Cryptography and Communications

, Volume 6, Issue 2, pp 137–155 | Cite as

Secret sharing schemes based on graphical codes

  • Ying GaoEmail author
  • Romar dela Cruz


We study the access structure and multiplicativity of linear secret sharing schemes based on codes from complete graphs. First, we describe the access structure of the schemes based on cut-set and cycle codes. Second, we show that the class of access structures based on odd cycles cannot be realized by ideal multiplicative linear secret sharing schemes over any finite field. This can be seen as a contribution to the characterization of access structures of ideal multiplicative schemes. The access structure based on odd cycles corresponds to the scheme based on the dual of the extended cycle code. Finally, we show that we can obtain ideal multiplicative linear secret sharing scheme based on the dual of an augmented extended cycle code.


Secret sharing Linear code Matroid Graph 

Mathematics Subject Classifications (2000)

94A62 94B05 05C50 



The work of Y. Gao is supported in part by the National Natural Science Foundation of China by Grant 11101019 and the Fundamental Research Funds for the Central Universities in China (No. YWF-10-02-072). Part of the work was done while she was visiting Nanyang Technological University. The work of R. dela Cruz is supported in part by the NTU PhD Research Scholarship and the Merlion PhD Grant of the French Embassy in Singapore. He would like to thank Telecom-ParisTech for its hospitality. The authors would like to thank Carles Padró and Huaxiong Wang for some helpful discussions, and the anonymous reviewers for their valuable comments and suggestions.


  1. 1.
    Ashikhmin, A., Barg, A.: Minimal vectors in linear codes. IEEE Trans. Inform. Theory IT-44, 2010–2017 (1998)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Beimel, A.: Secret sharing schemes: a survey. In: Coding and Cryptology, Third International Workshop, IWCC 2011. Lecture Notes in Computer Science, vol. 6639, pp. 11–46. Springer, New York (2011)Google Scholar
  3. 3.
    Beimel, A.: Secure Schemes for Secret Sharing and Key Distribution. Ph.D. dissertation, Technion-Israel Inst. Technol., Haifa, Israel (1996)Google Scholar
  4. 4.
    Beimel, A., Chor, B.: Universally ideal secret-secret sharing schemes. IEEE Trans. Inform. Theory IT-40, 786–794 (1994)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Blakley, G.R.: Safeguarding cryptographic keys. In: Proceedings of the 1979 AFIPS National Computer Conference, pp. 313–317. AFIPS Press, Monval, NJ (1979)Google Scholar
  6. 6.
    Brickell, E., Davenport, D.: On the classification of ideal secret sharing schemes. J. Cryptol. 4, 123–134 (1991)zbMATHGoogle Scholar
  7. 7.
    Chen, H., Cramer, R.: Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. In: Proceedings of 26th Annual IACR CRYPTO. Lecture Notes in Computer Science, vol. 4117, pp. 521–536. Springer, New York (2006)Google Scholar
  8. 8.
    Cramer, R., Damgärd, I., Maurer, U.: General secure multi-party computation from any linear secret-sharing schemes. In: Proceedings of 19th Annual IACR EUROCRYPT. Lecture Notes in Computer Science, vol. 1807, pp. 316–334. Springer, New York (2000)Google Scholar
  9. 9.
    Cramer, R., Daza, V., Gracia, I., Urroz, J., Leander, G., Martí-Farré, J., Padró, C.: On codes, matroids, and secure multi-party computation from linear secret-sharing schemes. IEEE Trans. Inform. Theory IT-54, 2644–2657 (2008)CrossRefGoogle Scholar
  10. 10.
    Ding, C., Yuan, J.: Covering and Secret Sharing with Linear Codes. In: Discrete Mathematics and Theoretical Computer Science. Lecture Notes in Computer Science, vol. 2731, pp. 11–25. Springer, New York (2003)CrossRefGoogle Scholar
  11. 11.
    Gerards, A., Schrijver, A.: Signed Graph – Regular Matroids – Grafts. Research Memorandum, Faculteit der Economische Wetenschappen, Tilburg University (1986)Google Scholar
  12. 12.
    Goldreich, O., Micali, S., Wigderson, A.: How to play ANY mental game. In: Proc. 19th annual ACM Symposium on Theory of Computing, STOC’87, pp. 218–229. New York (1987)Google Scholar
  13. 13.
    Hakimi, S., Bredeson, J.: Graph-theoretic error-correcting codes. IEEE Trans. Inform. Theory IT-14, 584–591 (1968)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Jungnickel, D., Vanstone, S.: Graphical codes revisited. IEEE Trans. Inform. Theory IT-43, 136–146 (1997)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Karchmer, M., Wigderson, A.: On span programs. In: Proc. 8th IEEE Structure in Complexity Theory, pp. 102–111. IEEE Computer Society Press, Los Alamitos, CA (1993)Google Scholar
  16. 16.
    Kasper, E., Nikova, S., Nikov, V.: Strongly multiplicative hierarchical threshold secret sharing. In: Proc. 2nd Int. Conf. on Information Theoretic Security. Lecture Notes in Computer Science, vol. 4883, pp. 148–168. Springer, New York (2007)CrossRefGoogle Scholar
  17. 17.
    Liu, M., Xiao, L., Zhang, Z.: Multiplicative linear secret sharing schemes based on connectivity of graphs. IEEE Trans. Inform. Theory IT-53, 3973–3978 (2007)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Massey, J.L.: Minimal codewords and secret sharing. In: Proc. 6th Joint Swedish-Russian Workshop Inf. Theory, pp. 276–279. Molle, Sweden (1993)Google Scholar
  19. 19.
    Nikova, S., Nikov, V.: On multiplicative secret sharing schemes realizing graph access structures. In: International Workshop on Optimal Codes and Related Topics, pp. 194–199. Balchik, Bulgaria (2007)Google Scholar
  20. 20.
    Oxley, J.: Matroid Theory. Oxford Science Publications, Oxford University Press, New York (1992)zbMATHGoogle Scholar
  21. 21.
    Padró, C., Gracia, I.: Representing small identically self-dual matroids by self-dual codes. SIAM J. Discrete Math. 20, 1046–1055 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Stinson, D.: An explication of secret sharing schemes. Des. Codes Cryptogr. 2, 357–390 (1992)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Stinson, D.: Cryptography Theory and Practice, 3rd edn. CRC Press, Boca Raton, FL (2005)Google Scholar
  25. 25.
    West, D.: Introduction to Graph Theory, 2nd edn. Prentice Hall, New York (2001)Google Scholar
  26. 26.
    Yao, A.: Protocols for secure computation. In: Proc. 23rd IEEE Symp. Foundation of Computer Science, FOCS ’82, IL, pp. 160–164. Chicago (1982)Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang University, LMIB of the Ministry of EducationBeijingPeople’s Republic of China
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Institute of Mathematics, College of ScienceUniversity of the Philippines DilimanQuezon CityPhilippines

Personalised recommendations