Skip to main content
SpringerLink
Log in

On the Dealer's Randomness Required in Secret Sharing Schemes

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper we provide upper and lower bounds on the randomness required by the dealer to set up a secret sharing scheme for infinite classes of access structures. Lower bounds are obtained using entropy arguments. Upper bounds derive from a decomposition construction based on combinatorial designs (in particular, t-(v,k,λ) designs). We prove a general result on the randomness needed to construct a scheme for the cycle Cn; when n is odd our bound is tight. We study the access structures on at most four participants and the connected graphs on five vertices, obtaining exact values for the randomness for all them. Also, we analyze the number of random bits required to construct anonymous threshold schemes, giving upper bounds. (Informally, anonymous threshold schemes are schemes in which the secret can be reconstructed without knowledge of which participants hold which shares.)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. Baker, Partitioning the Planes of AG 2m (2) into 2-designs, Discrete Math., Vol. 15 (1976), pp. 205–211.

    Google Scholar 

  2. J. C. Benaloh and J. Leichter, Generalized Secret Sharing and Monotone Functions, in Advances in Cryptology — Crypto '88, Lecture Notes in Computer Science, Vol. 403 (1990), pp. 27–35.

  3. T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Bibliographisches Institut, Zurich, 1985.

    Google Scholar 

  4. G. R. Blakley, Safeguarding Cryptographic Keys, Proceedings of AFIPS 1979 National Computer Conference, Vol. 48 (1979), pp. 313–317.

    Google Scholar 

  5. C. Blundo, A. De Santis, L. Gargano, and U. Vaccaro, On the Information Rate of Secret Sharing Schemes, Theoretical Computer Science, Vol. 154 (1996), 283–306.

    Google Scholar 

  6. C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro, Graph Decompositions and Secret Sharing Schemes, Journal of Cryptology, Vol. 8 (1995), pp. 39–64.

    Google Scholar 

  7. C. Blundo, A. De Santis, and U. Vaccaro, Randomness in Distribution Protocols, Technical Report, Università di Salerno. [A preliminary version appeared in 21st International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, Vol. 820 (dy1994), pp. 568–579.]}

  8. C. Blundo, A De Santis, and U. Vaccaro, On Secret Sharing Schemes, Technical Report, Università di Salerno, 1995.

  9. C. Blundo, A. Giorgio Gaggia, and D. R. Stinson, On the Dealer's Randomness Required in Secret Sharing Schemes, Advances in Cryptology — EUROCRYPT '94, Lecture Notes in Computer Science, Vol. 950 (1995), pp. 35–46.

  10. C. Blundo and D. R. Stinson, Anonymous Secret Sharing Schemes, submitted for publication.

  11. E. F. Brickell and D. M. Davenport, On the Classification of Ideal Secret Sharing Schemes, J. Cryptology, Vol. 4 (1991), pp. 123–134.

    Google Scholar 

  12. E. F. Brickell and D. R. Stinson, Some Improved Bounds on the Information Rate of Perfect Secret Sharing Schemes, J. Cryptology, Vol. 5 (1992), pp. 153–166.

    Google Scholar 

  13. R. M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro, A Note on Secret Sharing Schemes, Sequences II: Methods in Communication, Security and Computer Science, Springer-Verlag, 1991, pp. 335–344.

  14. R. M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro, On the Size of Shares for Secret Sharing Schemes, J. Cryptology, Vol. 6 (1993), pp. 157–169.

    Google Scholar 

  15. C. J. Colbourn and J. H. Dinitz, CRC Handbook on Combinatorial Designs, CRC Press, Inc, 1996.

  16. T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley & Sons, 1991.

  17. O. Goldreich, S. Micali, and A. Wigderson, How to Play any Mental Game, Proceedings of 19th ACM Symposium on Theory of Computing, 1987, pp. 218–229.

  18. H. Hanani, On Quadruple Systems, Canad. J. Math., Vol. 12 (1960), pp. 145–157.

    Google Scholar 

  19. H. Hanani, On Some Tactical Configurations, Canad. J. Math., Vol. 15 (1963) pp. 702–722.

    Google Scholar 

  20. H. Hanani, D. K. Ray-Chaudhuri, and R. M. Wilson, On Resolvable Designs, Discrete Math., Vol. 3 (1972), pp. 343–357.

    Google Scholar 

  21. R. Impagliazzo and D. Zuckerman, How to Recycle Random Bits, Proceedings 30th IEEE Symposium on Foundations of Computer Science, 1989, pp. 248–255.

  22. M. Ito, A. Saito, and T. Nishizeki, Secret Sharing Scheme Realizing General Access Structure, Globecom '87, 1987, pp. 99–102.

  23. E. D. Karnin, J. W. Greene, and M. E. Hellman, On Secret Sharing Systems, IEEE Trans. Inform. Theory, Vol. 29 (1983), pp. 35–41.

