Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction

Xiao-Wen Chang Damien Stehlé 1, 2, * Gilles Villard 1, 2, *
* Corresponding author
2 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : In 1982, Arjen Lenstra, Hendrik Lenstra Jr. and László Lovász introduced an efficiently computable notion of reduction of basis of a Euclidean lattice that is now commonly referred to as LLL-reduction. The precise definition involves the R-factor of the QR factorisation of the basis matrix. A natural mean of speeding up the LLL reduction algorithm is to use a (floating-point) approximation to the R-factor. In the present article, we investigate the accuracy of the factor R of the QR factorisation of an LLL-reduced basis. Our main contribution is the first fully rigorous perturbation analysis of the R-factor of LLL-reduced matrices under column-wise perturbations. Our results should be very useful to devise LLL-type algorithms relying on floating-point approximations.
Complete list of metadatas

Cited literature [21 references]  Display  Hide  Download
Contributor : Gilles Villard <>
Submitted on : Thursday, April 7, 2011 - 4:28:46 PM
Last modification on : Thursday, February 7, 2019 - 4:59:27 PM
Long-term archiving on : Friday, July 8, 2011 - 2:38:04 AM


Files produced by the author(s)


  • HAL Id : ensl-00529425, version 2



Xiao-Wen Chang, Damien Stehlé, Gilles Villard. Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction. Mathematics of Computation, American Mathematical Society, 2012, 81 (279), pp.1487-1511. ⟨ensl-00529425v2⟩



Record views


Files downloads