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Designs, Codes and Cryptography

, Volume 74, Issue 3, pp 719–729 | Cite as

Secret sharing on the d-dimensional cube

  • László CsirmazEmail author
Article

Abstract

We prove that for \(d>1\) the best information ratio of the perfect secret sharing scheme based on the edge set of the d-dimensional cube is exactly d/2. Using the technique developed, we also prove that the information ratio of the infinite \(d\)-dimensional lattice is d.

Keywords

Secret sharing scheme Polymatroid Information theory 

Mathematics Subject Classification

94A60 94A17 52B40 68R10 

Notes

Acknowledgments

This research was partially supported by TAMOP-4.2.2.C-11/1/KONV-2012-0001 and the Lendulet Program of the Hungarian Academy of Sciences.

References

  1. 1.
    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 (2011).Google Scholar
  2. 2.
    Beimel A., Farràs O., Mintz Y.: Secret sharing schemes for very dense graphs. In: Advances in Cryptology (CRYPTO 2012). Lecture Notes in Computer Science, vol. 7417, pp. 144–161 (2012).Google Scholar
  3. 3.
    Blundo C., De Santis A., Stinson D.R., Vaccaro U.: Graph decomposition and secret sharing schemes. J. Cryptol. 8, 39–64 (1995).Google Scholar
  4. 4.
    Blundo C., De Santis A., De Simone R., Vaccaro U.: Tight bounds on the information rate of secret sharing schemes. Des. Codes Cryptogr. 11, 107–122 (1997).Google Scholar
  5. 5.
    Brickell E.F., Davenport D.M.: On the classification of ideal. J. Cryptol. 4, 123–134 (1991).Google Scholar
  6. 6.
    Brickell E.F., Stinson D.R.: Some improved bounds on the information rate of perfect secret sharing schemes. J. Cryptol. 5, 153–166 (1992).Google Scholar
  7. 7.
    Csirmaz L.: Secret sharing schemes on graphs. Stud. Math. Hung. 44, 297–306. Available as IACR preprint. http://eprint.iacr.org/2005/059 (2007). Accessed 1 Oct 2013.
  8. 8.
    Csirmaz L.: Secret sharing on infinite graphs. Tatra Mt. Math. Publ. 41, 1–18 (2008).Google Scholar
  9. 9.
    Csirmaz L., Ligeti P.: On an infinite family of graphs with information ratio \(2-1/k.\) Computing 85, 127–136 (2009).Google Scholar
  10. 10.
    Csirmaz L., Tardos G.: Optimal information rate of secret sharing schemes on trees. IEEE Trans. Inf. Theory 59(4), 2527–2530 (2013).Google Scholar
  11. 11.
    Csiszár I., Körner J.: Information Theory. Coding Theorems for Discrete Memoryless Systems. Academic Press, New York (1981).Google Scholar
  12. 12.
    Erdős P., Pyber L.: Covering a graph by complete bipartite. Discret. Math. 170, 249–251 (1997).Google Scholar
  13. 13.
    Jackson W.-A., Martin K.M.: Perfect secret sharing schemes on five participants. Des. Codes Cryptogr. 9, 267–286 (1996).Google Scholar
  14. 14.
    Padró C.: Lecture notes in secret sharing. Available as IACR preprint. http://eprint.iacr.org/2012/674. Accessed 1 Oct 2013.
  15. 15.
    Stinson D.R.: Decomposition construction for secret sharing schemes. IEEE Trans. Inf. Theory 40, 118–125 (1994).Google Scholar
  16. 16.
    van Dijk M.: On the information rate of perfect secret sharing schemes. Des. Codes Cryptogr. 6, 143–160 (1995).Google Scholar
  17. 17.
    Wang W., Li Z., Song Y.: The optimal information rate of perfect secret sharing schemes. In: 2011 International Conference on Business Management and Electronic Information (BMEI), vol. 2, pp. 207–212 (2011).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Central European UniversityBudapestHungary

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