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

- [Show abstract] [Hide abstract]

**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).CoRR. 01/2010; abs/1011.5666. -
##### Conference Paper: Deterministic construction of an approximate M-ellipsoid and its applications to derandomizing lattice algorithms.

[Show abstract] [Hide abstract]

**ABSTRACT:**We give a deterministic O(logn)n-time and space algorithm for the Shortest Vector Problem (SVP) of a lattice under any norm, improving on the previous best deterministic nO(n)-time algorithms for general norms. This approaches the 2O(n)-time and space complexity of the randomized sieve based SVP algorithms (Arvind and Joglekar, FSTTCS 2008), first introduced by Ajtai, Kumar and Sivakumar (STOC 2001) for l2-SVP, and the M-ellipsoid covering based SVP algorithm of Dadush et al. (FOCS 2011). Here we continue with the covering approach of Dadush et al., and our main technical contribution is a deterministic approximation of an M-ellipsoid for any convex body. To achieve this we exchange the M-position of a convex body by a related position, known as the minimal mean width position of the polar body. We reduce the task of computing this position to solving a semi-definite program whose objective is a certain Gaussian expectation, which we show can be approximated deterministically.Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011; 01/2012 - [Show abstract] [Hide abstract]

**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.Proceedings of the National Academy of Sciences 09/2013; · 9.81 Impact Factor

Data provided are for informational purposes only. Although carefully collected, accuracy cannot be guaranteed. The impact factor represents a rough estimation of the journal's impact factor and does not reflect the actual current impact factor. Publisher conditions are provided by RoMEO. Differing provisions from the publisher's actual policy or licence agreement may be applicable.