Improved hardness results for unique shortest vector problem

Abstract

The unique shortest vector problem on a rational lattice is the problem of finding the shortest non-zero vector under the promise that it is unique (up to multiplication by −1). We give several incremental improvements on the known hardness of the unique shortest vector problem (uSVP) using standard techniques. This includes a deterministic reduction from the shortest vector problem to the uSVP, the NP-hardness of uSVP on $(1+\frac{1}{\mathrm{poly}\left(n\right)})$-unique lattices, and a proof that the decision version of uSVP defined by Cai [4] is in $\mathrm{co}\text{-}\mathrm{NP}$ for $n^{1/4}$-unique lattices.