### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Improving Visual Quality for Share Images

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## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Basic Extended VC

Ordinary VC produces meaningless and noise-like shares, which makes it difficult to manage the shares if more than one shares need to be stored. It is difficult for the share manager to determine which share belongs to which secret image. Furthermore, during transmission, these meaningless shares may arouse the suspicions from potential attackers.

An extended visual cryptography (extended VC), as proposed in [8], produces meaningful shares (also called shadows). By using a tag image as the cover image, the meaningful shares are easier to store or manage. Furthermore, the meaningful shares may also act as a steganographic mechanism in visual cryptography and try to hide the fact that the transmitted image is a share from visual cryptography. As a result, the shares in extended VC are less likely to arouse suspicion from attackers if their quality is high enough. For this purpose, the share should be perceptually as close to the cover image as possible.

A $(2,2)$-threshold extended VC was first introduced in Naor and Shamir’s 1994 seminal paper [8]. Let $\mathbf{S}$ be a secret image and $\mathbf{C}{1}$ and $\mathbf{C}{2}$ be two cover images. The secret $\mathbf{S}$ is split into the two shares $\mathbf{B}{1}$ and $\mathbf{B}{2}$, where $\mathbf{B}{i}$ has the appearance of corresponding cover image $\mathbf{C}{i}$, respectively. In Naor and Shamir’s scheme, each secret pixel is expanded to a $2 \times 2$ block. Let’s focus on one secret pixel $S[i, j]$. Let $\left(c_{1}, c_{2}\right)$ be the two corresponding pixels on the cover images. To encode a white secret pixel $S[i, j]=0$, the encoder uses one of the following four basis matrices:

\begin{aligned} &\mathbf{M}{00}^{0}=\left[\begin{array}{llll} 0 & 0 & 1 & 1 \ 1 & 0 & 1 & 0 \end{array}\right], \quad \mathbf{M}{01}^{0}=\left[\begin{array}{llll} 0 & 0 & 1 & 1 \ 1 & 0 & 1 & 1 \end{array}\right] \ &\mathbf{M}{10}^{0}=\left[\begin{array}{llll} 1 & 0 & 1 & 1 \ 0 & 0 & 1 & 1 \end{array}\right], \quad \mathbf{M}{11}^{0}=\left[\begin{array}{llll} 1 & 0 & 1 & 1 \ 1 & 0 & 1 & 1 \end{array}\right] \end{aligned}
where the basis matrix $\mathbf{M}{c{1} c_{2}}^{s}$ is for the case when the secret pixel equals to $s$ and the two corresponding cover pixels equal to $c_{1}$ and $c_{2}$, respectively. Similarly, to encode a black secret pixel $S[i, j]=1$, the encoder uses one of the following four basis matrices:
$$\begin{array}{ll} \mathbf{M}{00}^{1}=\left[\begin{array}{llll} 0 & 0 & 1 & 1 \ 1 & 1 & 0 & 0 \end{array}\right], & \mathbf{M}{01}^{1}=\left[\begin{array}{llll} 0 & 0 & 1 & 1 \ 1 & 1 & 1 & 0 \end{array}\right], \ \mathbf{M}{10}^{1}=\left[\begin{array}{llll} 1 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \end{array}\right], & \mathbf{M}{11}^{1}=\left[\begin{array}{llll} 0 & 1 & 1 & 1 \ 1 & 1 & 1 & 0 \end{array}\right], \end{array}$$
After a basis matrix is chosen by the combination of $S[i, j]=s, C_{1}[i, j]=c_{1}$ and $C_{2}[i, j]=c_{2}$, the columns of this matrix are permuted. Then, the first row is reorganized into a $2 \times 2$ block and assigned to share $\mathbf{B}{1}$, while the second row is re-organized into a $2 \times 2$ block and assigned to share $\mathbf{B}{2}$. By stacking two rows of matrices $\mathbf{M}{c{1}, c_{2}}^{0}$, the blackness is 3 , while stacking two rows of matrices $\mathbf{M}{c{1}, c_{2}}^{1}$ produces blackness 4 . So, the target image will reveal the secret image. Furthermore, for $c_{i}=1$, the $i$-th row of corresponding $\mathbf{M}^{s}$ contains three black pixels, while for $c_{i}=0$, the $i$-th row of corresponding $\mathbf{M}^{s}$ contains two black pixels. So, the shares will resemble corresponding cover images.

