Almost Optimum t-Cheater Identifiable Secret Sharing Schemes

  • Satoshi Obana
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6632)


In Crypto’95, Kurosawa, Obana and Ogata proposed a k-out-of-n secret sharing scheme capable of identifying up to t cheaters with probability 1 − ε under the condition \(t \leq \lfloor\)(k–1)/3. The size of share \(|{\cal V}_i|\) of the scheme satisfies \(|{\cal V}_i|\) = \(|{\cal S}|/\epsilon^{t+2}\), which was the most efficient scheme known so far. In this paper, we propose new k-out-of-n secret sharing schemes capable of identifying cheaters. The proposed scheme possesses the same security parameters t,ε as those of Kurosawa et al.. The scheme is surprisingly simple and its size of share is \(|{\cal V}_i|=|{\cal S}|/\epsilon\), which is much smaller than that of Kurosawa et al. and is almost optimum with respect to the size of share; that is, the size of share is only one bit longer than the existing bound. Further, this is the first scheme which can identify cheaters, and whose size of share is independent of any of n,k and t. We also present schemes which can identify up to \(\lfloor{(k-2)/2}\), and \(\lfloor{(k-1)/2}\) cheaters whose sizes of share can be approximately written by \(|{\cal V}_i|\approx (n\cdot(t+1)\cdot 2^{3t-1}\cdot|{\cal S}|)/\epsilon\) and \(|{\cal V}_i|\approx ((n\cdot t\cdot 2^{3t})^2\cdot|{\cal S}|)/\epsilon^2\), respectively. The number of cheaters that the latter two schemes can identify meet the theoretical upper bound.


Secret Sharing Cheater Identification Reed-Solomon Code Universal Hash 


  1. 1.
    Araki, T.: Efficient (k,n) Threshold Secret Sharing Scheme Secure against Cheating from n − 1 Cheaters. In: Pieprzyk, J., Ghodosi, H., Dawson, E. (eds.) ACISP 2007. LNCS, vol. 4586, pp. 133–142. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Araki, T., Obana, S.: Flaws in Some Secret Sharing Schemes against Cheating. In: Pieprzyk, J., Ghodosi, H., Dawson, E. (eds.) ACISP 2007. LNCS, vol. 4586, pp. 122–132. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  3. 3.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. AFIPS 1979, National Computer Conference, vol. 48, pp. 137–313 (1979)Google Scholar
  4. 4.
    Brickell, E.F., Stinson, D.R.: The Detection of Cheaters in Threshold Schemes. SIAM Journal on Discrete Mathematics 4(4), 502–510 (1991)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Carpentieri, M.: A Perfect Threshold Secret Sharing Scheme to Identify Cheaters. Designs, Codes and Cryptography 5(3), 183–187 (1995)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Cramer, R., Dodis, Y., Fehr, S., Padró, C., Wichs, D.: Detection of Algebraic Manipulation with Applications to Robust Secret Sharing and Fuzzy Extractors. In: Smart, N.P. (ed.) EUROCRYPT 2008. LNCS, vol. 4965, pp. 471–488. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Carpentieri, M., De Santis, A., Vaccaro, U.: Size of Shares and Probability of Cheating in Threshold Schemes. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 118–125. Springer, Heidelberg (1994)Google Scholar
  8. 8.
    Cramer, R., Damgård, I., Fehr, S.: On the Cost of Reconstructing a Secret, or VSS with Optimal Reconstruction Phase. In: Kilian, J. (ed.) CRYPTO 2001. LNCS, vol. 2139, pp. 503–523. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Cabello, S., Padró, C., Sáez, G.: Secret Sharing Schemes with Detection of Cheaters for a General Access Structure. Designs, Codes and Cryptography 25(2), 175–188 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    den Boer, B.: A Simple and Key-Economical Unconditional Authentication Scheme. Journal of Computer Security 2, 65–71 (1993)Google Scholar
  11. 11.
    Dolev, D., Dwork, C., Waarts, O., Yung, M.: Perfectly Secure Message Transmission. Journal of the ACM 40(1), 17–47 (1993)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Kurosawa, K., Obana, S., Ogata, W.: t-Cheater Identifiable (k,n) Secret Sharing Schemes. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 410–423. Springer, Heidelberg (1995)Google Scholar
  13. 13.
    Kurosawa, K., Suzuki, K.: Almost Secure (1-Round, n-Channel) Message Transmission Scheme. In: Desmedt, Y. (ed.) ICITS 2007. LNCS, vol. 4883, pp. 99–112. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    MacWilliams, F., Sloane, N.: The Theory of Error Correcting Codes. North Holland, Amsterdam (1977)MATHGoogle Scholar
  15. 15.
    McEliece, R.J., Sarwate, D.V.: On Sharing Secrets and Reed-Solomon Codes. Communications of the ACM 24(9), 583–584 (1981)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Obana, S., Araki, T.: Almost Optimum Secret Sharing Schemes Secure Against Cheating for Arbitrary Secret Distribution. In: Lai, X., Chen, K. (eds.) ASIACRYPT 2006. LNCS, vol. 4284, pp. 364–379. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Ogata, W., Kurosawa, K.: Optimum Secret Sharing Scheme Secure against Cheating. In: Maurer, U.M. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 200–211. Springer, Heidelberg (1996)Google Scholar
  18. 18.
    Ogata, W., Kurosawa, K.: Provably Secure Metering Scheme. In: Okamoto, T. (ed.) ASIACRYPT 2000. LNCS, vol. 1976, pp. 388–398. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Ogata, W., Kurosawa, K., Stinson, D.R.: Optimum Secret Sharing Scheme Secure against Cheating. SIAM Journal on Discrete Mathematics 20(1), 79–95 (2006)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Pedersen, T.: Non-interactive and Information-Theoretic Secure Verifiable Secret Sharing. In: Feigenbaum, J. (ed.) CRYPTO 1991. LNCS, vol. 576, pp. 129–140. Springer, Heidelberg (1992)Google Scholar
  21. 21.
    Rabin, T., Ben-Or, M.: Verifiable Secret Sharing and Multiparty Protocols with Honest Majority. In: Proc. STOC 1989, pp. 73–85 (1989)Google Scholar
  22. 22.
    Rabin, T.: Robust Sharing of Secrets When the Dealer is Honest or Cheating. Journal of the ACM 41(6), 1089–1109 (1994)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Shamir, A.: How to Share a Secret. Communications of the ACM 22(11), 612–613 (1979)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Stinson, D.R.: On the Connections between Universal Hashing, Combinatorial Designs and Error-Correcting Codes. Congressus Numerantium 114, 7–27 (1996)MathSciNetMATHGoogle Scholar
  25. 25.
    Tompa, M., Woll, H.: How to Share a Secret with Cheaters. Journal of Cryptology 1(3), 133–138 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© International Association for Cryptologic Research 2011

Authors and Affiliations

  • Satoshi Obana
    • 1
  1. 1.NECJapan

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