We show the first dimension-preserving search-to-decision reductions for approximate SVP and CVP. In particular, for any @math , we obtain an efficient dimension-preserving reduction from @math -SVP to @math -GapSVP and an efficient dimension-preserving reduction from @math -CVP to @math -GapCVP. These results generalize the known equivalences of the search and decision versions of these problems in the exact case when @math . For SVP, we actually obtain something slightly stronger than a search-to-decision reduction---we reduce @math -SVP to @math -unique SVP, a potentially easier problem than @math -GapSVP.
We rely heavily on Lyubashevsky and Micciancio's dimension-preserving reduction from @math -unique SVP to @math -GapSVP @cite_37 . Their result is necessary to prove Theorem , and our deterministic bit-by-bit SVP reduction is very similar to Lyubashevsky and Micciancio's reduction. The main difference between our deterministic SVP reduction and that of Lyubashevsky and Micciancio is that @cite_37 work only with lattices that satisfy the promise of @math -unique SVP. They show that this promise is enough to guarantee that the @math -GapSVP oracle essentially behaves as an GapSVP oracle. In contrast, our reduction works over general lattices, so we have to worry about accumulating error. (We also use a different method to reduce the dimension of the lattice. )
@cite_37
2069278600 1971445617 2146007300 2949722426
We present a new efficient algorithm for the search version of the approximate Closest Vector Problem with Preprocessing (CVPP). Our algorithm achieves an approximation factor of O(n sqrt log n ), improving on the previous best of O(n^ 1.5 ) due to Lag arias, Lenstra, and Schnorr hkzbabai . We also show, somewhat surprisingly, that only O(n) vectors of preprocessing advice are sufficient to solve the problem (with the slightly worse approximation 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(sqrt n log n ) due to Aharonov and Regev AharonovR04 . To achieve these results, we show a reduction to the same problem restricted to target points that are close to the lattice and a more efficient reduction to a harder problem, Bounded Distance Decoding with preprocessing (BDDP). Combining either reduction with the previous best-known algorithm for BDDP by Liu, Lyubashevsky, and Micciancio LiuLM06 gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+eps)gamma approximate CVP to gamma approximate CVP where the target is at distance at most 1+1 eps times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun DadushK13 . As our final and most technical contribution, we 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 MR04 . We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide tilde Omega (sqrt n ) factor larger. We give various deterministic polynomial time reductions among approximation problems on point lattices. Our reductions are both efficient and robust, in the sense that they preserve the rank of the lattice and approximation factor achieved. Our main result shows that for any γ ≥ 1, approximating all the successive minima of a lattice (and, in particular, approximately solving the Shortest Independent Vectors Problem, SIVPγ) within a factor γ reduces under deterministic polynomial time rank-preserving reductions to approximating the Closest Vector Problem (CVP) within the same factor γ. This solves an open problem posed by Blomer in (ICALP 2000). As an application, we obtain faster algorithms for the exact solution of SIVP that run in time n! · sO(1) (where n is the rank of the lattice, and s the size of the input,) improving on the best previously known solution of Blomer (ICALP 2000) by a factor 3n. We also show that SIVP, CVP and many other lattice problems are equivalent in their exact version under deterministic polynomial time rank-preserving reductions. We give a deterministic algorithm for solving the (1 + e) approximate Closest Vector Problem (CVP) on any n dimensional lattice and any norm in 2O(n) (1 + 1 e)n time and 2n poly(n) space. Our algorithm builds on the lattice point enumeration techniques of Micciancio and Voulgaris (STOC 2010) and Dadush, Peikert and Vempala (FOCS 2011), and gives an elegant, deterministic alternative to the "AKS Sieve" based algorithms for (1 + e)-CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). Furthermore, assuming the existence of a poly(n)-space and 2O(n) time algorithm for exact CVP in the l2 norm, the space complexity of our algorithm can be reduced to polynomial. Our main technical contribution is a method for "spar-sifying" any input lattice while approximately maintaining its metric structure. To this end, we employ the idea of random sublattice restrictions, which was first employed by Khot (FOCS 2003) for the purpose of proving hardness for Shortest Vector Problem (SVP) under lp norms. Improving on the Voronoi cell based techniques of Micciancio and Voulgaris (SIAM J. Comp. 13), and Sommer, Feder and Shalvi (SIAM J. Disc. Math. 09), we give a Las Vegas @math expected time and space algorithm for CVPP (the preprocessing version of the Closest Vector Problem, CVP). This improves on the @math deterministic runtime of the Micciancio Voulgaris algorithm, henceforth MV, for CVPP (which also solves CVP) at the cost of a polynomial amount of randomness (which only affects runtime, not correctness). As in MV, our algorithm proceeds by computing a short path on the Voronoi graph of the lattice, where lattice points are adjacent if their Voronoi cells share a common facet, from the origin to a closest lattice vector. Our main technical contribution is a randomized procedure that given the Voronoi relevant vectors of a lattice - the lattice vectors inducing facets of the Voronoi cell - as preprocessing and any "close enough" lattice point to the target, computes a path to a closest lattice vector of expected polynomial size. This improves on the @math path length given by the MV algorithm. Furthermore, as in MV, each edge of the path can be computed using a single iteration over the Voronoi relevant vectors. As a byproduct of our work, we also give an optimal relationship between geometric and path distance on the Voronoi graph, which we believe to be of independent interest.