LLL Algorithm. A lattice reduction algorithm, named after discoverers Lenstra, Lenstra, and Lovasz (1982), that produces a lattice basis of short vectors. It was noticed by Lenstra et al. (1982) that the algorithm could be used to obtain factors of univariate polynomials, which amounts to the determination of integer relations the LLL algorithm, gives a (p2 3)n approximation ratio, where n is the dimension of the lattice. In many of the applications, this algorithm is applied for a constant n; in such cases, we obtain a constant approximation factor. In 1801, Gauss gave an algorithm that can be viewed as an algorithm for solving SVP in two dimensions A central tool in the algorithmic study of lattices (and their applications) is the LLL algorithm of Lenstra, Lenstra and Lovasz. The LLL algorithm runs in polynomial time and ﬁnds an approximate solution x to the shortest vector problem, in the sense that the length of the solution x found by the algorithm is a Lenstra-Lenstra-Lovasz (LLL) Algorithm is an approximation algorithm of the shortest vector problem, which runs in polynomial time and nds an approximation within an exponential factor of the correct answer The LLL Algorithm. The LLL algorithm was first realized in the 1980s by Lenstra, Lenstra, and Lovasz. Its original intent was not to break any cryptosystems, but to factor polynomials with rational coefficients. It also improved upon the lattice reduction algorithm in order to solve integer linear programming

* role in LLL basis reduction algorithm*. We start with a basis fb 1;b 2gand we try to reduce it. If b 1 is shorter than b 2 the intuitive approach is to substract from b 2 an integer multiple zof b 1. We want to choose zsuch that the new vector b 2 zb 1 is as short as possible. To solve this problem we take for zthe coe cient uof the orthogonal projection of b 2 on b 1 (c Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a spigot algorithm for . Although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm [2] can find a short (not necessarily shortest) basis in polynomial time with guaranteed worst-case performance PLL Algorithms (Permutation of Last Layer) Developed by Feliks Zemdegs and Andy Klise Algorithm Presentation Format Suggested algorithm here Alternative algorithms here PLL Case Name - Probability = 1/x Permutations of Edges Only R2 U (R U R' U') R' U' (R' U R') y2 (R' U R' U') R' U' (R' U R U) R2' Ub - Probability = 1/1

There are 57 different OLL variations, therefore needed 57 different algorithms to learn in order to complete the OLL step in just 1 algorithm. It is best to start with 2 look OLL and navigate your way around the full OLL (Learn 2 Look OLL). The algorithms are divided into groups based on the shapes they form on the U face Computational aspects of geometry of numbers have been revolutionized by the Lenstra-Lenstra-Lov´asz lattice reduction algorithm (LLL), which has led to break- throughs in ﬁelds as diverse as computer algebra, cryptology, and algorithmic number theory 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

Pris: 1769 kr. Häftad, 2012. Skickas inom 7-10 vardagar. Köp The LLL Algorithm av Phong Q Nguyen, Brigitte Vallee på Bokus.com The LLL basis reduction algorithm was the ﬁrst polynomial-time algorithm to compute a reducedbasisofagivenlattice,andhencealsoashortvectorinthelattice.Itapproximatesan NP-hard problem where the approximation quality solely depends on the dimension of the lattice, but not the lattice itself. The algorithm has applications in number theory, compute 1 [LLL 1982] uses a p-adic root, while [Sch onhage 1984] uses a real or complex root. Both work in polynomial time. 2 Neither was usedin computer algebra systems; [Zassenhaus 1969] (not polynomial time!) is usually much faster. 3 Faster algorithm [vH 2002]: apply LLL to a much smaller lattice The LLL algorithm works as follows: given an integral input basis B 2Zn n (the integrality condition is without loss of generality), do the following: 1.Compute Be, the Gram-Schmidt orthogonalized vectors of B. 2.Let B SizeReduce(B). (This algorithm, deﬁned below, ensures that the basis is size reduced, and does not change L(B) or Be.) 3

