Lower Bounds for Visual Cryptography Schemes

A visual cryptography scheme is a method to secretly share an image among a given group of participants. In this paper, we investigate the best pixel expansion of different models of visual cryptography schemes. In this regard, we consider visual cryptography schemes satisfying the model introduced by Tzeng and Hu [13]. In such a model the recovered secret image can be darker or lighter than the background. Blundo et al. [4] introduced a lower bound for the best pixel expansion of this scheme in terms of minimal qualified subsets. We present another lower bound for the best pixel expansion of the scheme. As a corollary, we introduce a lower bound based on an induced matching of hypergraph of qualified sets. Finally, we study access structures based on graphs and we present an upper bound for the smallest pixel expansion in terms of strong chromatic index.