# On the reduction of a random basis

• Volume: 13, page 437-458
• ISSN: 1292-8100

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## Abstract

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For p ≤ n, let b1(n),...,bp(n) be independent random vectors in ${ℝ}^{n}$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If ${\stackrel{^}{b}}_{1}^{\left(n\right)},...,{\stackrel{^}{b}}_{p}^{\left(n\right)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors ${r}_{j}^{\left(n\right)}=\parallel {\stackrel{^}{b}}_{n-j+1}^{\left(n\right)}{\parallel }^{2}/{\parallel {\stackrel{^}{b}}_{n-j}^{\left(n\right)}\parallel }^{2}$, j = 1,...,p - 1. We show that as n → +∡ the process $\left({r}_{j}^{\left(n\right)}-1,j\ge 1\right)$ tends in distribution in some sense to an explicit process $\left({ℛ}_{j}-1,j\ge 1\right)$; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.

## How to cite

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Akhavi, Ali, Marckert, Jean-François, and Rouault, Alain. "On the reduction of a random basis." ESAIM: Probability and Statistics 13 (2009): 437-458. <http://eudml.org/doc/250640>.

@article{Akhavi2009,
abstract = {For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb\{R\}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_\{1\}^\{(n)\},\ldots, \widehat b_p^\{(n)\}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors $r_j^\{(n)\} = \Vert \widehat b^\{(n)\}_\{n-j+1\}\Vert^2 / \Vert \widehat b^\{(n)\}_\{n-j\} \Vert^2$, j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^\{(n)\}-1,j\geq 1)$ tends in distribution in some sense to an explicit process $(\{\mathcal R\}_j -1,j\geq 1)$; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞. },
author = {Akhavi, Ali, Marckert, Jean-François, Rouault, Alain},
journal = {ESAIM: Probability and Statistics},
keywords = {Random matrices; random basis; orthogonality index; determinant; lattice reduction.; random matrices; lattice reduction; beta distribution; Gram-Schmidt orthogonalization},
language = {eng},
month = {9},
pages = {437-458},
publisher = {EDP Sciences},
title = {On the reduction of a random basis},
url = {http://eudml.org/doc/250640},
volume = {13},
year = {2009},
}

TY - JOUR
AU - Akhavi, Ali
AU - Marckert, Jean-François
AU - Rouault, Alain
TI - On the reduction of a random basis
JO - ESAIM: Probability and Statistics
DA - 2009/9//
PB - EDP Sciences
VL - 13
SP - 437
EP - 458
AB - For p ≤ n, let b1(n),...,bp(n) be independent random vectors in $\mathbb{R}^n$ with the same distribution invariant by rotation and without mass at the origin. Almost surely these vectors form a basis for the Euclidean lattice they generate. The topic of this paper is the property of reduction of this random basis in the sense of Lenstra-Lenstra-Lovász (LLL). If $\widehat b_{1}^{(n)},\ldots, \widehat b_p^{(n)}$ is the basis obtained from b1(n),...,bp(n) by Gram-Schmidt orthogonalization, the quality of the reduction depends upon the sequence of ratios of squared lengths of consecutive vectors $r_j^{(n)} = \Vert \widehat b^{(n)}_{n-j+1}\Vert^2 / \Vert \widehat b^{(n)}_{n-j} \Vert^2$, j = 1,...,p - 1. We show that as n → +∡ the process $(r_j^{(n)}-1,j\geq 1)$ tends in distribution in some sense to an explicit process $({\mathcal R}_j -1,j\geq 1)$; some properties of the latter are provided. The probability that a random random basis is s-LLL-reduced is then showed to converge for p = n - g, and g fixed, or g = g(n) → +∞.
LA - eng
KW - Random matrices; random basis; orthogonality index; determinant; lattice reduction.; random matrices; lattice reduction; beta distribution; Gram-Schmidt orthogonalization
UR - http://eudml.org/doc/250640
ER -

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