Secret Sharing Schemes for Dense Forbidden Graphs

A secret-sharing scheme realizes a given graph if every two vertices connected by an edge can reconstruct the secret and every independent set in the graph does not get any information about the secret. A secret-sharing scheme realizes a forbidden graph if every two vertices connected by an edge can reconstruct the secret and every two vertices which are not connected by an edge do not get any information about the secret. Similar to secret-sharing schemes for general access structures, there are gaps between the known lower bounds and upper bounds on the total share size for graphs and for forbidden graphs. Following [Beimel et al. CRYPTO 2012], our goal in this paper is to understand how the total share size increases by removing few edges from a graph that can be realized by an efficient secret-sharing scheme. We show that if a graph with n vertices contains at least $$\left {\begin{array}{c}n\\ 2\end{array}}\right -n^{1+\beta }$$ edges for some $$0 \le \beta < \frac{1}{2}$$, i.e., it is obtained by removing few edges from the complete graph, then there is a scheme realizing its forbidden graph in which the total share size is $$On^{7/6+2\beta /3}$$. This should be compared to $$On^{3/2}$$, the best known upper bound for the total share size in general forbidden graphs. Additionally, we show that a forbidden graph access structure obtained by removing few edges from an arbitrary graph G can be realized by a secret-sharing scheme with total share size of $$Om + n^{7/6+2\beta /3}$$, where m is the total size of the shares in a secret-sharing scheme realizing G and $$n^{1+\beta }$$ is the number of the removed edges. We also show that for a graph obtained by removing few edges from an arbitrary graph G with n vertices, if the chromatic number of the graph that contains the removed edges is small, then there is a fairly efficient scheme realizing the resulting graph; specifically, we construct a secret-sharing scheme with total share size of $$\tilde{O}m^{2/3} n^{2/3+2\beta /3} c^{1/3}$$, where m is the total size of the shares in a secret-sharing scheme realizing G, the value $$n^{1+\beta }$$ is an upper bound on the number of the removed edges, and c is the chromatic number of the graph of the removed edges. This should be compared to $$On^2/\log n$$, the best known upper bound for the total share size for general graphs.