Conference Paper

Some Sieving Algorithms for Lattice Problems.

DOI: 10.4230/LIPIcs.FSTTCS.2008.1738 Conference: IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2008, December 9-11, 2008, Bangalore, India
Source: OAI


We study the algorithmic complexity of lattice problems based on the sieving technique due to M. Ajtai, R. Kumar and D. Sivakumar [“A sieve algorithm for the shortest lattice vector”, in: Proceedings of the thirty-third annual ACM symposium on theory of computing (STOC 2001). New York, NY: Association for Computing Machinery (ACM). 601–610 (2001; doi:10.1145/380752.380857)]. Given a k-dimensional subspace M⊆ℝ n and a full rank integer lattice ℒ⊆ℚ n , the subspace avoiding problem SAP, defined by J. Blömer and S. Naewe [Lect. Notes Comput. Sci. 4596, 65–77 (2007; Zbl 1171.11328)], is to find a shortest vector in ℒ∖M. We first give a 2 O(n+klogk) time algorithm to solve the subspace avoiding problem. Applying this algorithm we obtain the following results. 1. We give a 2 O(n) time algorithm to compute ith successive minima of a full rank lattice ℒ⊂ℚ n if i is O(n logn). 2. We give a 2 O(n) time algorithm to solve a restricted closest vector problem (CVP), where the inputs fulfil a promise about the distance of the input vector from the lattice. 3. We also show that unrestricted CVP has a 2 O(n) exact algorithm if there is a 2 O(n) time exact algorithm for solving CVP with additional input v i ∈ℒ, 1≤i≤n, where ∥v i ∥ p is the ith successive minima of ℒ for each i. We also give a new approximation algorithm for SAP and the convex body avoiding problem which is a generalization of SAP. Several of our algorithms work for gauge functions as metric, where the gauge function has a natural restriction and is accessed by an oracle.

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    • "These problems are central to the geometry of numbers and have applications to integer programming, factoring polynomials, cryptography etc. The fastest known algorithms for solving SVP in general norms, are 2 O(n) time randomized algorithms based on the AKS sieve [1] [2]. Finding deterministic algorithms of this complexity for both SVP and CVP has been an important open problem. "
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    ABSTRACT: We give a deterministic 2(o(n))algorithm for computing an M-ellipsoid of a convex body, matching a known lower bound. This leads to a nearly optimal deterministic algorithm for estimating the volume of a convex body and improved deterministic algorithms for fundamental lattice problems under general norms.
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    ABSTRACT: We give a novel algorithm for enumerating lattice points in any convex body, and give applications to several classic lattice problems, including the Shortest and Closest Vector Problems (SVP and CVP, respectively) and Integer Programming (IP). Our enumeration technique relies on a classical concept from asymptotic convex geometry known as the M-ellipsoid, and uses as a crucial subroutine the recent algorithm of Micciancio and Voulgaris (STOC 2010) for lattice problems in the l_2 norm. As a main technical contribution, which may be of independent interest, we build on the techniques of Klartag (Geometric and Functional Analysis, 2006) to give an expected 2^O(n)-time algorithm for computing an M-ellipsoid for any n-dimensional convex body. As applications, we give deterministic 2^{O(n)}-time and -space algorithms for solving exact SVP, and exact CVP when the target point is sufficiently close to the lattice, on n-dimensional lattices in any (semi-)norm given an M-ellipsoid of the unit ball. In many norms of interest, including all l_p norms, an M-ellipsoid is computable in deterministic poly(n) time, in which case these algorithms are fully deterministic. Here our approach may be seen as a derandomization of the "AKS sieve" for exact SVP and CVP (Ajtai, Kumar, and Sivakumar; STOC 2001 and CCC 2002). As a further application of our SVP algorithm, we derive an expected O(f*(n))^n-time algorithm for Integer Programming, where f*(n) denotes the optimal bound in the so-called "flatness theorem," which satisfies f*(n) = O(n^{4/3} \polylog(n)) and is conjectured to be f*(n)=\Theta(n). Our runtime improves upon the previous best of O(n^{2})^{n} by Hildebrand and Koppe (2010).
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