Visual Cryptography for General Access Structures

A visual cryptography scheme for a set ℘ of n participants is a method of encoding a secret image SI into n shadow images called shares, where each participant in ℘ receives one share. Certain qualified subsets of participants can "visually" recover the secret image, but other, forbidden, sets of participants have no information (in an information-theoretic sense) on SI. A "visual" recovery for a set X ⊆ ℘ consists of xeroxing the shares given to the participants in X onto transparencies, and then stacking them. The participants in a qualified set X will be able to see the secret image without any knowledge of cryptography and without performing any cryptographic computation. In this paper we propose two techniques for constructing visual cryptography schemes for general access structures. We analyze the structure of visual cryptography schemes and we prove bounds on the size of the shares distributed to the participants in the scheme. We provide a novel technique for realizing k out of n threshold visual cryptography schemes. Our construction for k out of n visual cryptography schemes is better with respect to pixel expansion than the one proposed by M. Naor and A. Shamir (Visual cryptography, in "Advances in Cryptology - Eurocrypt '94" CA. De Santis, Ed.), Lecture Notes in Computer Science, Vol. 950, pp. 1-12, Springer-Verlag, Berlin, 1995) and for the case of 2 out of n is the best possible. Finally, we consider graph-based access structures, i.e., access structures in which any qualified set of participants contains at least an edge of a given graph whose vertices represent the participants of the scheme.