Closest Vector Problem With Preprocessing Research Articles

Approximate Voronoi cells for lattices, revisited

Abstract We revisit the approximate Voronoi cells approach for solving the closest vector problem with preprocessing (CVPP) on high-dimensional lattices, and settle the open problem of Doulgerakis–Laarhoven–De Weger [PQCrypto, 2019] of determining exact asymptotics on the volume of these Voronoi cells under the Gaussian heuristic. As a result, we obtain improved upper bounds on the time complexity of the randomized iterative slicer when using less than 2 0.076 d + o ( d ) $2^{0.076d + o(d)}$ memory, and we show how to obtain time–memory trade-offs even when using less than 2 0.048 d + o ( d ) $2^{0.048d + o(d)}$ memory. We also settle the open problem of obtaining a continuous trade-off between the size of the advice and the query time complexity, as the time complexity with subexponential advice in our approach scales as d d / 2 + o ( d ) $d^{d/2 + o(d)}$ matching worst-case enumeration bounds, and achieving the same asymptotic scaling as average-case enumeration algorithms for the closest vector problem.

A Deterministic Single Exponential Time Algorithm for Most Lattice Problems Based on Voronoi Cell Computations

We give deterministic $\tilde{O}(2^{2n})$-time $\tilde{O}(2^n)$-space algorithms to solve all the most important computational problems on point lattices in NP, including the shortest vector problem (SVP), closest vector problem (CVP), and shortest independent vectors problem (SIVP). This improves the $n^{O(n)}$ running time of the best previously known algorithms for CVP [R. Kannan, Math. Oper. Res., 12 (1987), pp. 415--440] and SIVP [D. Micciancio, Proceedings of the $19$th Annual ACM-SIAM Symposium on Discrete Algorithms, 2008, pp. 84--93] and gives a deterministic and asymptotically faster alternative to the $2^{O(n)}$-time (and space) randomized algorithm for SVP of Ajtai, Kumar, and Sivakumar [Proceedings of the $33$rd Annual ACM Symposium on Theory of Computing, 2001, pp. 266--275]. The core of our algorithm is a new method to solve the closest vector problem with preprocessing (CVPP) that uses the Voronoi cell of the lattice (described as intersection of half-spaces) as the result of the preproces...

The Hardness of the Closest Vector Problem With Preprocessing Over$ell_infty$Norm

We show that the closest vector problem with preprocessing (CVPP) over <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$ell_infty$</tex> norm <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$(hboxCVPP_infty)$</tex> is NP-hard. The result is obtained by the reduction from the subset sum problem with preprocessing to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$hbox CVPP_infty$</tex> . The reduction also shows the NP-hardness of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$hbox CVP_infty$</tex> , which is much simpler than all previously known proofs. In addition, we also give a direct reduction from exact 3-sets cover problem to <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$hbox CVPP_infty$</tex> .

Improved Inapproximability of Lattice and Coding Problems With Preprocessing

We show that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate to within /spl radic/3-/spl epsi/ for any /spl epsi/>0. In addition, we show that the nearest codeword problem with preprocessing (NCPP) is NP-hard to approximate to within 3-/spl epsi/. These results improve previous results of Feige and Micciancio. We also present the first inapproximability result for the relatively nearest codeword problem with preprocessing (RNCP). Finally, we describe an n-approximation algorithm to CVPP.

The inapproximability of lattice and coding problems with preprocessing

We prove that the closest vector problem with preprocessing (CVPP) is NP-hard to approximate within any factor less than 5 3 . More specifically, we show that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely in the target vector. It follows that there are lattices for which the closest vector problem cannot be approximated within factors γ< 5 3 in polynomial time, no matter how the lattice is represented, unless NP is equal to P (or NP is contained in P/poly, in case of nonuniform sequences of lattices). The result easily extends to any ℓ p norm, for p⩾1, showing that CVPP in the ℓ p norm is hard to approximate within any factor γ< 5 3 p . As an intermediate step, we establish analogous results for the nearest codeword problem with preprocessing (NCPP), proving that for any finite field GF(q), NCPP over GF( q) is NP-hard to approximate within any factor less than 5 3 .