Simultaneous reduction of a lattice basis and its reciprocal basis


Given a latticeL we are looking for a basisB=[b 1, ...b n ] ofL with the property that bothB and the associated basisB *=[b *1 , ...,b * n ] of the reciprocal latticeL * consist of short vectors. For any such basisB with reciprocal basisB * let\(S(B) = \mathop {\max }\limits_{1 \leqslant i \leqslant n} (|b_i | \cdot |b_i^ * |)\). Håstad and Lagarias [7] show that each latticeL of full rank has a basisB withS(B)≤exp(c 1·n 1/3) for a constantc 1 independent ofn. We improve this upper bound toS(B)≤exp(c 2·(lnn)2) withc 2 independent ofn.

We will also introduce some new kinds of lattice basis reduction and an algorithm to compute one of them. The new algorithm proceeds by reducing the quantity\(\sum\limits_{i = 1}^n {|b|^2 } \cdot |b_i^ * |^2 \). In combination with an exhaustive search procedure, one obtains an algorithm to compute the shortest vector and a Korkine-Zolotarev reduced basis of a lattice that is efficient in practice for dimension up to 30.

This is a preview of subscription content, access via your institution.


  1. [1]

    J. H. Conway, andN. J. A. Sloane:Sphere packings, lattices and groups, Springer Verlag, New York, 1988.

    Google Scholar 

  2. [2]

    M. J. Coster, B. A. LaMacchia, A. M. Odlyzko, andC. P. Schnorr: An improved low-density subset sum algorithm, to appear inComputational Complexity.

  3. [3]

    U. Dieter: How to compute the shortest vector in a lattice,Math. Comp. 29 (1975), 827–833.

    Google Scholar 

  4. [4]

    M. Euchner, andC. P. Schnorr: Lattice basis reduction: improved practical algorithms and solving subset sum problems,Proceedings of Fundamentals of Computation Theory, FTC '91, Ed. L. Budach, Springer LNCS529 (1991) 68–85.

  5. [5]

    P. M. Gruber, andJ. Lekkerkerker:Geometry of numbers, North Holland, Amsterdam, 1987.

    Google Scholar 

  6. [6]

    Johan Håstad, B. Just, J. C. Lagarias, andC. P. Schnorr, Polynomial time algorithms for finding integer relations among real numbers,SIAM J. Comput. 18 (1989), 859–881.

    Google Scholar 

  7. [7]

    Johan Håstad, andJ. C. Lagarias: Simultaneously good bases of a lattice and its reciprocal lattice,Math. Ann. 287 (1990), 163–174.

    Google Scholar 

  8. [8]

    R. Kannan: Improved algorithms on integer programming and related lattice problems,Proc. 15th Annual ACM Symp. on Theory of Computing (1983), 293–206.

  9. [9]

    D. E. Knuth:The art of computer programming, Vol. 2: Seminumerical algorithms, 2nd edition, Addison-Wesley, 1981.

  10. [10]

    A. Korkine, andG. Zolotarev: Sur les formes quadratiques,Math. Ann. 6 (1873), 366–389.

    Google Scholar 

  11. [11]

    J. C. Lagarias, H. W. Lenstra, Jr., andC. P. Schnorr: Korkine Zolotarev bases and successive minima of a lattice and its reciprocal,Combinatorica 10 (1990), 333–348.

    Google Scholar 

  12. [12]

    J. C. Lagarias, andA. M. Odlyzko: Solving low-density subset sum problems,J. Assoc. Comp. Mach. 32 (1985), 229–246.

    Google Scholar 

  13. [13]

    B. A. LaMacchia: Basis reduction algorithms and subset sum problems. Thesis for the degree of Master of Science, Department of Electrical engineering and Computer Science, Massachusetts Institute mof Technology, May 1991.

  14. [14]

    A. K. Lenstra, H. W. Lenstra, Jr., andL. Lovász: Factoring polynomials with rational coefficients,Math. Ann. 261 (1982), 515–534.

    Google Scholar 

  15. [15]

    H. W. Lenstra, Jr.: Integer programming with a fixed number of variables,Math. Oper. Res. 8 (1983), 538–548.

    Google Scholar 

  16. [16]

    C. P. Schnorr: A hierarchy of polynomial time lattice basis reduction algorithms,Theor. Comp. Sci. 53 (1987), 201–227.

    Google Scholar 

  17. [17]

    C. P. Schnorr: A more efficient algorithm for lattice basis reduction,J. Algorithms 9 (1988), 47–62.

    Google Scholar 

  18. [18]

    C. P. Schnorr: Factoring integers and computing discrete logarithms via diophantine approximation,Proceedings of Eurocrypt '91, Brighton, May 1991, to appear in Springer LNCS.

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Seysen, M. Simultaneous reduction of a lattice basis and its reciprocal basis. Combinatorica 13, 363–376 (1993).

Download citation

AMS subject classification code (1991)

  • 11 H 55