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