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Optimal complexity of secret sharing schemes with four minimal qualified subsets

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Abstract

The complexity of a secret sharing scheme is defined as the ratio between the maximum length of the shares and the length of the secret. This paper deals with the open problem of optimizing this parameter for secret sharing schemes with general access structures. Specifically, our objective is to determine the optimal complexity of the access structures with exactly four minimal qualified subsets. Lower bounds on the optimal complexity are obtained by using the known polymatroid technique in combination with linear programming. Upper bounds are derived from decomposition constructions of linear secret sharing schemes. In this way, the exact value of the optimal complexity is determined for several access structures in that family. For the other ones, we present the best known lower and upper bounds.

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References

  1. 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 

  2. Beimel A., Livne N., Padró C.: Matroids can be far from ideal secret sharing. Fifth Theory of Cryptography Conference, TCC 2008. Lecture Notes in Computer Science, vol. 4948, pp. 194–212 (2008).

  3. 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 

  4. Benaloh J., Leichter J.: Generalized secret sharing and monotone functions. Advances in Cryptology, CRYPTO’88. Lecture Notes in Computer Science, vol. 403, pp. 27–35 (1990).

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

  6. 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 Computer Science, vol. 740, pp. 148–167 (1993).

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

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

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. 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 

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

  16. Farràs O., Padró C.: Ideal hierarchical secret sharing schemes. Seventh IACR Theory of Cryptography Conference, TCC 2010, Lecture Notes in Computer Science, vol. 5978, pp. 219–236 (2010).

  17. Farràs O., Martí-Farré J., Padró C.: Ideal multipartite secret sharing schemes. Advances in Cryptology, Eurocrypt 2007, Lecture Notes in Computer Science, vol. 4515, pp. 448–465 (2007).

  18. Farràs O., Metcalf-Burton J.R., Padró C., Vázquez L.: On the optimization of bipartite secret sharing schemes. Fourth International Conference on Information Theoretic Security ICITS 2009, Lecture Notes in Computer Science, vol. 5973, pp. 93–109 (2010).

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

    Article  MathSciNet  MATH  Google Scholar 

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

  21. 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 

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

    MathSciNet  MATH  Google Scholar 

  23. Karnin E.D., Greene J.W., Hellman M.E.: On secret sharing systems. IEEE Trans. Inform. Theory 29, 35–41 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  24. Martí-Farré J., Padró C.: Secret sharing schemes with three or four minimal qualified subsets. Des. Codes Cryptogr. 34, 17–34 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martí-Farré J., Padró C.: Secret sharing schemes on access structures with intersection number equal to one. Discrete Appl. Math. 154, 552–563 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Padró C., Sáez G.: Lower bounds on the information rate of secret sharing schemes with homogeneous access structure. Inform. Process. Lett. 83, 345–351 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  30. 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).

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. Stinson D.R.: New general lower bounds on the information rate of secret sharing schemes. Advances in Cryptology—CRYPTO’92. Lecture Notes in Computer Science, vol. 740, pp. 168–182 (1993).

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  36. van Dijk M.: A linear construction of secret sharing schemes. Des. Codes Cryptogr. 12, 161–201 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. 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 

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Authors and Affiliations

  1. Technical University of Catalonia, C/Jordi Girona, 1-3, 08034, Barcelona, Catalonia, Spain

    Jaume Martí-Farré

  2. Nanyang Technological University, 21 Nanyang Link, Singapore, 637371, Singapore

    Carles Padró

  3. Instituto Politécnico Nacional, ESCOM - IPN, Av. Juan de Dios Bátiz s/n, 07738, México D. F., Mexico

    Leonor Vázquez

Authors
  1. Jaume Martí-Farré
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  2. Carles Padró
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  3. Leonor Vázquez
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Corresponding author

Correspondence to Carles Padró.

Additional information

Communicated by H. van Tilborg.

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Martí-Farré, J., Padró, C. & Vázquez, L. Optimal complexity of secret sharing schemes with four minimal qualified subsets. Des. Codes Cryptogr. 61, 167–186 (2011). https://doi.org/10.1007/s10623-010-9446-0

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  • DOI: https://doi.org/10.1007/s10623-010-9446-0

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