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

2022年7月6日

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## 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

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

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

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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。