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- W2606133313 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." @default.
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- W2606133313 date "2017-04-12" @default.
- W2606133313 modified "2023-09-27" @default.
- W2606133313 title "On the Quantitative Hardness of CVP" @default.
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