Article

A sieve algorithm for the shortest lattice vector problem

Authors:

Miklós Ajtai,

Ravi Kumar,

D. SivakumarAuthors Info & Claims

STOC '01: Proceedings of the thirty-third annual ACM symposium on Theory of computing

Pages 601 - 610

https://doi.org/10.1145/380752.380857

Published: 06 July 2001 Publication History

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Abstract

We present a randomized 2^{O(n)} time algorithm to compute a shortest non-zero vector in an n-dimensional rational lattice. The best known time upper bound for this problem was 2^{O(n\log n)} first given by Kannan [7] in 1983. We obtain several consequences of this algorithm for related problems on lattices and codes, including an improvement for polynomial time approximations to the shortest vector problem. In this improvement we gain a factor of log log n in the exponent of the approximating factor.

References

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R. Kumar and D. Sivakumar. On polynomial approximations to the shortest lattice vector length. Proc. 12th Symposium on Discrete Algorithms, 2001. To appear.

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Index Terms

A sieve algorithm for the shortest lattice vector problem
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Randomness, geometry and discrete structures

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STOC '01: Proceedings of the thirty-third annual ACM symposium on Theory of computing

July 2001

755 pages

ISBN:1581133499

DOI:10.1145/380752

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 06 July 2001

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Hardness of approximating the shortest vector problem in lattices

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Hardness of approximating the shortest vector problem in lattices

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