# 数学代写|离散数学作业代写discrete mathematics代考|MPCS50103

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## 数学代写|离散数学作业代写discrete mathematics代考|Cyclic Groups

These examples suggest that a periodical structure, which we observed in this particular case, should exist in general when we deal with the comparison of the integers. This periodicity is explored in more detail now. It is clear that an order of any element in a finite group cannot be bigger than the order of the group itself. Group elements having this largest possible order $\operatorname{ord}(G)=|G|$ are called primitive elements or generators of the group. Thus, as we saw above, the group $\mathbb{Z}{7}^{}$ has two generators 3 and 5 . since both these elements have the largest possible order $6=7-1$. We also see that the powers of both these elements, 3 and 5 , – of course, modulo 7 , are $3,2,6,4,5,1$ and $5,4,6,2,3,1$, that is, each sequence is the entire group $\mathbb{Z}{7}^{}$; the elements with the maximal order are called generators of a group, and these groups are called cyclic groups.

Problem 132. Are $\mathbb{Z}{11}^{}$ and $\mathbb{Z}{13}^{}$ cyclic groups? Compute the orders of elements of these groups and find, if any, their generators.

Problem 133. The additive group of all the integers is an infinite cyclic group.
We omit a proof of the next statement, important in cryptography.
Theorem 10. (1) If d is prime, then $\mathbb{Z}_{d}^{*}$ with a congruence as a group operation, is a commutative cyclic group.
(2) If $|G|$ is prime, then all elements $a \in G, a \neq 1$, are primitive.
The next property says that all cyclic groups of a given order are, in a sense, the same.
(3) All the finite cyclic groups of the given order n are isomorphic to each other. All infinite cyclic groups are isomorphic to one another.

## 数学代写|离散数学作业代写discrete mathematics代考|THE DISCRETE LOGARITHM PROBLEM

While developing the Affine Ciphers, we had to find the inverse elements of some group elements. It is ensy if we work with renl numbers, since $x^{-1}$ exists for every real $x \neq 0$. However, in cryptography $x$ is supposed to be integer, and its reciprocal must be also integer; hence the problem of finding the reciprocal may have no solution. Moreover, the exponent does not have to be – 1. Again, solving an equation $a^{x}=b$, when $a>0$ and $b$ is a real number, straightforwardly leads to logarithms. Therefore, we have to extend that notion to a discrete setting.

Consider the finite cyclic group $\mathbb{Z}{p}^{}$ with prime $p$, its order is $p-1$, and let $g \in \mathbb{Z}{p}^{}$ be a generator of this group. Let also another element be $h \in \mathbb{Z}_{p}^{*}$. The Discrete Logarithm Problem (DLP) requires finding the integer $x, 1 \leq x \leq p-1$, such that $g^{x}=h(\bmod p)$. We denote the solution of this congruence, if it exists, as $x=\log h(\bmod p)$.

For example, computations in Example 11 tell that $5^{\circ 4}(\bmod 7)=2$, therefore, we set $g=5, h=2$, and get $x=\log _{5} 2(\bmod 7)=2$, which has nothing in common with $\ln 2 / \ln 5 \approx 0.43$. We can straightforwardly check that $5^{* 4}=625$ and $625=7 \times 89+2$.

This example shows why DLP is used in cryptography. We deal with a one-way function – see Def. 68 (p. 167). Given the value of the discrete logarithm, the verification is straightforward and fast. But the computations for finding this value currently, for really large parameters, are infeasible. For more about the DLP, the reader can consult, for example, [40] and the references therein.

## 数学代写|离散数学作业代写discrete mathematics代考| Cyclic Groups

（2）如果 $|G|$ 是栍数，则所有元溸 $a \in G, a \neq 1$ ，是原始的。

（3）给定阶 $n$ 的所有有限旿环群彼此同构。所有无限楿环群彼此同构。

## 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。

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