Designs, Codes and Cryptography

, Volume 72, Issue 2, pp 345–367 | Cite as

Combinatorial solutions providing improved security for the generalized Russian cards problem

  • C. M. Swanson
  • D. R. Stinson


We present the first formal mathematical presentation of the generalized Russian cards problem, and provide rigorous security definitions that capture both basic and extended versions of weak and perfect security notions. In the generalized Russian cards problem, three players, Alice, Bob, and Cathy, are dealt a deck of \(n\) cards, each given \(a\), \(b\), and \(c\) cards, respectively. The goal is for Alice and Bob to learn each other’s hands via public communication, without Cathy learning the fate of any particular card. The basic idea is that Alice announces a set of possible hands she might hold, and Bob, using knowledge of his own hand, should be able to learn Alice’s cards from this announcement, but Cathy should not. Using a combinatorial approach, we are able to give a nice characterization of informative strategies (i.e., strategies allowing Bob to learn Alice’s hand), having optimal communication complexity, namely the set of possible hands Alice announces must be equivalent to a large set of \(t-(n, a, 1)\)-designs, where \(t=a-c\). We also provide some interesting necessary conditions for certain types of deals to be simultaneously informative and secure. That is, for deals satisfying \(c = a-d\) for some \(d \ge 2\), where \(b \ge d-1\) and the strategy is assumed to satisfy a strong version of security (namely perfect \((d-1)\)-security), we show that \(a = d+1\) and hence \(c=1\). We also give a precise characterization of informative and perfectly \((d-1)\)-secure deals of the form \((d+1, b, 1)\) satisfying \(b \ge d-1\) involving \(d-(n, d+1, 1)\)-designs.


Combinatorial designs Russian cards problem Information-theoretic cryptography Protocols 

Mathematics Subject Classification (2000)

05B05 94A60 



D. Stinson’s research is supported by NSERC discovery grant 203114-11.


  1. 1.
    Albert M., Aldred R., Atkinson M., van Ditmarsch H., Handley C.: Safe communication for card players by combinatorial designs for two-step protocols. Australas. J. Comb. 33, 33–46 (2005).Google Scholar
  2. 2.
    Albert M., Cordón-Franco A., van Ditmarsch H., Fernández-Duque D., Joosten J., Soler-Toscano F.: Secure communication of local states in interpreted systems. In: International Symposium on Distributed Computing and Artificial Intelligence. Advances in Intelligent and Soft Computing, vol. 91, pp. 117–124. Springer, Berlin (2011).Google Scholar
  3. 3.
    Atkinson M., van Ditmarsch H.: Avoiding bias in cards cryptography. Australas. J. Comb. 44, 3–18 (2009).Google Scholar
  4. 4.
    Chouinard II L.: Partitions of the 4-subsets of a 13-set into disjoint projective planes. Discret. Math. 45, 297–300 (1983).Google Scholar
  5. 5.
    Colbourn C., Dinitz J.: The CRC Handbook of Combinatorial Designs. CRC Press, New York (1996).Google Scholar
  6. 6.
    Cordón-Franco A., Ditmarsch H., Fernández-Duque D., Joosten J., Soler-Toscano F.: A secure additive protocol for card players. Australas. J. Comb. 54, 163–176 (2012).Google Scholar
  7. 7.
    Cordón-Franco A., van Ditmarsch H., Fernández-Duque D., Soler-Toscano F.: A geometric protocol for cryptography with cards. arXiv:1207.5216.
  8. 8.
    Cyriac A., Krishnan K.M., Lower bound for the communication complexity of the Russian cards problem. arXiv:0805.1974.
  9. 9.
    Dembowski P.: Finite Geometries. Springer, New York (1968).Google Scholar
  10. 10.
    Duan Z., Yang C.: Unconditional secure communication: a Russian cards protocol. J. Comb. Optim. 19, 501–530 (2010).Google Scholar
  11. 11.
    Fischer M., Wright R.: Multiparty secret key exchange using a random deal of cards. Adv. Cryptol. (Crypto ’91). LNCS 576, 141–155 (1992).Google Scholar
  12. 12.
    Fischer M., Wright R.: An application of game-theoretic techniques to cryptography. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 13, pp. 99–118. AMS, Providence (1993).Google Scholar
  13. 13.
    Fischer M., Wright R.: An efficient protocol for unconditional secure secret key exchange. In: Proceedings of 4th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA ’93), pp. 475–483. Society for Industrial and Applied Mathematics, New Orleans (1993).Google Scholar
  14. 14.
    Fischer M., Wright R.: Bounds on secret key exchange using a random deal of cards. J. Cryptol. 9, 71–99 (1996).Google Scholar
  15. 15.
    Fischer M., Paterson M., Rackoff C.: Secret bit transmission using a random deal of cards. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 2, pp. 173–181. AMS, Providence (1991).Google Scholar
  16. 16.
    He J., Duan Z.: Public Communication Based on Russian Cards Protocol: A Case Study. LNCS, vol. 6831, pp. 192–206. Springer, Berlin (2011).Google Scholar
  17. 17.
    Koizumi K., Mizuki T., Nishizeki T.: Necessary and sufficient numbers of cards for the transformation protocol. In: Computing and Combinatorics 10th Annual International Conference (COCOON 2004). LNCS, vol. 3106, pp. 92–101. Springer, Heidelberg (2004).Google Scholar
  18. 18.
    Mizuki T., Shizuya H., Nishizeki T.: A complete characterization of a family of key exchange protocols. Int. J. Inf. Secur. 1, 131–142 (2002).Google Scholar
  19. 19.
    van Ditmarsch H.: The Russian cards problem. The dynamics of knowledge. Studia Logica 75, 31–62 (2003).Google Scholar
  20. 20.
    van Ditmarsch H.: The case of the hidden hand. J. Appl. Non-Classical Log. 15(4), 437–452 (2005).Google Scholar
  21. 21.
    van Ditmarsch H., SolerToscano F.: Three steps. In: 12th International Workshop on Computational Logic in Multi-Agent Systems (CLIMA XII). LNCS, vol. 6814, pp. 41–57. Springer, Heidelberg (2011).Google Scholar
  22. 22.
    van Ditmarsch H., van der Hoek W., van der Meyden R., Ruan J.: Model checking Russian cards. Electron. Notes Theor. Comput. Sci. 149, 105–123 (2006).Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada

Personalised recommendations