Collaborative Visual Cryptography Schemes

A (<inline-formula> <tex-math notation="LaTeX">$k,n$ </tex-math></inline-formula>)-conventional visual cryptography (VC) scheme is designed to share one secret and each participant takes one share. When some common participants are involved in multiple VC schemes for multiple secrets, each needs to take multiple shares. This procedure needs more shares, which is inconvenient. It is desirable that the collaboration between the VC schemes can allow each common participant to keep only one share. Simply merging or gluing together two traditional (<inline-formula> <tex-math notation="LaTeX">$k_{1}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$n_{1}$ </tex-math></inline-formula>)- and (<inline-formula> <tex-math notation="LaTeX">$k_{2}$ </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">$n_{2}$ </tex-math></inline-formula>)-VC schemes, after making their pixel expansions the same, might be able to facilitate the collaboration and allow each common participant to keep only one share. But there is a security risk that when a subset of <inline-formula> <tex-math notation="LaTeX">$k_{1}$ </tex-math></inline-formula> participants are from the collection of noncommon participants, some from scheme 1 and some from scheme 2, they can reconstruct secret 1, which is inconsistent with the intention of the original scheme. Similarly, <inline-formula> <tex-math notation="LaTeX">$k_{2}$ </tex-math></inline-formula> noncommon participants could reconstruct secret 2. This shortcoming is inherited from the brute-force combination of traditional schemes. Therefore, a more sophisticated mechanism is required; this is the main task of this paper. In this paper, we first transform collaborative VC (CVC) schemes into the multiple secrets VC scheme with a general access structure. The construction of the basis matrices in CVC scheme between two VC schemes is formulated into an integer linear programming problem that minimizes the pixel expansion under the corresponding security and contrast constraints. Also the collaboration among more VC schemes is constructed. Finally, the experimental results illustrate the construction procedure of the CVC scheme and demonstrate the effectiveness of the CVC scheme.

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