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We show the first dimension-preserving search-to-decision reductions forapproximate SVP and CVP. In particular, for any $\gamma \leq 1 + O(\log n/n)$,we obtain an efficient dimension-preserving reduction from $\gamma^{O(n/\logn)}$-SVP to $\gamma$-GapSVP and an efficient dimension-preserving reductionfrom $\gamma^{O(n)}$-CVP to $\gamma$-GapCVP. These results generalize the knownequivalences of the search and decision versions of these problems in the exactcase when $\gamma = 1$. For SVP, we actually obtain something slightly strongerthan a search-to-decision reduction---we reduce $\gamma^{O(n/\log n)}$-SVP to$\gamma$-unique SVP, a potentially easier problem than $\gamma$-GapSVP.

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