Local bounds for the optimal information ratio of secret sharing schemes

  • Oriol Farràs
  • Jordi Ribes-González
  • Sara Ricci


The information ratio of a secret sharing scheme \(\varSigma \) is the ratio between the length of the largest share and the length of the secret, and it is denoted by \(\sigma (\varSigma )\). The optimal information ratio of an access structure \(\varGamma \) is the infimum of \(\sigma (\varSigma )\) among all schemes \(\varSigma \) with access structure \(\varGamma \), and it is denoted by \(\sigma (\varGamma )\). The main result of this work is that for every two access structures \(\varGamma \) and \(\varGamma '\), \(|\sigma (\varGamma )-\sigma (\varGamma ')|\le |\varGamma \cup \varGamma '|-|\varGamma \cap \varGamma '|\). We prove it constructively. Given any secret sharing scheme \(\varSigma \) for \(\varGamma \), we present a method to construct a secret sharing scheme \(\varSigma '\) for \(\varGamma '\) that satisfies that \(\sigma (\varSigma ')\le \sigma (\varSigma )+|\varGamma \cup \varGamma '|-|\varGamma \cap \varGamma '|\). As a consequence of this result, we see that close access structures admit secret sharing schemes with similar information ratio. We show that this property is also true for particular classes of secret sharing schemes and models of computation, like the family of linear secret sharing schemes, span programs, Boolean formulas and circuits. In order to understand this property, we also study the limitations of the techniques for finding lower bounds on the information ratio and other complexity measures. We analyze the behavior of these bounds when we add or delete subsets from an access structure.


Cryptography Secret sharing Information ratio Monotone span program Monotone Boolean formula 

