Paper 2012/404
Secret Sharing Schemes for Very Dense Graphs
Amos Beimel, Oriol Farràs, and Yuval Mintz
Abstract
A secretsharing scheme realizes a graph if every two vertices connected by an edge can reconstruct the secret while every independent set in the graph does not get any information on the secret. Similar to secretsharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the share size for graphs. Motivated by the question of what makes a graph ``hard'' for secretsharing schemes (that is, require large shares), we study very dense graphs, that is, graphs whose complement contains few edges. We show that if a graph with $n$ vertices contains $\binom{n}{2}n^{1+\beta}$ edges for some constant $0\leq\beta <1$, then there is a scheme realizing the graph with total share size of $\tilde{O}(n^{5/4+3\beta/4})$. This should be compared to $O(n^2/\log n)$  the best upper bound known for the share size in general graphs. Thus, if a graph is ``hard'', then the graph and its complement should have many edges. We generalize these results to nearly complete $k$homogeneous access structures for a constant $k$. To complement our results, we prove lower bounds for secretsharing schemes realizing very dense graphs, e.g., for linear secretsharing schemes we prove a lower bound of $\Omega(n^{1+\beta/2})$ for a graph with $\binom{n}{2}n^{1+\beta}$ edges.
Metadata
 Available format(s)
 Publication info
 Published elsewhere. A prelimenary version of this paper appears in the Proceedings of Crypto 2012.
 Keywords
 Secret sharingshare sizegraph access structuresequivalence cover number
 Contact author(s)
 oriol farras @ urv cat
 History
 20120724: received
 Short URL
 https://ia.cr/2012/404
 License

CC BY
BibTeX
@misc{cryptoeprint:2012/404, author = {Amos Beimel and Oriol Farràs and Yuval Mintz}, title = {Secret Sharing Schemes for Very Dense Graphs}, howpublished = {Cryptology ePrint Archive, Paper 2012/404}, year = {2012}, note = {\url{https://eprint.iacr.org/2012/404}}, url = {https://eprint.iacr.org/2012/404} }