Advertisement

Combinatorial interpretation of secret sharing schemes

  • Kaoru Kurosawa
  • Koji Okada
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 917)

Abstract

In a perfect secret sharing scheme, it is known that log2 ¦V i ¦ ≥H(S), where S is a secret and V i is the share of user i. On the other hand, log2 ¦Ŝ¦ ≥H(S), where Ŝ is the domain of S. The equality holds if and only if S is uniformly distributed. Therefore, if S is uniformly distributed, we have ¦V i ¦≥¦Ŝ¦. However, if S is not uniformly distributed, log2 ¦Ŝ¦> H(S). In this case, we have log2¦V i ¦≥H(S) <log2¦Ŝ¦. Then, which is bigger, ¦Vi¦ or ¦Ŝ¦? The answer is not known.

In this paper, we first prove that ¦V i ¦ >-¦Ŝ¦ for any distribution of S by using a combinatorial argument. This is a more sharp lower bound on ¦V i ¦ for not uniformly distributed S. Our proof makes it intuitively clear why ¦V i ¦ must be so large, also. Further, we show an extension of our combinatorial technique for some access structures.

Keywords

Secret Sharing Access Structure Information Rate Secret Sharing Scheme Minimal Access 
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.
    G.R.Blakley: Safeguarding cryptographic keys. Proc. of the AFIPS 1979 National Computer Conference, vol. 48, pp. 313–317 (1979)Google Scholar
  2. 2.
    A.Shamir: How to share a secret. Communications of the ACM, 22, (11), pp. 612–613 (1979)Google Scholar
  3. 3.
    R.M.Capocelli, A.De Santis, L.Gargano, U.Vaccaro: On the size of shares for secret sharing schemes. Crypto'91, pp.101–113 (1991)Google Scholar
  4. 4.
    E.D.Karnin, J.W.Green, M.E.Hellman: On secret sharing systems. IEEE Trans. IT-29, No.1, pp. 35–41 (1982)Google Scholar
  5. 5.
    E.F.Brickell, D.M.Davenport: On the classification of ideal secret sharing schemes. Journal of Cryptology, vol. 4, No.2, pp. 123–134 (1991)Google Scholar
  6. 6.
    E.F.Brickell, D.R.Stinson: Some improved bounds on the information rate of perfect secret sharing schemes. Journal of Cryptology, vol. 5, No.3, pp. 153–166 (1992)Google Scholar
  7. 7.
    C.Blund, A.De Santis, D.R.Stinson, U.Vaccaro: Graph decomposition and secret sharing schemes. Eurocrypt'92, pp.1–20 (1992)Google Scholar
  8. 8.
    Y.Frankel, Y.Desmedt: Classification of ideal homomorphic threshold schemes over finite Abelian groups. Eurocrypt'92, pp.21–29 (1992)Google Scholar
  9. 9.
    C.Blund, A.De Santis, L.Gargano, U.Vaccaro: On the information rate of secret sharing schemes. Crypto'92 (1992)Google Scholar
  10. 10.
    D.R.Stinson: New general bounds on the information rate of secret sharing schemes. Crypto'92 (1992)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Kaoru Kurosawa
    • 1
  • Koji Okada
    • 1
  1. 1.Department of Electrical and Electronic Engineering, Faculty of EngineeringTokyo Institute of TechnologyTokyoJapan

Personalised recommendations