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Linear algebraic techniques to construct monochrome visual cryptographic schemes for general access structure and its applications to color images

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Abstract

Though the monochrome (black and white) visual cryptography has a very rich literature, a very few papers have been published for the construction of general access structure. In this paper we put forward a method of construction of a strong monochrome visual cryptographic scheme (VCS) for general access structure using linear algebra. As a particular case of general access structure, \((k,n)\)-VCS for \(2 \le k \le n\) is obtained. The \((n,n)\)-VCS obtained from the scheme attains the optimal pixel expansion as well as optimal relative contrast. We provide an efficient construction of \((n-1,n)\)-VCS. We further extend our monochrome VCS to color VCS for restricted access structures. Finally, we provide some interesting examples that will lead to some future research directions in the area of VCS.

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Acknowledgments

Firstly, we would like to thank the anonymous reviewers for their important and helpful comments. We are also grateful to Professor B K Roy and Professor T K Dutta of Indian Statistical Institute, Kolkata for their important discussions which helped us to a great extent in writing this paper.

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Authors and Affiliations

  1. Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata, 700019, India

    Avishek Adhikari

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  1. Avishek Adhikari
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Correspondence to Avishek Adhikari.

Additional information

Communicated by C. Blundo.

Appendices

Appendix 1

In this section we are going to compare our scheme for \((n-1,n)\) threshold case with the schemes proposed in [14] and [10] in terms of pixel expansion and relative contrasts. Let \((m_D, \alpha _D\)), \((m_B, \alpha _B\)) and \((m, \alpha \)) denote respectively the (pixel expansion, relative contrast) pairs of the schemes proposed in [14], Table 1], [10, Lemma 4.4] and this paper respectively. The Table 1 provides a comparison of these parameters.

Table 1 Comparison of pixel expansions and relative contrasts of different VCS
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1.1 \((5,6)\)-VCS

The two matrices represent basis matrices for the (5,6)-VCS obtained using Theorem 2.4.

$$\begin{aligned} S^0= \left[ \begin{array}{lll} \leftarrow \text{ Col } \text{1 } \text{ to } \text{16 } \rightarrow &{} \leftarrow \text{ Col } \text{17 } \text{ to } \text{32 } \rightarrow &{} \leftarrow \text{ Col } \text{33 } \text{ to } \text{48 } \rightarrow \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 \\ \end{array} \right] . \end{aligned}$$
$$\begin{aligned} S^1= \left[ \begin{array}{lll} \leftarrow \text{ Col } \text{1 } \text{ to } \text{16 } \rightarrow &{} \leftarrow \text{ Col } \text{17 } \text{ to } \text{32 } \rightarrow &{} \leftarrow \text{ Col } \text{33 } \text{ to } \text{48 } \rightarrow \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 \end{array} \right] . \end{aligned}$$

Both \(S^0\) and \(S^1\) contain some common columns. To obtain better pixel expansion, we delete the common columns. In Table 2, we point out the column numbers to locate the common columns. For example, the column 2 in \(S^0\) is same as column 40 in \(S^1\) and so on.

Table 2 List of 18 deleted common columns from \(S^{0} \) and \(S^{1}\) for (5, 6)-VCS
Full size table

1.2 \((6,7)\)-VCS

The following two matrices, \(S^0=S^0_1 || S^0_2\) and \(S^1=S^1_1 || S^1_2\) represent basis matrices for the (6,7)-VCS obtained using Theorem 2.4.

