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Volume 8 (2012) Article 23 pp. 513-531
Tensor-based Hardness of the Shortest Vector Problem to within Almost Polynomial Factors
Received: August 26, 2011
Published: September 25, 2012
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Keywords: lattices, shortest vector problem, NP-hardness, hardness of approximation
ACM Classification: F.2.2, F.1.3, G.1.6
AMS Classification: 68Q17, 52C07, 11H06, 11H31, 05B40

Abstract: [Plain Text Version]

$ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RP}{\mathsf{RP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\poly}{\mathop{\mathrm{poly}}} \newcommand{\eps}{\varepsilon} $

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 novel part 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.