We present a substantially more efficient variant, both in terms of running
time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky,
and Micciancio for solving CVPP (the preprocessing version of the Closest
Vector Problem, CVP) with a distance guarantee. For instance, for any $\alpha <
1/2$, our algorithm finds the (unique) closest lattice point for any target
point whose distance from the lattice is at most $\alpha$ times the length of
the shortest nonzero lattice vector, requires as preprocessing advice only $N
\approx \widetilde{O}(n \exp(\alpha^2 n /(1-2\alpha)^2))$ vectors, and runs in
time $\widetilde{O}(nN)$.
  As our second main contribution, we present reductions showing that it
suffices to solve CVP, both in its plain and preprocessing versions, when the
input target point is within some bounded distance of the lattice. The
reductions are based on ideas due to Kannan and a recent sparsification
technique due to Dadush and Kun. Combining our reductions with the LLM
algorithm gives an approximation factor of $O(n/\sqrt{\log n})$ for search
CVPP, improving on the previous best of $O(n^{1.5})$ due to Lagarias, Lenstra,
and Schnorr. When combined with our improved algorithm we obtain, somewhat
surprisingly, that only O(n) vectors of preprocessing advice are sufficient to
solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance
  Decoding and the Closest Vector Problem with Preprocessing". Conference on
  Computational Complexity (2014

Dadush, Daniel

Regev, Oded

Stephens-Davidowitz, Noah

English

arXiv

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/\xe2\x88\x9alog n), improving on the previous best of O(n1.5) due to Lagarias, Lenstra, and Schnorr [1]. 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(\xe2\x88\x9an/log n) due to Aharonov and Regev [2]. 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 [3] gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+\xe2\x88\x88)\xce\xb3 approximate CVP to \xce\xb3 approximate CVP where the target is at distance at most 1+1/\xe2\x88\x88 times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun [4]. 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 [5]. We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide \xce\xa9(\xe2\x88\x9an) factor larger

Dadush, D.N. (Daniel)

Regev, O. (Oded)

Stephens-Davidowitz, N. (Noah)

CWI's Institutional Repository

On the Closest Vector Problem with a distance guarantee

Daniel Dadush

Oded Regev

Noah Stephens-Davidowitz

Crossref

On the Closest Vector Problem with a Distance Guarantee

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/√log n), improving on the previous best of O(n1.5) due to Lagarias, Lenstra, and Schnorr [1]. 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(√n/log n) due to Aharonov and Regev [2]. 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 [3] gives our main result. In the setting of CVP without preprocessing, we also give a reduction from (1+∈)γ approximate CVP to γ approximate CVP where the target is at distance at most 1+1/∈ times the minimum distance (the length of the shortest non-zero vector) which relies on the lattice sparsification techniques of Dadush and Kun [4]. 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 [5]. We show that this is never smaller than the LLM decoding radius, and that it can be up to an wide Ω(√n) factor larger

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On the Closest Vector Problem with a Distance Guarantee

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