Finding Lower Bounds on the Complexity of Secret Sharing Schemes by Linear Programming | SpringerLink
Skip to main content
Springer Nature Link
Log in

Finding Lower Bounds on the Complexity of Secret Sharing Schemes by Linear Programming

  • Conference paper
LATIN 2010: Theoretical Informatics (LATIN 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6034))

Included in the following conference series:

Abstract

Determining the optimal complexity of secret sharing schemes for every given access structure is a difficult and long-standing open problem in cryptology. Lower bounds have been found by a combinatorial method that uses polymatroids. In this paper, we point out that the best lower bound that can be obtained by this method can be determined by using linear programming, and this can be effectively done for access structures on a small number of participants. By applying this linear programming approach, we present better lower bounds on the optimal complexity and the optimal average complexity of several access structures. Finally, by adding the Ingleton inequality to the previous linear programming approach, we find a few examples of access structures for which there is a gap between the optimal complexity of linear secret sharing schemes and the combinatorial lower bound.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
JPY 3498
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
JPY 11439
Price includes VAT (Japan)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
JPY 14299
Price includes VAT (Japan)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Anderson, I.: Combinatorics of Finite Sets. Oxford University Press, Oxford (1987)

    MATH  Google Scholar 

  2. Bazaraa, M., Jarvis, J., Sherali, H.: Linear Programming and Network Flows, 2nd edn. John Wiley& Sons, Chichester (1990)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Beimel, A., Livne, N.: On matroids and non-ideal secret sharing. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 482–501. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  5. Beimel, A., Livne, N., Padró, C.: Matroids can be far from ideal secret sharing. In: Canetti, R. (ed.) TCC 2008. LNCS, vol. 4948, pp. 194–212. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  6. Beimel, A., Orlov, I.: Secret Sharing and Non-Shannon Information Inequalities. In: Reingold, O. (ed.) TCC 2009. LNCS, vol. 5444, pp. 539–557. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Benaloh, J., Leichter, J.: Generalized secret sharing and monotone functions. In: Goldwasser, S. (ed.) CRYPTO 1988. LNCS, vol. 403, pp. 27–35. Springer, Heidelberg (1990)

    Google Scholar 

  9. Blakley, G.R.: Safeguarding cryptographic keys. In: AFIPS Conf. P., vol. 48, pp. 313–317 (1979)

    Google Scholar 

  10. Blundo, C., de Santis, A., de Simone, R., Vaccaro, U.: Tight bounds on the information rate of secret sharing schemes. Design Code Cryptogr. 11, 107–122 (1997)

    Article  MATH  Google Scholar 

  11. Chan, T.H., Guillé, L., Grant, A.: The minimal set of Ingleton inequalities (2008), http://arXiv.org/abs/0802.2574

  12. Chen, B.-L., Sun, H.-M.: Weighted Decomposition Construction for Perfect Secret Sharing Schemes. Comput. Math. Appl. 43(6-7), 877–887 (2002)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  14. Csirmaz, L.: Secret sharing on the d-dimensional cube. Cryptology ePrint Archive. Report 2005/177 (2005), http://eprint.iacr.org

  15. Csirmaz, L.: An impossibility result on graph secret sharing. Designs, Codes and Cryptography 53, 195–209 (2009)

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

  18. van Dijk, M.: More information theoretical inequalities to be used in secret sharing? Inform. Process. Lett. 63, 41–44 (1997)

    Article  Google Scholar 

  19. van Dijk, M., Jackson, W.-A., Martin, K.M.: A note on duality in linear secret sharing schemes. Bull. Inst. Combin. Appl. 19, 93–101 (1997)

    MATH  MathSciNet  Google Scholar 

  20. van Dijk, M., Kevenaar, T., Schrijen, G., Tuyls, P.: Improved constructions of secret sharing schemes by applying-(λ, ω)-decompositions. Inform. Process. Lett. 99 (4), 154–157 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  21. Dougherty, R., Freiling, C., Zeger, K.: Six new non-Shannon information in- equalities. In: ISIT 2006, pp. 233–236 (2006)

    Google Scholar 

  22. Fehr, S.: Linear VSS and Distributes Commitments Based on Secret Sharing and Pairwise Checks. In: Yung, M. (ed.) CRYPTO 2002. LNCS, vol. 2442, pp. 565–580. Springer, Heidelberg (2002)

    Google Scholar 

  23. Fujishige, S.: Polymatroidal Dependence Structure of a Set of Random Variables. Inform. and Control. 39, 55–72 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  24. Ingleton, A.W.: Conditions for representability and transversability of matroids. In: Proc. Fr. Br. Conf. 1970, pp. 62–27 (1971)

    Google Scholar 

  25. Ito, M., Saito, A., Nishizeki, T.: Secret sharing scheme realizing any access structure. In: Proc. IEEE Globecom 1987, pp. 99–102 (1987)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  29. Martí-Farré, J., Padró, C.: On Secret Sharing Schemes, Matroids and Polymatroids. In: Vadhan, S.P. (ed.) TCC 2007. LNCS, vol. 4392, pp. 273–290. Springer, Heidelberg (2007), Cryptology ePrint Archive, http://eprint.iacr.org/2006/077

    Chapter  Google Scholar 

  30. Matúš, F.: Piecewise linear conditional information inequality. On IEEE Trans. Inform. Theory 44, 236–238 (2006)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  32. Matúš, F.: Infinitely many information inequalities. In: IEEE International Symposium on Information Theory 2007, pp. 41–44 (2007)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Yeung, R.: A First Course in Information Theory. Kluwer Academic/Plenum Publishers (2002)

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universitat Politècnica de Catalunya,  

    Carles Padró & Leonor Vázquez

Authors
  1. Carles Padró
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leonor Vázquez
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Faculty of Mathematics, Department of Computer Science, University of Waterloo, N2L 3G1, Waterloo, Ontario, Canada

    Alejandro López-Ortiz

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Padró, C., Vázquez, L. (2010). Finding Lower Bounds on the Complexity of Secret Sharing Schemes by Linear Programming. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_31

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-12200-2_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-12199-9

  • Online ISBN: 978-3-642-12200-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation