Abstract
We investigate graph based secret sharing schemes and its information ratio, also called complexity, measuring the maximal amount of information the vertices has to store. It was conjectured that in large girth graphs, where the interaction between far away nodes is restricted to a single path, this ratio is bounded. This conjecture was supported by several result, most notably by a result of Csirmaz and Ligeti (Computing 85(1):127–136, 2009) saying that the complexity of graphs with girth at least six and no neighboring high degree vertices is strictly below 2. In this paper we refute the above conjecture. First, a family of d-regular graphs is defined iteratively such that the complexity of these graphs is the largest possible (d + 1)/2 allowed by Stinson’s bound (IEEE Trans. Inf. Theory 40(1):118–125, 1994). This part extends earlier results of van Dijk (Des. Codes Crypt. 6(2):143–169, 1995) and Blundo et al. (Des. Codes Crypt. 11(2):107–110, 1997), and uses the so-called entropy method. Second, using combinatorial arguments, we show that this family contains graphs with arbitrary large girth. In particular, we obtain the following purely combinatorial result, which might be interesting on its own: there are d-regular graphs with arbitrary large girth such that any fractional edge-cover by stars (or by complete multipartite graphs) must cover some vertex (d + 1)/2 times.
Similar content being viewed by others
References
Beimel, A.: Secret-sharing schemes: a survey. In: Chee, Y.M., et al. (eds.) Coding and Cryptology, IWCC 2011 LNCS, vol. 6639, Springer (2011)
Blundo, C., De Santis, A., De Simone, R., Vaccaro, U.: Graph decomposition and secret sharing schemes. J. Crypt. 8, 39–64 (1995)
Blundo, C., De Santis, A., De Simone, R., Vaccaro, U.: Tight bounds on the information rate of secret sharing schemes. Des. Codes Crypt. 11(2), 107–110 (1997)
Brickell, E.F., Stinson, D.R.: Some improved bounds on the information rate of perfect secret sharing schemes. J. Crypt. 5, 153–166 (1992)
Csirmaz, L.: The size of a share must be large. J. Crypt. 10(4), 223–231 (1997)
Csirmaz, L.: Secret sharing schemes on graphs. Studia Math. Hung. 44, 297–306 (2007)
Csirmaz, L., Ligeti, P.: On an infinite family of graphs with information ratio 2 − 1/k. Computing 85(1), 127–136 (2009)
Csirmaz, L., Ligeti, P., Tardos, G.: Erdos-pyber theorem for hypergraphs and secret sharing. Graphs and Combinatorics 31(5), 1335–1346 (2015)
Csirmaz, L., Tardos, G.: Optimal information rate of secret sharing schemes on trees. IEEE Trans. Inf. Theory 59(4), 2527–2530 (2013)
van Dijk, M.: On the information rate of perfect secret sharing schemes. Des. Codes Crypt. 6(2), 143–169 (1995)
Erdos, P., Lovasz, L.: Problems and results on 3-chromatic hypergraphs and some related questions. In: Infinite and Finite Sets, volume 11 of Colloq. Math. Soc. J. Bolyai, pp. 609–627 (1975)
Erdos, P., Pyber, L.: Covering a graph by complete bipartite graphs. Disc. Math. 170(1-3), 249–251 (1997)
Stinson, D.R.: Decomposition construction for secret sharing schemes. IEEE Trans. Inf. Theory 40(1), 118–125 (1994)
Acknowledgements
This research was partially supported by the ÚNKP-17-4 new National Excellence Program of the Ministry of Human Capacities and the Lendület program of the Hungarian Academy of Sciences. The authors thank the members of the Crypto Group of the Rényi Institute, and especially Gábor Tardos, the numerous insightful discussions on the topic of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the Topical Collection on Special Issue: Mathematical Methods for Cryptography
Rights and permissions
About this article
Cite this article
Csirmaz, L., Ligeti, P. Secret sharing on large girth graphs. Cryptogr. Commun. 11, 399–410 (2019). https://doi.org/10.1007/s12095-018-0338-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-018-0338-x