refinements of the LLL algorithm that are beyond the scope of the brief summary in this book. We list some of these now. r As mentioned earlier, it is necessary to use floating-point ...

Author: Steven D. Galbraith

Publisher: Cambridge University Press

ISBN: 9781107013926

Category: Computers

Page: 615

View: 416

This advanced graduate textbook gives an authoritative and insightful description of the major ideas and techniques of public key cryptography.
2012-03-15

# Elliptic Curves To explain the reduction procedure mentioned in Section A.2 we have to introduce the LLL-algorithm which, among other things, is very useful in solving numerical diophantine approximation problems. It was designed by Lenstra, ...

Author: Susanne Schmitt

Publisher: Walter de Gruyter

ISBN: 9783110168082

Category: Mathematics

Page: 367

View: 689

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.
2003

# Public Key Infrastructures Services and Applications Experimental parameters Input parameters of Algorithm 1 Selection of lattice Key dimension n bit length t frequency l reduction in Step 4 ... Similarly to the LLL algorithm, there are a number of variants for the BKZ algorithm.

Author: Sabrina De Capitani di Vimercati

Publisher: Springer

ISBN: 9783642400124

Category: Computers

Page: 195

View: 300