Journal of Cryptology

, Volume 29, Issue 2, pp 336–362

Secret-Sharing Schemes for Very Dense Graphs

Article

Abstract

A secret-sharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph “hard” for secret-sharing schemes (that is, they require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with \(n\) vertices contains \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n^{1+\beta }\) edges for some constant \(0 \le \beta <1\), then there is a scheme realizing the graph with total share size of \(\tilde{O}(n^{5/4+3\beta /4})\). This should be compared to \(O(n^2/\log (n))\), the best upper bound known for the total share size in general graphs. Thus, if a graph is “hard,” then the graph and its complement should have many edges. We generalize these results to nearly complete \(k\)-homogeneous access structures for a constant \(k\). To complement our results, we prove lower bounds on the total share size for secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes, we prove a lower bound of \(\Omega (n^{1+\beta /2})\) for a graph with \(\left( {\begin{array}{c}n\\ 2\end{array}}\right) -n^{1+\beta }\) edges.

Keywords

Secret-sharing Share size Graph access structures Complete bipartite covers Equivalence covers 

References

  1. 1.
    N. Alon. Covering graphs by the minimum number of equivalence relations. Combinatorica, 6(3):201–206, 1986.Google Scholar
  2. 2.
    N. Alon and J. H. Spencer. The Probabilistic Method. John Wiley & Sons, 3rd edition, 2008.Google Scholar
  3. 3.
    L. Babai, A. Gál, and A. Wigderson. Superpolynomial lower bounds for monotone span programs. Combinatorica, 19(3):301–319, 1999.Google Scholar
  4. 4.
    A. Beimel. Secret-sharing schemes: A survey. In IWCC 2011, volume 6639 of Lecture Notes in Computer Science, pages 11–46, 2011.Google Scholar
  5. 5.
    A. Beimel and B. Chor. Universally ideal secret sharing schemes. IEEE Trans. on Information Theory, 40(3):786–794, 1994.Google Scholar
  6. 6.
    A. Beimel, A. Gál, and M. Paterson. Lower bounds for monotone span programs. Computational Complexity, 6(1):29–45, 1997. Conference version: FOCS ’95.Google Scholar
  7. 7.
    A. Beimel, Y. Ishai, R. Kumaresan, and E. Kushilevitz. On the cryptographic complexity of the worst functions. In Y. Lindell, editor, Proc. of the Eleventh Theory of Cryptography Conference– TCC 2014, volume 8349 of Lecture Notes in Computer Science, pages 317–342. Springer-Verlag, 2014.Google Scholar
  8. 8.
    M. Ben-Or, S. Goldwasser, and A. Wigderson. Completeness theorems for non cryptographic fault-tolerant distributed computations. In Proc. of the 20th ACM Symp. on the Theory of Computing, pages 1–10, 1988.Google Scholar
  9. 9.
    J. Benaloh and J. Leichter. Generalized secret sharing and monotone functions. In S. Goldwasser, editor, Advances in Cryptology – CRYPTO ’88, volume 403 of Lecture Notes in Computer Science, pages 27–35. Springer-Verlag, 1990.Google Scholar
  10. 10.
    G. R. Blakley. Safeguarding cryptographic keys. In R. E. Merwin, J. T. Zanca, and M. Smith, editors, Proc. of the 1979 AFIPS National Computer Conference, volume 48 of AFIPS Conference proceedings, pages 313–317. AFIPS Press, 1979.Google Scholar
  11. 11.
    G. R. Blakley and C. Meadows. The security of ramp schemes. In G. R. Blakley and D. Chaum, editors, Advances in Cryptology – CRYPTO ’84, volume 196 of Lecture Notes in Computer Science, pages 242–268. Springer-Verlag, 1985.Google Scholar
  12. 12.
    C. Blundo, A. De Santis, R. de Simone, and U. Vaccaro. Tight bounds on the information rate of secret sharing schemes. Designs, Codes and Cryptography, 11(2):107–122, 1997.Google Scholar
  13. 13.
    C. Blundo, A. De Santis, L. Gargano, and U. Vaccaro. On the information rate of secret sharing schemes. Theoretical Computer Science, 154(2):283–306, 1996.Google Scholar
  14. 14.
    C. Blundo, A. De Santis, D. R. Stinson, and U. Vaccaro. Graph decomposition and secret sharing schemes. J. of Cryptology, 8(1):39–64, 1995.Google Scholar
