### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|CIS556

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• 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|Trapdoor One-Way Functions

Definition 1.16. A trapdoor one-way function is defined as a one-way function $f: A \rightarrow B$ with the additional property that by having extra information (known as trapdoor information) it will become feasible to find and identify any given $b \in \operatorname{Im}(f)$, with an $a \in A$ in such way that $f(a)=b$.

In Example 1.15, we show the concept of a trapdoor one-way function. With extra information about the factors of $n=2957524163$ it will become much easier to invert the function. The factors of 2957524163 are large enough that finding them by hand computation would be difficult. With the help of any computer software we can find the factors quite quickly. If, for example, we have very large distinct prime numbers (each number having around 200 decimal digits), $p$ and $q$, with today’s technologies, it’s quite difficult even with the most powerful computers to find $p$ and $q$ from $n$. This is the wellknown problem entitled as integer factorization problem, which for quantum computers will not represent an issue.

One-way and trapdoor one-way functions represent the basic foundation for publickey cryptography. These concepts are very important and they will become much clearer later when their application to cryptographic techniques are implemented and discussed. It is quite important to keep these abstract concepts from this section in mind as the concrete methods and the main foundation for the cryptography algorithms that will be implemented later within this book.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Permutations

Permutations represent functions that are in cryptographic constructs.
Definition 1.17. Consider $S$ to be a finite set formed of elements. A permutation $p$ on $S$ represents a bijection as defined in Definition 1.8. The bijection is represented from $S$ to itself as $p: S \rightarrow S$.

Example 1.18. This example represents a permutation example. Let’s consider the following permutation: $S={1,2,3,4,5}$. The permutation $p: S \rightarrow S$ is defined as follows:
$$p(1)=2, p(2)=5, p(3)=4, p(4)=2, p(5)=1$$
A permutation can be described in different ways. It can be written as above or as an array as
$$p=\left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \ 3 & 5 & 4 & 2 & 1 \end{array}\right)$$
in which the top row of the array is represented by the domain and the bottom row is represented by the image under $p$ as mapping.
As the permutations are bijections, they have inverses. If the permutation is written as an away (second form), its inverse will be very easily to find by interchanging the rows in the array and reordering the elements from the new top row and the bottom row. In this case, the inverse of $p$ is defined as follows:
$$p^{-1}=\left(\begin{array}{ccccc} 1 & 2 & 3 & 4 & 5 \ 5 & 4 & 1 & 3 & 2 \end{array}\right)$$
Example 1.19. This example represents a permutation example. Let’s consider $A$ to be the set of integers ${0,1,2, \ldots, p \cdot q-1}$ where $p$ and $q$ represent two distinct large primes. We need to suppose also that neither $p-1$ nor $q-1$ can be divisible by 3 . The function $p(a)=r_{a}$, in which $r_{a}$ represents the remainder when $a^{3}$ is divided by $p q$ can be demonstrated and shown as being the inverse perumutation. The inverse permutation is computationally infeasible by computers nowadays, unless $p$ and $q$ are known.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|Permutations

$$p(1)=2, p(2)=5, p(3)=4, p(4)=2, p(5)=1$$

## 有限元方法代写

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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。