### 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|CS171

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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|(One-to-One) Functions

Definition 1.6. We can say that a function or transformation is $1-1$ (one-to-one) if each of the elements found within the codomain $B$ are represented as the image of at most one element in the domain $A$.

Definition 1.7. We can say that a function or transformation is onto if each of the elements found within the codomain $B$ represents the image of at least one element that can be found in the domain. At the same time, a function $f: A \rightarrow B$ is known as being onto if $\operatorname{Im}(f)=B$.
Definition 1.8. If function $f: A \rightarrow B$ is to be considered $1-1$ and $\operatorname{Im}(f)=B$, and then the function $f$ is called bijection.
Conclusion 1.9. If $f: A \rightarrow B$ is considered $1-1$, then $f: A \rightarrow \operatorname{Im}(f)$ represents the bijection. In special cases, if $f: A \rightarrow B$ is represented as $1-1$, and $A$ and $B$ are represented as finite sets with the same size, then $f$ represents a bijection.
Based on the scheme and its representation, if $f$ represents a bijection, then each element from $B$ has exactly one line that is incidental with it. The function shown and described in Examples $1.3$ and $1.4$ does not represent bijections. As you can see in Example 1.3, element 3 doesn’t have the image of any other element that can be found within the domain. In Example 1.4, each element from the codomain is identified with two preimages.
Definition 1.10. If $f$ is a bijection from $A$ to $B$, then it is a quite simple matter to define a bijection $g$ from $B$ to $A$ as follows: for each $b \in B$ we will define $g(b)=a$ where $a \in A$ and $f(a)=b$. The function $g$ is obtained from $f$ and it is called an inverse function of $f$ and is denoted as $g=f^{-1}$.

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|One-Way Functions

In cryptography, there are a certain types of functions that play an important role. Due to the rigor, a definition for one-way function is given as follows.
Definition 1.12. Let’s consider function $f$ from set $A$ to set $B$ that is called a oneway function if $f(a)$ proves to be simple and easy to be computed for all $a \in A$ but for “essentially all” elements $b \in \operatorname{Im}(f)$ it is computationally infeasible to manage to find any $a \in A$ in such way that $f(a)=b$.
Note 1.13. This note represents some additional notes and clarifications of the terms used in Definition1.12.

1. For the terms easy and computationally infeasible a rigorous definition is necessary but it will distract the attention from the general idea that is being agreed. For the goal of this chapter, the simple and intuitive meaning is sufficient.
2. The phrase “essentially all” refers to the idea that there are a couple of values $b \in B$ for which it is easy to find an $a \in A$ in such way that $b=f(a)$. As an example, one may compute $b=f(a)$ for a small number of $a$ values and then for these, the inverse is known by a table look-up. A different way to describe this property of a one-way function is as follows: for any random $b \in \operatorname{Im}(f)$, it is computationally feasible to have and find any $a \in A$ in such way that $f(a)=b$.

The following examples show the concept behind a one-way function.
Example 1.14. (one-way function) Consider $A={1,2,3, \ldots, 16}$ and let’s define $f(a)=r_{a}$ for all the elements $a \in A$ where $r_{a}$ represents the remainder when $3^{x}$ will be divided with 17 .

## 数学代写|密码学作业代写Cryptography & Cryptanalysis代考|One-Way Functions

1. 对于简单且在计算上不可行的术语，需要严格定义，但这会分散人们对正在商定的一般概念的注意力。对于 本章的目标，简单直观的含义就足够了。
2. 短语“基本上全部”指的是有几个值的想法 $b \in B$ 很容易找到 $a \in A$ 以这样的方式 $b=f(a)$. 例如，可以计算 $b=f(a)$ 对于少数 $a$ 值，然后对于这些值，通过查表可以知道倒数。描述单向函数的这一性质的另一种方法 如下: 对于任何随机 $b \in \operatorname{Im}(f)$ ，在计算上是可行的 $a \in A$ 以这样的方式 $f(a)=b$.
以下示例显示了单向函数背后的概念。
示例 1.14。(单向函数) 考虑 $A=1,2,3, \ldots, 16$ 让我们定义 $f(a)=r_{a}$ 对于所有元素 $a \in A$ 在哪里 $r_{a}$ 表示余 数时 $3^{x}$ 将除以 17 。

## 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。