  15. 15.
    E. F. Brickell. Some ideal secret sharing schemes. Journal of Combin. Math. and Combin. Comput., 6:105–113, 1989.Google Scholar
  16. 16.
    E. F. Brickell and D. M. Davenport. On the classification of ideal secret sharing schemes. J. of Cryptology, 4(73):123–134, 1991.Google Scholar
  17. 17.
    S. Bublitz. Decomposition of graphs and monotone formula size of homogeneous functions. Acta Informatica, 23:689–696, 1986.Google Scholar
  18. 18.
    R. M. Capocelli, A. De Santis, L. Gargano, and U. Vaccaro. On the size of shares for secret sharing schemes. J. of Cryptology, 6(3):157–168, 1993.Google Scholar
  19. 19.
    D. Chaum, C. Crépeau, and I. Damgård. Multiparty unconditionally secure protocols. In Proc. of the 20th ACM Symp. on the Theory of Computing, pages 11–19, 1988.Google Scholar
  20. 20.
    B. Chor and E. Kushilevitz. Secret sharing over infinite domains. J. of Cryptology, 6(2):87–96, 1993.Google Scholar
  21. 21.
    R. Cramer, I. Damgård, and U. Maurer. General secure multi-party computation from any linear secret-sharing scheme. In B. Preneel, editor, Advances in Cryptology – EUROCRYPT 2000, volume 1807 of Lecture Notes in Computer Science, pages 316–334. Springer-Verlag, 2000.Google Scholar
  22. 22.
    G. Di Crescenzo and C. Galdi. Hypergraph decomposition and secret sharing. Discrete Applied Mathematics, 157(5):928–946, 2009.Google Scholar
  23. 23.
    L. Csirmaz. The size of a share must be large. J. of Cryptology, 10(4):223–231, 1997.Google Scholar
  24. 24.
    L. Csirmaz. Secret sharing schemes on graphs. Technical Report 2005/059, Cryptology ePrint Archive, 2005. eprint.iacr.org/.
  25. 25.
    L. Csirmaz. An impossibility result on graph secret sharing. Designs, Codes and Cryptography, 53(3):195–209, 2009.CrossRefMathSciNetMATHGoogle Scholar
  26. 26.
    L. Csirmaz, P. Ligeti, and G. Tardos. Erdös-pyber theorem for hypergraphs and secret sharing. Graphs and Combinatorics, 2014.Google Scholar
  27. 27.
    L. Csirmaz and G. Tardos. Secret sharing on trees: problem solved. IACR Cryptology ePrint Archive, 2009:71, 2009.Google Scholar
  28. 28.
    Y. Desmedt and Y. Frankel. Shared generation of authenticators and signatures. In J. Feigenbaum, editor, Advances in Cryptology – CRYPTO ’91, volume 576 of Lecture Notes in Computer Science, pages 457–469. Springer-Verlag, 1992.Google Scholar
  29. 29.
    M. van Dijk. On the information rate of perfect secret sharing schemes. Designs, Codes and Cryptography, 6:143–169, 1995.Google Scholar
  30. 30.
    P. Erdös and L. Pyber. Covering a graph by complete bipartite graphs. Discrete Mathematics, 170(1-3):249–251, 1997.Google Scholar
  31. 31.
    O. Farràs, J. Martí-Farré, and C. Padró. Ideal multipartite secret sharing schemes. J. of Cryptology, 25(1):434–463, 2012.Google Scholar
  32. 32.
    A. Gál. A characterization of span program size and improved lower bounds for monotone span programs. In Proc. of the 30th ACM Symp. on the Theory of Computing, pages 429–437, 1998.Google Scholar
  33. 33.
    A. Gál and P. Pudlák. A note on monotone complexity and the rank of matrices. Inform. Process. Lett., 87:321–326, 2003.Google Scholar
  34. 34.
    V. Goyal, O. Pandey, A. Sahai, and B. Waters. Attribute-based encryption for fine-grained access control of encrypted data. In Proc. of the 13th ACM Conference on Computer and Communications Security, pages 89–98, 2006.Google Scholar
  35. 35.
    M. Ito, A. Saito, and T. Nishizeki. Secret sharing schemes realizing general access structure. In Proc. of the IEEE Global Telecommunication Conf., Globecom 87, pages 99–102, 1987. Journal version: Multiple assignment scheme for sharing secret. J. of Cryptology, 6(1):15–20, 1993.Google Scholar
  36. 36.
    M. Jerrum. A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures & Algorithms, 7:157–166, 1995.Google Scholar
  37. 37.
    S. Jukna. On set intersection representations of graphs. Journal of Graph Theory, 61:55–75, 2009.CrossRefMathSciNetMATHGoogle Scholar
  38. 38.
