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Revision #1 to TR10-014 | 3rd March 2012 02:35

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

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Abstract:

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 (Kannan, Math. Operation Research 12(3):415-440, 1987) and SIVP (Micciancio, SODA 2008), 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, STOC 2001). 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 preprocessing function. A direct consequence of our results is a derandomization of the best current polynomial time approximation algorithms for SVP and CVP achieving $2^{O(n \log\log n / \log n)}$ approximation factor.



Changes to previous version:

This is a substantial revision of the previous version. New improved algorithms reduce the worst-case running time for all problems to $\tilde O(2^{2n})$.


Paper:

TR10-014 | 2nd February 2010 00:51

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





TR10-014
Authors: Daniele Micciancio, Panagiotis Voulgaris
Publication: 2nd February 2010 08:41
Downloads: 1376
Keywords: 


Abstract:

We give deterministic $2^{O(n)}$-time 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 (Kannan, Math. Operation Research 12(3):415-440, 1987) and SIVP (Micciancio, Proc. of SODA, 2008), and gives a deterministic alternative to the $2^{O(n)}$-time (and space) randomized algorithm for SVP of (Ajtai, Kumar and Sivakumar, STOC 2001). 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 preprocessing function. In the process, we also give algorithms for several other lattice problems, including computing the kissing number of a lattice, and computing the set of all Voronoi relevant vectors. All our algorithms are deterministic, and have $2^{O(n)}$ time and space complexity.



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