Measure-independent characterization of contrast optimal visual cryptography schemes

Visual cryptography has been studied in two models and visual cryptography schemes have been evaluated using different contrast measures. Naor and Shamir introduced the deterministic model while Kafri and Keren introduced the random grid model. In the deterministic model, three different measures of contrast have been proposed, γ_ns, γ_vv and γ_es, although only γ_ns, has been thoroughly studied. Tight bounds on γ_ns are known for several classes of schemes. In the random grid model the contrast is γ_rg.In this paper we focus the attention on the deterministic model and follow a measure-independent approach, which, by using the structural properties of the schemes, enables us to provide a characterization of optimal schemes that is independent of the specific measure used to assess the contrast. In particular we characterize and provide constructions of optimal schemes for the cases of (2, n)-threshold and (n, n)-threshold schemes. Then, we apply the measure-independent results to the three measures γ_ns, γ_vv and γ_es, that have been used in the literature obtaining both new characterizations and constructions of optimal schemes as well as alternative proofs of known results.Finally we provide a connection between the deterministic and the random grid models showing that γ_es is the equivalent of γ_rg. This opens up a door between the two models which have been so far treated separately.