Chinese Remaindering for Algebraic Numbers in a Hidden Field

  • Igor E. Shparlinski
  • Ron Steinfeld
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2369)

Abstract

We use lattice reduction to obtain a polynomial time algorithm for Chinese Remaindering in algebraic number fields in the case when the field itself is unknown.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Ajtai, R. Kumar and D. Sivakumar, ‘A sieve algorithm for the shortest lattice vector problem’, Proc. 33rd ACM Symp. on Theory of Comput., Crete, Greece, July 6–8, 2001, 601–610.Google Scholar
  2. 2.
    H. Cohen, A course in computational algebraic number theory, Springer-Verlag, Berlin, 1993.MATHGoogle Scholar
  3. 3.
    J. H. Conway and N. J. A. Sloan, Sphere packings, lattices and groups, Springer-Verlag, Berlin, 1998.Google Scholar
  4. 4.
    C. Ding, D. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in computing, coding, cryptography, World Scientific, Singapore, 1996.MATHGoogle Scholar
  5. 5.
    J. von zur Gathen, ‘Irreducibility of multivariate polynomials’, J. Comp. and Syst. Sci., 31 (1985), 225–264.MATHCrossRefGoogle Scholar
  6. 6.
    J. von zur Gathen and E. Kaltofen, ‘Factoring sparse multivariate polynomials’, J. Comp. and Syst. Sci., 31 (1985), 265–287.MATHCrossRefGoogle Scholar
  7. 7.
    R. T. Gregory and E. V. Krishnamurthy, Methods and applications of error-free computation, Springer-Verlag, Berlin, 1984.MATHGoogle Scholar
  8. 8.
    M. Grötschel, L. Lovász and A. Schrijver, Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin, 1993.MATHGoogle Scholar
  9. 9.
    A. Joux and J. Stern, ‘Lattice reduction: A toolbox for the cryptanalyst’, J. Cryptology, 11 (1998), 161–185.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    R. Kannan, ‘Algorithmic geometry of numbers’, Annual Review of Comp. Sci., 2 (1987), 231–267.CrossRefMathSciNetGoogle Scholar
  11. 11.
    W. Narkiewicz, Elementary and analytic theory of algebraic numbers, Polish Sci. Publ., Warszawa, 1990.MATHGoogle Scholar
  12. 12.
    P. Q. Nguyen and J. Stern, ‘Lattice reduction in cryptology: An update’, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 1838 (2000), 85–112.Google Scholar
  13. 13.
    P. Q. Nguyen and J. Stern, ‘The two faces of lattices in cryptology’, Lect. Notes in Comp. Sci., Springer-Verlag, Berlin, 2146 (2001), 146–180.Google Scholar
  14. 14.
    C. P. Schnorr, ‘A hierarchy of polynomial time basis reduction algorithms’, Theor. Comp. Sci., 53 (1987), 201–224.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    V. Shoup, ‘NTL: A library for doing number theory (version 5.2b)’, http://-shoup.net/ntl/, 2001.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Igor E. Shparlinski
    • 1
  • Ron Steinfeld
    • 2
  1. 1.Department of ComputingMacquarie University SydneyAustralia
  2. 2.School of Network ComputingMonash UniversityFrankstonAustralia

Personalised recommendations