$$\begin{aligned} S^0_1= \left[ \begin{array}{llll} \leftarrow \text{ Col } \text{1 } \text{ to } \text{16 } \rightarrow &{} \leftarrow \text{ Col } \text{17 } \text{ to } \text{32 } \rightarrow &{} \leftarrow \text{ Col } \text{33 } \text{ to } \text{48 } \rightarrow &{} \leftarrow \text{ Col } \text{49 } \text{ to } \text{64 } \rightarrow \\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 \\ \end{array} \right] \!. \end{aligned}$$
$$\begin{aligned} S^0_2= \left[ \begin{array}{llll} \leftarrow \text{ Col } \text{65 } \text{ to } \text{80 } \rightarrow &{} \leftarrow \text{ Col } \text{81 } \text{ to } \text{96 } \rightarrow &{} \leftarrow \text{ Col } \text{97 } \text{ to } \text{112 } \rightarrow &{} \leftarrow \text{ Col } \text{113 } \text{ to } \text{128 } \rightarrow \\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 \\ 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1\\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1\\ \end{array} \right] . \end{aligned}$$
$$\begin{aligned} S^1_1= \left[ \begin{array}{llll} \leftarrow \text{ Col } \text{1 } \text{ to } \text{16 } \rightarrow &{} \leftarrow \text{ Col } \text{17 } \text{ to } \text{32 } \rightarrow &{} \leftarrow \text{ Col } \text{33 } \text{ to } \text{48 } \rightarrow &{} \leftarrow \text{ Col } \text{49 } \text{ to } \text{64 } \rightarrow \\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 \\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1\\ \end{array} \right] . \end{aligned}$$
$$\begin{aligned} S^1_2= \left[ \begin{array}{llll} \leftarrow \text{ Col } \text{65 } \text{ to } \text{80 } \rightarrow &{} \leftarrow \text{ Col } \text{81 } \text{ to } \text{96 } \rightarrow &{} \leftarrow \text{ Col } \text{97 } \text{ to } \text{112 } \rightarrow &{} \leftarrow \text{ Col } \text{113 } \text{ to } \text{128 } \rightarrow \\ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 \\ 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 &{} 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 &{} 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 &{} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 \\ 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 &{} 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 \\ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 &{} 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 \\ 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 &{} 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 \\ 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 &{} 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1\\ \end{array} \right] . \end{aligned}$$

In Table 3, we point out the column numbers to locate the common columns.

Table 3 List of 58 deleted common columns from \(S^0\) and \(S^1\) for (6, 7)-VCS
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Table 4 Comparison of pixel expansions of different access structures having at most four participants as given in Table 1 of [5]
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Appendix 2

Example 7.1

Let us consider the access structure with \(\Gamma _0=\{ \{1,2 \}, \{1,3 \}, \{1,4\},\{2,3,4\}\}\) on the set \(\mathcal P =\{1,2,3,4 \}\) of four participants. We construct the basis matrices by using Theorem 2.4 and illustrate the scheme through Figs. 1, 2, 3, 4, 5, 6, 7, and 8.

Fig. 1
figure 1

The secret image

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Fig. 2
figure 2

Share 1

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Fig. 3
figure 3

Share 2

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Fig. 4
figure 4

Share 3

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Fig. 5
figure 5

Share 4

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Fig. 6
figure 6

Superimposed image share 1 + share 2

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Fig. 7
figure 7

Share 2 + share 3 + share 4 Superimposed image

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Fig. 8
figure 8

Share 2 + share 3 No information

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Appendix 3

Example 8.1

Let us consider the access structure with \(\Gamma _0=\{ \{1,2 \}, \{1,3 \}\}\) on the set \(\mathcal P =\{1,2,3 \}\) of three participants with color set \(\mathcal C =\{ C,Y,G\}\). We construct the basis matrices by using Theorems 2.4 and 4.2 and illustrate the scheme through Figs. 9, 10, 11, 12, 13, 14, and 15.

Fig. 9
figure 9

The secret image

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Fig. 10
figure 10

Share 1

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Fig. 11
figure 11

Share 2

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Fig. 12
figure 12

Share 3

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Fig. 13
figure 13

Superimposed image share 1 + share 2

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Fig. 14
figure 14

Superimposed image share 1 + share 3

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Fig. 15
figure 15

Superimposed image share 2 + share 3

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Table 5 Comparison of contrast ratios for different access structures having at most four participants as given in Table 1 of [5]
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Adhikari, A. Linear algebraic techniques to construct monochrome visual cryptographic schemes for general access structure and its applications to color images. Des. Codes Cryptogr. 73, 865–895 (2014). https://doi.org/10.1007/s10623-013-9832-5

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