Segment LLL-Reduction with Floating Point Orthogonalization
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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.
KeywordsLLL-reduction Householder reflexion floating point arithmetic stability scaled basis segment LLL-reduction local LLL-reduction
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- 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
- KS01.H. Koy and C.P. Schnorr, Segment LLL-Reduction of Lattice Bases. Proceedings CaLC 2001, pp. 67–80.Google Scholar
- LH95.C.L. Lawson and R.J. Hanson, Solving Least Square Problems, SIAM, Philadelphia, 1995.Google Scholar
- NTL.NTL homepage: http://www.shoup.net/ntl/, 2000.
- 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
- 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
- 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