Games on line graphs and sand piles
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
Random lattices, threshold phenomena and efficient reduction algorithms
Theoretical Computer Science
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
Cryptanalysis of RSA with private key d less than N0.292
IEEE Transactions on Information Theory
Analyzing blockwise lattice algorithms using dynamical systems
CRYPTO'11 Proceedings of the 31st annual conference on Advances in cryptology
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The LLL algorithm aims at finding a “reduced” basis of a Euclidean lattice and plays a primary role in many areas of mathematics and computer science. However, its general behaviour is far from being well understood. There are already many experimental observations about the number of iterations or the geometry of the output, that raise challenging questions which remain unanswered and lead to natural conjectures which are yet to be proved. However, until now, there exist few experimental observations about the precise execution of the algorithm. Here, we provide experimental results which precisely describe an essential parameter of the execution, namely the “logarithm of the decreasing ratio”. These experiments give arguments towards a “regularity” hypothesis (R). Then, we propose a simplified model for the LLL algorithm based on the hypothesis (R), which leads us to discrete dynamical systems, namely sandpiles models. It is then possible to obtain a precise quantification of the main parameters of the LLL algorithm. These results fit the experimental results performed on general input bases, which indirectly substantiates the validity of such a regularity hypothesis and underlines the usefulness of such a simplified model.