Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-24T01:24:10.232Z Has data issue: false hasContentIssue false
  • English
  • Français

Block Reduced Lattice Bases and Successive Minima

Published online by Cambridge University Press:  12 September 2008

C. P. Schnorr
C. P. Schnorr
Affiliation:
Fachbereich Mathematik/Informatik, Universität Frankfurt, Postfach 111932, D–60054 Frankfurt a.M., Germany. email: schnorr@informatik.uni-frankfurt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

A lattice basis bi,…,bm is called block reduced with block size β if for every β consecutive vectors bi,…,bi+β−1, the orthogonal projections of bi,…,bi+β−1 in span(bi,…,bi−1)⊥ are reduced in the sense of Hermite and Korkin–Zolotarev. Let λi denote the successive minima of lattice L, and let b1,…,bm be a basis of L that is block reduced with block size β. We prove that for i = 1,…, m

where γβ is the Hermite constant for dimension β. For block size β = 3 and odd rank m ≥ 3, we show that

where the maximum is taken over all block reduced bases of all lattices L. We present critical block reduced bases achieving this maximum. Using block reduced bases, we improve Babai's construction of a nearby lattice point. Given a block reduced basis with block size β of the lattice L, and given a point x in the span of L, a lattice point υ can be found in time βO(β) satisfying

These results also give improvements on the method of solving integer programming problems via basis reduction.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 3 , Issue 4 , December 1994 , pp. 507 - 522
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Babai, L. (1986) On Lovász' lattice reduction and the nearest lattice point problem. Combinatorica 6 1–13.Google Scholar
[2]Bachem, A. and Kannan, R. (1984) Lattices and the basis reduction algorithm. TR, Carnegie Mellon University.Google Scholar
[3]Cassels, J. W. S. (1971) An introduction to the geometry of numbers, Springer-Verlag.Google Scholar
[4]Conway, J. H. and Sloane, H. J. A. (1988) Sphere Packings, Lattices and Groups, Springer-Verlag.CrossRefGoogle Scholar
[5]Coster, M. J., Joux, A., LaMacchia, B. A., Odlyzko, A. M., Schnorr, C. P. and Stern, J. (1992) An improved low-density subset sum algorithm. Computational Complexity 2 111–128.CrossRefGoogle Scholar
[6]Grötschel, M., Lovász, L. and Schrijver, A. (1988) Geometric algorithms and combinatorial optimization, Springer-Verlag.Google Scholar
[7]Hermite, C. (1850) Extraits de lettres de M. Ch. Hermite à M. Jacobi sur différents objets de la théorie des nombres, Deuxième lettre du 6 août 1845. J. Reine Angew. Math. 40 279–290.Google Scholar
[8]Kannan, R. (1987) Minkowski's convex body theorem and integer programming. Math. Oper. Res. 12 415–440.CrossRefGoogle Scholar
[9]Korkin, A. and Zolotarev, G. (1873) Sur les formes quadratiques. Math. Ann. 6 366–389.CrossRefGoogle Scholar
[10]Lagarias, J. C., Lenstra, H. W. Jr. and Schnorr, C. P. (1990) Korkin–Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10 333–348.Google Scholar
[11]Lagrange, J. L. (1773) Recherches d'arithmétique, Nouv. Mém. Acad. Berlin 265–312. Œvres, vol. viii, 693–753.Google Scholar
[12]Lenstra, A. K., Lenstra, H. W. Jr. and Lovász, L. (1982) Factoring polynomials with rational coefficients. Math. Ann. 261 515–534.CrossRefGoogle Scholar
[13]Lenstra, H. W. Jr. (1983) Integer programming with a fixed number of variables. Math. Oper. Res. 8 538–548.Google Scholar
[14]Lovász, L. (1986) An algorithmic theory of numbers, graphs and convexity. CBMS–NSF Regional Conference Series in Applied Mathematics 50, SIAM, Philadelphia, Pennsylvania.Google Scholar
[15]Lovász, L. (1988) Geometry of numbers and integer programming. Proceedings 13th Annual Symp. on Mathematical Programming 177–201.Google Scholar
[16]Mahler, K. (1938) A theorem on inhomogeneous diophantine inequalities. Nederl. Akad. Wetensch., Proc. 41 634–637.Google Scholar
[17]Schnorr, C. P. (1987) A hierarchy of polynomial time lattice basis reduction algorithms. Theoret. Comput. Sci. 53 201–224.Google Scholar
[18]Schnorr, C. P. (1993) Factoring Integers and Computing Discrete Logarithms via Diophantine Approximation. In: Cai, J.-Y. (ed.) Advances in Computational Complexity Theory. AMS DIMACS Series in Discr. Math, and Theor. Comp. Sc. 13 171–182.Google Scholar
[19]Schnorr, C. P. and Euchner, M. (1991) Lattice basis reduction: improved algorithms and solving subset sum problems. In: Budach, L. (ed.) Proceedings of Fundamentals of Computation Theory, FCT91. Springer-Verlag Lecture Notes in Computer Science 529 68–85. (Complete paper to appear in Mathematical Programming Studies.)Google Scholar