Posted on Categories:abstract algebra, 抽象代数, 数学代写

## 数学代写|抽象代数作业代写ALGEBRA代考|MTH309 Applications

avatest.org抽象代数abstract algebra代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的抽象代数abstract algebra作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此抽象代数abstract algebra作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在抽象代数abstract algebra代写方面经验极为丰富，各种抽象代数abstract algebra相关的作业也就用不着 说。

## 数学代写|抽象代数作业代写ALGEBRA代考|Data Security

Because computers are built from two-state electronic components, it is natural to represent information as strings of $0 \mathrm{~s}$ and $1 \mathrm{~s}$ called binary strings. A binary string of length $n$ can naturally be thought of as an element of $Z_{2} \oplus Z_{2} \oplus \cdots \oplus Z_{2}$ (n copies) where the parentheses and the commas have been deleted. Thus the binary string 11000110 corresponds to the element $(1,1,0,0,0,1,1,0)$ in $Z_{2} \oplus Z_{2} \oplus Z_{2} \oplus Z_{2} \oplus Z_{2} \oplus Z_{2} \oplus Z_{2} \oplus Z_{2}$.Similarly, two binary strings $a_{1} a_{2} \cdots a_{n}$ and $b_{1} b_{2} \cdots b_{n}$ are added componentwise modulo 2 just as their corresponding elements in $Z_{2} \oplus Z_{2} \oplus \cdots \oplus Z_{2}$ are. For example,
$$11000111+01110110=10110001$$
and
$$10011100+10011100+00000000 .$$
The fact that the sum of two binary sequences $a_{1} a_{2} \cdots a_{n}+b_{1} b_{2} \cdots b_{n}=00 \cdots 0$ if and only if the sequences are identical is the basis for a data security system used to protect Internet transactions.

## 数学代写|抽象代数作业代写ALGEBRA代考|Public Key Cryptography

Unlike auctions such as those on eBay, where each bid is known by everyone, a silent auction is one in which each bid is secret. Suppose that you wanted to use your Twitter account to run a silent auction. How could a scheme be devised so that users could post their bids in such a way that the amounts are intelligible only to the account holder? In the mid-1970s, Ronald Rivest,bio]Rivest, Ronald Adi Shamir,bio]Shamir, Adi and Leonard Adlemanbio]Adleman, Leonard devised an ingenious method that permits each person who is to receive a secret message to tell publicly how to scramble messages sent to him or her. And even though the method used to scramble the message is known publicly, only the person for whom it is intended will be able to unscramble the message. The idea is based on the fact that there exist efficient methods for finding very large prime numbers (say about 100 digits long) and for multiplying large numbers, but no one knows an efficient algorithm for factoring large integers (say about 200 digits long). The person who is to receive the message chooses a pair of large primes $p$ and $q$ and chooses an integer $e$ (called the encryption exponent) with $1<e<m$, where $m=\operatorname{lcm}(p-1, q-1)$, such that $e$ is relatively prime to $m$ (any such $e$ will do). This person calculates $n=$ $p q$ ( $n$ is called the key) and announces that a message $M$ is to be sent to him or her publicly as $M^{e} \bmod n$. Although $e, n$, and $M^{e}$ are available to everyone, only the person who knows how to factor $n$ as $p q$ will be able to decipher the message.
To present a simple example that nevertheless illustrates the principal features of the method, say we wish to send the messages “YES.” We convert the message into a string of digits by replacing A by 01 , B by $02, \ldots, Z$ by 26 , and a blank by 00 . So, the message YES becomes 250519 . To keep the numbers involved from becoming too unwieldy, we send the message in blocks of four digits and fill in with blanks when needed. Thus, the messages YES is represented by the two blocks 2505 and 1900 . The person to whom the message is to be sent has picked two primes $p$ and $q$, say $p=37$ and $q=73$, and a number $e$ that has no prime divisors in common with $\operatorname{lcm}(p-1, q-1)=72$, say $e=5$, and has published $n=37 \cdot 73=2701$ and $e=5$ in a public forum. We will send the “scrambled” numbers $(2505)^{5} \bmod 2701$ and $(1900)^{5}$ mod 2701 rather than 2505 and 1900 , and the receiver will unscramble them. We show the work involved for us and the receiver only for the block 2505 . We determine $(2505)^{5} \bmod 2701=2415$ by using a modular arithmetic calculator such as the one at planetcalc.com/8326/. ${ }^{3}$

