## 2 Preliminaries
### A \((k,n)\)-Ovc Scheme  
When a binary secret image is shared in a \((k,n)\) VC scheme, each pixel in the secret image is encoded into \(m\) subpixels in each one of the \(n\) shadow images that are printed on transparencies. The encoding rule is represented by two \(n\times m\) Boolean matrices \(B_{1}\) and \(B_{0}\), whose elements are \(1\) and \(0\) denoting black and white subpixels, respectively. A white pixel lets through the light, and a black pixel blocks the light. Two families of basis matrices \(C_{1}\) (resp. \(C_{0}\)) include all matrices obtained by permuting the columns in \(B_{1}\) (resp. \(B_{0}\)). When a black (resp. white) pixel is taken from the secret image, one row of the basis matrix \(B_{1}\) (resp. \(B_{0}\)) in the family \(C_{1}\) (resp. \(C_{0}\)) is distributed to a related shadow image. Stacking shadows \(i_{1},\ldots,i_{r}\) together, we obtain the OR-ed (\"OR\" operation) \(m\)-vector being \(V=\oplus(i_{1},\ldots,i_{r})\) and \(``\oplus\"\) denotes the \"OR\" operation for each secret pixel. The gray-level of each secret pixel in the recovered secret image is proportional to Hamming weight (the number of \(1\)s in the vector \(V\)) \(H(V)\). Let \(\oplus(B_{i}|U)\) denote the Boolean OR-ing of the row \(j\), \(j\in U\) of \(B_{i}\), \(i=0,1\). Let \(|U|\) denote the cardinality of participant set \(U\). The formal definition of \((k,n)\)-OVC scheme is given below [26].  
**Definition 1** ([26]).: _With a set of two \(n\times m\) Boolean basis matrices, \(B_{0}\) and \(B_{1}\), the result of a column permutation of \(B_{i}\) defines the color of the \(m\) subpixels in each one of the \(n\) shadows when encoding a pixel \(i\in\{0,1\}\). Such basis matrices constitute a \((k,n)\)-OVC scheme if the following conditions are met._  
**(1)**: _Contrast condition:_ \(H(\oplus(B_{1}|U))\geq h\) _and_ \(H(\oplus(B_{0}|U))\leq l\) _for_ \(|U|\geq k\)_, where_ \(0\leq l<h\leq m\)_._
**(2)**: _Security condition:_ \(H(\oplus(B_{1}|U))=H(\oplus(B_{0}|U))\) _for_ \(|U|\leq k-1\)_._  
Condition (1) is called contrast condition: it requires that the secret image can be visually recognized by human eyes as the different Hamming weights of black and white colors. The contrast is generally defined as \(\alpha=(H(\oplus(B_{1}|U))-H(\oplus(B_{0}|U)))/m\), and it uses the relative difference of the gray levels of the two pixel to represent the visual quality of the recovered secret image. Condition (2) shows the perfect security of a \((k,n)\)-OVC scheme such that any less than \(k\) shadows obtains nothing about the secret images. \(m\) is the pixel expansion.  
The following is a more intuitive description of a \((k,n)\)-OVC scheme through a simple example of a \((2,2)\)-OVC scheme.  
**Example 1**.: _The form of \(B_{0}\) and \(B_{1}\) is as follows when \(n=2\) and \(k=2\),_  
\[B_{0}=\left[\begin{array}{cc}1&0\\ 1&0\end{array}\right]\ \ \text{and}\ \ B_{1}=\left[\begin{array}{cc}1&0\\ 0&1\end{array}\right].\]  
_The encoding and decoding process of the shadows is shown below in Table 1._  
The white (resp. black) subpixels in pre-pixel are expressed by \(\phi W\) (resp. \(\varphi B\)); here, \(\phi\) (resp. \(\varphi\)) is the number of white (resp. black) subpixels in a pixel. It is observed that the \((2,2)\)-VC scheme has perfect security through Example 1. When stacking less than two shadows, all black and white pixels are encoded as 1B1W or 1W1B. We have 2B0W and 1B1W for black and white secret pixels when stacking two shadows. The contrast is \(\alpha=1/2\).