A more efficient algorithm for lattice basis reduction
Journal of Algorithms
A set of level 3 basic linear algebra subprograms
ACM Transactions on Mathematical Software (TOMS)
A componentwise perturbation analysis of the QR decomposition
SIAM Journal on Matrix Analysis and Applications
On the perturbation of LU, Cholesky, and QR factorizations
SIAM Journal on Matrix Analysis and Applications
Lattice basis reduction: improved practical algorithms and solving subset sum problems
Mathematical Programming: Series A and B
A course in computational algebraic number theory
A course in computational algebraic number theory
Perturbation Analyses for the QR Factorization
SIAM Journal on Matrix Analysis and Applications
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Public-Key Cryptosystems from Lattice Reduction Problems
CRYPTO '97 Proceedings of the 17th Annual International Cryptology Conference on Advances in Cryptology
NTRU: A Ring-Based Public Key Cryptosystem
ANTS-III Proceedings of the Third International Symposium on Algorithmic Number Theory
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the international conference on linear algebra and arithmetic, Rabat, Morocco, 28-31 May 2001
Fast LLL-type lattice reduction
Information and Computation
Super-fast validated solution of linear systems
Journal of Computational and Applied Mathematics - Special issue: Scientific computing, computer arithmetic, and validated numerics (SCAN 2004)
Efficient polynomial L-approximations
ARITH '07 Proceedings of the 18th IEEE Symposium on Computer Arithmetic
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
Numerical techniques for computing the inertia of products of matrices of rational numbers
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Rigorous and Efficient Short Lattice Vectors Enumeration
ASIACRYPT '08 Proceedings of the 14th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
H-LLL: using householder inside LLL
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation
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Given a lattice basis of n vectors in Zn, we propose an algorithm using 12n3+O(n2) floating point operations for checking whether the basis is LLL-reduced. If the basis is reduced then the algorithm will hopefully answer "yes". If the basis is not reduced, or if the precision used is not sufficient with respect to n, and to the numerical properties of the basis, the algorithm will answer "failed". Hence a positive answer is a rigorous certificate. For implementing the certificate itself, we propose a oating point algorithm for computing (certified) error bounds for the R factor of the QR factorization. This algorithm takes into account all possible approximation and rounding errors. The certificate may be implemented using matrix library routines only. We report experiments that show that for a reduced basis of adequate dimension and quality the certificate succeeds, and establish the effectiveness of the certificate. This effectiveness is applied for certifying the output of fastest existing floating point heuristics of LLL reduction, without slowing down the whole process.