Fast LLL-type lattice reduction

Information and Computation

Volumn 204, Issue 1, 2006, Pages 1-25

  Schnorr, Claus Peter ^a  

^a GOETHE UNIVERSITY   (Germany)

Author keywords

Floating point errors; Householder reflection; Length defect; LLL reduction; Local LLL reduction; QR decomposition; Segments; SLLL reduction

Indexed keywords

COMMUNICATION CHANNELS (INFORMATION THEORY);

FLOATING POINT ERRORS; HOUSEHOLDER REFLECTIONS; LENGTH DEFECTS; LLL-REDUCTION; LOCAL LLL-REDUCTION; Q R DECOMPOSITION; SEGMENTS; SLLL-REDUCTION;

DIGITAL ARITHMETIC;

EID: 84855206587     PISSN: 08905401     EISSN: 10902651     Source Type: Journal    
DOI: 10.1016/j.ic.2005.04.004     Document Type: Article

References (40)

1. 2 is NP-hard for randomized reductions, Proc. 30th STOC (1998) 10-19.
2. M. Ajtai, The Worst-case behavior of Schnorr's algorithm approximating the shortest nonzero vector in lattice, Proc. 35th STOC (2003) 396-406.
3. M. Ajtai, R. Kumar, D. Sivakumar, A sieve algorithm for the shortest lattice vector problemn, Proc. 33th STOC (2001) 601-610. (Pubitemid 32865777)
4. A. Akhavi, Random lattices threshold phenomena and efficient reduction algorithms, Theoret. Comput. Sci. 287(2002) 359-385.
5. J. Blömer, A. May, New partial key exposure attacks on RSA, in: Proc. Crypto'2003, Lecture Notes in Computer Science, vol. 2729, vol. 2729, Springer, New York, 2003, pp. 27-43. (Pubitemid 137636933)
6. D. Bleichenbacher, P. Q. Nguyen, Noisy polynomial interpolation and noisy Chinese remaindering, in: Eurocrypt 2000, Lecture Notes in Computer Science, vol. 1807, vol. 1807, Springer, New York, 2000, pp. 53-69.
7. J. Blömer, J. P. Seifert, On the complexity of Computing short linearly independent vectors and short bases in lattice, Proc. 31th STOC (1999) 711-720.
8. D. Boneh, Finding smooth integers in small intervals using CRT decoding, Proc. 32th STOC (2000) 265-272.
9. D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities, J. Cryptol. 10(1997) 233-260. (Pubitemid 127754574)
10. D. Coppersmith, Finding small solutions to small degree polynomials, in: Cryptography and Lattices, Lecture Notes in Computer Sciences, vol. 2146, Springer, New York, 2001, pp. 20-31. (Pubitemid 33329083)
11. H. Daudé, B. Vallée, An upper bound on the average number of iterations of the LLL algorithm, Theoret. Comput. Sci. 123(1994) 115-395.
12. P. van Emde Boas, Another NP-Complete partition problem and the complexity of computing short vectors in a lattice, Mathematics Department, University of Amsterdam, TR 81-04, 1981..
13. C. F. Gauss, Disquisitiones Arithmeticae. 1801; English transl, Yale Univ Press, New Haven, CT, 1966..
14. B. Helfrich, Algorithms to construct Minkowski reduced and Hermite reduced bases, Theor. Comput. Sci. 41(1985) 125-139. (Pubitemid 16593835)
15. J. Håstad, B. Just, J. C. Lagarias, C. P. Schnorr, Polynomial time algorithms for finding integer relations among real numbers, SIAM J. Comput. 18(1989) 859-881.
16. R. Kannan, Minkowski's Convex Body theorem and integer programming, Math. Oper. Res. 12(1984) 415-440.
17. H. Koy, C. P. Schnorr, Segment LLL-Reduction, in: Cryptography and Lattices, Lecture Notes in Computer Sciences, vol. 2146, Springer, New York, 2001, pp. 67-80, //www.mi.informatik.uni-frankfurt.de/research/papers.html.
18. H. Koy, C. P. Schnorr, Segment LLL-Reduction with Floating Point Orthogonalization, in: Cryptography and Lattices, Lecture Notes in Computer Sciences, vol. 2146, Springer, New York, 2001, pp. 81-96, //www.mi.informatik. uni-frankfurt.de/research/papers.html. (Pubitemid 33329088)
