Lattice Sparsification and the Approximate Closest Vector Problem

Dadush, Daniel; Kun, Gábor

doi:10.1137/1.9781611973105.78

Cited by 11 publications

(24 citation statements)

References 47 publications

(61 reference statements)

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“…See Corollary 6.2 for the precise hardness result. Our reduction relies on a lattice sparsification technique introduced by Dadush and Kun [DK13] who used it to develop deterministic single-exponential time algorithms for approximate CVP under general norms.…”

Section: Our Contributionsmentioning

confidence: 99%

“…We follow the sparsification idea of [DK13]. Basically, given a target t, we try to find a sublattice L of L, such that L has minimum distance proportional to dist(t, L ) with dist(t, L ) not much larger than dist(t, L).…”

Section: Reduction To Bounded Distance Using Sparsificationmentioning

confidence: 99%

“…The reductions are based on ideas due to Kannan [Kan87] and a recent sparsification technique [DK13]. Combining our reductions with the LLM algorithm gives an approximation factor of O(n/ log n) for search CVPP, improving on the previous best of O(n 1.5 ) due to Lagarias, Lenstra, and Schnorr [LLS90].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Closest Vector Problem with a Distance Guarantee

Dadush

Regev

Stephens-Davidowitz

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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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 [LLM06] for solving CVPP (the preprocessing version of the Closest Vector Problem, CVP) with a distance guarantee. For instance, for any α < 1/2, our algorithm finds the (unique) closest lattice point for any target point whose distance from the lattice is at most α times the length of the shortest nonzero lattice vector, requires as preprocessing advice only N ≈ O(n exp(α 2 n/(1 − 2α) 2 )) vectors, and runs in time 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 [Kan87] and a recent sparsification technique [DK13]. Combining our reductions with the LLM algorithm gives an approximation factor of O(n/ log n) for search CVPP, improving on the previous best of O(n 1.5 ) due to Lagarias, Lenstra, and Schnorr [LLS90]. 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).

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Section: Our Contributionsmentioning

confidence: 99%

Section: Reduction To Bounded Distance Using Sparsificationmentioning

confidence: 99%

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On the Closest Vector Problem with a Distance Guarantee

Dadush

Regev

Stephens-Davidowitz

2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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“…This concludes the proof for strictly convex and smooth norms. We only used the smoothness assumption to show (7) with the help of Proposition 20. For two-dimensional lattices, we obtain (7) from Proposition 16 without smoothness.…”

Section: Strictly Convex Normsmentioning

confidence: 99%

Voronoi Cells of Lattices with Respect to Arbitrary Norms

Blömer¹,

Kohn²

2018

SIAM J. Appl. Algebra Geometry

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We study the geometry and complexity of Voronoi cells of lattices with respect to arbitrary norms. On the positive side, we show for strictly convex and smooth norms that the geometry of Voronoi cells of lattices in any dimension is similar to the Euclidean case, i.e., the Voronoi cells are defined by the so-called Voronoi-relevant vectors and the facets of a Voronoi cell are in one-to-one correspondence with these vectors. On the negative side, we show that Voronoi cells are combinatorially much more complicated for arbitrary strictly convex and smooth norms than in the Euclidean case. In particular, we construct a family of three-dimensional lattices whose number of Voronoi-relevant vectors with respect to the 3norm is unbounded. Our result indicates, that the break through single exponential time algorithm of Micciancio and Voulgaris for solving the shortest and closest vector problem in the Euclidean norm cannot be extended to achieve deterministic single exponential time algorithms for lattice problems with respect to arbitrary p-norms. In fact, the algorithm of Micciancio and Voulgaris and its run time analysis crucially depend on the fact that for the Euclidean norm the number of Voronoi-relevant vectors is single exponential in the lattice dimension. arXiv:1512.00720v5 [math.MG]

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“…For example, the (1 + ε)-approximate closest vectors to t can be grouped into 2 O(n) (1 + 1/ε) n clusters of diameter ε · dist(t, L) (see, e.g., [37], [47]). While the clustering bound that we obtain is both stronger and simpler to prove (using an elementary parity argument), we are unaware of prior work mentioning this particular bound.…”

Section: Related Workmentioning

confidence: 99%

Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

Aggarwal

Dadush

Stephens-Davidowitz

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We give a 2 n+o(n) -time and space randomized algorithm for solving the exact Closest Vector Problem (CVP) on n-dimensional Euclidean lattices. This improves on the previous fastest algorithm, the deterministic O(4 n )-time and O(2 n )-space algorithm of Micciancio and Voulgaris [1]. We achieve our main result in three steps. First, we show how to modify the sampling algorithm from [2] to solve the problem of discrete Gaussian sampling over lattice shifts, L − t, with very low parameters. While the actual algorithm is a natural generalization of [2], the analysis uses substantial new ideas. This yields a 2 n+o(n) -time algorithm for approximate CVP with the very good approximation factor γ = 1 + 2 −o(n/ log n) . Second, we show that the approximate closest vectors to a target vector t can be grouped into "lowerdimensional clusters," and we use this to obtain a recursive reduction from exact CVP to a variant of approximate CVP that "behaves well with these clusters." Third, we show that our discrete Gaussian sampling algorithm can be used to solve this variant of approximate CVP.The analysis depends crucially on some new properties of the discrete Gaussian distribution and approximate closest vectors, which might be of independent interest.

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Lattice Sparsification and the Approximate Closest Vector Problem

Cited by 11 publications

References 47 publications

On the Closest Vector Problem with a Distance Guarantee

On the Closest Vector Problem with a Distance Guarantee

Voronoi Cells of Lattices with Respect to Arbitrary Norms

Solving the Closest Vector Problem in 2^n Time -- The Discrete Gaussian Strikes Again!

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