On the optimization of bipartite secret sharing schemes

Abstract

Optimizing the ratio between the maximum length of the shares and the length of the secret value in secret sharing schemes for general access structures is an extremely difficult and long-standing open problem. In this paper, we study it for bipartite access structures, in which the set of participants is divided in two parts, and all participants in each part play an equivalent role. We focus on the search of lower bounds by using a special class of polymatroids that is introduced here, the tripartite ones. We present a method based on linear programming to compute, for every given bipartite access structure, the best lower bound that can be obtained by this combinatorial method. In addition, we obtain some general lower bounds that improve the previously known ones, and we construct optimal secret sharing schemes for a family of bipartite access structures.

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References

  1. 1.

    Beimel A., Ishai Y.: On the power of nonlinear secret sharing schemes. SIAM J. Discrete Math. 19, 258–280 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    Beimel A., Weinreb E.: Separating the power of monotone span programs over different fields. SIAM J. Comput. 34, 1196–1215 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. 3.

    Beimel A., Weinreb E.: Monotone circuits for monotone weighted threshold functions. Inform. Process. Lett. 97, 12–18 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. 4.

    Beimel A., Livne N., Padró C.: Matroids can be far from ideal secret sharing. Theory of Cryptography Conference, TCC 2008. Lect. Notes Comput. Sci. vol. 4948, pp. 194–212 (2008).

  5. 5.

    Beimel A., Tassa T., Weinreb E.: Characterizing ideal weighted threshold secret sharing. SIAM J. Discrete Math. 22, 360–397 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  6. 6.

    Blakley G.R.: Safeguarding cryptographic keys. AFIPS Conference Proceedings, vol. 48, pp. 313–317 (1979).

  7. 7.

    Blundo C., De Santis A., Gargano L., Vaccaro U.: On the information rate of secret sharing schemes. Advances in Cryptology—CRYPTO’92. Lecture Notes in Comput. Sci. vol. 740, pp. 148–167 (1993).

  8. 8.

    Blundo C., De Santis A., Stinson D.R., Vaccaro U.: Graph decompositions and secret sharing schemes. J. Cryptol. 8, 39–64 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    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)

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Brickell E.F.: Some ideal secret sharing schemes. J. Combin. Math. Combin. Comput. 9, 105–113 (1989)

    MathSciNet  Google Scholar 

  11. 11.

    Brickell E.F., Davenport D.M.: On the classification of ideal secret sharing schemes. J. Cryptol. 4, 123–134 (1991)

    MATH  Google Scholar 

  12. 12.

    Brickell E.F., Stinson D.R.: Some improved bounds on the information rate of perfect secret sharing schemes. J. Cryptol. 5, 153–166 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    Capocelli R.M., De Santis A., Gargano L., Vaccaro U.: On the size of shares of secret sharing schemes. J. Cryptol. 6, 157–168 (1993)

    Article  MATH  Google Scholar 

  14. 14.

    Cover T.M., Thomas J.A.: Elements of Information Theory, 2nd edn. Wiley, New York (2006)

    Google Scholar 

  15. 15.

    Csirmaz L.: The size of a share must be large. J. Cryptol. 10, 223–231 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. 16.

    Csirmaz L.: An impossibility result on graph secret sharing. Des. Codes Cryptogr. 53, 195–209 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. 17.

    Csirmaz L., Ligeti P.: On an infinite family of graphs with information ratio 2 − 1/k. Computing 85, 127–136 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. 18.

    Csirmaz L., Tardos G.: Secret sharing on trees: problem solved. Preprint (2009). Available at Cryptology ePrint Archive. http://eprint.iacr.org/2009/071.

  19. 19.

    Farràs O., Martí-Farré J., Padró C.: Ideal multipartite secret sharing schemes. Advances in Cryptology, EUROCRYPT 2007, Lecture Notes in Comput. Sci., vol. 4515, pp. 448–465 (2007). The full version of this paper is available at the Cryptology ePrint Archive, Report 2006/292, http://eprint.iacr.org/2006/292.

  20. 20.

    Farràs O., Padró C.: Ideal hierarchical secret sharing schemes. Seventh IACR Theory of Cryptography Conference, TCC 2010, Lecture Notes in Comput. Sci., vol. 5978, pp. 219–236 (2010). The full version of this paper is available at the Cryptology ePrint Archive, Report 2009/141, http://eprint.iacr.org/2009/141

  21. 21.

    Fujishige S.: Polymatroidal dependence structure of a set of random variables. Inform. Control. 39, 55–72 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  22. 22.

