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Segment LLL-Reduction with Floating Point Orthogonalization

  • Henrik Koy
  • Claus Peter Schnorr
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2146)

Abstract

We associate with an integer lattice basis a scaled basis that has orthogonal vectors of nearly equal length. The orthogonal vectors or the QR-factorization of a scaled basis can be accurately computed up to dimension 216 by Householder reflexions in floating point arithmetic (fpa) with 53 precision bits.

We develop a highly practical fpa-variant of the new segment LLL- reduction of Koy and Schnorr [KS01]. The LLL-steps are guided in this algorithm by the Gram-Schmidt coefficients of an associated scaled basis. The new reduction algorithm is much faster than previous codes for LLL-reduction and performs well beyond dimension 1000.

Keywords

LLL-reduction Householder reflexion floating point arithmetic stability scaled basis segment LLL-reduction local LLL-reduction 

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References

  1. GGH.
    O. Goldreich, S. Goldwasser, and S. Halevi, Public-key cryptosystems from lattice reduction problems. Proc. Crypto’97, LNCS 1294, Springer-Verlag, pp. 112–131, 1997.Google Scholar
  2. KS01.
    H. Koy and C.P. Schnorr, Segment LLL-Reduction of Lattice Bases. Proceedings CaLC 2001, pp. 67–80.Google Scholar
  3. LLL82.
    A.K. Lenstra, H. W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann.261, pp. 515–534, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  4. LH95.
    C.L. Lawson and R.J. Hanson, Solving Least Square Problems, SIAM, Philadelphia, 1995.Google Scholar
  5. NTL.
    NTL homepage: http://www.shoup.net/ntl/, 2000.
  6. RS96.
    C. Rössner and C.P. Schnorr, An optimal stable continued fraction algorithm for arbitrary dimension. 5.-th IPCO, LNCS 1084, pp. 31–43, Springer-Verlag, 1996.Google Scholar
  7. S87.
    C.P. Schnorr, A hierarchy of polynomial time lattice basis reduction algorithms, Theoretical Computer Science53, pp. 201–224, 1987.zbMATHMathSciNetCrossRefGoogle Scholar
  8. S88.
    C.P. Schnorr, A more efficient algorithm for lattice basis reduction, J. Algorithms9, pp. 47–62, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  9. SE91.
    C.P. Schnorr and M. Euchner, Lattice Basis Reduction: Improved Practical Algorithms and Solving Subset Sum Problems, Proc. Fundamentals of Computation Theory’91, L. Budach, ed., LNCS 529, Springer-Verlag, pp. 68–85, 1991. (Complete paper in Mathematical Programming Studies 66A, No 2, pp. 181–199, 1994.)Google Scholar
  10. Sc84.
    A. Schönhage, Factorization of univariate integer polynomials by diophantine approximation and improved lattice basis reduction algorithm, Proc. 11-th Coll. Automata, Languages and Programming, Antwerpen 1984, LNCS 172, Springer-Verlag, pp. 436–447, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Henrik Koy
    • 1
  • Claus Peter Schnorr
    • 2
  1. 1.Deutsche Bank AG, Frankfurt am MainGermany
  2. 2.Fachbereiche Mathematik und InformatikUniversität FrankfurtFrankfurt am MainGermany

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