- Building Lattice Reduction (LLL) Intuition. 2017-07-25. The Lenstra-Lenstra-Lovász (LLL) algorithm is an algorithm that efficiently transforms a bad basis for a lattice L into a pretty good basis for the same lattice. This transformation of a bad basis into a better basis is known as lattice reduction, and it has useful applications
- The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors
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- 1 The History of the LLL-Algorithm . 2 Hermite's Constant and Lattice Algorithms. 3 Probabilistic Analyses of Lattice Reduction Algorithms. 4 Progress on LLL and Lattice Reduction . 5 Floating-Point LLL: Theoretical and Practical Aspects. 6 LLL: A Tool for Effective Diophantine Approximation. 7 Selected Applications of LLL in Number Theor
- Der LLL-Algorithmus ist ein nach Arjen Lenstra, Hendrik Lenstra und László Lovász benannter Algorithmus, der für ein Gitter eine Basis aus möglichst kurzen Vektoren berechnet. Diese Vektoren sind Approximationen für die kürzesten voneinander linear unabhängigen Vektoren des Gitters. Bei seiner Entdeckung war der LLL-Algorithmus der erste effiziente Gitterreduktionsalgorithmus

- The LLL Algorithm [Elektronisk resurs] Survey and Applications / edited by Phong Q. Nguyen, Brigitte Vallée. Nguyen, Phong Q. (författare) Vallée, Brigitte. (författare) SpringerLink (Online service) ISBN 9783642022951 Publicerad: Berlin, Heidelberg : Springer Berlin Heidelberg, 201
- L'algorithme LLL procède à une réduction de base de réseau. Il prend en entrée un nombre d de vecteurs de base d'un réseau, tels que ces vecteurs soient de dimension n et de norme inférieure à B, et retourne en sortie une base de réseau LLL-réduite, c'est-à-dire presque orthogonale, en temp
- Lee1, Alice Pellet-Mary2, Damien Stehl e1;3, and Alexandre Wallet4 1 Univ. Lyon, EnsL, UCBL, CNRS, Inria, LIP, F-69342 Lyon Cedex 07, France fchang
- The Formalized LLL Algorithm In this section we brie y review the existing Isabelle/HOL formalization of the LLL algorithm [5], focusing only on the process of formally verifying its correctness; for an explanation of the algorithm itself, we refer to [25] or [14]. The algorithm to be formalized is given as pseudo-code in Algorithm1. Here, bxe.
- LLL algorithm in the case when the approximation parameter t of the algorithm has its extreme value 1. This case is of interest because the output is then the strongest Lova´sz-reduced basis. Experiments reported by Lagarias and Odlyzko [LO83] seem to show that the algorithm remain polynomial in average

Algorithmic LLL Here we discuss an algorithm for applications of the LLL, found in the breakthrough work of [Moser and Tardos (2010)]. The setup. the algorithm is a stable set sequence (note that it must be nite, since for a xed root E(t) i the induced subgraph R(t) i is nite) 3. The LLL algorithm The LLL algorithm alternates two steps, aimed at achieving the two properties of an LLL reduced basis. Once we have size-reduced the input basis B, there is only one way B can fail to be LLL reduced: violate the second condition, i.e., kˇ i(b i)k2 >kˇ i(b i+1)k2 for some index i. If this happens, the algorithm swaps b i. The LLL Algorithm This book constitutes the refereed proceedings of the 4th International Algorithmic Number Theory Symposium, ANTS-IV, held in Leiden, The Netherlands, in July 2000. The book presents 36 contributed papers which have gone through a thorough round of reviewing, selection and revision. Also included are 4 invited survey papers 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.

The LLL algorithm introduced by Lenstra, Lenstra and Lov´asz in [20] was the ﬁrst algorithm to allow for an efﬁcient computation of a fairly well-reduced lattice basis in theory but suffered from stability and performance issues in practice L, then the running time of the standard LLL algorithm is O(nq5m3). Albeit polynomial, this is not negligible and does not allow any e cient im-plementation. Following a suggestion made by Odlyzko and independently by Schnorr, actual implementations of LLL reductions, including the one used by the authors, substitut

