(Gap/S)ETH Hardness of SVP

TL;DR

This paper establishes strong computational hardness results for the Shortest Vector Problem in various $oldsymbol{ ext{l}_p}$ norms, showing no efficient algorithms exist under standard complexity hypotheses for almost all $p > 2$, including approximation variants.

Contribution

It provides the first comprehensive hardness results for $ ext{SVP}_p$ across the entire range of $p$, linking algorithmic limits to the Strong Exponential Time Hypothesis and Gap-ETH.

Findings

No $2^{n/C_p}$-time algorithms for almost all $p > p_0 eq 2$ unless SETH is false.

No $2^{o(n)}$-time algorithms for $p > 2$ unless Gap-ETH is false.

No $2^{o(n)}$-time algorithms for $1 \\leq p \\leq 2$ unless Gap-ETH is false or certain lattice properties fail.

Abstract

$ \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\SVP}{\problem{SVP}} \newcommand{\ensuremath}[1]{#1} $We prove the following quantitative hardness results for the Shortest Vector Problem in the $\ell_p$ norm ($\SVP_p$), where $n$ is the rank of the input lattice. $\bullet$ For "almost all" $p > p_0 \approx 2.1397$, there no $2^{n/C_p}$-time algorithm for $\SVP_p$ for some explicit constant $C_p > 0$ unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. $\bullet$ For any $p > 2$, there is no $2^{o(n)}$-time algorithm for $\SVP_p$ unless the (randomized) Gap-Exponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each $p > 2$, there exists a constant $\gamma_p > 1$ such that the same result holds even for $\gamma_p$-approximate $\SVP_p$. $\bullet$ There is no $2^{o(n)}$-time algorithm for $\SVP_p$ for any $1 \leq p \leq 2$ unless either (1) (non-uniform) Gap-ETH is false; or (2) there is no family of lattices with exponential kissing number in the $\ell_2$ norm. Furthermore, for each $1 \leq p \leq 2$, there exists a constant $\gamma_p > 1$ such that the same result holds even for $\gamma_p$-approximate $\SVP_p$.