New insight into linear algebraic technique to construct visual cryptography scheme for general access structure

Abstract

The most essential advantage of applying linear algebra to construct visual cryptography scheme (VCS) lies in that it only requires solving linear equations in the construction of initial basis matrices, which are the basis matrices before removing the common columns. In this paper, we give some new insight into linear algebraic technique to construct VCS, where we can take more equations simultaneously. Then based on this knowledge, we propose a construction of VCS for general access structure. The construction is efficient in the sense that it gets the smallest initial pixel expansion compared with some well-known constructions. At the same time, by using the technique of deleting common columns from the initial basis matrices, the proposed construction achieves the optimal pixel expansions in most cases according to our experimental results. However, finding exact number of common columns in the initial basis matrices is a challenging issue. Then we deal with this issue and find out that the exact number of common columns is n − 2 for (2, n) threshold access structures. Finally, we provide some future research directions in the algebraic aspect of VCS.

Appendices

Appendix A

Let us first group the set Γ₀ = {{1, 2}, {1, 3}, {1, 4}} into two collections, namely ${{\Gamma }^{1}_{0}}=\{\{1,2\},\{1,3\}\}$ and ${{\Gamma }^{2}_{0}}=\{\{1,4\}\}$. For ${{\Gamma }^{1}_{0}}$, consider the following two systems of two linear equations over the binary field:

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=0\\ x_{1}+x_{3}=0 \end{array} \right. $$

(15)

and

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=1\\ x_{1}+x_{3}=1 \end{array} \right. $$

(16)

Let ${S^{0}_{1}}$ and ${S^{1}_{1}}$ be the Boolean matrices whose columns are just all possible solutions of (15) and (16), respectively. Thus, ${S^{0}_{1}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 1\\ 0 & 1\\ 0 & 0 \end {array} \right ]$ and ${S^{1}_{1}}=\left [ \begin {array}{cc} 0 & 1\\ 1 & 0\\ 1 & 0\\ 0 & 0 \end {array} \right ]$. Note that the fourth participant is non-essential for the strong access structure determined by ${{\Gamma }^{1}_{0}}$, so we assign it the values (00).

For ${{\Gamma }^{2}_{0}}$, consider the following two systems of one linear equations over the binary field:

$$ \begin{array}{l} x_{1}+x_{4}=0 \end{array} $$

(17)

and

$$ \begin{array}{l} x_{1}+x_{4}=1 \end{array} $$

(18)

Let ${S^{0}_{2}}$ and ${S^{1}_{2}}$ be the Boolean matrices whose columns are just all possible solutions of (17) and (18), respectively. Thus, ${S^{0}_{2}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 0\\ 0 & 0\\ 0 & 1 \end {array} \right ]$ and ${S^{1}_{2}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 0\\ 0 & 0\\ 1 & 0 \end {array} \right ]$. Note that the second and third participants are non-essential for the strong access structure determined by ${{\Gamma }^{2}_{0}}$, so we assign both of them the values (00).

Finally, by Lemma 2 we construct a (Γ _{Q u a l} , Γ _{F o r b} )-VCS on a set of four participants having Γ₀ = {{1, 2}, {1, 3}, {1, 4}} with the basis matrices $S^{0}=\left [ \begin {array}{cccc} 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end {array} \right ]$ and $S^{1}=\left [ \begin {array}{cccc} 0 & 1 & 0 & 1\\ 1 & 0 & 0 & 0\\ 1 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 \end {array} \right ]$. It gives the pixel expansion 4.

Appendix B

Let us first group the collection Γ₀={{1,2},{1,3},{2,3},{1,4}} into two collections, namely ${{\Gamma }^{1}_{0}}=\{\{1,2\},\{1,3\}\}$ and ${{\Gamma }^{2}_{0}}=\{\{2,3\},\{1,4\}\}$. For ${{\Gamma }^{1}_{0}}$, consider the following two systems of two linear equations over the binary field:

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=0\\ x_{1}+x_{3}=0 \end{array} \right. $$

(19)

and

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=1\\ x_{1}+x_{3}=1 \end{array} \right. $$

(20)

Let ${S^{0}_{1}}$ and ${S^{1}_{1}}$ be the Boolean matrices whose columns are just all possible solutions of (19) and (20), respectively. Thus, ${S^{0}_{1}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 1\\ 0 & 1\\ 0 & 0 \end {array} \right ]$ and ${S^{1}_{1}}=\left [ \begin {array}{cc} 0 & 1\\ 1 & 0\\ 1 & 0\\ 0 & 0 \end {array} \right ]$. Note that the fourth participant is non-essential for the strong access structure determined by ${{\Gamma }^{1}_{0}}$, so we assign it the values (00).

