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

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

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 $$ \hat{B}[\ell, j]=B\left[i{\ell}, j\right],
where $\ell \in{1, \ldots, q}, i_{\ell} \in \mathrm{P}$, and $j \in{1, \ldots, n}$.
Another formal operation introduced by Iwamoto is stacking of rows of a matrix. Let a matrix $\mathbf{B}$ be partitioned as
\mathbf{b}{1} \ \vdots \ \mathbf{b}{n}
\end{array}\right] \in \mathbb{Z}{2}^{n \times m} $$ 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| Security Issue in Visual Cryptography


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

作为一种密码方案,安全要求通常是强制性的。因此,在设计 VC 算法时,通常会在安全性的约束下尝试最大化对比度。一些研究人员还建议在安全性、对比度和像素扩展之间进行权衡。如果我们放宽一个方面的要求,那么也许可以提高另外两个方面的性能。例如,如果我们放宽对安全性的要求,则可以提高对比度并减少像素扩展[6,10,12].

对于严格意义上的VC方案,通常将份额打印在透明胶片上,通过堆叠实现解码,解码不需要计算。在 VC 的定义(定义 2.1)中,对于q<ķ股,两组子矩阵C^0和C^1是等价的。因此,攻击者在获得C^0和C^1, 无法判断是否

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。

但是,考虑到共享的媒体和解码机制,可以合理地假设,如果攻击者仅使用他的视觉系统从q<ķ共享,则“计算”设备是堆叠操作和 HVS。那么如果从堆叠结果q<ķ矩阵的行,无法推断s,我们的算法在这些假设下是安全的。我们称之为弱安全。

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(ķ,n)-阈值方案,一组参与者可以表示为磷= \left{i_{1}, \ldots, i_{q}\right} \subset{1, \ldots, n}\left{i_{1}, \ldots, i_{q}\right} \subset{1, \ldots, n}. 给定一个基矩阵乙∈从2n×米,可以通过限制乙到 set 中指定的行磷. 该操作表示为


在哪里ℓ∈1,…,q,一世ℓ∈磷, 和j∈1,…,n.
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

乙=[b1 ⋮ bn]∈从2n×米然后,堆积成排的乙表示为


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

大多数打印设备和一些显示设备只能渲染有限数量的颜色 [9]。例如,典型的激光打印机一次只能输出一个黑点或“无点”。为了打印灰度图像,我们必须将输入颜色量化为更粗的尺度,使得人眼观察时感知到的颜色仍然与原始颜色相似。

在本章中,我们假设输入图像是灰度图像,并且每个像素都归一化到范围[0,1]. 如果像素被量化,则假定为 8 位量化。假设打印机只能产生两种颜色:黑色(黑点)和白色(无点)。

图像量化问题可以表述如下: 给定一个输入灰度图像X[一世,j],量化器产生二值图像是[一世,j]∈0,1使其在视觉上与输入图像相似X[一世,j]从足够的距离观看时。让H是表示人类视觉感知的系统,则数字半色调问题可以表述为优化问题:

图像的大小米×ñ. 由于具有高维搜索空间,直接解决这个优化问题通常是不切实际的。2米×ñ可行的解决方案。


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

最简单的方法是使用双层量化器 [6] 独立地量化每个样本:

是[一世,j]={1, 如果 X[一世,j]>吨, 0, 除此以外 
在哪里吨是一个常数阈值。对于这个量化器,两个重建级别是(是0,是1)=(0,1)决策边界是(d0,d1,d2)=(0,吨,1). 门槛吨可以通过最小化均方误差来设计:

在哪里pX(X)是输入图像样本的 PDFX. 它可以建模为区间上的均匀分布[0,1],考虑到自然图像的直方图种类繁多。通过解决∂米小号和(吨)∂吨=0, 很容易发现吨=1/2.

在 Lena 图像上使用双电平量化器的半色调结果如图 3.1 所示。可以看出,只保留了一些高对比度的边缘和纹理,丢失了很多低对比度的细节。半色调图像在视觉上与原始灰度图像不相似。

灰度图像和半色调图像的几个重要特性在简单的二级量化半色调中没有被利用。首先,灰度图像通常由边缘分隔的平滑区域组成。其次,HVS 只能感知小区域的半色调图像的局部平均,或者等效地,HVS 是一个低通系统。利用这些属性的一种方法是使用分布式多级量化器,其中量化级在空间上分布在一个小区域中。这个想法导致了有序抖动。

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多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


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