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On the Information Rate of Secret Sharing Schemes

Extended Abstract
  • C. Blundo
  • A. De Santis
  • L. Gargano
  • U. Vaccaro
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 740)

Abstract

We derive new limitations on the information rate and the average information rate of secret sharing schemes for access structure represented by graphs. We give the first proof of the existence of access structures with optimal information rate and optimal average information rate less that 1/2 + ε, where ε is an arbitrary positive constant. We also provide several general lower bounds on information rate and average information rate of graphs. In particular, we show that any graph with n vertices admits a secret sharing scheme with information rate Ω((log n)/n).

Keywords

Access Structure Information Rate Conditional Entropy Complete Bipartite Graph Secret Share Scheme 
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.

References

  1. 1.
    J. C. Benaloh and J. Leichter, Generalized Secret Sharing and Monotone Functions, in “Advances in Cryptology-CRYPTO 88”, Ed. S. Goldwasser, vol. 403 of “Lecture Notes in Computer Science”, Springer-Verlag, pp. 27–35.CrossRefGoogle Scholar
  2. 2.
    G. R. Blakley, Safeguarding Cryptographic Keys, Proceedings AFIPS 1979 National Computer Conference, pp.313–317, June 1979.Google Scholar
  3. 3.
    C. Blundo Secret Sharing Schemes for Access Structures based on Graphs, Tesi di Laurea, University of Salerno, Italy, 1991, (in Italian).Google Scholar
  4. 4.
    C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decomposition and Secret Sharing Schemes, Eurocrypt 1992, Hungary.Google Scholar
  5. 5.
    E. F. Brickell and D. M. Davenport, On the classification of ideal secret sharing schemes, J. Cryptology, 4:123–134, 1991.MATHGoogle Scholar
  6. 6.
    E. F. Brickell and D. R. Stinson, Some Improved Bounds on the Information Rate of Perfect Secret Sharing Schemes, Lecture Notes in Computer Science, 537:242–252, 1991. To appear in J. Cryptology.Google Scholar
  7. 7.
    R. M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro, On the Size of Shares for Secret Sharing Schemes, in “Advances in Cryptology-CRYPTO 91”, Ed. J. Feigenbaum, vol. 576 of “Lecture Notes in Computer Science”, Springer-Verlag, pp. 101–113. To appear in J. Cryptology.Google Scholar
  8. 8.
    I. Csiszár and J. Körner, Information Theory. Coding theorems for discrete memoryless systems, Academic Press, 1981.Google Scholar
  9. 9.
    P. Erdös and L. Pyber, unpublished.Google Scholar
  10. 10.
    T. Feder and R. Motwani, Clique Partition, Graph Compression and Speeding-up Algorithms, Proceedings of the 23rd Annual ACM Sympsoium on Theory of Computing, New Orleans, 1991, pp. 123–133.Google Scholar
  11. 11.
    R. G. Gallager, Information Theory and Reliable Communications, John Wiley & Sons, New York, NY, 1968.Google Scholar
  12. 12.
    M. Garey and D. Johnson, Computers and Intractability: a Guide to the Theory of NP-Completeness, W. H. Freeman & Co., New York, 1979.MATHGoogle Scholar
  13. 13.
    E. D. Karnin, J. W. Greene, and M. E. Hellman, On Secret Sharing Systems, IEEE Trans. on Inform. Theory, vol. IT-29, no. 1, Jan. 1983, pp. 35–41.CrossRefMathSciNetGoogle Scholar
  14. 14.
    S. C. Kothari, Generalized Linear Threshold Schemes, in “Advances in Cryptology-CRYPTO 84”, G. R. Blakley and D. Chaum Eds., vol. 196 of “Lecture Notes in Computer Science”, Springer-Verlag, pp. 231–241.CrossRefGoogle Scholar
  15. 15.
    M. Ito, A. Saito, and T. Nishizeki, Secret Sharing Scheme Realizing General Access Structure, Proc. IEEE Global Telecommunications Conf., Globecom 87, Tokyo, Japan, 1987.Google Scholar
  16. 16.
    K. M. Martin, Discrete Structures in the Theory of Secret Sharing, PhD Thesis, University of London, 1991.Google Scholar
  17. 17.
    K. M. Martin, New secret sharing schemes from old, submitted to Journal of Combin. Math. and Combin. Comput.Google Scholar
  18. 18.
    L. Pyber, Covering the Edges of a Graph by..., in Sets, Graphs and Numbers, Colloquia Mathematica Soc. János Bolyai, L. Lovász, D. Miklós, T. Szönyi, Eds., (to appear).Google Scholar
  19. 19.
    A. Shamir, How to Share a Secret, Communications of the ACM, vol. 22, n. 11, pp. 612–613, Nov. 1979.CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    G. J. Simmons, An Introduction to Shared Secret and/or Shared Control Schemes and Their Application, Contemporary Cryptology, IEEE Press, pp. 441–497, 1991.Google Scholar
  21. 21.
    D. R. Stinson, An Explication of Secret Sharing Schemes, Technical Report UNL-CSE-92-004, Department of Computer Science and Engineering, University of Nebraska, February 1992.Google Scholar
  22. 22.
    Z. Tuza, Covering of Graphs by Complete Bipartite Subgraphs; Complexity of 0–1 matrices, Combinatorica, vol. 4, n. 1, pp. 111–116, 1984.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • C. Blundo
    • 1
  • A. De Santis
    • 1
  • L. Gargano
    • 1
  • U. Vaccaro
    • 1
  1. 1.Dipartimento di Informatica ed ApplicazioniUniversità di SalernoBaronissi (SA)Italy

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