Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One

We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any $\gamma \leq 1 + O(\log n/n)$, we obtain an efficient dimension-preserving reduction from $\gamma^{O(n/\log n)}$-SVP to $\gamma$-GapSVP and an efficient dimension-preserving reduction from $\gamma^{O(n)}$-CVP to $\gamma$-GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when $\gamma = 1$. For SVP, we actually obtain something slightly stronger than a search-to-decision reduction---we reduce $\gamma^{O(n/\log n)}$-SVP to $\gamma$-unique SVP, a potentially easier problem than $\gamma$-GapSVP.Comment: Updated to acknowledge additional prior wor

Improved hardness results for unique shortest vector problem

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 [4] is in co-NP for n(1/4)-unique lattices. (C) 2016 Published by Elsevier B.V