Abstract
In this paper we introduce several new heuristics as to speed up known lattice basis reduction methods and improve the quality of the computed reduced lattice basis in practice. We analyze substantial experimental data and to our knowledge, we are the first to present a general heuristic for determining which variant of the reduction algorithm, for varied parameter choices, yields the most efficient reduction strategy for reducing a particular problem instance.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ajtai, M.: Generating Hard Instances of Lattice Problems. In: Proceedings of the 28th ACM Symposium on Theory of Computing, pp. 99–108 (1996)
Ajtai, M., Dwork, C.: A Public-Key Cryptosystem with Worst-Case/Average- Case Equivalence. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 284–293 (1997)
Biehl, I., Buchmann, J., Papanikolaou, T.: LiDIA: A Library for Computational Number Theory. Technical Report 03/95, SFB 124, Universität des Saarlandes, Saarbrücken, Germany (1995)
Cohen, H.: A Course in Computational Algebraic Number Theory, 2nd edn. Springer, Heidelberg (1993)
Coppersmith, D.: Finding a Small Root of a Univariate Modular Equation. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 155–165. Springer, Heidelberg (1996)
Coster, M.J., Joux, A., LaMacchia, B.A., Odlyzko, A.M., Schnorr, C.P., Stern, J.: Improved Low-Density Subset Sum Algorithms. Journal of Computational Complexity 2, 111–128 (1992)
Domich, P.D., Kannan, R., Trotter, L.E.: Hermite Normal Form Computation using Modulo Determinant Arithmetic. Mathematics Operations Research 12(1), 50–59 (1987)
Goldreich, O., Goldwasser, S., Halevi, S.: Public-Key-Cryptosystems from Lat- tice Reduction Problems. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 112–131. Springer, Heidelberg (1997)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combina- torial Optimization, 2nd edn. Springer, Heidelberg (1993)
Havas, G., Majewski, B.S., Matthews, K.R.: Extended GCD Algorithms. Technical Report TR0302, The University of Queensland, Brisbane, Australia (1994)
Joux, A., Stern, J.: Lattice Reduction: A Toolbox for the Cryptanalyst. Journal of Cryptology 11(3), 161–185 (1998)
Knuth, D.E.: The Art of Computer Programming. Seminumerical algo- rithms, 2nd edn., vol. 2. Addison-Wesley, Reading (1981)
Lenstra, A.K., Lenstra, H.W., Lovász, L.: Factoring Polynomials with Rational Coefficients. Math. Ann. 261, 515–534 (1982)
LiDIA Group: LiDIA Manual. Universitát des Saarlandes/TU Darmstadt, Germany, see LiDIA homepage (1999), http://www.informatik.tu-darmstadt.de/TI/LiDIA
Magma homepage (1999), http://www.maths.usyd.edu.au:8000/comp/magma/Overview.html
Nguyen, P.: Cryptanalysis of the Goldreich-Goldwasser-Halevi Cryptosystem from Crypto 1997. In: Wiener, M. (ed.) CRYPTO 1999. LNCS, vol. 1666, pp. 288–304. Springer, Heidelberg (1999)
Nguyen, P., Stern, J.: Cryptanalysis of a Fast Public Key Cryptosystem Pre- sented at SAC 1997. In: Tavares, S., Meijer, H. (eds.) SAC 1998. LNCS, vol. 1556, p. 213. Springer, Heidelberg (1999)
NTL homepage (1999), http://www.cs.wisc.edu/~shoup/ntl
Pohst, M.E., Zassenhaus, H.J.: Algorithmic Algebraic Number Theory. Cambridge University Press, Cambridge (1989)
Radziszowski, S., Kreher, D.L.: Solving Subset Sum Problems with the L 3 Algorithm. J. Combin. Math. Combin. Computation 3, 49–63 (1988)
Rickert, N.W.: Efficient Reduction of Quadratic Forms. In: Proceedings of Computers and Mathematics 1989, pp. 135–139 (1989)
Schnorr, C.P., Euchner, M.: Lattice Basis Reduction: Improved Practical Al- gorithms and Solving Subset Sum Problems. In: Budach, L. (ed.) FCT 1991. LNCS, vol. 529, pp. 68–85. Springer, Heidelberg (1991)
de Weger, B.: Algorithms for Diophantine Equations. PhD Thesis, Centrum voor Wiskunde en Informatica, Amsterdam, Netherlands (1988)
Wetzel, S.: Lattice Basis Reduction Algorithms and their Applications. PhD Thesis, Universitát des Saarlandes, Saarbrúcken, Germany (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Backes, W., Wetzel, S. (2000). New Results on Lattice Basis Reduction in Practice. In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_7
Download citation
DOI: https://doi.org/10.1007/10722028_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67695-9
Online ISBN: 978-3-540-44994-2
eBook Packages: Springer Book Archive