- code for LLL algorithm in pari/gp. Ask Question Asked 6 years, 9 months ago. Active 6 years ago. Viewed 726 times 1. I know that in PARI/GP the function qflll performs LLL algorithm on a set of bases. However, is it.
- This is the MATLAB code for the complex LLL (CLLL) algorithm: Ying Hung Gan, Cong Ling, and Wai Ho Mow, Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection, IEEE Trans. Signal Processing, vol. 57, pp. 2701-2710, July 2009
- The LLL algorithm for lattice reduction proved to be of a great deal of practical interest in many areas of compu-tational number theory and cryptography, as it (amongst other things) gives an approximate solution to the shortest vector problem, which is NP-hard [2], in polynomial time. I
- Présentation. L'algorithme LLL procède à une réduction de base de réseau.Il prend en entrée un nombre d de vecteurs de base d'un réseau, tels que ces vecteurs soient de dimension n et de norme inférieure à B, et retourne en sortie une base de réseau LLL-réduite, c'est-à-dire presque orthogonale, en temps ().. Pseudo-code. L'algorithme LLL repose sur l'algorithme de réduction.
- algorithm of Lenstra, Lenstra and Lovasz (the LLL algorithm). The upper bound is of the form O(n* log n), where n is the dimension of the problem. It is essentially independent of the length of the input vectors, so that, in any fixed dimension, the LLL algorithm turns out to be of complexity O(1
- The LLL algorithm has received a lot of attention as an effective numerical tool for preconditioning an integer least squares problem. However, the workings of the algorithm are not well understood. In this paper, we present a new way to look at the LLL reduction, which leads to a new implementation metho

LLL--Attack. LLL algorithm implementation using C++ LLL is used here to attack Merkele-Hellman cryptosystem which is based on knapsack problem. LLL is based on Network reduction method which transforms the base of an ititial network to a reduced base and also uses Gram-Schmidt process. useful link The LLL Algorithm: Survey and Applications December 2009. December 2009. Read More. Authors: Phong Q. Nguyen, ; Brigitte Vall

The Lenstra-Lenstra-Lovász lattice basis reduction algorithm (called LLL or ${\rm L}^3$) is a fundamental tool in computational number theory and theoretical computer science, which can be viewed as an efficient algorithmic version of Hermite's inequality on Hermite's constant The LLL algorithm is to Arjen Lenstra, Hendrik Lenstra and László Lovász named algorithm, which for a grid calculates a base of the shortest possible vectors. These vectors are approximations for the shortest linearly independent vectors of the grid. When it was discovered, the LLL algorithm was the first efficient lattice reduction algorithm LLL algorithm for codes appears to be an algorithmic realisation of the classic bound of Griesmer [Gri60], in the same way that LLL for lattices realizes Hermite's bound. These contributions establish an initial dictionary between reduction for codes and for lattices, summarised in Table 1 LLL algorithm. In this paper, we ﬁrst present the original LLL algorithm in Section 2 and the LLL algorithm with delayed size-reduction in Section 3, then propose a parallel LLL algorithm in Section 4 and its Pthread implementation in Section 5. Permission to make digital or hard copies of all or part of this work fo In the present paper we present a new fp LLL algorithm that relies on the computation of the QR-factorization of the basis using Householder's algorithm. H-LLL computes fp approximations to the coeﬃcients of the R-factor and uses them to perform exact operations on the basis. We prove that if the precision is large enough, then H-LLL runs.

** applications de LLL, on peut consulter par exemple [15] et [4]**. Pour la th eorie de la r eduction des r eseaux et de l'algorithme LLL, on peut consulter [7] ou encore [2]. Dans ce mini-court, on s'interesse a l'application de l'algorithme LLL en cryp-tographie et plus particuli erement a la cryptanalyse du cryptosyst eme propos e e LLL algorithm in NTL library. Ask Question Asked 6 years, 10 months ago. Active 6 years, 10 months ago. Viewed 1k times 2. I am learning to use the NTL library currently, especially for the LLL algorithm later. Does anyone have any idea of the usage of the LLL function in the NTL library? Thanks in. Algorithm LLL abbreviation meaning defined here. What does LLL stand for in Algorithm? Top LLL abbreviation related to Algorithm: Lenstra, Lenstra and Lovás The LLL algorithm is a well-known and widely used lattice basis reduction algorithm. In many applications, its speed is critical. Parallel computing can improve speed. However, the original LLL is sequential in nature

