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Mathematics of Computation (2012) 0-0
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Perturbation Analysis of the QR Factor R in the Context of LLL Lattice Basis Reduction
Xiao-Wen Chang1, Damien Stehlé2, Gilles Villard3, 4

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.
1 :  SOCS - School of Computer Science [Quebec]
2 :  Macquarie University
3 :  LIP - Laboratoire de l'Informatique du Parallélisme
4 :  Inria Grenoble Rhône-Alpes / LIP Laboratoire de l'Informatique du Parallélisme - ARENAIRE
Lattice reduction – LLL – QR factorization – Perturbation analysis