Short Bases of Lattices over Number Fields

  • Claus Fieker
  • Damien Stehlé
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)

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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    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)Google Scholar
  2. 2.
    Belabas, K.: Topics in computational algebraic number theory. J. théorie des nombres de Bordeaux 16, 19–63 (2004)MATHMathSciNetGoogle Scholar
  3. 3.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bosma, W., Pohst, M.: Computations with finitely generated modules over Dedekind domains. In: Proc. ISSAC 1991, pp. 151–156. ACM, New York (1991)CrossRefGoogle Scholar
  5. 5.
    Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, Heidelberg (1995)Google Scholar
  6. 6.
    Cohen, H.: Hermite and Smith normal form algorithms over Dedekind domains. Math. Comp. 65, 1681–1699 (1996)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Cohen, H.: Advanced topics in Computational Number Theory. Springer, Heidelberg (2000)MATHGoogle Scholar
  8. 8.
    Evertse, J.-H.: Reduced bases of lattices over number fields. Indag. Mathem. N.S. 2(3), 153–168 (1992)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Fieker, C.: Minimizing representations over number fields II: Computations in the Brauer group. J. Algebra 3(322), 752–765 (2009)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Fieker, C., Pohst, M.E.: Lattices over number fields. In: Cohen, H. (ed.) ANTS 1996. LNCS, vol. 1122, pp. 147–157. Springer, Heidelberg (1996)Google Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Hoppe, A.: Normal forms over Dedekind domains, efficient implementation in the computer algebra system KANT. PhD thesis, Technical University of Berlin (1998)Google Scholar
  13. 13.
    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)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    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)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lenstra, A.K., Lenstra Jr., H.W., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lovász, L.: An Algorithmic Theory of Numbers, Graphs and Convexity. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1986)Google Scholar
  17. 17.
    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)CrossRefGoogle Scholar
  18. 18.
    Magma. The Magma computational algebra system for algebra, number theory and geometry, http://magma.maths.usyd.edu.au/magma/
  19. 19.
    Micciancio, D.: Generalized compact knapsacks, cyclic lattices, and efficient one-way functions. Comput. Complexity 16(4), 365–411 (2007)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Micciancio, D., Goldwasser, S.: Complexity of lattice problems: a cryptographic perspective. Kluwer Academic Press, Dordrecht (2002)MATHGoogle Scholar
  21. 21.
    Mollin, R.A.: Algebraic Number Theory. Chapman and Hall/CRC Press (1999)Google Scholar
  22. 22.
    Napias, H.: A generalization of the LLL-algorithm over Euclidean rings or orders. J. théorie des nombres de Bordeaux 2, 387–396 (1996)MathSciNetGoogle Scholar
  23. 23.
    O’Meara, O.T.: Introduction to Quadratic Forms. In: Grundlehren der Mathematischen Wissenschaften, vol. 117. Springer, Heidelberg (1963)Google Scholar
  24. 24.
    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)CrossRefGoogle Scholar
  25. 25.
    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)Google Scholar
  26. 26.
    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)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Claus Fieker
    • 1
  • Damien Stehlé
    • 1
    • 2
  1. 1.Magma Computer Algebra Group, School of Mathematics and StatisticsUniversity of SydneyAustralia
  2. 2.CNRS and Macquarie University 

Personalised recommendations