Advertisement

Random permutations from logarithmic signatures

  • Spyros S. Magliveras
  • Nasir D. Memon
Track 4: Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 507)

Abstract

A cryptographic system, called pgm, was invented in the late 1970's by S. Magliveras. pgm is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. Logarithmic signatures were initially motivated by C. Sims' bases and strong generators. A logarithmic signature α, for a given group G, induces a mapping \(\hat \alpha\) from Z |G| to G. Hence it would be natural to use logarithmic signatures for generating random elements in a group. In this paper we focus on generating random permutations in the symmetric group S n. Random permutations find applications in design of experiments, simulation, cryptology, voice-encryption etc. Given a logarithmic signature α for S n and a seed s 0, we could efficiently compute the following sequence : \(\hat \alpha\)(s 0), \(\hat \alpha\)(s 0 + 1), ..., \(\hat \alpha\)(s 0 + r - 1) of r permutations. We claim that this sequence behaves like a sequence of random permutations. We undertake statistical tests to substantiate our claim.

Keywords

Conjugacy Class Symmetric Group Permutation Group Random Permutation Random Element 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    M. Furst, J. E. Hopcroft, and E. Luks. Polynomial-time algorithms for permutation groups. In Proceedings of the 21'st IEEE Symposium on Foundations of Computation of Computer Science, pages 36–41, 1980.Google Scholar
  2. [2]
    D. E. Knuth. The Art of Computer Programming. Addison-Wesley, 2'nd edition, 1981.Google Scholar
  3. [3]
    S. S. Magliveras. A cryptosystem from logarithmic signatures of finite groups. In Proceedings of the 29'th Midwest Symposium on Circuits and Systems. Elsevier Publishing Company, August 1986.Google Scholar
  4. [4]
    S. S. Magliveras and N. D. Memon. Algebraic properties of cryptosystem PGM. In Advances in Crptology, Crypto 89. Springer-Verlag, 1989.Google Scholar
  5. [5]
    N. D. Memon. On logarithmic signatures and applications. Master's thesis, University of Nebraska at Lincoln, May 1989.Google Scholar
  6. [6]
    E. S. Page. A note on generating random permutations. Applied Statistics, 16:273–274, 1967.CrossRefGoogle Scholar
  7. [7]
    C. R. Rao. Generation of random permutations of given number of elements using random sampling numbers. Sankhya, 23:305–307, 1961.zbMATHGoogle Scholar
  8. [8]
    C. C. Sims. Some group-theoretic algorithms. In M. F. Newman, editor, Topics in Algebra, pages 108–124. Springer-Verlag, 1978. Springer Lecture notes in Math. Vol 697.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Spyros S. Magliveras
    • 1
  • Nasir D. Memon
    • 1
  1. 1.University of NebraskaLincoln

Personalised recommendations