## 数学代写|抽象代数作业代写ALGEBRA代考| Data Security

$$11000111+01110110=10110001$$

$$10011100+10011100+00000000 .$$

## 数学代写|抽象代数作业代写ALGEBRA代考| Public Key Cryptography

Adlemanbio]Adleman，Leonard设计了一种巧妙的方法，允许每个收到秘密信息的人公开告诉如何扰乱发送给他或她 的消息。即使用于扰乱消息的方法是公开的，也只有预期的人才能解开消息的混乱。这个想法是基于这样一个事实， 即存在有效的方法来查找非常大的素数 (例如大约 100 位长) 和乘以大数，但是没有人知道一种有效的算法来分解大 整数 (比如大约200位长)。接收消息的人选择一对大素数 $p$ 和 $q$ 并选择一个整数 $e$ (称为加密指数) 与 $1<e<m$ 哪里 $m=\operatorname{lcm}(p-1, q-1)$ ，使得 $e$ 是相对质数 $m$ （任何此类 $e$ 会做的）。此人计算 $n=p q(n$ 称为密钥），并宣布消息 $M$ 将公开发送给他或她 $M^{e} \bmod n$.虽然 $e, n$ 和 $M^{e}$ 每个人都可以使用，只有知道如何分解的人 $n$ 如 $p q$ 将能够破译消息。

$(2505)^{5} \bmod 2701=2415$ 通过使用模块化算术计算器，例如 planetcalc.com $/ 8326 \%^{3}$

## 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。

Posted on Categories:abstract algebra, 抽象代数, 数学代写

## 数学代写|抽象代数作业代写ALGEBRA代考|MATH355 External Direct Products

avatest.org抽象代数abstract algebra代写，免费提交作业要求， 满意后付款，成绩80\%以下全额退款，安全省心无顾虑。专业硕 博写手团队，所有订单可靠准时，保证 100% 原创。avatest.org™， 最高质量的抽象代数abstract algebra作业代写，服务覆盖北美、欧洲、澳洲等 国家。 在代写价格方面，考虑到同学们的经济条件，在保障代写质量的前提下，我们为客户提供最合理的价格。 由于统计Statistics作业种类很多，同时其中的大部分作业在字数上都没有具体要求，因此抽象代数abstract algebra作业代写的价格不固定。通常在经济学专家查看完作业要求之后会给出报价。作业难度和截止日期对价格也有很大的影响。

avatest.org™ 为您的留学生涯保驾护航 在数学Mathematics作业代写方面已经树立了自己的口碑, 保证靠谱, 高质且原创的数学Mathematics代写服务。我们的专家在抽象代数abstract algebra代写方面经验极为丰富，各种抽象代数abstract algebra相关的作业也就用不着 说。

## 数学代写|抽象代数作业代写ALGEBRA代考|Definition and Examples

In this chapter, we show how to piece together groups to make larger groups. In Chapter 9, we will show that we can often start with one large group and decompose it into a product of smaller groups in much the same way as a composite positive integer can be broken down into a product of primes. These methods will later be used to give us a simple way to construct all finite Abelian groups.
Definition
External Direct Product
Let $G_{1}, G_{2}, \ldots, G_{n}$ be a finite collection of groups. The external direct product of $G_{1}, G_{2}, \ldots, G_{n}$, written as $G_{1} \oplus G_{2} \oplus \cdots \oplus G_{n}$, is the set of all $n$-tuples for which the ith component is an element of $G_{i}$ and the operation is componentwise.
In symbols,
$$G_{1} \oplus G_{2} \oplus \cdots \oplus G_{n}=\left{\left(g_{1}, g_{2}, \ldots, g_{n}\right) \mid g_{i} \in G_{i}\right}$$
where $\left(g_{1}, g_{2}, \ldots, g_{n}\right)\left(g_{1}^{\prime}, g_{2}^{\prime}, \ldots, g_{n}^{\prime}\right)$ is defined to be $\left(g_{1} g_{1}^{\prime}, g_{2} g_{2}^{\prime}, \ldots, g_{n} g_{n}^{\prime}\right)$. It is understood that each product $g_{i} g_{i}^{\prime}$ is performed with the operation of $G_{i}$. Note that in the case that each $G_{i}$ is finite, we have by properties of sets that $\left|G_{1} \oplus G_{2} \oplus \cdots \oplus G_{n}\right|=\left|G_{1}\right|\left|G_{2}\right| \cdots\left|G_{n}\right|$. We leave it to the reader to show that the external direct product of groups is itself a group (Exercise 1).

