On the Lattice Distortion Problem

TL;DR

This paper introduces the Lattice Distortion Problem (LDP), analyzing how similar two lattices are through minimal distortion, providing approximation algorithms, and proving NP-hardness of approximation.

Contribution

It establishes approximation bounds for lattice distortion using basis reduction and relates LDP to the Seysen basis, also proving NP-hardness of approximation.

Findings

Approximate lattice distortion within a polynomial factor using constructive methods

Relate Seysen basis reduction to lattice distortion measures

Prove NP-hardness of approximating LDP within any constant factor

Abstract

We introduce and study the \emph{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.