Tensor-based Hardness of the Shortest Vector Problem to within Almost Polynomial Factors

TL;DR

This paper proves stronger hardness of approximation results for the Shortest Vector Problem in lattices using tensor products, improving previous bounds and analyzing lattice behavior under tensorization.

Contribution

The paper introduces a novel analysis of lattice tensorization to establish tighter hardness of approximation bounds for SVP.

Findings

Hardness factor of $2^{(\log n)^{1-\epsilon}}$ under NP $ ot ightarrow$ RTIME

Hardness factor of $n^{c/\log\log n}$ under NP $ ot ightarrow$ RSUBEXP

Lattices behave well under tensorization, enabling improved hardness proofs.

Abstract

$ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\eps}{\epsilon} \newcommand{\poly}{\mathop{\mathrm{poly}}} $We show that unless $\NP \subseteq \RTIME (2^{\poly(\log{n})})$, there is no polynomial-time algorithm approximating the Shortest Vector Problem ($\SVP$) on $n$-dimensional lattices in the $\ell_p$ norm ($1 \leq p< \infty$) to within a factor of $2^{(\log{n})^{1-\eps}}$ for any $\eps > 0$. This improves the previous best factor of $2^{(\log{n})^{1/2-\eps}}$ under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption $\NP \nsubseteq \RSUBEXP$, we obtain a hardness factor of $n^{c/\log\log{n}}$ for some $c> 0$. Our proof starts with Khot's $\SVP$ instances that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novelty is in the analysis, where we show that the lattices of Khot behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski (2006).