On Bounded Distance Decoding and the Closest Vector Problem with Preprocessing

We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). This is the problem of finding a lattice vector whose distance from the target point is within some factor γ of the closest lattice vector, where the algorithm is allowed to take polynomial-length advice about the lattice from an unbounded preprocessing algorithm. Our algorithm achieves an approximation factor of O(n/ √ log n), improving on the previous best of O(n1.5) due to Lagarias, Lenstra, and Schnorr [LLS90]. We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse factor of O(n)). We remark that this still leaves a large gap with respect to the decisional version of CVPP, where the best known approximation factor is O( √ n/ log n) due to Aharonov and Regev [AR05]. To achieve these results, we show a reduction to the same problem restricted to target points that are very close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). BDDP is the problem of finding the unique closest lattice point for target vectors that are very close to the lattice (with polynomial-length advice from preprocessing on the lattice). Combining either reduction with the previous bestknown algorithm for BDDP by Liu, Lyubashevsky, and Micciancio [LLM06] gives our main result. We also present a substantially more efficient variant of the LLM algorithm (both in terms of run-time and amount of preprocessing advice), and via an improved analysis, show that it can decode up to a distance proportional to the reciprocal of the smoothing parameter of the dual lattice [MR07]. We show that this is never smaller than the LLM decoding radius, and that it can be up to an Ω̃( √ n) factor larger. ∗Courant Institute of Mathematical Sciences, New York University. †Supported by the National Science Foundation (NSF) under Grant No. CCF-1320188. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.