For ${{\Gamma }^{2}_{0}}$, consider the following two systems of two linear equations over the binary field:

$$ \begin{array}{l} x_{2}+x_{3}=0\\ x_{1}+x_{4}=0 \end{array} $$

(21)

and

$$ \begin{array}{l} x_{2}+x_{3}=1\\ x_{1}+x_{4}=1 \end{array} $$

(22)

Let ${S^{0}_{2}}$ and ${S^{1}_{2}}$ be the Boolean matrices whose columns are just all possible solutions of (21) and (22), respectively. Thus, ${S^{0}_{2}}=\left [ \begin {array}{cccc} 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 1 \end {array} \right ]$ and ${S^{1}_{2}}=\left [ \begin {array}{cccc} 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0\\ 1 & 1 & 0 & 0 \end {array} \right ]$.

Finally, by Lemma 2 we construct a (Γ _{Q u a l} , Γ _{F o r b} )-VCS on a set of four participants having Γ₀={{1,2},{1,3},{2,3},{1,4}} with the basis matrices $S^{0}=\left [ \begin {array}{cccccc} 0 & 1 & 0 & 0 & 1 & 1\\ 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 0 & 1 & 1 \end {array} \right ]$ and $S^{1}=\left [ \begin {array}{cccccc} 0 & 1 & 0 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & 0 & 1\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 1 & 0 & 0 \end {array} \right ]$. It gives the pixel expansion 5, obtained by deleting the common column (0110)^T from the initial basis matrices S ⁰ and S ¹.

Appendix C

Let us first group the set Γ₀={{1,2},{1,3},{2,3},{1,4}} into two collections, namely ${{\Gamma }^{1}_{0}}=\{\{1,2\},\{2,3\}\}$ and ${{\Gamma }^{2}_{0}}=\{\{1,3\},\{1,4\}\}$. For ${{\Gamma }^{1}_{0}}$, consider the following two systems of two linear equations over the binary field:

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=0\\ x_{2}+x_{3}=0 \end{array} \right. $$

(23)

and

$$ \left\{ \begin{array}{l} x_{1}+x_{2}=1\\ x_{2}+x_{3}=1 \end{array} \right. $$

(24)

Let ${S^{0}_{1}}$ and ${S^{1}_{1}}$ be the Boolean matrices whose columns are just all possible solutions of (23) and (24), respectively. Thus, ${S^{0}_{1}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 1\\ 0 & 1\\ 0 & 0 \end {array} \right ]$ and ${S^{1}_{1}}=\left [ \begin {array}{cc} 1 & 0\\ 0 & 1\\ 1 & 0\\ 0 & 0 \end {array} \right ]$. Note that the fourth participant is non-essential for the strong access structure determined by ${{\Gamma }^{1}_{0}}$, so we assign it the values (00).

For ${{\Gamma }^{2}_{0}}$, consider the following two systems of two linear equations over the binary field:

$$ \begin{array}{l} x_{1}+x_{3}=0\\ x_{1}+x_{4}=0 \end{array} $$

(25)

and

$$ \begin{array}{l} x_{1}+x_{3}=1\\ x_{1}+x_{4}=1 \end{array} $$

(26)

Let ${S^{0}_{2}}$ and ${S^{1}_{2}}$ be the Boolean matrices whose columns are just all possible solutions of (25) and (26), respectively. Thus, ${S^{0}_{2}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 0\\ 0 & 1\\ 0 & 1 \end {array} \right ]$ and ${S^{1}_{2}}=\left [ \begin {array}{cc} 0 & 1\\ 0 & 0\\ 1 & 0\\ 1 & 0 \end {array} \right ]$. Note that the second participant is non-essential for the strong access structure determined by ${{\Gamma }^{2}_{0}}$, so we assign it the values (00).

Finally, by Lemma 2 we construct a (Γ _{Q u a l} , Γ _{F o r b} )-VCS on a set of four participants having Γ₀ = {{1, 2}, {3, 4}, {2, 3}, {2, 4}} with the basis matrices $S^{0}=\left [ \begin {array}{cccc} 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 0\\ 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 1 \end {array} \right ]$ and $S^{1}=\left [ \begin {array}{cccc} 1 & 0 & 0 & 1\\ 0 & 1 & 0 & 0\\ 1 & 0 & 1 & 0\\ 0 & 0 & 1 & 0 \end {array} \right ]$. It gives the pixel expansion 4.