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 γ ≤ 1+O(logn/n), we obtain an efficient dimension-preserving reduction from γO(n/ log n)-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 γunique SVP, a potentially easier problem than γ-GapSVP.