eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2016-08-18
9:1
9:17
10.4230/LIPIcs.ESA.2016.9
article
On the Lattice Distortion Problem
Bennett, Huck
Dadush, Daniel
Stephens-Davidowitz, Noah
We introduce and study the Lattice Distortion Problem (LDP). LDP asks how "similar" two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal
distortion is one.
As our first contribution, we show that the distortion between any two lattices is approximated up to a n^{O(log(n))} factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a 2^{O(n*log(log(n))/log(n))} factor of optimal in polynomial time and within a n^{O(log(n))} factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol057-esa2016/LIPIcs.ESA.2016.9/LIPIcs.ESA.2016.9.pdf
lattices
lattice distortion
lattice isomoprhism
geometry of numbers
basis reduction