Bounds for visual cryptography schemes

In this paper, we investigate the best pixel expansion of various models of visual cryptography schemes. In this regard, we consider visual cryptography schemes introduced by Tzeng and Hu (2002) [13]. In such a model, only minimal qualified sets can recover the secret image and the recovered secret image can be darker or lighter than the background. Blundo et al. (2006) [4] introduced a lower bound for the best pixel expansion of this scheme in terms of minimal qualified sets. 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, for the best pixel expansion of the aforementioned model and the traditional model of visual cryptography scheme realized by basis matrices. 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.