Implementations of extended LLL In systems like GP/PARI one uses an implementation of the LLL lattice reduction algorithm which allows dependent vectors and which also returns a transformation matrix. However, no good complexity analysis is available for their qflll algorithm (and similar algorithms). The method of the original LLL paper does not apply in any obvious way 1 The Nearest Plane Algorithm The algorithm has two main steps. First, it applies the LLL reduction to the input lattice. It then looks for an integer combination of the basis vectors that is close to the target vector t. This step is essentially the same as one inner loop in the reduction step of the LLL algorithm. INPUT: Basis B 2 Zm£n, t 2 Z

Computational aspects of geometry of numbers have been revolutionized by the Lenstra-Lenstra-Lovasz lattice reduction algorithm (LLL), which has led to bre- t ** Computational aspects of geometry of numbers have been revolutionized by the LenstraLenstraLovasz lattice reduction algorithm (LLL)**, which has led to bre- throughs in elds as diverse as computer algebra, cryptology, and algorithmic number theory LLL算法在密码分析中有许多应用，例如针对knapsack非对称密码算法的攻击、针对基于格的非对称密码算法（ Implementing LLL algorithm in Haskell. Debugging, haskell / By zjs. I'm implementing the LLL basis reduction algorithm in Haskell. I'm basing my code on the pseudocode on Wikipedia. Here is what I have so far. Apologies for the code dump; I strongly suspect the issue lies in lll but I'm giving everything just in case In other hand, these massive data needs processing speed. To meet these requirements, this paper provides modifications on NTRU public key cryptosystem to prevent LLL algorithm from recover the plain text from known cipher text.The method is based on a specific swapping process in the encryption process to thwart it

The LLL algorithm is a polynomial-time lattice reduction algorithm, named after its inventors, Arjen Lenstra, Hendrik Lenstra and Lszl Lovsz. 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 ** The LLL Algorithm von Nguyen, Phong Q**.|VallÃ Â©e, Brigitte und eine große Auswahl ähnlicher Bücher, Kunst und Sammlerstücke erhältlich auf AbeBooks.de The LLL algorithm has received a lot of attention as an effective numerical tool for preconditioning an integer least squares problem. However, the workings of the algorithm are not well understood. In this paper, we present a new way to look at the LLL reduction, which leads to a new implementation method that performs better than the original LLL scheme Among the existing LR algorithms, the fixed... Computationally efficient fixed complexity LLL algorithm for lattice‐reduction‐aided multiple‐input-multiple‐output precoding - Wang - 2016 - IET Communications - Wiley Online Librar Posts about LLL algorithm written by Michael N Powers. An Inequality for Polynomials Posted: June 25, 2012 | Author: Michael N Powers | Filed under: factoring algorithm | Tags: lattice, LLL algorithm, norm of a polynomial | Leave a comment It is natural to view the set of polynomials with real coefficients and degree less than or equal to as forming a real vector space with basis The.

- Summary changed from LLL algorithm not available for matrices over QQ or RR to LLL algorithm for matrices over QQ comment:14 Changed 6 years ago by cremona Under review
- 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
- g Karen Aardal, Friedrich Eisenbrand. 10. Using LLL-Reduction for Solving RSA and Factorization Problems Alexander May. 11. Practical Lattice-Based Cryptography: NTRUEncrypt and NTRUSign Jeff Hoffstein, Nick Howgrave-Graham, Jill Pipher, William Whyte
- The
**LLL****Algorithm**von Phong Q. Nguyen und eine große Auswahl ähnlicher Bücher, Kunst und Sammlerstücke erhältlich auf AbeBooks.de

The LLL Algorithm por Phong Q. Nguyen, 9783642022944, disponible en Book Depository con envío gratis Evolving from an elementary discussion, this book develops the Euclidean algorithm to a very powerful tool to deal with general continued fractions, non-normal tables, look-ahead algorithms for Hankel and Toeplitz matrices, and for Krylov subspace methods