Mathematics Subject Classification

94A60 68P30 94A62 52B40 



  1. 1.
    Alon N., Spencer J.H.: The Probabilistic Method, 3rd edn. John Wiley & Sons, New York (2008).CrossRefzbMATHGoogle Scholar
  2. 2.
    Babai L., Gál A., Wigderson A.: Superpolynomial lower bounds for monotone span programs. Combinatorica 19(3), 301–319 (1999).MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Beimel A.: Secret-sharing schemes: a survey. In: Third International Workshop on Coding and Cryptology (IWCC 2011). Lecture Notes in Computer Science, vol. 6639, pp. 11–46 (2011).Google Scholar
  4. 4.
    Beimel A., Orlov I.: Secret sharing and non-Shannon information inequalities. IEEE Trans. Inf. Theory 57, 5634–5649 (2011).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beimel A., Gál A., Paterson M.: Lower bounds for monotone span programs. In: 36th Annual Symposium on Foundations of Computer Science (STOC), pp. 674–681 (1995).Google Scholar
  6. 6.
    Beimel A., Farràs O., Mintz Y.: Secret sharing schemes for very dense graphs. J. Cryptol. 29(2), 336–362 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Beimel A., Farràs O., Peter N.: Schemes secret sharing, for dense forbidden graphs. In: Security and Cryptography for Networks (SCN 2016). Lecture Notes in Computer Science, vol. 9841, pp. 509–528 (2016)Google Scholar
  8. 8.
    Beimel A., Farràs O., Mintz Y., Peter N.: Theory of Cryptography (TCC 2017). Lecture Notes in Computer Science, vol. 10678, pp. 394–423 (2017).Google Scholar
  9. 9.
    Bellare M., Rogaway P.: Robust computational secret sharing and a unified account of classical secret-sharing goals. In: Proceedings of the 2007 ACM Conference on Computer and Communications Security (CCS 2007), pp. 172–184 (2007).Google Scholar
  10. 10.
    Benaloh J., Leichter J.: Generalized secret sharing and monotone functions. In: Advances in Cryptology (CRYPTO 1988). Lecture Notes in Computer Science, vol. 403, pp. 27–35 (1988).Google Scholar
  11. 11.
    Blakley G.R.: Safeguarding cryptographic keys. In: 1979 AFIPS National Computer Conference, pp. 313–317 (1979).Google Scholar
  12. 12.
    Brickell E.F.: Some ideal secret sharing schemes. J. Combin. Math. Combin. Comput. 6, 105–113 (1989).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Brickell E.F., Davenport D.M.: On the classification of ideal secret sharing schemes. J. Cryptol. 4(73), 123–134 (1991).zbMATHGoogle Scholar
  14. 14.
    Cramer R., Damgård I., Maurer U.: Computation general secure multi-party, from any linear secret-sharing scheme. In: Advances in Cryptology—EUROCRYPT 2000. Lecture Notes in Computer Science, vol. 1807, pp. 316–334 (2000).Google Scholar
  15. 15.
    Csirmaz L.: The size of a share must be large. J. Cryptol. 10, 223–231 (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Csirmaz L.: Secret sharing on the d-dimensional cube. Des. Codes Cryptogr. 74(3), 719–729 (2015).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Farràs O., Martí-Farré J., Padró C.: Ideal multipartite secret sharing schemes. J. Cryptol. 25(1), 434–463 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Farràs O., Metcalf-Burton J.R., Padró C., Vázquez L.: On the optimization of bipartite secret sharing schemes. Des. Codes Cryptogr. 63(2), 255–271 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Farràs O., Ribes-González J., Ricci S.: Local bounds for the optimal information ratio of secret sharing schemes. IACR Cryptol. ePrint Archive 2016, 726 (2016).Google Scholar
  20. 20.
    Farràs O., Hansen T., Kaced T., Padró C.: On the information ratio of non-perfect secret sharing schemes. Algorithmica 79, 987–1013 (2017).MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Farràs O., Ribes-González J., Ricci S.: Privacy-preserving data splitting: a combinatorial approach. arXiv:1801.05974 (2018).
  22. 22.
    Farràs O., Kaced T., Martín S., Padró C.: Improving the linear programming technique in the search for lower bounds in secret sharing. In: Advances in Cryptology (EUROCRYPT 2018). Lecture Notes in Computer Science, vol. 10820, pp. 597–621 (2018).Google Scholar
  23. 23.
    Frankl P.: Extremal set systems. In: Handbook of Combinatorics, vol. II, pp. 1293–1329. Elsevier, Amsterdam (1995).Google Scholar
  24. 24.
    Gál A.: A characterization of span program size and improved lower bounds for monotone span programs. Comput. Complex. 10(4), 277–296 (2001).MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Goyal V., Pandey O., Sahai A., Waters B.: Attribute-based encryption for fine-grained access control of encrypted data. In: 13th CCS, pp. 89–98 (2006).Google Scholar
  26. 26.
    Ito M., Saito A., Nishizeki T.: Secret sharing scheme realizing any access structure. In: Proceedings of IEEE Globecom’87, pp. 99–102 (1987).Google Scholar
  27. 27.
    Jha M., Raskhodnikova S.: Testing and reconstruction of Lipschitz functions with applications to data privacy. SIAM J. Comput. 42(2), 700–731 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Jukna S.: Boolean Function Complexity. Advances and Frontiers. Springer, Berlin (2012).CrossRefzbMATHGoogle Scholar
  29. 29.
    Karchmer M., Wigderson A.: On span programs. In: 8th Structure in Complexity Theory, pp. 102–111 (1993).Google Scholar
  30. 30.
    Komargodski I., Naor M., Yogev E.: Secret-sharing for NP. In: Advances in Cryptology (ASIACRYPT 2014). Lecture Notes in Computer Science, vol. 8874, pp. 254–273 (2014).Google Scholar
  31. 31.
    Liu T., Vaikuntanathan V., Wee H.: Conditional disclosure of secrets via non-linear reconstruction. In: Advances in Cryptology (CRYPTO 2017). Lecture Notes in Computer Science, vol. 10401, pp. 758–790 (2017).Google Scholar
  32. 32.
    Martí-Farré J., Padró C.: On secret sharing schemes, matroids and polymatroids. J. Math. Cryptol. 4, 95–120 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Martín S., Padró C., Yang A.: Secret sharing, rank inequalities, and information inequalities. IEEE Trans. Inf. Theory 62, 599–609 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Padró C.: Lecture Notes in Secret Sharing. Cryptology ePrint Archive 2012/674.Google Scholar
  35. 35.
    Padró C., Vázquez L., Yang A.: Finding lower bounds on the complexity of secret sharing schemes by linear programming. Discret. Appl. Math. 161, 1072–1084 (2013).MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Pitassi T., Robere R.: Lifting Nullstellensatz to monotone span programs over any field. In: Electronic Colloquium on Computational Complexity (ECCC), p. 165 (2017).Google Scholar
  37. 37.
    Razborov A.A.: Applications of matrix methods to the theory of lower bounds in computational complexity. Combinatorica 10(1), 81–93 (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Razborov A.A.: On submodular complexity measures. In: Proceedings of the London Mathematical Society Symposium on Boolean Function Complexity, pp. 76–83 (1992).Google Scholar
  39. 39.
    Shamir A.: How to share a secret. Commun. ACM 22, 612–613 (1979).MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Vaikuntanathan V., Vasudevan P.N.: Sharing secret, advances statistical zero knowledge. In: Cryptology (ASIACRYPT 2015). Lecture Notes in Computer Science, vol. 9452, pp. 656–680 (2015).Google Scholar
  41. 41.
    Wegener I.: The Complexity of Boolean Functions. Wiley-Teubner, New York (1987).zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversitat Rovira i VirgiliTarragonaSpain

Personalised recommendations