# 数学代写|密码学作业代写Cryptography代考|Kerckhoffs’s Principle/Shannon’s Maxim

#### Doug I. Jones

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## 数学代写|密码学作业代写Cryptography代考|Information Diversity

Hartley entropy was introduced by Ralph Hartley in 1928 and is often referred to as the Hartley function. If a sample is randomly elected from a finite set A, the information revealed after the outcome is known is given by the Hartley function, shown in the following equation (Kakihara 2016):
$$H_0(A):=\log _b|A|$$
In the Hartley function, the $H_0(A)$ is the measure of uncertainty in set $A$. The cardinality of $A$ is denoted by LAI. If the base of the logarithm is 2 , then the unit of uncertainty is referred to as the Shannon. However, if the natural logarithm is used instead, then the unit is the nat. When Hartley first published this function, he utilized a base-ten logarithm. Since that time, if a base 10 is used, then the unit of information is called the Hartley. Hartley entropy tends to be a large estimate of entropy and is thus often referred to as max entropy.

Another measure of uncertainty in a message is the min-entropy (Wan et al. 2018). This measure of entropy tends to yield a small information entropy value, never greater than the standard Shannon entropy. This formula is often used to find a lower bound for information entropy. The formula is shown in the following equation:
$$H=\min _{1 \text { sisk }}\left(-\log _2 p_i\right)$$
In the equation above there exists some set $\mathrm{A}$ with an independent discrete random variable $X$ taken from that set and a probability that $X=x_i$ of $p_i$ for $i-1, \ldots . k$.

When the random variable $X$ has a min-entropy (i.e., an $H$ value), then the probability of observing any particular value for $X$ is less than or equal to $2^{-H}$.

Another measure of information entropy is the Rényi entropy (Linke et al. 2018). The Rényi entropy is used to generalize four separate entropy formulas: Hartley entropy, the Shannon entropy, the collision entropy, and the min-entropy (Hayashi 2012), as shown in the following equation:
$$H \alpha(x)=\frac{1}{1-\alpha} \log \left(\sum_{i=1}^n p_i^\alpha\right)$$
In the equation above $X$ denotes discrete random variable with possible outcomes $(1,2, \ldots . n)$ with probabilities $p_i$ for $i=1$ to $n$. Unless otherwise stated, the logarithm is base 2 , and the order, $\alpha$, is $0<\alpha<1$. The aforementioned collision entropy is simply the Rényi entropy in the special case that $\alpha=2$, and not using logarithm base 2 .

## 数学代写|密码学作业代写Cryptography代考|Complex Numbers

Imaginary numbers developed in response to a specific problem. The problem begins with the essential rules of multiplications. If you multiple a negative with a negative, you get a positive. For example, $-1 *-1=1$. This becomes a problem if one considers the square root of a negative number. Clearly the square root of a positive number is also positive. $\sqrt{4}=2, \sqrt{ } 1=1$, etc. But what is the $\sqrt{ }-1$ ? If you answer that it is -1 , that won’t work. $-1 *-1$ gives us positive 1 . This conundrum led to the development of imaginary numbers. Imaginary numbers are defined as follows: $i^2=-1$ (or conversely $\sqrt{ }-1=i$ ). So that the square root of any negative number can be expressed as some integer multiplied by $i$. A real number combined with an imaginary number is referred to as a complex number.

Imaginary numbers work in much the same manner as real numbers do. If you see the expression $4 i$, that denotes $4 * \sqrt{-1}$. It should also be noted that the name imaginary number is unfortunate. These numbers do indeed exist and are useful in a number of contexts. For example, such numbers are often used in quantum physics and quantum computing.

Complex numbers are simply real numbers and imaginary together in an expression. For example:
$$3+4 i$$
This is an example of a complex number. There is the real number 4 , combined with the imaginary number $i$ to produce $4 i$ or $4 * \sqrt{-1}$. Let us put this a bit more formally. A complex number is a polynomial with real coefficients and $i$ for which $i^2+1=0$ is true. You can perform all the usual arithmetic operations with complex numbers that you have performed with real numbers (i.e., rational numbers, irrational numbers, integers, etc.). Let us consider a basic example:
\begin{aligned} & (3+2 i)+(1+1 i) \ & =(3+1)+(2+1) i \ & =4+3 i \end{aligned}
Here is another example, this time subtracting.
\begin{aligned} & (1+2 i)-(3+4 i) \ & =(1-3)-(2 i-4 i) \ & =2-2 i \end{aligned}
As you can see, basic addition and subtraction with complex numbers are very easy. Multiplying complex numbers is also quite straightforward. We use the same method you probably learned in elementary algebra as a youth: FOIL or First, Outer, Inner, Last. This is shown in Fig. 4.1.

# 密码学代写

## 数学代写|密码学作业代写Cryptography代考|Information Diversity

$$H_0(A):=\log b|A|$$ 在哈特利函数中， $H_0(A)$ 是集合中不确定性的度量 $A$. 的基数 $A$ 由 LAI 表示。如果对数的底是 2，则不确定性单 位称为香农。但是，如果改为使用自然对数，则单位为 nat。当 Hartley 首次发布此函数时，他使用了以 10 为 底的对数。从那时起，如果使用基数 10，则信息单位称 为哈特利。哈特利熵往往是熵的一个大估计，因此通常 被称为最大墒。 消息中不确定性的另一个度量是最小熵 (Wan 等人， 2018 年)。这种熵度量往往会产生一个小的信息熵值， 永远不会大于标准香农熵。这个公式经常被用来寻找信 息熵的下界。公式如下式所示: $$H=\min {1 \text { sisk }}\left(-\log 2 p_i\right)$$ 在上面的等式中存在一些集合 $\mathrm{A}$ 具有独立的离散随机变 量 $X$ 从那个集合中取出的概率 $X=x_i$ 的 $p_i$ 为了 $i-1, \ldots k$. 当随机变量 $X$ 有一个最小熵（即 $H$ 值)，然后观察到任 何特定值的概率 $X$ 小于或等于 $2^{-H}$. 信息熵的另一种度量是 Rényi 熵（Linke 等人， 2018 年)。Rényi 熵用于推广四个独立的熵公式：哈特利 熵、香农熵、碰撞熵和最小熵 (Hayashi 2012)，如下式 所テ的 $$H \alpha(x)=\frac{1}{1-\alpha} \log \left(\sum{i=1}^n p_i^\alpha\right)$$

## 数学代写|密码学作业代写Cryptography代考|Complex Numbers

$$3+4 i$$

$$(3+2 i)+(1+1 i) \quad=(3+1)+(2+1) i=4$$

$$(1+2 i)-(3+4 i)=(1-3)-(2 i-4 i)=2$$

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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