    M. Karchmer and A. Wigderson. On span programs. In Proc. of the 8th IEEE Structure in Complexity Theory, pages 102–111, 1993.Google Scholar
  39. 39.
    E. D. Karnin, J. W. Greene, and M. E. Hellman. On secret sharing systems. IEEE Trans. on Information Theory, 29(1):35–41, 1983.Google Scholar
  40. 40.
    J. Kilian and N. Nisan. Private communication, 1990.Google Scholar
  41. 41.
    J. Martí-Farré and C. Padró. Secret sharing schemes on sparse homogeneous access structures with rank three. Electr. J. Comb., 11(1), 2004.Google Scholar
  42. 42.
    J. Martí-Farré and C. Padró. Secret sharing schemes with three or four minimal qualified subsets. Designs, Codes and Cryptography, 34(1):17–34, 2005.Google Scholar
  43. 43.
    J. Martí-Farré and C. Padró. On secret sharing schemes, matroids and polymatroids. Journal of Mathematical Cryptology, 4(2):95–120, 2010.Google Scholar
  44. 44.
    Y. Mintz. Information ratios of graph secret-sharing schemes. Master’s thesis, Dept. of Computer Science, Ben Gurion University, 2012.Google Scholar
  45. 45.
    M. Mitzenmacher and E. Upfal. Probability and Computing. Cambridge University Press, 2005.Google Scholar
  46. 46.
    M. Naor and A. Wool. Access control and signatures via quorum secret sharing. In 3rd ACM Conf. on Computer and Communications Security, pages 157–167, 1996.Google Scholar
  47. 47.
    C. Padró and G. Sáez. Secret sharing schemes with bipartite access structure. IEEE Trans. on Information Theory, 46:2596–2605, 2000.Google Scholar
  48. 48.
    C. Padró and G. Sáez. Lower bounds on the information rate of secret sharing schemes with homogeneous access structure. Inf. Process. Lett., 83(6):345–351, 2002.Google Scholar
  49. 49.
    J. Salas and A. D. Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statist. Phys., 86:551–579, 1997.Google Scholar
  50. 50.
    A. Shamir. How to share a secret. Communications of the ACM, 22:612–613, 1979.Google Scholar
  51. 51.
    B. Shankar, K. Srinathan, and C. Pandu Rangan. Alternative protocols for generalized oblivious transfer. In Proceedings of the 9th International Conference on Distributed Computing and Networking (ICDCN’08), volume 4904 of Lecture Notes in Computer Science, pages 304–309. Springer-Verlag, 2008.Google Scholar
  52. 52.
    G. J. Simmons, W. Jackson, and K. M. Martin. The geometry of shared secret schemes. Bulletin of the ICA, 1:71–88, 1991.Google Scholar
  53. 53.
    D. R. Stinson. New general lower bounds on the information rate of secret sharing schemes. In E. F. Brickell, editor, Advances in Cryptology – CRYPTO ’92, volume 740 of Lecture Notes in Computer Science, pages 168–182. Springer-Verlag, 1993.Google Scholar
  54. 54.
    D. R. Stinson. Decomposition construction for secret sharing schemes. IEEE Trans. on Information Theory, 40(1):118–125, 1994.Google Scholar
  55. 55.
    H. Sun and S. Shieh. Secret sharing in graph-based prohibited structures. In INFOCOM ’97, pages 718–724, 1997.Google Scholar
  56. 56.
    H.-M. Sun, H. Wang, B.-H. Ku, and J. Pieprzyk. Decomposition construction for secret sharing schemes with graph access structures in polynomial time. SIAM J. Discret. Math., 24:617–638, 2010.Google Scholar
  57. 57.
    T. Tassa. Generalized oblivious transfer by secret sharing. Des. Codes Cryptography, 58(1):11–21, 2011.Google Scholar
  58. 58.
    B. Waters. Ciphertext-policy attribute-based encryption: an expressive, efficient, and provably secure realization. In Proc. of the 14th International Conference on Practice and Theory in Public Key Cryptography, volume 6571 of Lecture Notes in Computer Science, pages 53–70. Springer-Verlag, 2011.Google Scholar
  59. 59.
    I. Wegener. The Complexity of Boolean Functions. Wiley-Teubner, 1987.Google Scholar

Copyright information

© International Association for Cryptologic Research 2014

Authors and Affiliations

  1. 1.Ben Gurion University of the NegevBe’er ShevaIsrael
  2. 2.Universitat Rovira i VirgiliTarragonaSpain

Personalised recommendations