    Google Scholar 

  24. D. E. Knuth and A.C. Yao, The Complexity of Nonuniform Random Number Generation, Algorithms and Complexity, Academic Press, 1976, pp. 357–428.

  25. D. Krizanc, D. Peleg, and E. Upfal, A Time-Randomness Tradeoff for Oblivious Routing, Proceedings 20th ACM Symposium on Theory of Computing, 1988, pp. 93–102.

  26. J. X. Lu, On Large Sets of Disjoint Steiner Triple Systems I, II and II, J. Combin. Theory A, Vol. 34 (1983), pp. 140–182.

    Google Scholar 

  27. J. X. Lu, On Large Sets of Disjoint Steiner Triple Systems IV, V and VI, J. Combin. Theory A, Vol. 37 (1984), pp. 136–192.

    Google Scholar 

  28. K. M. Martin, New Secret Sharing Schemes from Old, J. Combin. Math. Combin. Comput. Vol. 14 (1993), pp. 65–77.

    Google Scholar 

  29. S. J. Phillips and N. C. Phillips, Strongly Ideal Secret Sharing Schemes, J. Cryptology, Vol. 5 (1992), pp. 185–191.

    Google Scholar 

  30. D. K. Ray-Chaudhuri and R. M. Wilson, Solution of Kirkman's Schoolgirl Problem, Amer. Math. Soc. Proc. Symp. Pure Math., Vol. 19 (1971), pp. 187–204.

    Google Scholar 

  31. P. J. Schellenberg and D. R. Stinson, Threshold Schemes from Combinatorial Designs, J. Combin. Math. Combin. Comput., Vol. 5 (1989), pp. 143–160.

    Google Scholar 

  32. A. Shamir, How to Share a Secret, Communications of the ACM, Vol. 22 (1979), pp. 612–613.

    Google Scholar 

  33. G. J. Simmons, An Introduction to Shared Secret and/or Shared Control Schemes and Their Application, Contemporary Cryptology, IEEE Press, 1991, pp. 441–497.

  34. D. R. Stinson, New General Lower Bounds on the Information Rate of Secret Sharing Schemes, Advances in Cryptology-CRYPTO '92, Lecture Notes in Computer Science, Vol. 740 (1993), pp. 170–184.

  35. D. R. Stinson, An Explication of Secret Sharing Schemes, Designs, Codes and Cryptography, Vol. 2 (1992), pp. 357–390.

    Google Scholar 

  36. D. R. Stinson, Decomposition Constructions for Secret Sharing Schemes, IEEE Trans. Inform. Theory, Vol. 40 (1994), pp. 118–125.

    Google Scholar 

  37. D. R. Stinson, Combinatorial Designs and Cryptography, Surveys in Combinatorics, 1993, Cambridge Univ. Press, 1993, pp. 257–287.

  38. D. R. Stinson, Bibilography on Secret Sharing Schemes, available online at the following URL: http://bibd.unl.edu/∼stinson/ssbib.html

  39. D. R. Stinson and A. Vanstone, A Combinatorial Approach to Threshold Schemes, SIAM J. Discrete. Math., Vol. 1 (1988), pp. 230–236.

    Google Scholar 

  40. L. Teirlinck, A Completion of Lu's Determination of the Spectrum for Large Sets of Disjoint Steiner Triple Systems, J. Combin. Theory A, Vol. 57 (1991), pp. 302–305.

    Google Scholar 

  41. L. Teirlinck, Some New 2-resolvable Steiner Quadruple Systems, Designs, Codes and Cryptography, Vol, 4 (1994), pp. 5–10.

    Google Scholar 

  42. L. Teirlinck, Large Sets of Disjoint Designs and Related Structures, Contemporary Design Theory: A Collection of Surveys, John Wiley & Sons, 1992 pp. 561–592.

  43. G. V. Zaicev, V. A. Zinoviev, and N. V. Semakov, Interrelation of Preparata and Hamming Codes and Extension of Hamming Codes to New Double Error-Correcting Codes, Proc. 2nd International Symp. Inform. Theory, 1973, pp. 257–263.

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Informatica ed Applicazioni, Università di Salerno, 84081, Baronissi (SA), Italy

    C. Blundo & A. Giorgio Gaggia

  2. Department of Computer Science and Engineering, University of Nebraska, Lincoln, NE, 68588

    D. R. Stinson

Authors
  1. C. Blundo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Giorgio Gaggia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. D. R. Stinson
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blundo, C., Gaggia, A.G. & Stinson, D.R. On the Dealer's Randomness Required in Secret Sharing Schemes. Designs, Codes and Cryptography 11, 235–259 (1997). https://doi.org/10.1023/A:1008242111272

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008242111272

Search

Navigation