Secret Sharing Schemes for Very Dense Graphs

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 (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 n 2 − n 1+β edges for some constant 0 ≤ β < 1, then there is a scheme realizing the graph with total share size of O(n 5/4+3β/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 secret-sharing schemes realizing very dense graphs, e.g., for linear secret-sharing schemes we prove a lower bound of Ω(n 1+β/2 ) for a graph with n 2 − n 1+β edges.

On the Cryptographic Complexity of the Worst Functions

“…The best lower bound for total share size required to realize a graph access structure by a linear secret-sharing scheme is Ω(N 3/2 ) [7]. The problem of secret sharing for dense graphs was studied in [8]. Additional references on secret sharing of graph access structures can be found in [8].…”

“…The problem of secret sharing for dense graphs was studied in [8]. Additional references on secret sharing of graph access structures can be found in [8].…”

Recursive General Secret Sharing Scheme Based on Authorized Subsets

“…Farrás and Padró formalized the concept of hierarchical access structure [7]. Secret sharing schemes based on graph access structures were also proposed [8]- [10]. These schemes obtain the optimal information rates for some access structures, but these schemes cannot be applied to many access structures.…”

Secret sharing on the d-dimensional cube

“…It is an open problem to narrow this gap. It is interesting to note that the lower bound comes from a sparse graph (the maximal degree is o(n)), while the upper bound requires dense (Ω(n) average degree), but not very dense (n − o(n) average degree) graphs, see [2].…”