On the robustness of visual cryptographic schemes

In this paper, we consider the robustness of a special type of secret sharing scheme known as visual cryptographic scheme in which the secret reconstruction is done visually without any mathematical computation unlike other secret sharing schemes. Initially, secret sharing schemes were considered with the presumption that the corrupted participants involved in a protocol behave in a passive manner and submit correct shares during the reconstruction of secret. However, that may not be the case in practical situations. A minimal robust requirement, when a fraction of participants behave maliciously and submit incorrect shares, is that, the set of all shares, some possibly corrupted, can recover the correct secret. Though the concept of robustness is well studied for secret sharing schemes, it is not at all common in the field of visual cryptography. We, for the first time in the literature of visual cryptography, formally define the concept of robustness and put forward (2, n)-threshold visual cryptographic schemes that are robust against deterministic cheating. In the robust secret sharing schemes it is assumed that the number of cheaters is always less than the threshold value so that the original secret is not recovered by the coalition of cheaters only. In the current paper, We consider three different scenarios with respect to the number of cheaters controlled by a centralized adversary. We first consider the existence of only one cheater in a (2, n)-threshold VCS so that the secret image is not recovered by the cheater. Next we consider two different cases, with number of cheaters being greater than 2, with honest majority and without honest majority.