**Secret Sharing Schemes for
Very Dense Graphs**

Amos Beimel (

Yuval Mintz (

Oriol Farras (Universitat Rovira i

**Abstract:**

A secret-sharing 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 secret-sharing 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 secret-sharing schemes, 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 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 secret-sharing schemes realizing very dense graphs, e.g., for
linear secret-sharing schemes we prove a lower bound of $\Omega(n^{1+\beta/2})$.