Abstract
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.
Avishek Adhikari—Research is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India, Grant No. 2/48(10)/2013/NBHM(R.P.)/R&D II/695. The authors are also thankful to DST, Govt. of India and JSPS, Govt. of Japan for providing partial support for this collaborative research work under India Japan Cooperative Science Programme (vide Memo no. DST/INT/JSPS/P-191/2014 dated May 27, 2014.
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Appendix
Appendix
For illustration we provide the following Figures. The Fig. 1 shows the secret black and whie image while the Fig. 2 shows three shares S1, S2, S3 obtained through the construction method of Naor-Shamir [16] for a (2, 3)-VCS and MS3 is the fake but valid share produced by the third participant. Stacking of the shares are shown in Fig. 3. It shows that while superimposing the Shares 1, 2 and 3, we get the secret image back. However, if we replace S3 by MS3, we do not get the secret back. Figure 4 considers the same access structure but with our modified basis matrix construction method. Note that in the modified scheme, the superimposition of the two shares S1 and S2 over the modified share MS3 provides the secret even though MS3 produces fake but valid share.
Secret image
4 Shares: S1, S2, S3, MS3
Superimposed images
Robust VCS
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Dutta, S., Roy, P.S., Adhikari, A., Sakurai, K. (2017). On the Robustness of Visual Cryptographic Schemes. In: Shi, Y., Kim, H., Perez-Gonzalez, F., Liu, F. (eds) Digital Forensics and Watermarking. IWDW 2016. Lecture Notes in Computer Science(), vol 10082. Springer, Cham. https://doi.org/10.1007/978-3-319-53465-7_19
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