### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考| Security Issue in Visual Cryptography

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• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Strong Security and Weak Security

As a cryptography scheme, the security requirement is usually mandatory. So, when designing a VC algorithm, one usually tries to maximize the contrast, under the constraint of security. Some researchers also suggest that there are tradeoffs between security, contrast and pixel expansion. If we relax the requirement on one aspect, then maybe it is possible to improve the performance of the other two. For example, if we relax requirement on security, then it is possible to improve contrast and reduce pixel expansion $[6,10,12]$.

For a VC scheme in its strict sense, the shares are usually printed on transparencies and decoding is realized by stacking, and no computation is required for decoding. In the definition of VC (Definition 2.1), for $q<k$ shares, the two sets of sub-matrices $\hat{\mathrm{C}}{0}$ and $\hat{\mathrm{C}}{1}$ are equivalent. So, an attacker, after obtaining $\hat{\mathrm{C}}{0}$ and $\hat{\mathrm{C}}{1}$, cannot tell if

$s=0$ or $s=1$, no matter the computational abilities he may have at his disposal. This is called strict sense security or unconditional security [6, 10]. For this security, we assume that the attacker has infinite computational abilities.

However, considering the media of the shares and the decoding mechanism, it is reasonable to assume that if the attacker only uses his vision system to find clue of secret from the $q<k$ shares, then the ‘computation’ devices are stacking operation and HVS. Then if from the stacking result of the $q<k$ rows of the matrices, one cannot infer $s$, our algorithm is safe under these assumptions. We call this the weak security.

Weak security was proposed by Liu [12] and Iwamoto [10] independently for different types of VC systems. Liu’s work focuses on block encoding approach to size-invariant VC, while Iwamoto’s works focus on size-expanded VC (deterministic VC) for color image. A key conclusion from their work is that, by relaxing the security level to weak security, it is possible to improve the quality of the target image and reduce the pixel expansion.
In what follows, we introduce three security issues:

1. Iwamoto’s weak security.
2. Liu’s weak security.
3. Replacement attack.
In order to introduce the concept of weak security, we need to formalize some operations on the basis matrices: row restriction and row stacking, and introduce the concept of equivalence between two sets of matrices [10].

For $(k, n)$-threshold scheme, a set of participants can be represented by $\mathrm{P}=$ $\left{i_{1}, \ldots, i_{q}\right} \subset{1, \ldots, n}$. Given a basis matrix $\mathbf{B} \in \mathbb{Z}{2}^{n \times m}$, one can make another matrix by restricting the rows of $\mathbf{B}$ to the rows specified in set $P$. This operation is denoted by $$\hat{\mathbf{B}}=\mathbf{B} \llbracket \mathrm{P} \rrbracket$$ with $$\hat{B}[\ell, j]=B\left[i{\ell}, j\right],$$
where $\ell \in{1, \ldots, q}, i_{\ell} \in \mathrm{P}$, and $j \in{1, \ldots, n}$.
Another formal operation introduced by Iwamoto is stacking of rows of a matrix. Let a matrix $\mathbf{B}$ be partitioned as
$$\mathbf{B}=\left[\begin{array}{c} \mathbf{b}{1} \ \vdots \ \mathbf{b}{n} \end{array}\right] \in \mathbb{Z}{2}^{n \times m}$$ Then, the stacking of rows of $\mathbf{B}$ is denoted as $$\eta(\mathbf{B})=\mathbf{b}{1} \vee \ldots \vee \mathbf{b}_{n},$$

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Introduction to Digital Halftoning

Most printing devices and some display devices can only render limited number of colors [9]. For example, a typical laser printer can only output a black dot or ‘no dot’ at a time. In order to print grayscale images, we must quantize the input color to coarser scales, such that the perceived color is still similar to the original color when observed by human eyes.

