The computational complexity of simultaneous diophantine approximation problems
SIAM Journal on Computing
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
An upper bound on the average number of iterations of the LLL algorithm
Theoretical Computer Science - Special issue on number theory, combinatorics and applications to computer science
An introduction to the analysis of algorithms
An introduction to the analysis of algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical 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
Proceedings of the 11th Colloquium on Automata, Languages and Programming
Threshold Phenomena in Random Lattices and Efficient Reduction Algorithms
ESA '99 Proceedings of the 7th Annual European Symposium on Algorithms
Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems
FCT '91 Proceedings of the 8th International Symposium on Fundamentals of Computation Theory
Improved algorithms for integer programming and related lattice problems
STOC '83 Proceedings of the fifteenth annual ACM symposium on Theory of computing
Fast LLL-type lattice reduction
Information and Computation
Fast LLL-type lattice reduction
Information and Computation
Speeding-up lattice reduction with random projections
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
Modelling the LLL algorithm by sandpiles
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
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Two new lattice reduction algorithms are presented and analyzed. These algorithms, called the Schmidt reduction and the Gram reduction, are obtained by relaxing some of the constraints of the classical LLL algorithm. By analyzing the worst case behavior and the average case behavior in a tractable model, we prove that the new algorithms still produce "good" reduced basis while requiring fewer iterations on average. In addition, we provide empirical tests on random lattices coming from applications, that confirm our theoretical results about the relative behavior of the different reduction algorithms.