Using the images in Fig.4.1 as secret and cover images, we get the experimental result in Fig. 4.2. While this algorithm shows acceptable result for binary cover images, it is not directly applicable to halftone image.

The binary secret image and two halftone cover images are shown in Fig. 4.3. The two share images and recovered target image are shown in Fig. 4.4. The share images show reduced contrast. Some low contrast details are also lost.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|User-Friendly Random Grid

Naor and Shamir’s scheme is not size-invariant, because the size of the shares is four times of the size of the secret and cover images. Chen proposed a random grid based scheme, friendly random grid visual secret sharing (FRGVSS), which is size-invariant [1].

Note that in an extended VC, each share has to carry two type of pixels: the secret pixels and the cover pixels. The FRGVSS algorithm uses a probabilistic approach to determine which pixel on a share is to carry the secret pixel and which pixel to carry the corresponding cover pixel. With probability $\beta$, the share pixel carries a secret pixel. When a secret pixel is chosen, an ordinary random grid algorithm is used to generate two share pixels on two share images. With probabilities $\frac{1-\beta}{2}$, a cover pixel is chosen (assuming $(2,2)$-threshold VC), then the corresponding share pixel will reflect this cover pixel, and the other share will generate a black pixel to ensure that the stacking result is black.

The parameter $0<\beta<1$ controls the tradeoff between the quality of the share images and the quality of the target image. Using a small $\beta$, only a small number of pixels on a share are used to carrier the secret pixel, and a large number of pixels are used to carry the corresponding cover pixels. So, we may expect that the share images have a good quality (higher fidelity with the cover image), while the target image has a worse quality.
Chen’s algorithm is summarized in Algorithm 10 .
The experimental results for share images and target images are shown in Fig. $4.5$ and Fig.4.6, respectively, for different $\beta$. Obviously, we observe improved quality for the share images for a smaller $\beta$. But this comes with the price of a lower contrast in target image.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Pixel Swapping Algorithm

Another effort in improving the quality of share image is Lou’s pixel swapping algorithm [7]. This algorithm is block-based and tries to re-arrange the locations

of black pixels within a small block to make the target block as close to the secret block as possible. If the secret block is a black block (i.e., containing only black pixels), then among the two corresponding share blocks, we choose the one having more black pixels to modify. The black pixels in this block is re-arranged, so that the stacking result contains as many black pixels as possible. If the secret block is a white block (i.e., containing only white pixels), then among the two corresponding share blocks, we choose the one having more black pixels to modify. The black pixels in this block is re-arranged, so that the stacking result contains as many white pixels as possible. Ōbviously, this re-arrangement only guarantees best-effort approximation, so the target image is only partially reconstructed.

The pixels are only moved around in a small block and the proportion of black pixels is not changed, so the share image has a good visual quality. However, only the secret block with all black pixels and the secret block with all white pixels are approximated with best effort. From the stacking result, we may see interferences from the cover image. Lou’s algorithm is summarized in Algorithm $11 .$

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Basic Extended VC

[8] 中提出的扩展视觉密码学（扩展 VC）产生有意义的共享（也称为影子）。通过使用标签图像作为封面图像，有意义的共享更易于存储或管理。此外，有意义的共享也可以作为视觉密码学中的一种隐写机制，并试图隐藏传输的图像是视觉密码学的共享的事实。因此，扩展 VC 中的份额如果质量足够高，则不太可能引起攻击者的怀疑。为此，分享应该在感知上尽可能接近封面图像。

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|User-Friendly Random Grid

Naor 和 Shamir 的方案不是大小不变的，因为共享的大小是秘密和覆盖图像大小的四倍。Chen 提出了一种基于随机网格的方案，友好的随机网格视觉秘密共享 (FRGVSS)，它是大小不变的 [1]。

Chen 的算法总结在算法 10 中。

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