## 数学代写|抽象代数作业代写ALGEBRA代考|Properties of External Direct Products

The order of an element in a direct product of a finite number of finite groups is the least common multiple of the orders of the components of the element. In symbols,
$$\left|\left(g_{1}, g_{2}, \ldots, g_{n}\right)\right|=\operatorname{lcm}\left(\left|g_{1}\right|,\left|g_{2}\right|, \ldots,\left|g_{n}\right|\right)$$
PROOF Denote the identity of $G_{i}$ by $e_{i}$. Let $s=\operatorname{lcm}\left(\left|g_{1}\right|,\left|g_{2}\right|, \ldots,\left|g_{n}\right|\right)$ and $t=\left|\left(g_{1}, g_{2}, \ldots, g_{n}\right)\right|$. Because $s$ is a multiple of each $\left|g_{i}\right|$ implies that $\left(g_{1}, g_{2}, \ldots, g_{n}\right)^{s}=\left(g_{1}^{s}, g_{2}^{s}, \ldots, g_{n}^{s}\right)=\left(e_{1}, e_{2}, \ldots, e_{n}\right)$,we know that $t \leq s$. On the other hand, from $\left(g_{1}^{t}, g_{2}^{t}, \ldots, g_{n}^{t}\right)=\left(g_{1}, g_{2}, \ldots, g_{n}\right)^{t}=\left(e_{1}, e_{2}, \ldots, e_{n}\right)$ we see that $t$ is a common multiple of $\left|g_{1}\right|,\left|g_{2}\right|, \ldots,\left|g_{n}\right|$.Thus, $s \leq t$.
The next three examples are applications of Theorem 8.1.

• EXAMPLE 4 Examples of groups of order 100 include
$Z_{100} ; Z_{25} \oplus Z_{2} \oplus Z_{2} ; Z_{5} \oplus Z_{5} \oplus Z_{4} ; Z_{5} \oplus Z_{5} \oplus Z_{2} \oplus Z_{2} ; D_{50} ; D_{10} \oplus Z_{5} ; D_{5} \oplus Z_{10} ;$ and $D_{5} \oplus D_{5}$. That these are not isomorphic is an easy consequence of Theorem 8.1.
• EXAMPLE 5 Let $m$ and $n$ be positive integers that are divisible by 5 . We determine the number of elements of order 5 in $Z_{m} \oplus Z_{n}$. By Theorem $8.1$, we need only count the number of elements $(a, b)$ in $Z_{m} \oplus Z_{n}$ with the property that $5=|(a, b)|=\operatorname{lcm}(|a|,|b|)$. Clearly this requires that $|a|=$ 1or 5 and $|b|=1$ or 5 , but not $(a, b)=(0,0)$. Since both $Z_{m}$ and $Z_{n}$ each have a unique subgroup of order 5 there are exactly five choices for $a$ and 5 choices for $b$ and therefore 25 choices for $(a, b)$, including $(0,0)$. So, there are exactly 24 elements in $Z_{m} \oplus Z_{n}$ of order 5 .

The identical argument shows that if $m$ and $n$ be positive integers that are divisible by a prime $p$, then the number of elements of order $p$ in $Z_{m} \oplus Z_{n}$ is $p^{2}-1$.

## 数学代写|抽象代数作业代写ALGEBRA代考| Properties of External Direct Products

$$\left|\left(g_{1}, g_{2}, \ldots, g_{n}\right)\right|=\operatorname{lcm}\left(\left|g_{1}\right|,\left|g_{2}\right|, \ldots,\left|g_{n}\right|\right)$$

• 例4 订单 100 组的示例包括
$Z_{100} ; Z_{25} \oplus Z_{2} \oplus Z_{2} ; Z_{5} \oplus Z_{5} \oplus Z_{4} ; Z_{5} \oplus Z_{5} \oplus Z_{2} \oplus Z_{2} ; D_{50} ; D_{10} \oplus Z_{5} ; D_{5} \oplus Z_{10} ;$ 和 $D_{5} \oplus D_{5}$. 这些不是同构的， 这是定理8.1的简单结果。
• 示例 5 让 $m$ 和 $n$ 是可被 5 整除的正整数。我们确定 5 阶的元素数 $Z_{m} \oplus Z_{n}$.按定理 $8.1$ ，我们只需要计算元素的数量 $(a, b)$ 在 $Z_{m} \oplus Z_{n}$ 具有属性 $5=|(a, b)|=\operatorname{lcm}(|a|,|b|)$. 显然，这需要 $|a|=1$ 或 5 和 $|b|=1$ 或 5 ，但不是 $(a, b)=(0,0)$. 由于两者兼而有之 $Z_{m}$ 和 $Z_{n}$ 每个都有一个唯一的 5 阶子群，正好有五个选择 $a$ 和 5 选择 $b$ 因此有 25 种选择 $(a, b)$ 包括 $(0,0)$.因此，正好有 24 个元素 $Z_{m} \oplus Z_{n}$ 订单 5 .
相同的参数表明，如果 $m$ 和 $n$ 是可被素数整除的正整数 $p$ ，然后是元素的顺序数 $p$ 在 $Z_{m} \oplus Z_{n}$ 是 $p^{2}-1$.

## 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。