Abstract
The Gaussian algorithm for lattice reduction in dimension 2 (under both the standard version and the centered version) is analysed. It is found that, when applied to random inputs, the complexity is asymptotically constant, the probability distribution decays geometrically, and the dynamics is characterized by a conditional invariant measure. The proofs make use of connections between lattice reduction, continued fractions, continuants, and functional operators. Detailed numerical data are also presented.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Babenko, K. I. On a problem of Gauss. Soviet Mathematical Doklady 19, 1 (1978), 136–140.
Berndt, B. C. Ramanujan's Notebooks, Part I. Springer Verlag, 1985.
Borwein, J. M., and Borwein, P. B. Strange series and high precision fraud. American Mathematical Monthly 99, 7 (Aug. 1992), 622–640.
Cohen, H. A Course in Computational Algebraic Number Theory. No. 138 in Graduate Texts in Mathematics. Springer-Verlag, 1993.
Daudé, H. Des fractions continues à la réduction des réseaux: analyse en moyenne. PhD thesis, Université de Caen, 1993.
Daudé, H., and Vallée, B. An upper bound on the average number of iterations of the LLL algorithm. Theoretical Computer Science 123, 1 (1994), 95–115.
Edwards, H. M. Riemann's Zeta Function. Academic Press, 1974.
Hensley, D. The number of steps in the Euclidean algorithm. Preprint, 1993.
Kaib, M., and Schnorr, C. P. A sharp worst-case analysis of the Gaussian lattice basis reduction algorithm for any norm. Preprint, 1992. To appear in J. of Algorithms.
Knuth, D. E. The Art of Computer Programming, 2nd ed., vol. 2: Seminumerical Algorithms. Addison-Wesley, 1981.
Lagarias, J. C. Worst-case complexity bounds for algorithms in the theory of integral quadratic forms. Journal of Algorithms 1, 2 (1980), 142–186.
Lenstra, A. K., Lbnstra, H. W., and Lovász, L. Factoring polynomials with rational coefficients. Mathematische Annalen 261 (1982), 513–534.
Mahler, K. Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen. Mathematische Annalen 101 (1929), 342–366.
Mayer, D., and Roepstorff, G. On the relaxation time of Gauss's continued fraction map. I. The Hilbert space approach. Journal of Statistical Physics 47, 1/2 (Apr. 1987), 149–171.
Mayer, D., and Roepstorff, G. On the relaxation time of Gauss's continued fraction map. II. The Banach space approach (transfer operator approach). Journal of Statistical Physics 50, 1/2 (Jan. 1988), 331–344.
Mayer, D. H. On a ζ function related to the continued fraction transformation. Bulletin de la Société Mathématique de France 104 (1976), 195–203.
Mayer, D. H. Continued fractions and related transformations. In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, M, K. Tim Bedford and C. Series, Eds. Oxford University Press, 1991, pp. 175–222.
Rockett, A., and Szüsz, P.Continued Fractions. World Scientific, Singapore, 1992.
Scharlau, W., and Opolka, H. From Fermat to Minkowski, Lectures on the Theory of Numbers and its Historical Developments. Undergraduate Texts in Mathematics. Springer-Verlag, 1984.
Serre, J.-P. A Course in Arithmetic. Graduate Texts in Mathematics. Springer Verlag, 1973.
Vallée, B. Gauss' algorithm revisited. Journal of Algorithms 12 (1991), 556–572.
Vallée, B., and Flajolet, P. Gauss' reduction algorithm: An average case analysis. In Proceedings of the 31st Symposium on Foundations of Computer Science (Oct. 1990), IEEE Computer Society Press, pp. 830–839.
Vardi, I. Computational Recreations in Mathernatica. Addison Wesley, 1991.
Wirsing, E. On the theorem of Gauss-Kusmin-Lévy and a Frobenius-type theorem for function spaces. Acta Arithmetica 24 (1974), 507–528.
Zippel, R.Effective Polynomial Computations. Kluwer Academic Publishers, Boston, 1953.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Daudé, H., Flajolet, P., Vallée, B. (1994). An analysis of the Gaussian algorithm for lattice reduction. In: Adleman, L.M., Huang, MD. (eds) Algorithmic Number Theory. ANTS 1994. Lecture Notes in Computer Science, vol 877. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-58691-1_52
Download citation
DOI: https://doi.org/10.1007/3-540-58691-1_52
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58691-3
Online ISBN: 978-3-540-49044-9
eBook Packages: Springer Book Archive