Deterministic Hardness of Approximation For SVP in all Finite $\ell_p$ Norms

TL;DR

This paper proves that, assuming a certain complexity hypothesis, approximating the shortest vector problem in any finite 3p norm within a super-polynomial factor is deterministically hard, extending known results beyond the Euclidean case.

Contribution

It establishes the first deterministic hardness of approximation results for SVP in all finite p norms, generalizing prior Euclidean-specific findings.

Findings

SVP in all finite p norms is hard to approximate within 2^{(\,log n)^{1 - o(1)}} under certain complexity assumptions.

The result applies to lattices of rank n and extends deterministic hardness from Euclidean to all finite p norms.

Previously, only Euclidean case p=2 was known for exact SVP hardness under deterministic reductions.

Abstract

We show that, assuming NP $\not\subseteq$ $\cap_{\delta > 0}$DTIME$\left(\exp{n^\delta}\right)$, the shortest vector problem for lattices of rank $n$ in any finite $\ell_p$ norm is hard to approximate within a factor of $2^{(\log n)^{1 - o(1)}}$, via a deterministic reduction. Previously, for the Euclidean case $p=2$, even hardness of the exact shortest vector problem was not known under a deterministic reduction.