In this chapter, we assume that the input image is a grayscale image and each pixel is normalized to the range $[0,1]$. If the pixel is quantized, 8-bit quantization is assumed. The printer is assumed to be able to produce only two colors: black (black dot) and white (no dot).

The image quantization problem then can be stated as follows: Given an input grayscale image $x[i, j]$, the quantizer produces a binary image $y[i, j] \in{0,1}$ such that it is visually similar to the input image $x[i, j]$ when viewing from sufficient distance. Let $\mathcal{H}$ be a system representing the human visual perception, then the digital halftoning problem can be formulated as an optimization problem:
$$\min {\mathbf{y} \in \mathbb{Z}{2}^{M \times N}} \mathcal{H}{\mathbf{x}-\mathbf{y}}$$
where the images are of size $M \times N$. Solving this optimization problem directly is usually impractical due to the high dimensional searching space that has $2^{M \times N}$ feasible solutions.

Many heuristic approaches are proposed for solving the halftoning problem in (3.1), including constant threshold bi-level quantization, ordered dithering, error diffusion and direct binary search (DBS).

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Bi-level Quantization

The simplistic approach is to quantize each sample independently using a bi-level quantizer [6]:
$$y[i, j]= \begin{cases}1, & \text { if } x[i, j]>T, \ 0, & \text { otherwise }\end{cases}$$
where $T$ is a constant threshold. For this quantizer, the two reconstruction levels are $\left(y_{0}, y_{1}\right)=(0,1)$ and the decision boundaries are $\left(d_{0}, d_{1}, d_{2}\right)=(0, T, 1)$. The threshold $T$ can be designed by minimizing the mean-squared error:
$$\operatorname{MSE}(T)=\sum_{i=0}^{1} \int_{d_{i}}^{d_{i+1}}\left(x-y_{i}\right)^{2} p_{X}(x) d x$$
where $p_{X}(x)$ is the PDF of the samples of the input image $x$. It can be modeled as uniform distribution over the interval $[0,1]$, considering the wide varieties of the histogram of natural images. By solving $\frac{\partial \mathrm{MSE}(T)}{\partial T}=0$, one can easily find that $T=1 / 2$.

A halftoning result using bi-level quantizer on Lena image is shown in Fig. 3.1. As can be seen, only some high contrast edges and textures are preserved, a lot of low contrast details are lost. The halftone image is not visually similar to the original grayscale image.

Several important properties of the grayscale images and halftone images are not utilized in simple bi-level quantization halftoning. First, a grayscale image usually consists of smooth regions separated by edges. Second, the HVS can only perceive a local average of the halftone image in a small region, or equivalently, the HVS is a low-pass system. One way to utilize these properties is to use a distributed multilevel quantizer, where the quantization levels are spatially distributed in a small region. This idea leads to ordered dithering.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Strong Security and Weak Security

s=0或者s=1，无论他可能拥有的计算能力如何。这称为严格意义上的安全性或无条件安全性 [6, 10]。对于这种安全性，我们假设攻击者具有无限的计算能力。

Liu [12] 和 Iwamoto [10] 分别针对不同类型的 VC 系统提出了弱安全性。Liu 的工作侧重于尺寸不变 VC 的块编码方法，而 Iwamoto 的工作侧重于彩色图像的尺寸扩展 VC（确定性 VC）。他们工作的一个关键结论是，通过将安全级别放宽到弱安全，可以提高目标图像的质量并减少像素扩展。

1. 岩本的弱安全性。
2. 刘的弱安全。
3. 替换攻击。
为了引入弱安全性的概念，我们需要形式化一些基于矩阵的操作：行限制和行堆叠，并引入两组矩阵之间的等价概念[10]。

Iwamoto 介绍的另一个正式操作是矩阵行的堆叠。让一个矩阵乙被划分为

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Bi-level Quantization

MSE⁡(吨)=∑一世=01∫d一世d一世+1(X−是一世)2pX(X)dX

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## MATLAB代写

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