On the quantitative hardness of CVP. Bennett, H., Golovnev, A., & In FOCS, 2017.
$\newcommand{\eps}{ε} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{§VP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1}$For odd integers $p ≥ 1$ (and $p = ∞$), we show that the Closest Vector Problem in the $\ell_p$ norm ($\CVP_p$) over rank $n$ lattices cannot be solved in $2^{(1-\eps) n}$ time for any constant $\eps > 0$ unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of $p ≥ 1$, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of $\CVP_2$ (i.e., $\CVP$ in the Euclidean norm), for which a $2^{n +o(n)}$-time algorithm is known. In particular, our result applies for any $p = p(n) \neq 2$ that approaches $2$ as $n \to ∞$. We also show a similar SETH-hardness result for $§VP_∞$; hardness of approximating $\CVP_p$ to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for $\CVP_p$ and $\CVPP_p$ for any $1 ≤ p < ∞$ under different assumptions.
@inproceedings{BGSQuantitativeHardness17,
title = {On the quantitative hardness of {CVP}},
abstract = {$\newcommand{\eps}{\epsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1}$For odd integers $p \geq 1$ (and $p = \infty$), we show that the Closest Vector Problem in the $\ell_p$ norm ($\CVP_p$) over rank $n$ lattices cannot be solved in $2^{(1-\eps) n}$ time for any constant $\eps > 0$ unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of $p \geq 1$, not including the even integers. This comes tantalizingly close to settling the quantitative time complexity of the important special case of $\CVP_2$ (i.e., $\CVP$ in the Euclidean norm), for which a $2^{n +o(n)}$-time algorithm is known. In particular, our result applies for any $p = p(n) \neq 2$ that approaches $2$ as $n \to \infty$. We also show a similar SETH-hardness result for $\SVP_\infty$; hardness of approximating $\CVP_p$ to within some constant factor under the so-called Gap-ETH assumption; and other quantitative hardness results for $\CVP_p$ and $\CVPP_p$ for any $1 \leq p < \infty$ under different assumptions.},
url = {http://arxiv.org/abs/1704.03928},
booktitle = {FOCS},
author = {Bennett, Huck and Golovnev, Alexander and {Stephens-Davidowitz}, Noah},
year = {2017},
}