# 数学代写|抽象代数作业代写abstract algebra代考|MATH2701

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes-lab™ 为您的留学生涯保驾护航 在代写抽象代数abstract algebra方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写抽象代数abstract algebra代写方面经验极为丰富，各种代写抽象代数abstract algebra相关的作业也就用不着说。

• 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
couryes™为您提供可以保分的包课服务

## 数学代写|抽象代数作业代写abstract algebra代考|A Brief Background on Cryptography

In this section, we will study an application of group theory to cryptography, the science of keeping information secret.

Cryptography has a long history, with one of the first documented uses of cryptography attributed to Caesar. When writing messages he wished to keep in confidence, the Roman emperor would shift each letter by 3 to the right, assuming the alphabet wraps around. In other words, he would substitute a letter of $\mathrm{A}$ with $\mathrm{D}, \mathrm{B}$ with $\mathrm{E}$ and so forth, down to replacing $\mathrm{Z}$ with $\mathrm{C}$. To anyone who intercepted the modified message, it would look like nonsense. This was particularly valuable if Caesar thought there existed a chance that an enemy could intercept orders sent to his military commanders.

After Caesar’s cipher, there came letter wheels in the early Renaissance, letter codes during the American Civil War, the Navajo windtalkers during World War II, the Enigma machine used by the Nazis, and then a whole plethora of techniques since then. Military uses, protection of financial data, and safety of intellectual property have utilized cryptographic techniques for centuries. For a long time, the science of cryptography remained the knowledge of a few experts because both governments and companies held that keeping their cryptographic techniques secret would make it even harder for “an enemy” to learn one’s information security tactics.

Today, electronic data storage, telecommunication, and the Internet require increasingly complex cryptographic algorithms. Activities that are commonplace like conversing on a cellphone, opening a car remotely, purchasing something online, all use cryptography so that a conversation cannot be intercepted, someone else cannot easily unlock your car, or an eavesdropper cannot intercept your credit card information.

Because of the proliferation of applications of cryptography in modern society, no one should assume that the cryptographic algorithm used in any given instance remains secret. In fact, modern cryptographers do not consider an information security algorithm at all secure if part of its effectiveness relies on the algorithm remaining secret. But not everything about a cryptographic algorithm can be known to possible eavesdroppers if parties using the algorithm hope to keep some message secure. Consequently, most, if not all, cryptographic techniques involve an algorithm but also a “key,” which can be a letter, a number, a string of numbers, a string of bits, a matrix or some other mathematical object. The security of the algorithm does not depend on the algorithm staying secret but rather on the key remaining secret. Users can change keys from time to time without changing the algorithm and have confidence that their messages remain secure.

## 数学代写|抽象代数作业代写abstract algebra代考|Fast Exponentiation

Let $G$ be a group, let $g$ be an element in $G$, and let $n$ be a positive integer. To calculate the power $g^{n}$, one normally must calculate
$$g^{n}=\overbrace{g \cdot g \cdots g}^{n \text { times }},$$
which involves $n-1$ operations. (If fact, when we implement this into a computer algorithm, since we must take into account the operation of incrementing a counter, the above direct calculation takes a minimum of $2 n-1$ computer operations.) If the order $|g|$ and the power $n$ are large, one may not notice any patterns in the powers of $g$ that would give us any shortcuts to determining $g^{n}$ with fewer than $n-1$ group operations.

The Fast Exponentiation Algorithm allows one to calculate $g^{n}$ with many fewer group operations than $n$, thus significantly reducing the calculation time.

The reason that $x$ has the value of $g^{n}$ at the end of the for loop is because when the algorithm terminates,
$$x=g^{b_{k} 2^{k}+b_{k-1} 2^{k-1}+\cdots+b_{1} 2+b_{0}},$$
which is precisely $g^{n}$. Note that in the binary expansion $n=\left(b_{k} b_{k-1} \cdots b_{1} b_{0}\right){2}$, there is an assumption that $b{k}=1$.

## 数学代写|抽象代数作业代写abstract algebra代考|Fast Exponentiation

$$g^{n}=\overbrace{g \cdot g \cdots g}^{n \text { times }}$$

$$x=g^{b_{k} 2^{k}+b_{k-1} 2^{k-1}+\cdots+b_{1} 2+b_{0}}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)