Abstract
Lattices over number fields arise from a variety of sources in algorithmic algebra and more recently cryptography. Similar to the classical case of ℤ-lattices, the choice of a nice, “short” (pseudo)-basis is important in many applications. In this article, we provide the first algorithm that computes such a “short” (pseudo)-basis. We utilize the LLL algorithm for ℤ-lattices together with the Bosma-Pohst-Cohen Hermite Normal Form and some size reduction technique to find a pseudo-basis where each basis vector belongs to the lattice and the product of the norms of the basis vectors is bounded by the lattice determinant, up to a multiplicative factor that is a field invariant. As it runs in polynomial time, this provides an effective variant of Minkowski’s second theorem for lattices over number fields.
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., Kumar, R., Sivakumar, D.: A sieve algorithm for the shortest lattice vector problem. In: Proc. STOC 2001, pp. 601–610. ACM, New York (2001)
Belabas, K.: Topics in computational algebraic number theory. J. théorie des nombres de Bordeaux 16, 19–63 (2004)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)
Bosma, W., Pohst, M.: Computations with finitely generated modules over Dedekind domains. In: Proc. ISSAC 1991, pp. 151–156. ACM, New York (1991)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1995)
Cohen, H.: Hermite and Smith normal form algorithms over Dedekind domains. Math. Comp. 65, 1681–1699 (1996)
Cohen, H.: Advanced topics in Computational Number Theory. Springer, Heidelberg (2000)
Evertse, J.-H.: Reduced bases of lattices over number fields. Indag. Mathem. N.S. 2(3), 153–168 (1992)
Fieker, C.: Minimizing representations over number fields II: Computations in the Brauer group. J. Algebra 3(322), 752–765 (2009)
Fieker, C., Pohst, M.E.: Lattices over number fields. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 147–157. Springer, Heidelberg (1996)
Gan, Y.H., Ling, C., Mow, W.H.: Complex lattice reduction algorithm for low-complexity full-diversity MIMO detection. IEEE Trans. Signal Processing 57, 2701–2710 (2009)
Hoppe, A.: Normal forms over Dedekind domains, efficient implementation in the computer algebra system KANT. PhD thesis, Technical University of Berlin (1998)
Kannan, R., Bachem, A.: Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix. SIAM J. Comput. 8(4), 499–507 (1979)
Lagarias, J.C., Lenstra Jr., H.W., Schnorr, C.P.: Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10, 333–348 (1990)
Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)
Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1986)
Lyubashevsky, V., Micciancio, D.: Generalized compact knapsacks are collision resistant. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 144–155. Springer, Heidelberg (2006)
Magma. The Magma computational algebra system for algebra, number theory and geometry, http://magma.maths.usyd.edu.au/magma/
Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Comput. Complexity 16(4), 365–411 (2007)
Micciancio, D., Goldwasser, S.: Complexity of lattice problems: a cryptographic perspective. Kluwer Academic Press, Dordrecht (2002)
Mollin, R.A.: Algebraic Number Theory. Chapman and Hall/CRC Press (1999)
Napias, H.: A generalization of the LLL-algorithm over Euclidean rings or orders. J. théorie des nombres de Bordeaux 2, 387–396 (1996)
O’Meara, O.T.: Introduction to Quadratic Forms. In: Grundlehren der Mathematischen Wissenschaften, vol. 117. Springer, Heidelberg (1963)
Peikert, C., Rosen, A.: Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 145–166. Springer, Heidelberg (2006)
Peikert, C., Rosen, A.: Lattices that admit logarithmic worst-case to average-case connection factors. In: Proc. STOC 2007, pp. 478–487. ACM, New York (2007)
Stehlé, D., Steinfeld, R., Tanaka, K., Xagawa, K.: Efficient public key encryption based on ideal lattices. In: Matsui, M. (ed.) ASIACRYPT 2009. LNCS, vol. 5912, pp. 617–635. Springer, Heidelberg (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fieker, C., Stehlé, D. (2010). Short Bases of Lattices over Number Fields. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-14518-6_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-14517-9
Online ISBN: 978-3-642-14518-6
eBook Packages: Computer ScienceComputer Science (R0)