# 数学代写|密码学作业代写Cryptography代考|Essential Number Theory and Discrete Math

#### 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 在代写密码学Cryptography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。我们的专家在代写密码学Cryptography代写方面经验极为丰富，各种代写密码学Cryptography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|密码学作业代写Cryptography代考|Relatively Prime

The concept of prime numbers, and how to generate them, is critical to your study of cryptography. There is another topic you should be familiar with; that is the concept of numbers that are relatively prime, also often called co-prime. The concept is actually pretty simple. Two numbers $\mathrm{x}$ and $\mathrm{y}$ are relatively prime (co-prime) if they have no common divisors other than 1 . So, for example, the number 8 and 15 are co-prime. The factors of 8 are 1,2, and 4 . The factors of 15 are 1,3 , and 5 . Since these two numbers ( 8 and 15 ) have no common factors other than 1 , they are relatively prime. There are lots of interesting facts associated with co-prime numbers. Many of these facts were discovered by Leonhard Euler.

Leonhard Euler asked a few simple questions about co-prime numbers. The first question he asked was, given an integer $n$, how many numbers are co-prime to $n$ ? That number is called Euler’s totient or simply the totient. The symbol for the totient of a number is shown in Fig. 4.6.

The term for the integers that are smaller than $\mathrm{n}$ and have no common factors with $\mathrm{n}$ (other than 1 ) is totative. For example, 8 is a totative of 15 . Another term for Euler’s totient is the Euler phi function.

The story of Euler’s totient is emblematical of much of mathematics. Euler introduced this function in 1793 . For many years after his death, there was no practical application for this concept. In fact, there was no practical use for it until modern asymmetric cryptography in the 1970 s, almost 200 years later. This is one example of pure mathematics that later is found to have very important practical applications.
The next question Euler asked was, what if a given integer, $n$, is a prime number?
How many totatives will it have (i.e., what is the totient of $n$ )? Let us take a prime number as an example, for example, 7. Given that it is prime, we know that none of the numbers smaller than it have any common factors with it. Thus, $2,3,4,5$, and 6 are all totatives of 7 . And since 1 is a special case, it is also a totative of 7 , so we find there are six numbers that are totatives of 7. It turns out that for any $n$ that is prime, the Euler’s totient of $n$ is $n-1$. Therefore, if $n=13$, then the totient of $\mathrm{n}$ is 12 . And then Euler asked yet another question about co-prime numbers. Let us assume you have two prime numbers $m$ and $n$. We already know that the totient of each is $m-1$ and $n-1$, respectively. But what happens if we multiply the two numbers together? If we multiply $m * n$ and get $k$, what is the totient of $k$ ? Well, you can simply test every number less than $k$ and find out. That might work for small values of $k$ but would be impractical for larger values. Euler discovered a nice shortcut. He found that if you have two prime numbers ( $m$ and $n$ ) and multiplied them together to get $k$, the totient of that $\mathrm{k}$ was simply the totient if $m$ * the totient of $n$. Put another way, the totient of $k$ is $(m-1)(n-1)$.

Let us look at an example. Assume we have to prime numbers $m=3$ and $m=5$.

## 数学代写|密码学作业代写Cryptography代考|Modulus Operations

Modulus operations are important in cryptography, particularly in RSA. Let us begin with the simplest explanation and then later delve into more details. To use the modulus operator, simply divide $\mathrm{A}$ by $\mathrm{N}$ and return the remainder.
Thus $5 \bmod 2=1$
And $12 \bmod 5=2$
This explains how to use the modulus operator, and in many cases, this is as far as many programming textbooks go. But this still leaves the question of why are we doing this? Exactly what is a modulus? One way to think about modulus arithmetic is to imagine doing any integer math you might normally do but bound your answers by some number. A classic example is the clock. It has numbers 1 through 12 , so any arithmetic operation you do has to have an answer that is 12 or less. If it is currently 4 o’clock and I ask you to meet me in 10 hours, simple math would say I am asking you to meet me at 14 o’clock. But that is not possible; our clock is bounded by the number 12 ! The answer is simple, take your answer and use the mod operator with a modulus of 12 and look at the remainder.
$$14 \bmod 12=2$$
I am actually asking you to meet me at 2 o’clock (whether that is a.m. or p.m. depends on which the original 4 o’clock was, but that is really irrelevant to understanding the concepts of modular arithmetic). This is an example of how you use modular arithmetic every day. The clock example is widely used; in fact, most textbooks rely on that example. It is intuitive, and most readers will readily grasp the essential point of modular arithmetic. You are performing arithmetic bounded by some number, which is called the modulus. In the clock example, the modulus is 12 , but one can perform modular arithmetic using any integer as the modulus.While the basic concepts of modular arithmetic dates back to Euclid who wrote about modular arithmetic in his book Elements, the modern approach to modular arithmetic was published by Carl Gauss in 1801

# 密码学代写

## 数学代写|密码学作业代写Cryptography代考|Modulus Operations

$5 \bmod 2=1$
$12 \bmod 5=2$

$$14 \bmod 12=2$$

## 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 :)