Cheating Detectable Secret Sharing Schemes Supporting an Arbitrary Finite Field

  • Satoshi Obana
  • Kazuya Tsuchida
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8639)


In this paper, we present k-out-of-n threshold secret sharing scheme which can detect share forgery by at most k − 1 cheaters. Though, efficient schemes with such a property are presented so far, some schemes cannot be applied when a secret is an element of \(\mathbb{F}_{2^N}\) and some schemes require a secret to be an element of a multiplicative group. The schemes proposed in the paper possess such a merit that a secret can be an element of arbitrary finite field. Let \(|\mathcal{S}|\) and ε be the size of secret and successful cheating probability of cheaters, respectively. Then the sizes of share \(|\mathcal{V}_i|\) of two proposed schemes respectively satisfy \(|\mathcal{V}_i|=(2\cdot|\mathcal{S}|)/\epsilon\) and \(|\mathcal{V}_i|=(4\cdot|\mathcal{S}|)/\epsilon\) which are only 2 and 3 bits longer than the existing lower bound.


Secret Sharing Cheating Detection Arbitrary Finite Field 


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  1. 1.
    Araki, T.: Efficient (k,n) Threshold Secret Sharing Schemes 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.
    Araki, T., Ogata, W.: A Simple and Efficient Secret Sharing Scheme Secure against Cheating. IEICE Trans. Fundamentals E94-A(6), 1338–1345 (2011)CrossRefGoogle Scholar
  4. 4.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proc. AFIPS 1979, National Computer Conference, vol. 48, pp. 313–317 (1979)Google Scholar
  5. 5.
    Brickell, E.F., Stinson, D.R.: The Detection of Cheaters in Threshold Schemes. SIAM Journal on Discrete Mathematics 4(4), 502–510 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Carpentieri, M.: A Perfect Threshold Secret Sharing Scheme to Identify Cheaters. Designs, Codes and Cryptography 5(3), 183–187 (1995)CrossRefzbMATHMathSciNetGoogle 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)CrossRefGoogle Scholar
  8. 8.
    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)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cevallos, A., Fehr, S., Ostrovsky, R., Rabani, Y.: Unconditionally-secure Robust Secret Sharing with Compact Shares. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 195–208. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  10. 10.
    Choudhury, A.: Brief announcement: Optimal Amortized Secret Sharing with Cheater Identification. In: Proc. PODC 2012, p. 101. ACM (2012)Google Scholar
  11. 11.
    Cramer, R., Damgård, I.B., 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
  12. 12.
    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
  13. 13.
    Kurosawa, K., Obana, S., Ogata, W.: t-Cheater Identifiable (k, n) Threshold Secret Sharing Schemes. In: Coppersmith, D. (ed.) CRYPTO 1995. LNCS, vol. 963, pp. 410–423. Springer, Heidelberg (1995)Google Scholar
  14. 14.
    McEliece, R.J., Sarwate, D.V.: On Sharing Secrets and Reed-Solomon Codes. Communications of the ACM 24(9), 583–584 (1981)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Obana, S.: Almost Optimum t-Cheater Identifiable Secret Sharing Schemes. In: Paterson, K.G. (ed.) EUROCRYPT 2011. LNCS, vol. 6632, pp. 284–302. Springer, Heidelberg (2011)CrossRefGoogle 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., Araki, T.: Cheating Detectable Secret Sharing Schemes for Random Bit String. IEICE Trans. Fundamentals E96-A(11), 2230–2234 (2013)CrossRefGoogle Scholar
  18. 18.
    Ogata, W., Eguchi, H.: Cheating Detectable Threshold Scheme against Most Powerful Cheaters for Long Secrets. Designs, Codes and Cryptography (published online, October 2012)Google 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)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Pedersen, T.P.: 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)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Shamir, A.: How to Share a Secret. Communications of the ACM 22(11), 612–613 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Tompa, M., Woll, H.: How to Share a Secret with Cheaters. Journal of Cryptology 1(3), 133–138 (1989)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Xu, R., Morozov, K., Takagi, T.: On Cheater Identifiable Secret Sharing Schemes Secure against Rushing Adversary. In: Sakiyama, K., Terada, M. (eds.) IWSEC 2013. LNCS, vol. 8231, pp. 258–271. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Satoshi Obana
    • 1
  • Kazuya Tsuchida
    • 1
  1. 1.Hosei UniversityJapan

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