# On the Quantitative Hardness of CVP

## TL;DR

This paper establishes strong conditional lower bounds on the time complexity of solving the Closest Vector Problem (CVP) in various $oldsymbol{ extit{l}_p}$ norms, showing it cannot be done in subexponential time unless widely believed complexity hypotheses fail.

## Contribution

It proves SETH-based hardness for $ ext{CVP}_p$ for odd $p
eq 2$, extending to almost all $p
eq 2$, and provides related hardness results for $ ext{SVP}_ ext{infinity}$ and approximation problems.

## Key findings

- $ ext{CVP}_p$ cannot be solved in $2^{(1- ext{eps})n}$ time for odd $p
eq 2$ unless SETH fails.
- Hardness results extend to almost all $p
eq 2$, approaching the Euclidean case.
- Additional hardness results for $ ext{SVP}_ ext{infinity}$ and approximation variants under different hypotheses.

## Abstract

$ \newcommand{\eps}{\varepsilon} \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.