Improved hardness results for unique shortest vector problem

TL;DR

This paper advances the understanding of the computational hardness of the unique shortest vector problem (uSVP) by providing new reductions, hardness results, and complexity class implications, including NP-hardness and coNP membership under various conditions.

Contribution

It introduces new deterministic and randomized reductions for uSVP, establishes NP-hardness in specific norms, and simplifies previous complexity class results related to uSVP.

Findings

Deterministic reduction from SVP to uSVP.

NP-hardness of uSVP in the ty norm.

uSVP is NP-hard under randomized reductions.

Abstract

We give several improvements on the known hardness of the unique shortest vector problem. - We give a deterministic reduction from the shortest vector problem to the unique shortest vector problem. As a byproduct, we get deterministic NP-hardness for unique shortest vector problem in the $\ell_\infty$ norm. - We give a randomized reduction from SAT to uSVP_{1+1/poly(n)}. This shows that uSVP_{1+1/poly(n)} is NP-hard under randomized reductions. - We show that if GapSVP_\gamma \in coNP (or coAM) then uSVP_{\sqrt{\gamma}} \in coNP (coAM respectively). This simplifies previously known uSVP_{n^{1/4}} \in coAM proof by Cai \cite{Cai98} to uSVP_{(n/\log n)^{1/4}} \in coAM, and additionally generalizes it to uSVP_{n^{1/4}} \in coNP. - We give a deterministic reduction from search-uSVP_\gamma to the decision-uSVP_{\gamma/2}. We also show that the decision-uSVP is {\bf NP}-hard for randomized reductions, which does not follow from Kumar-Sivakumar \cite{KS01}.