An LLL-reduction algorithm with quasi-linear time complexity

Andrew Novocin 1, 2 Damien Stehlé 1, 2 Gilles Villard 1, 2
2 ARENAIRE - Computer arithmetic
Inria Grenoble - Rhône-Alpes, LIP - Laboratoire de l'Informatique du Parallélisme
Abstract : We devise an algorithm, L1 tilde, with the following specifications: It takes as input an arbitrary basis B=(b_i)_i in Z^{d x d} of a Euclidean lattice L; It computes a basis of L which is reduced for a mild modification of the Lenstra-Lenstra-Lovász reduction; It terminates in time O(d^{5+eps} beta + d^{omega+1+eps} beta^{1+eps}) where beta = log max ||b_i|| (for any eps > 0 and omega is a valid exponent for matrix multiplication). This is the first LLL-reducing algorithm with a time complexity that is quasi-linear in the bit-length beta of the entries and polynomial in the dimension d. The backbone structure of L1 tilde is able to mimic the Knuth-Schönhage fast gcd algorithm thanks to a combination of cutting-edge ingredients. First the bit-size of our lattice bases can be decreased via truncations whose validity are backed by recent numerical stability results on the QR matrix factorization. Also we establish a new framework for analyzing unimodular transformation matrices which reduce shifts of reduced bases, this includes bit-size control and new perturbation tools. We illustrate the power of this framework by generating a family of reduction algorithms.
Document type :
Conference papers
STOC'11 - 43rd annual ACM symposium on Theory of computing, 2011, San Jose, United States. ACM New York, NY, USA, pp.403-412, 2011, <10.1145/1993636.1993691>
Liste complète des métadonnées

https://hal-ens-lyon.archives-ouvertes.fr/ensl-00534899
Contributor : Gilles Villard <>
Submitted on : Thursday, April 7, 2011 - 4:13:04 PM
Last modification on : Monday, January 9, 2012 - 2:29:32 PM

File

L1-hal.pdf
Files produced by the author(s)

Identifiers

Collections

Citation

Andrew Novocin, Damien Stehlé, Gilles Villard. An LLL-reduction algorithm with quasi-linear time complexity. STOC'11 - 43rd annual ACM symposium on Theory of computing, 2011, San Jose, United States. ACM New York, NY, USA, pp.403-412, 2011, <10.1145/1993636.1993691>. <ensl-00534899v2>

Share

Metrics

Record views

759

Document downloads

847