Linear Secret-Sharing Schemes for Forbidden Graph Access Structures

A secret-sharing scheme realizes the forbidden graph access structure determined by a graph G = (V, E) if the parties are the vertices of the graph and the subsets that can reconstruct the secret are the pairs of vertices in E (i.e., the edges) and the subsets of at least three vertices. Secret-sharing schemes for forbidden graph access structures defined by bipartite graphs are equivalent to conditional disclosure of secrets (CDS) protocols. We study the complexity of realizing a forbidden graph access structure by linear secret-sharing schemes, which are schemes in which the secret can be reconstructed from the shares by a linear mapping. We provide efficient constructions and lower bounds on the share size of linear secret- sharing schemes for sparse and very dense graphs, closing the gap between upper and lower bounds. Given a sparse (resp. very dense) graph with n vertices and at most n(1+beta) edges (resp. at least ((n)(2)) - n(1+beta) edges), for some 0 <= beta < 1, we construct a linear secret-sharing scheme realizing its forbidden graph access structure with total share size <(O)over tilde>(n(1+beta/2)). Furthermore, we construct linear secretsharing schemes realizing these access structures in which the size of each share is (O) over tilde (n(1/4+beta/4)). We also provide constructions achieving different trade-offs between the size of each share and the total share size. We prove that almost all forbidden graph access structures require linear secret-sharing schemes with total share size Omega(n(3/2)); this shows that the construction of Gay, Kerenidis, and Wee [CRYPTO 2015] is optimal. Furthermore, we show that for every 0 <= beta < 1 there exist a graph with at most n(1+beta) edges and a graph with at least ((n)(2)) - n(1+beta) edges such that the total share size in any linear secret-sharing scheme realizing the associated forbidden graph access structures is Omega(n(1+beta/2)). Finally, we show that for every 0 <= beta < 1 there exist a graph with at most n(1+beta) edges and a graph with at least ((n)(2)) - n(1+beta) edges such that the size of the share of at least one party in any linear secret-sharing scheme realizing these forbidden graph access structures is Omega(n(1/4+beta/4)). This shows that our constructions are optimal (up to poly-logarithmic factors).