Algorithms to construct Minkowski reduced and Hermite reduced lattice bases
Theoretical Computer Science
Minkowski's convex body theorem and integer programming
Mathematics of Operations Research
Generating hard instances of lattice problems (extended abstract)
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Approximation algorithms for NP-hard problems
The hardness of approximate optima in lattices, codes, and systems of linear equations
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
A public-key cryptosystem with worst-case/average-case equivalence
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
On the limits of non-approximability of lattice problems
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
On the complexity of computing short linearly independent vectors and short bases in a lattice
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Some Recent Progress on the Complexity of Lattice Problems
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
Applications of a New Transference Theorem to Ajtai's Connection Factor
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
An Improved Worst-Case to Average-Case Connection for Lattice Problems
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Approximating-CVP to within Almost-Polynomial Factors is NP-Hard
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
E-SMART '01 Proceedings of the International Conference on Research in Smart Cards: Smart Card Programming and Security
Implicit Factoring: On Polynomial Time Factoring Given Only an Implicit Hint
Irvine Proceedings of the 12th International Conference on Practice and Theory in Public Key Cryptography: PKC '09
Sampling methods for shortest vectors, closest vectors and successive minima
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
Improvements in closest point search based on dual HKZ-bases
Theoretical Computer Science
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In this paper we introduce a new technique to solve lattice problems. The technique is based on dual HKZ-bases. Using this technique we show how to solve the closest vector problem in lattices with rank n in time n! ċ sO(1), where s is the input size of the problem. This is an exponential improvement over an algorithm due to Kannan and Helfrich [16,15]. Based on the new technique we also show how to compute the successive minima of a lattice in time n! ċ 3n ċ sO(1), where n is the rank of the lattice and s is the input size of the lattice. The problem of computing the successive minima plays an important role in Ajtai's worst-case to average-case reduction for lattice problems. Our results reveal a close connection between the closest vector problem and the problem of computing the successive minima of a lattice.