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 + 1/poly(n))-unique
lattices, and a proof that the decision version of uSVP defined by Cai \cite{Cai98} is in co-NP for
n^{1/4}-unique lattices.

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 \gamma.
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 + 1/poly(n). This improves the result of Kumar-Sivakumar\cite{KS01} showing that uSVP_{1+1/poly(n)} is NP-hard under randomized reductions.

Additionally, we show that if GapSVP_{\gamma} is in co-NP (or co-AM) then uSVP_{\sqrt{\gamma}} is in co-NP (co-AM respectively). This improves previously known uSVP_{n^{1/4}} \in co-AM proof by Cai \cite{Cai98} to uSVP_{(n/\log n)^{1/4}} \in co-AM, and additionally generalizes it to uSVP_{n^{1/4}} \in co-NP. Note that when we say that uSVP \in 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_{\gamma} to decision-uSVP_{\gamma/2}.