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Lattice reduction in two dimensions: analyses under realistic probabilistic models
, 2003
"... The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm “inside ” the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All th ..."
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Cited by 7 (1 self)
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The Gaussian algorithm for lattice reduction in dimension 2 is precisely analysed under a class of realistic probabilistic models, which are of interest when applying the Gauss algorithm “inside ” the LLL algorithm. The proofs deal with the underlying dynamical systems and transfer operators. All the main parameters are studied: execution parameters which describe the behaviour of the algorithm itself as well as output parameters, which describe the geometry of reduced bases.
FloatingPoint LLL: Theoretical and Practical Aspects
"... The textbook LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the GramSchmidt orthogonalisation by floatingpoint approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floating ..."
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Cited by 7 (2 self)
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The textbook LLL algorithm can be sped up considerably by replacing the underlying rational arithmetic used for the GramSchmidt orthogonalisation by floatingpoint approximations. We review how this modification has been and is currently implemented, both in theory and in practice. Using floatingpoint approximations seems to be natural for LLL even from the theoretical point of view: it is the key to reach a bitcomplexity which is quadratic with respect to the bitlength of the input vectors entries, without fast integer multiplication. The latter bitcomplexity strengthens the connection between LLL and Euclid’s gcd algorithm. On the practical side, the LLL implementer may weaken the provable variants in order to further improve their efficiency: we emphasise on these techniques. We also consider the practical behaviour of the floatingpoint LLL algorithms, in particular their output distribution, their runningtime and their numerical behaviour. After 25 years of implementation, many questions motivated by the practical side of LLL remain open.
Probabilistic analyses of the plain multiple gcd algorithm
"... Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ` − 1 phases ( ` is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing ..."
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Among multiple gcd algorithms on polynomials as on integers, one of the most natural ones performs a sequence of ` − 1 phases ( ` is the number of inputs), with each of them being the Euclid algorithm on two entries. We present here a complete probabilistic analysis of this algorithm, by providing both the averagecase and the distributional analysis, and by handling in parallel the integer and the polynomial cases, for polynomials with coefficients in a finite field. The main parameters under consideration are the number of iterations in each phase and the evolution of the size of the current gcd along the execution. Three phenomena are clearly emphasized through this analysis: the fact that almost all the computations are performed during the first phase, the great difference between the probabilistic behavior of the first phase compared to subsequent phases, and, as can be expected, the great similarity between the integer and the polynomial cases.