Solving low-density subset sum problems
Journal of the ACM (JACM)
The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Journal of Algorithms
The generalized Gauss reduction algorithm
Journal of Algorithms
The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract)
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Worst-Case Complexity of the Optimal LLL Algorithm
LATIN '00 Proceedings of the 4th Latin American Symposium on Theoretical Informatics
Some Recent Progress on the Complexity of Lattice Problems
COCO '99 Proceedings of the Fourteenth Annual IEEE Conference on Computational Complexity
The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant
FOCS '98 Proceedings of the 39th Annual Symposium on Foundations of Computer Science
Improved algorithms for integer programming and related lattice problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
An Average-Case Analysis of the Gaussian Algorithm for Lattice Reduction
Combinatorics, Probability and Computing
Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection
IEEE Transactions on Signal Processing
Gradual sub-lattice reduction and a new complexity for factoring polynomials
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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In this paper, we consider the open problem of the complexity of the LLL algorithm in the case when the approximation parameter of the algorithm has as its extreme value 1. This case is of interest because the output is then the strongest Lovász-reduced basis. Experiments reported by Lagarias and Odlyzko (J. ACM 32(1) (1985) 229) seem to show that the algorithm remains polynomial in average. However, no bound better than a naive exponential order one is established for the worst-case complexity of the optimal LLL algorithm, even for fixed small dimension (higher than 2). Here, we prove that, for any fixed dimension n, the number of iterations of the LLL algorithm is linear with respect to the size of the input. It is easy to deduce from Vallée (J. Algorithms 12 (1991) 556) that the linear order is optimal. Moreover in 3 dimensions, we give a tight bound for the maximum number of iterations and we characterize precisely the output basis. Our bound also improves the known one for the usual (non-optimal) LLL algorithm.