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
$\binom {n}{2} - n^{1+\beta }$
edges), for some
$0 \leq \beta < 1$
, we construct a linear secret-sharing scheme realizing its forbidden graph access structure with total share size
$\tilde {O} (n^{1+\beta /2})$
. Furthermore, we construct linear secret-sharing schemes realizing these access structures in which the size of each share is
$\tilde {O} (n^{1/4+\beta /4})$
. 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
$0 \leq \beta < 1$
there exist a graph with at most
$n^{1+\beta }$
edges and a graph with at least
$\binom {n}{2}-n^{1+\beta }$
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
$0 \leq \beta < 1$
there exist a graph with at most
$n^{1+\beta }$
edges and a graph with at least
$\binom {n}{2}-n^{1+\beta }$
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).