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A course in computational algebraic number theory
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CRYPTO'07 Proceedings of the 27th annual international cryptology conference on Advances in cryptology
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Analysis of gauss-sieve for solving the shortest vector problem in lattices
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
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IWCC'11 Proceedings of the Third international conference on Coding and cryptology
A O(1/ε2)n-time sieving algorithm for approximate integer programming
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
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We study four problems from the geometry of numbers, the shortest vector problem(Svp), the closest vector problem(Cvp), the successive minima problem(Smp), and the shortest independent vectors problem (Sivp). Extending and generalizing results of Ajtai, Kumar, and Sivakumar we present probabilistic single exponential time algorithms for all four problems for all @?"p norms. The results on Smp and Sivp are new for all norms. The results on Svp and Cvp generalize previous results of Ajtai et al. for the Euclidean @?"2 norm to arbitrary @?"p norms. We achieve our results by introducing a new lattice problem, the generalized shortest vector problem (GSvp). We describe a single exponential time algorithm for GSvp. We also describe polynomial time reductions from Svp,Cvp,Smp, and Sivp to GSvp, establishing single exponential time algorithms for the four classical lattice problems. This approach leads to a unified algorithmic treatment of the lattice problems Svp,Cvp,Smp, and Sivp.