Abstract
One of the main open problems in secret sharing is the characterization of the access structures of ideal secret sharing schemes. As a consequence of the results by Brickell and Davenport, every one of those access structures is related in a certain way to a unique matroid.
Matroid ports are combinatorial objects that are almost equivalent to matroid-related access structures. They were introduced by Lehman in 1964 and a forbidden minor characterization was given by Seymour in 1976. These and other subsequent works on that topic have not been noticed until now by the researchers interested on secret sharing.
By combining those results with some techniques in secret sharing, we obtain new characterizations of matroid-related access structures. As a consequence, we generalize the result by Brickell and Davenport by proving that, if the information rate of a secret sharing scheme is greater than 2/3, then its access structure is matroid-related. This generalizes several results that were obtained for particular families of access structures.
In addition, we study the use of polymatroids for obtaining upper bounds on the optimal information rate of access structures. We prove that every bound that is obtained by this technique for an access structure applies to its dual structure as well.
Finally, we present lower bounds on the optimal information rate of the access structures that are related to two matroids that are not associated with any ideal secret sharing scheme: the Vamos matroid and the non-Desargues matroid.
This work was partially supported by the Spanish Ministry of Education and Science under project TIC 2003-00866. This work was done while the second author was in a sabbatical stay at CWI, Amsterdam. This stay was funded by the Secretaría de Estado de Educación y Universidades of the Spanish Ministry of Education.
Chapter PDF
Similar content being viewed by others
Keywords
References
Beimel, A., Ishai, Y.: On the power of nonlinear secret sharing schemes. SIAM J. Discrete Math. 19, 258–280 (2005)
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)
Beimel, A., Tassa, T., Weinreb, E.: Characterizing Ideal Weighted Threshold Secret Sharing. In: Kilian, J. (ed.) TCC 2005. LNCS, vol. 3378, pp. 600–619. Springer, Heidelberg (2005)
Beimel, A., Weinreb, E.: Separating the power of monotone span programs over different fields. SIAM J. Comput. 34, 1196–1215 (2005)
Blakley, G.R.: Safeguarding cryptographic keys. AFIPS Conference Proceedings 48, 313–317 (1979)
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)
Blundo, C., De Santis, A., Gargano, L., Vaccaro, U.: On the information rate of secret sharing schemes. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 148–167. Springer, Heidelberg (1993)
Blundo, C., De Santis, A., Stinson, D.R., Vaccaro, U.: Graph decompositions and secret sharing schemes. J. Cryptology 8, 39–64 (1995)
Brickell, E.F.: Some ideal secret sharing schemes. J. Combin. Math. and Combin. Comput. 9, 105–113 (1989)
Brickell, E.F., Davenport, D.M.: On the classification of ideal secret sharing schemes. J. Cryptology 4, 123–134 (1991)
Brickell, E.F., Stinson, D.R.: Some improved bounds on the information rate of perfect secret sharing schemes. J. Cryptology 5, 153–166 (1992)
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)
Csirmaz, L.: The size of a share must be large. J. Cryptology 10, 223–231 (1997)
Fehr, S.: Efficient Construction of the Dual Span Program. Manuscript
Fujishige, S.: Entropy functions and polymatroids—combinatorial structures in information theory. Electron. Comm. Japan 61, 14–18 (1978)
Fujishige, S.: Polymatroidal Dependence Structure of a Set of Random Variables. Information and Control 39, 55–72 (1978)
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. ACM, New York (1998)
Ito, M., Saito, A., Nishizeki, T.: Secret sharing scheme realizing any access structure. In: Proc. IEEE Globecom’87, pp. 99–102. IEEE Computer Society Press, Los Alamitos (1987)
Jackson, W.-A., Martin, K.M.: Perfect secret sharing schemes on five participants. Des. Codes Cryptogr. 9, 267–286 (1996)
Karnin, E.D., Greene, J.W., Hellman, M.E.: On secret sharing systems. IEEE Trans. Inform. Theory 29, 35–41 (1983)
Lehman, A.: A solution of the Shannon switching game. J. Soc. Indust. Appl. Math. 12, 687–725 (1964)
Lehman, A.: Matroids and Ports. Notices Amer. Math. Soc. 12, 356–360 (1965)
Martí-Farré, J., Padró, C.: Secret sharing schemes on sparse homogeneous access structures with rank three. Electronic Journal of Combinatorics 11(1), Research Paper 72, 16 pp. (electronic) (2004)
Martí-Farré, J., Padró, C.: Secret sharing schemes with three or four minimal qualified subsets. Des. Codes Cryptogr. 34, 17–34 (2005)
Martí-Farré, J., Padró, C.: Ideal secret sharing schemes whose minimal qualified subsets have at most three participants. In: De Prisco, R., Yung, M. (eds.) SCN 2006. LNCS, vol. 4116, pp. 201–215. Springer, Heidelberg (2006)
Martí-Farré, J., Padró, C.: Secret sharing schemes on access structures with intersection number equal to one. Discrete Applied Mathematics 154, 552–563 (2006)
Matúš, F.: Matroid representations by partitions. Discrete Math. 203, 169–194 (1999)
Nikov, V., Nikova, S., Preneel, B.: On the Size of Monotone Span Programs. In: Blundo, C., Cimato, S. (eds.) SCN 2004. LNCS, vol. 3352, pp. 252–265. Springer, Heidelberg (2005)
Oxley, J.G.: Matroid theory. In: Oxford Science Publications, The Clarendon Press, New York (1992)
Padró, C., Sáez, G.: Secret sharing schemes with bipartite access structure. IEEE Trans. Inform. Theory 46, 2596–2604 (2000)
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), Quoting Marc Heymann Pignolo, mheymann@ma4.upc.edu
Seymour, P.D.: A forbidden minor characterization of matroid ports. Quart. J. Math. Oxford Ser. 27, 407–413 (1976)
Seymour, P.D.: On secret-sharing matroids. J. Combin. Theory Ser. B 56, 69–73 (1992)
Shamir, A.: How to share a secret. Commun. of the ACM 22, 612–613 (1979)
Simonis, J., Ashikhmin, A.: Almost affine codes. Des. Codes Cryptogr. 14, 179–197 (1998)
Stinson, D.R.: An explication of secret sharing schemes. Des. Codes Cryptogr. 2, 357–390 (1992)
Stinson, D.R.: New general lower bounds on the information rate of secret sharing schemes. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 168–182. Springer, Heidelberg (1993)
Stinson, D.R.: Decomposition constructions for secret-sharing schemes. IEEE Trans. Inform. Theory 40, 118–125 (1994)
Welsh, D.J.A.: Matroid Theory. Academic Press, London (1976)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Martí-Farré, J., Padró, C. (2007). On Secret Sharing Schemes, Matroids and Polymatroids. In: Vadhan, S.P. (eds) Theory of Cryptography. TCC 2007. Lecture Notes in Computer Science, vol 4392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70936-7_15
Download citation
DOI: https://doi.org/10.1007/978-3-540-70936-7_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70935-0
Online ISBN: 978-3-540-70936-7
eBook Packages: Computer ScienceComputer Science (R0)