Designs, Codes and Cryptography

, Volume 9, Issue 1, pp 29–38 | Cite as

Some t-Homogeneous Sets of Permutations

  • Jürgen Bierbrauer
  • Stephen Black
  • Yves Edel


Perpendicular Arrays are orderedcombinatorial structures, which recently have found applicationsin cryptography. A fundamental construction uses as ingredientscombinatorial designs and uniformly t-homogeneoussets of permutations. We study the latter type of objects. Thesemay also be viewed as generalizations of t-homogeneousgroups of permutations. Several construction techniques are given.Here we concentrate on the optimal case, where the number ofpermutations attains the lower bound. We obtain several new optimalsuch sets of permutations. Each example allows the constructionof infinite families of perpendicular arrays.

Permutation sets permutation groups perpendicular arrays authentication 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    J. Bierbrauer, The uniformly 3–homogeneous subsets of PGL 2(q), Journal of algebraic combinatorics, Vol. 4 (1995) pp. 99–102.Google Scholar
  2. 2.
    J. Bierbrauer and Y. Edel, Theory of perpendicular arrays, Journal of Combinatorial Designs, Vol. 6 (1994) pp. 375–406.Google Scholar
  3. 3.
    J. Bierbrauer and Y.Edel, Halving PSL 2(q), to appear in Journal of Geometry.Google Scholar
  4. 4.
    J. Bierbrauer and T. v. Tran, Halving PGL 2(2f), f odd: a Series of Cryptocodes, Designs, Codes and Cryptography, Vol. 1 (1991) pp. 141–148.Google Scholar
  5. 5.
    J. Bierbrauer, T. v. Tran, Some highly symmetric Authentication Perpendicular Arrays, Designs, Codes and Cryptography, Vol. 1 (1992) pp. 307–319.Google Scholar
  6. 6.
    E. S. Kramer, D. L. Kreher, R. Rees, and D. R. Stinson, On perpendicular arrays with t ≤ 3, Ars Combinatoria, Vol. 28 (1989) pp. 215–223.Google Scholar
  7. 7.
    C. R. Rao, Combinatorial Arrangements analogous to Orthogonal Arrays, Sankhya A, Vol. 23 (1961) pp. 283–286.Google Scholar
  8. 8.
    D. R. Stinson, The Combinatorics of Authentication and Secrecy Codes, Journal of Cryptology, Vol. 2 (1990) pp. 23–49.Google Scholar
  9. 9.
    D. R. Stinson and L. Teirlinck, A Construction for Authentication/Secrecy Codes from 3–homogeneous Permutation Groups, European Journal of Combinatorics, Vol. 11 (1990) pp. 73–79.Google Scholar

Copyright information

© Kluwer Academic Publishers 1996

Authors and Affiliations

  • Jürgen Bierbrauer
    • 1
  • Stephen Black
    • 2
  • Yves Edel
    • 3
  1. 1.Department of Mathematical SciencesMichigan Technological UniversityHoughtonUSA
  2. 2.IBM Heidelberg (Germany) Mathematisches Institut der UniversitätHeidelberg (Germany
  3. 3.Mathematisches Institut der UniversitätHeidelberg (Germany

Personalised recommendations