    Gál A.: A characterization of span program size and improved lower bounds for monotone span programs. In: Proceedings of 30th ACM Symposium on the Theory of Computing, STOC 1998, pp. 429–437 (1998).

  23. 23.

    Ingleton A.W.: Conditions for representability and transversability of matroids. In Proc. Fr. Br. Conf. 1970, pp. 62–67. Springer (1971).

  24. 24.

    Ito M., Saito A., Nishizeki T.: Secret sharing scheme realizing any access structure. In: Proc. IEEE Globecom’87, pp. 99–102 (1987).

  25. 25.

    Jackson W.-A., Martin K.M.: Geometric secret sharing schemes and their duals. Des. Codes Cryptogr. 4, 83–95 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  26. 26.

    Jackson W.-A., Martin K.M.: Perfect secret sharing schemes on five participants. Des. Codes Cryptogr. 9, 267–286 (1996)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Martí-Farré J., Padró C.: On secret sharing schemes, matroids and polymatroids. J. Math. Cryptol. 4, 95–120 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. 28.

    Martí-Farré J., Padró, C. Vázquez L.: Optimal complexity of secret sharing schemes with four minimal qualified subsets. Des. Codes Cryptogr. Online First (2010). doi:10.1007/s10623-010-9446-0.

  29. 29.

    Matúš F.: Adhesivity of polymatroids. Discrete Math. 307, 2464–2477 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  30. 30.

    Matúš F.: Two constructions on limits of entropy functions. IEEE Trans. Inform. Theory 53, 320–330 (2007)

    Article  MathSciNet  Google Scholar 

  31. 31.

    Metcalf-Burton J.R.: Information Rates of Minimal Non-Matroid-Related Access Structures. http://arxiv.org/pdf/0801.3642.

  32. 32.

    Metcalf-Burton J.R.: Improved upper bounds for the information rates of the secret sharing schemes induced by the vamos matroid. Discrete Math. 311, 651–662 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  33. 33.

    Padró C., Sáez G.: Secret sharing schemes with bipartite access structure. IEEE Trans. Inform. Theory 46, 2596–2604 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. 34.

    Padró C., Vázquez L.: Finding lower bounds on the complexity of secret sharing schemes by linear programming. Ninth Latin American Theoretical Informatics Symposium, LATIN 2010, Lecture Notes in Computer Science, vol. 6034, pp. 344–355 (2010).

  35. 35.

    Shamir A.: How to share a secret. Commun. ACM. 22, 612–613 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  36. 36.

    Simmons G.J.: How to (really) share a secret. Advances in Cryptology—CRYPTO’88, Lecture Notes in Comput. Sci., vol. 403, pp. 390–448 (1990).

  37. 37.

    Stinson D.R.: An explication of secret sharing schemes. Des. Codes Cryptogr. 2, 357–390 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  38. 38.

    Stinson D.R.: Decomposition constructions for secret-sharing schemes. IEEE Trans. Inform. Theory 40, 118–125 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. 39.

    Tassa T.: Hierarchical threshold secret sharing. J. Cryptol. 20, 237–264 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  40. 40.

    Tassa T., Dyn N.: Multipartite secret sharing by bivariate interpolation. J. Cryptol. 22, 227–258 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. 41.

    van Dijk M.: On the information rate of perfect secret sharing schemes. Des. Codes Cryptogr. 6, 143–169 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  42. 42.

    van Dijk M., Kevenaar T., Schrijen G., Tuyls P.: Improved constructions of secret sharing schemes by applying (λ, ω)-decompositions. Inform. Process. Lett. 99, 154–157 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  43. 43.

    Zhang Z., Yeung R.W.: On characterization of entropy function via information inequalities. IEEE Trans. Inform. Theory 44, 1440–1452 (1998)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Oriol Farràs.

Additional information

A preliminary version of this paper appeared in the Proceedings of the Fourth International Conference on Information Theoretic Security, ICITS 2009. The present journal version strengthens some of the previous results, and most of the proofs are now presented in a clearer and more elegant manner.

Communicated by K. Matsuura.

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Farràs, O., Metcalf-Burton, J.R., Padró, C. et al. On the optimization of bipartite secret sharing schemes. Des. Codes Cryptogr. 63, 255–271 (2012). https://doi.org/10.1007/s10623-011-9552-7

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Keywords

  • Cryptography
  • Secret sharing
  • Multipartite secret sharing
  • Polymatroids
  • Linear programming

Mathematics Subject Classification (2000)

  • 94A62
  • 05B35