The LLL algorithm A is a nonzero m × n matrix of integers, whose rows are LI and with at least two rows. Output: a unimodular matrix P and LLL(A), such that PA=LLL(A), where the rows of LLL(A) are LLL reduced. The LLL parameter α = m1/n1 has to satisfy 1/4 < α ≤ 1, but here the default value is 1 Lindep - LLL algorithm. In this version of the Lindep algorithm we use the lattice basis reduction algorithm of Lenstra-Lentsra-Lovasz (LLL). This algorithm finds a so called reduced basis of the lattice given by generators. The reduced basis contains an approximatio

1212 cases (including the solved / AUF states) which can be solved with 1211 algorithms. Since most speed solvers regard reflective symmetry and inversions as different cases (e.g. PLL is thought of as 21 algorithms rather than 13 algorithms plus mirrors and inverses), 1LLL should be thought of as 3915 algorithms Stanford Libraries' official online search tool for books, media, journals, databases, government documents and more 4 Look Last Layer Algorithms Developed by Feliks Zemdegs and Andy Klise Algorithm Presentation Format Edge Orientation F (U R U' R') F' Probability = 1/2 F (R U R' U') F' Probability = 1/4 F (R U R' U') F' U2 F (U R U' R') F' Probability = 1/8 Edges Already Oriented Probability = 1/8 Corner Orientation (R U R' U R U2 R') Probability = 4/2 The LLL algorithm [20] and its blockwise generalizations [36,8,10] are designed as polynomial-time Hermite-SVP algorithms. They achieve an approximation factor (1 + ε)n exponential in the lattice dimension n where ε > 0 depends on the algorithm and its parameters. This exponential factor can actually be made slightl

Introduction to LLL algorithm applied to linear modular inequalities. Ask Question Asked 5 months ago. Active 5 months ago. Viewed 460 times 7. 4 $\begingroup$ What is the Lenstra-Lenstra-Lovász lattice basis reduction algorithm about? How is it. The LLL algorithm is used to approximate the Shortest Vector Problem, i.e., it outputs a reduced basis. Such a basis will satisfy two conditions: $$ \forall i \gt j. \quad \lvert\mu_{ij}\rvert \le \frac{1}{2} \qquad\text{[size-reduced]} $$ $$ \forall i.\quad \lVert b_i^*\rVert^2 \le \lVert b_{i+1}^*\rVert^2 + \mu_{i+1,i}^2\lVert b_i^*\rVert^2 \qquad\text{[Lovász condition]}$ ** The bolded algorithm is the one that I use in my solving**. It doesn't say it's the best algorithm, just that I found it best working for me and my fingertricks, the other algorithms are also used by speedcubers. The sequence in is the last part of the solving, when the edge-corner pieces are being inserted to the block

The only available alternative is lattice-based schemes based on the LLL algorithm; no one has yet devised strategies, even using quantum computers, that would be able to crack them T1 - A verified LLL algorithm. AU - Divasón, Jose. AU - Joosten, Sebastiaan. AU - Thiemann, René. AU - Yamada, Akihisa. PY - 2018. Y1 - 2018. N2 - The Lenstra-Lenstra-Lovász basis reduction algorithm, also known as LLL algorithm, is an algorithm to find a basis with short, nearly orthogonal vectors of an integer lattice In the present paper we present a new fp **LLL** **algorithm** that relies on the computation of the QR-factorization of the basis using Householder's **algorithm**. H-LLL computes fp approximations to the coeﬃcients of the R-factor and uses them to perform exact operations on the basis. We prove that if the precision is large enough, then H-LLL runs.

