منابع مشابه
Adaptive precision LLL and Potential-LLL reductions with Interval arithmetic
Lattice reduction is fundamental in computational number theory and in computer science, especially in cryptography. The celebrated Lenstra–Lenstra–Lovász reduction algorithm (called LLL or L) has been improved in many ways through the past decades and remains one of the central tool for reducing lattice basis. In particular, its floating-point variants — where the long-integer arithmetic requi...
متن کاملLecture 3 Lll, Coppersmith
The idea behind the SizeReduce(B) subroutine is, in the Gram-Schmidt decomposition B = B̃ ·U, to shift the entries in the upper triangle of U by integers (via unimodular transformations), so that they lie in [−12 , 1 2). Because changing an entry of U may affect the ones above it (but not below it) in the same column, we must make the changes upward in each column. Formally, the algorithm works ...
متن کاملUnifying LLL inequalities
The Lenstra, Lenstra, and Lovász (abbreviated as LLL) basis reduction algorithm computes a basis of a lattice consisting of short, and near orthogonal vectors. The quality of an LLL reduced basis is expressed by three fundamental inequalities, and it is natural to ask, whether these have a common generalization. In this note we find unifying inequalities. Our main result is Theorem 1. Let b1, ....
متن کاملSVP , Gram - Schmidt , LLL
Last time we defined the minimum distance λ1(L) of a lattice L, and showed that it is upper bounded by √ n · det(L)1/n (Minkowski’s theorem), but this bound is often very loose. Some natural computational questions are: given a lattice (specified by some arbitrary basis), can we compute its minimum distance? Can we find a vector that achieves this distance? Can we find good approximations to th...
متن کاملAn improved LLL algorithm
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...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2004
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2004.02.003