Lower Bound for Visual Cryptography Schemes

For a given visual cryptography scheme, it is possible to present a basis matrices for it and most of constructions are based on basis matrices. In this paper we introduce a lower bound for the pixel expansion of visual cryptography schemes with basis matrices. To make the main theorem more flexible, we will introduce a lower bound based on induced matchings of hypergraph of qualified sets. As an application, we present an algebraic proof for the fact that the pixel expansion of basis matrices of any k out of k scheme is at least 2. In the sequel, we present a lower bound for the pixel expansion of a given graph access structure in term of maximum number of edges in an induced matching. Finally, we show that the minimum pixel expansion of basis matrices of graph access structure Pn is exactly ⌈ n+1 2 ⌉ and this shows the lower bound mentioned in the main theorem is sharp.