Abstract
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.
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., Dwork, C.: A public-key cryptosystem with worst-case/average-case equivalence. In: Proc. of STOC 1997, pp. 284–293. ACM, New York (1997)
Akhavi, A.: Random lattices, threshold phenomena and efficient reduction algorithms. TCS 287(2), 359–385 (2002)
Akhavi, A., Marckert, J.-F., Rouault, A.: On the reduction of a random basis. In: Proc. of the ANALCO 2007, New Orleans, SIAM, Philadelphia (2007)
Boneh, D., Durfee, G.: Cryptanalysis of RSA with private key d less than N 0.292. IEEE Trans Inform Theor 46(4), 233–260 (2000)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. JSC 24(3–4), 235–265 (1997)
Brisebarre, N., Chevillard, S.: Efficient polynomial L-approximations. In: Proc. of ARITH’18, pp. 169–176. IEEE, Los Alamitos (2007)
Brisebarre, N., Hanrot, G.: Floating-point L2-approximations to functions. In: Proc. of ARITH’18, pp. 177–186. IEEE, Los Alamitos (2007)
Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Springer, Heidelberg (1971)
Chen, Z., Dongarra, J.: Condition numbers of gaussian random matrices. SIAM J Matrix Anal A 27(3), 603–620 (2005)
Chen, Z., Storjohann, A.: A BLAS based C library for exact linear algebra on integer matrices. In: Proc. of ISSAC 2005, pp. 92–99. ACM, New York (2005)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1995)
Coppersmith, D.: Small solutions to polynomial equations, and low exponent RSA vulnerabilities. J. of Cryptology 10(4), 233–260 (1997)
Daudé, H., Vallée, B.: An upper bound on the average number of iterations of the LLL algorithm. TCS 123(1), 95–115 (1994)
von zur Gathen, J., Gerhardt, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (2003)
Johnson, W.B., Lindenstrauss, J.: Extension of Lipschitz mappings into a Hilbert space. Comm Contemp Math 26, 189–206 (1984)
Koy, H., Schnorr, C.P.: Segment LLL-reduction of lattice bases. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 67–80. Springer, Heidelberg (2001)
Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math Ann 261, 513–534 (1982)
Nguyen, P., Stehlé, D.: Floating-point LLL revisited. In: Cramer, R.J.F. (ed.) EUROCRYPT 2005. LNCS, vol. 3494, pp. 215–233. Springer, Heidelberg (2005)
Nguyen, P., Stern, J.: The two faces of lattices in cryptology. In: Silverman, J.H. (ed.) CaLC 2001. LNCS, vol. 2146, pp. 146–180. Springer, Heidelberg (2001)
Odlyzko, A.M., te Riele, H.J.J.: Disproof of Mertens conjecture. J reine angew Math 357, 138–160 (1985)
Rouault, A.: Asymptotic behavior of random determinants in the laguerre, gram and jacobi ensembles. Latin American Journal of Probability and Mathematical Statistics (ALEA) 3, 181–230 (2007)
Schnorr, C.P.: Progress on LLL and lattice reduction. In: Proc. of the LLL+25 conference (to appear)
Stehlé, D.: On the randomness of bits generated by sufficiently smooth functions. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 257–274. Springer, Heidelberg (2006)
Stehlé, D., Lefèvre, V., Zimmermann, P.: Searching worst cases of a one-variable function. IEEE Trans Comp. 54(3), 340–346 (2005)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Akhavi, A., Stehlé, D. (2008). Speeding-Up Lattice Reduction with Random Projections (Extended Abstract). In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds) LATIN 2008: Theoretical Informatics. LATIN 2008. Lecture Notes in Computer Science, vol 4957. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78773-0_26
Download citation
DOI: https://doi.org/10.1007/978-3-540-78773-0_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-78772-3
Online ISBN: 978-3-540-78773-0
eBook Packages: Computer ScienceComputer Science (R0)