Bounds for Visual Cryptography Schemes

TL;DR

This paper explores bounds on pixel expansion in various visual cryptography schemes, providing new lower bounds and analyzing access structures to optimize secret image sharing.

Contribution

It introduces new lower bounds for pixel expansion based on hypergraph matchings and access structures, advancing understanding of scheme efficiency.

Findings

New lower bounds for pixel expansion in visual cryptography schemes

Analysis of hypergraph matchings for scheme optimization

Upper bounds for pixel expansion using strong chromatic index

Abstract

In this paper, we investigate the best pixel expansion of the various models of visual cryptography schemes. In this regard, we consider visual cryptography schemes introduced by Tzeng and Hu [13]. In such a model, only minimal qualified sets can recover the secret image and that 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 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 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.