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Introduction to Secret-Sharing

  • Andreas BlassEmail author
Chapter

Abstract

This is the written form of a talk that I gave at the Dagstuhl seminar “Dependence Logic: Theory and Applications”. My purpose is to explain what the theory of secret-sharing is about; to point out its connections with the fundamental notions, dependence and independence, of dependence logic; and to indicate some of the results and open problems of this theory.

Keywords

Positive Probability Access Structure Assistant Manager Random String Ideal Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Beimel, A., Livne, N., Padró, C.: Matroids can be far from ideal secret sharing. In: Canetti, R. (ed.) Theory of Cryptography, pp. 194–212. Springer, Berlin (2008)CrossRefGoogle Scholar
  2. 2.
    Brickell, E.F., Davenport, D.M.: On the classification of ideal secret sharing schemes. J. Cryptology 4, 123–134 (1991)zbMATHGoogle Scholar
  3. 3.
    Brylawski, T.: Appendix of matroid cryptomorphisms. In: [21], pp. 298–312Google Scholar
  4. 4.
    Brylawski, T.: Constructions. In: [21], Chapter 7, pp. 127–223Google Scholar
  5. 5.
    Dougherty, R., Freiling, C., Zeger, K.: Linear rank inequalities on five or more variables. http://arxiv.org/abs/0910.0284
  6. 6.
    Dougherty, R., Freiling, C., Zeger, K.: Six new non-Shannon information inequalities. In: IEEE International Symposium on Information Theory, Seattle, WA (July 2006)Google Scholar
  7. 7.
    Ingleton, A.W.: Representation of matroids. In: Welsh, D.J.A. (ed.) Combinatorial Mathematics and Its Applications, Academic Press, London, pp. 149–167 (1971)Google Scholar
  8. 8.
    Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. Inf. Theory 29, 35–41 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Kinser, R.: New inequalities for subspace arrangements. J. Comb. Theory A 118, 152–161 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kurosawa, K., Okada, K., Sakano, K., Ogota, W., Tsujii, S.: Nonperfect secret sharing schemes and matroids. In: EUROCRYPT ’93, pp. 126–141 (1994)Google Scholar
  11. 11.
    Livne, N.: On matroids and non-ideal secret sharing. Master’s thesis, Ben-Gurion University (2005)zbMATHGoogle Scholar
  12. 12.
    Martí-Farré, J., Padró, C.: On secret sharing schemes, matroids, and polymatroids. Cryptology ePrint Archive, Report 2006/077 (2006). http://eprint.iacr.org/
  13. 13.
    Matuš, F.G.: Two constructions on limits of entropy functions. IEEE Trans. Inf. Theory 53, 320–330 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Metcalf-Burton, J.R.: Information rates for secret sharing over various access structures. Ph.D. thesis, University of Michigan (2009)Google Scholar
  15. 15.
    Seymour, P.D.: A forbidden minor characterization of matroid ports. Q. J. Math. 27, 407–41 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Seymour, P.D.: On secret-sharing matroids. J. Comb. Theory B 56, 69–73 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shamir, A.: How to share a secret. Commun. ACM 22, 612–613 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Stinson, D.R.: An explication of secret sharing schemes. Des. Codes Crypt. 2, 357–390 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Vámos, P.: On the In: representation of independence structures. Theory of Matroids, Encyclopedia of Mathematics, (1968)zbMATHGoogle Scholar
  21. 22.
    Zhang, Z., Yeung, R.W.: On characterization of entropy function via information inequalities. IEEE Trans. Inf. Theory 44, 1440–452 (1998)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Mathematics DepartmentUniversity of MichiganAnn ArborUSA

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