Paper 2013/685
Solving shortest and closest vector problems: The decomposition approach
Anja Becker, Nicolas Gama, and Antoine Joux
Abstract
In this paper, we present a heuristic algorithm for solving exact,
as well as approximate, shortest vector and closest vector problems on lattices. The algorithm can be seen
as a modified sieving algorithm for which the vectors of the intermediate
sets lie in overlattices or translated cosets of overlattices.
The key idea is hence to no longer work with a single lattice but to move the problems around in
a tower of related lattices. We initiate the algorithm by sampling
very short vectors in an overlattice of the original lattice that
admits a quasi-orthonormal basis and hence an efficient enumeration
of vectors of bounded norm. Taking sums of vectors in the sample, we
construct short vectors in the next lattice. Finally, we
obtain solution vector(s) in the initial lattice as a sum of vectors
of an overlattice. The complexity analysis relies on
the Gaussian heuristic. This heuristic is backed by experiments in
low and high dimensions that closely reflect these estimates when
solving hard lattice problems in the average case.
This new approach allows us
to solve not only shortest vector problems, but also closest vector
problems, in lattices of dimension
Note: The great part of the paper is published at ANTS 2014 under the title ``A Sieve Algorithm Based on Overlattices". We added here the section ``Example for co-cyclic lattices or q-ary lattices" which gives a concrete example of a tower of lattices one might consider at first trial.
Metadata
- Available format(s)
-
PDF
- Category
- Foundations
- Publication info
- Published elsewhere. Minor revision. ANTS2014: Eleventh Algorithmic Number Theory Symposium ANTS-XI
- Keywords
- latticeshortest vector problemclosest vector problemdecomposition techniquestructural reductionsieving
- Contact author(s)
- anja becker @ epfl ch
- History
- 2014-06-13: last of 3 revisions
- 2013-10-24: received
- See all versions
- Short URL
- https://ia.cr/2013/685
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2013/685, author = {Anja Becker and Nicolas Gama and Antoine Joux}, title = {Solving shortest and closest vector problems: The decomposition approach}, howpublished = {Cryptology {ePrint} Archive, Paper 2013/685}, year = {2013}, url = {https://eprint.iacr.org/2013/685} }