Lattice reduction algorithms have numerous applications in number theory, algebra, as well as in cryptanalysis. The most famous algorithm for lattice reduction is the LLL algorithm. In polynomial time it computes a reduced basis with provable output quality. One early improvement of the LLL algorithm was LLL with deep insertions (DeepLLL). The output of this version of LLL has higher quality in...

The LLL algorithm, named after its inventors, Lenstra, Lenstra and Lovász, is one of themost popular lattice reduction algorithms in the literature. In this paper, we propose the first variant of LLL algorithm that is dedicated for ideal lattices, namely, the iLLL algorithm. Our iLLL algorithm takes advantage of the fact that within LLL procedures, previously reduced vectors can be re-used for ...

We present an efficient variant of LLL-reduction of lattice bases in the sense of Lenstra, Lenstra, Lovász [LLL82]. We organize LLL-reduction in segments of size k. Local LLL-reduction of segments is done using local coordinates of dimension 2k. Strong segment LLL-reduction yields bases of the same quality as LLL-reduction but the reduction is n-times faster for lattices of dimension n. We exte...

In this paper, a modied version of LLL algorithm, which is a an algorithm with output-sensitivecomplexity, is presented to convert a given Grobner basis with respect to a specic order of a polynomialideal I in arbitrary dimensions to a Grobner basis of I with respect to another term order.Also a comparison with the FGLM conversion and Buchberger method is considered.

6 The LLL algorithm has received a lot of attention as an effective numerical tool for preconditioning 7 an integer least squares problem. However, the workings of the algorithm are not well understood. In this 8 paper, we present a new way to look at the LLL reduction, which leads to a new implementation method 9 that performs better than the original LLL scheme. 10 © 2007 Published by Elsevie...

This paper presents a study of the LLL algorithm from perspective statistical physics. Based on our experimental and theoretical results, we suggest that interpreting as sandpile model may help understand much its mysterious behavior. In language physics, work evidence certain 1-d models with simpler toppling rules belong to same universality class. consists three parts. First, introduce whose ...

Reduction can be important to aid quickly attaining the integer least squares (ILS) estimate from noisy data. We present an improved Lenstra-Lenstra-Lovasz (LLL) algorithm with fixed complexity by extending a parallel reduction method for positive definite quadratic forms to lattice vectors. We propose the minimum angle of a reduced basis as an alternative quality measure of orthogonality, whic...

We surview variants and extensions of the LLL-algorithm of Lenstra, Lenstra Lovász, extensions to quadratic indefinite forms and to faster and stronger reduction algorithms. The LLL-algorithm with Householder orthogonalisation in floating-point arithmetic is very efficient and highly accurate. We surview approximations of the shortest lattice vector by feasible lattice reduction, in particular ...

No efficient algorithm is known to find the shortest vector in a lattice (in arbitrary dimension), or even just computing its length λ1. 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 th...