We give several improvements on the known hardness of the unique shortest vector problem in lattices, i.e., the problem of finding a shortest vector in a given lattice given a promise that the shortest vector is unique upto a uniqueness factor . We give a deterministic reduction from the shortest vector problem to the unique shortest vector problem. Before this, only a randomized reduction \cite{KS01} was known. This implies that if SVP is NP-hard, then uSVP is NP-hard. We also give a (randomized) reduction from SAT to uSVP on an n-dimensional lattice with uniqueness factor 1+1poly(n) . This improves the result of Kumar-Sivakumar\cite{KS01} showing that uSVP1+1poly(n) is NP-hard under randomized reductions.

Additionally, we show that if GapSVP is in co-NP (or co-AM) then uSVP is in co-NP (co-AM respectively). This improves previously known uSVPn14 co-AM proof by Cai \cite{Cai98} to uSVP(nlogn)14 co-AM, and additionally generalizes it to uSVPn14 co-NP. Note that when we say that uSVP co-NP or co-AM , we mean the decision version that was implicitly defined in Cai \cite{Cai98}.

We also show that the decision-uSVP is {\bf NP}-hard for randomized reductions, which does not follow from Kumar-Sivakumar \cite{KS01}. We also give a deterministic reduction from search-uSVP to decision-uSVP2.