** However**, the known polynomial bounds for computational complexity are shown only for parameter $\delta < 1$; for \lq\lq optimal\rq\rq{} parameter $\delta = 1$ which ensures the best output quality, no polynomial bounds are known, and except for LLL algorithm, it is even not formally proved that the algorithm always halts within finitely many steps On Complex LLL Algorithm for Integer Forcing Linear Receivers A. Sakzad, J. Harshan, and E. Viterbo, Department of ECSE, Monash University, Australia {amin.sakzad, harshan.jagadeesh, and emanuele.viterbo}@monash.edu Abstract-Integer-forcing (IF) linear receiver has been recently introduced for multiple-input multiple-output (MIMO) fadin So, reading thisbook entitled Free Download The LLL Algorithm: Survey and Applications (Information Security and Cryptography) By does not need mush time. You will enjoy following this book while spent your free time. Theexpression in this word allows the ereader believe to scan and read this book again and also Integer Least Squares: Sphere Decoding and the LLL Algorithm ∗ Sanzheng Qiao† Department of Computing and Software, McMaster University, 1280 Main St. West Hamilton, Ontario, L8S 4L7, Canada. ABSTRACT This paper considers the problem of integer least squares, where the least squares solution is an integer vector, whereas the coeﬃcient. the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Zn to Kn, and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to ﬂoating-point approximations). 1 Introductio

This paper proposes a high-efficient preprocessing algorithm for 16 × 16 MIMO detections. The proposed algorithm combines a sorting-relaxed QR decomposition (SRQRD) and a modified greedy LLL (MGLLL) algorithm. First, SRQRD is conducted to decompose the channel matrices. This decomposition adopts a relaxed sorting strategy together with a paralleled Givens Rotation (GR) array scheme, which can. Matrices over QQ do not have a method LLL(...) like matrices over ZZ.Here is a simple implementation: get rid of the denominator call the LLL method of integer matrices ; set the denominator back. Note: in previous version it was suggested that such an approach might be useful for matrices over RR.But it is not completely clear how to do that efficiently The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lovász Variants of the LLL Algorithm in Digital Communications: Complexity Analysis and Fixed-Complexity Implementation. Wai Ho Mow. Related Papers. Dual-lattice ordering and partial lattice reduction for SIC-based MIMO detection. By Wai Ho Mow. Complex Lattice Reduction Algorithm for Low-Complexity MIMO Detection Boston University Libraries. Services . Navigate; Linked Data; Dashboard; Tools / Extras; Stats; Share . Social. Mai

Three Ls of an algorithm. Coming back to Lovász's work, let's have a look at one of his most famous results, which fits right in with the things we've explored so far: the LLL algorithm, named after Lovász and the brothers Arjen and Hendrik Lenstra. To see what it's about, imagine a dot on a piece of paper. Draw two arrows starting at that dot Authors: Changmin Lee Alice Pellet-Mary Damien Stehlé Alexandre Wallet: Download: DOI: 10.1007/978-3-030-34621-8_3 Search ePrint Search Google: Abstract: The LLL algorithm takes as input a basis of a Euclidean lattice, and, within a polynomial number of operations, it outputs another basis of the same lattice but consisting of rather short vectors Download PDF Abstract: The Lenstra-Lenstra-Lovász (LLL) algorithm is the most practical lattice reduction algorithm in digital communications. In this paper, several variants of the LLL algorithm with either lower theoretic complexity or fixed-complexity implementation are proposed and/or analyzed

A common technique to perform lattice basis reduction is the Lenstra, Lenstra, Lovasz (LLL) algorithm. An implementation of this algorithm in real-time systems suffers from the problem of variable run-time and complexity. This correspondence proposes a modification of the LLL algorithm. The signal flow is altered to follow a deterministic structure, which promises to obtain an easier. 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.. LLL algorithm possibly alters the lattice basis R~, Q~, and T (corresponding to a column swap), which involves a number of costly computations. Since the number of column swaps is approximately proportional to the time required for processing a matrix on a given architecture (because the computationa LLL algorithm, which takes an arbitrary lattice basis as its input, and returns a vector within a factor of 2O(n) of the lattice's shortest overall vector. 2 The Lenstra-Lenstra-Lovasz algorithm The Lenstra-Lenstra-Lovasz (LLL) algorithm [14] was originally presented as a method for factoring poly I don't know of any strict theoretical proof of this connection, but extensive experiments as made by Gama, Nguyen and Stehlé (cf. P. Nguyen, D. Stehlé, LLL on average, Algorithmic Number Theory 4076 (2006), pp.1-17 or N. Gama, P. Nguyen, Predicting lattice reduction, Proc. of the EUROCRYPT 2008, Vol. 4965, pp.31-51) show that for LLL with $\delta=0.999$ the average Hermite factor, taking.