19. H. Koy, C. P. Schnorr, Segment and strong segment LLL-reduction of lattice bases, TR Universität Franfurt, April 2002, www.mi.informatik.uni- frankfurt.de/research/papers.html.
20. H. W. Lenstra, Integer programming with a fixed number of variables, Math. Oper. Res. 8(1983) 538-548.
21. C. L. Lawson, R. J. Hanson, Solving least square problems, SIAM Publications, 1995 (first published by Prentice Hall, New Jersey 1974)..
22. A. K. Lenstra, H. W. Lenstra, L. Lovász, Factoring Polynomials with Rational Coefficients, Math. Ann. 261(1982) 515-534.
23. L. Lovász, An algorithmic theory of numbers, in: Graphs and Convexity, CBMS-NSF Regional Conference Series in Applied Mathematics, 50, SIAM Publications, Philadelphia, 1986..
24. A. May, New RSA vulnerabilities using lattice reduction methods. Dissertation Thesis, University of Paderborn, October 2003..
25. D. Micciancio, The shortest vector problem is NP-hard to approximate within some constant, in: Proceedings FOCS 1998, Final version: SIAM J. Comput. 30(2001) 2008-2035. (Pubitemid 34125447)
26. D. Micciancio, A linear space algorithm for computing the Hermite normal form, in: Proceedings ISSAC 2001, Lecture Notes in Computer Sciences, vol. 2146, Springer, New York, 2001, pp. 126-145. (Pubitemid 33329091)
27. D. Micciancio, S. Goldwasser, Complexity of Lattice Problems, A Cryptographic Perspective, Kluwer Academic Publishers, London, 2002.
28. S. Mehrotra, Z. Li, Reduction of lattice bases using modular arithmetic. TR. Dept. of Industrial Engeneering and Management Sciences, Northwestern University, Evanston, Il. Oct. 2001, mehrotra, zhifeng@iems.nwu.edu..
29. P. Q. Nguyen, J. Stern, Lattice reduction in cryptology, an update. Algorithmic number theory, in: Lecture Notes in Computer Sciences, vol. 1838, Springer, New York, 2000, pp. 85-112, full version http://www.di.ens.fr/pnguyen, stern/.
30. P. Q. Nguyen, D. Stehle, Floating-Point LLL Revisited, To appear in Proc. Eurocrypt'05..
31. C. P. Schnorr, A hierarchy of polynomial time lattice basis reduction algorithms, Theoret. Comput. Sci. 53(1987) 201-224. (Pubitemid 18536041)
32. C. P. Schnorr, A more efficient algorithm for lattice reduction, J. Algor. 9(1988) 47-62.
33. C. P. Schnorr, M. Euchner, Lattice basis reduction and solving subset sum problems. Fundamentals of Comput. Theory, Lecture Notes in Computer Sciences, vol. 591, Springer, New York, pp. 68-85, 1991.
34. The complete paper appeared in Math. Programming Studies, 66A, 2, 1994, pp. 181-199..
35. C. P. Schnorr, Lattice reduction by random sampling and birthday methods, in:H. Alt, M. Habib (Eds.), Proc. STACS 2003, Lecture Notes in Computer Sciences, vol. 2607, Springer, New York, 2003, pp. 145-156.
36. C. P. Schnorr, Gittertheorie und Algorithmische Geometrie. Lecture Notes Universität Frankfurt, Frankfurt, 2004. http://www.mi.informatik.uni- frankfurt.de/index.html#teaching..
37. 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, Lecture Notes in Computer Sciences vol. 172, Springer, New York, pp. 436-447, 1984..
38. V. Shoup, NTL, Number Theory C++Library. http://www.shoup. net/ntl/..
39. A. Storjohann, Faster Algorithms for Integer Lattice Basis Reduction. TR 249, Swiss Federal Institute of Technology, ETH-Zurich, Department of Computer Science, Zurich, Switzerland, July 1996. www.inf.ethz.ch/research/publications/ html..
40. J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1963.