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
A course in computational algebraic number theory
A course in computational algebraic number theory
A public-key cryptosystem with worst-case/average-case equivalence
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
Random lattices, threshold phenomena and efficient reduction algorithms
Theoretical Computer Science
Modern Computer Algebra
Searching Worst Cases of a One-Variable Function Using Lattice Reduction
IEEE Transactions on Computers
A BLAS based C library for exact linear algebra on integer matrices
Proceedings of the 2005 international symposium on Symbolic and algebraic computation
Condition Numbers of Gaussian Random Matrices
SIAM Journal on Matrix Analysis and Applications
Efficient polynomial L-approximations
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
Floating-point L^2 -approximations to functions
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
On the randomness of bits generated by sufficiently smooth functions
ANTS'06 Proceedings of the 7th international conference on Algorithmic Number Theory
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Cryptanalysis of RSA with private key d less than N0.292
IEEE Transactions on Information Theory
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Lattice reduction algorithms such as LLL and its floating-point variants have a very wide range of applications in computational mathematics and in computer science: polynomial factorization, cryptology, integer linear programming, etc. It can occur that the lattice to be reduced has a dimension which is small with respect to the dimension of the space in which it lies. This happens within LLL itself. We describe a randomized algorithm specifically designed for such rectangular matrices. It computes bases satisfying, with very high probability, properties similar to those returned by LLL. It significantly decreases the complexity dependence in the dimension of the embedding space. Our technique mainly consists in randomly projecting the lattice on a lower dimensional space, by using two different distributions of random matrices.