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# 数学代写|离散数学代写DISCRETE MATHEMATICS代写|MAT132 Public Key Cryptography; The RSA System

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## 数学代写|离散数学代写DISCRETE MATHEMATICS代写|Public Key Cryptography; The RSA System

Ever since written communication was used, people have been interested in trying to conceal the content of their messages from their adversaries. This has led to the development of techniques of secret communication, a science known as cryptography.

The basic situation is that one party, A, say Albert, wants to send a message to another party, J, say Julia. However, there is a danger that some ill-intentioned third party, Machiavelli, may intercept the message and learn things that he is not supposed to know about and as a result do evil things. The original message, understandable to all parties, is known as the plain text (or plaintext). To protect the content of the message, Albert encrypts his message. When Julia receives the encrypted message, she must decrypt it in order to be able to read it. Both Albert and Julia share some information that Machiavelli does not have, a key. Without a key, Machiavelli, is incapable of decrypting the message and thus, to do harm.

There are many schemes for generating keys to encrypt and decrypt messages. We are going to describe a method involving public and private keys known as the RSA Cryptosystem, named after its inventors, Ronald Rivest, Adi Shamir, and Leonard Adleman (1978), based on ideas by Diffie and Hellman (1976). We highly recommend reading the orginal paper by Rivest, Shamir, and Adleman [12]. It is beautifully written and easy to follow. A very clear, but concise exposition can also be found in Koblitz [7]. An encyclopedic coverage of cryptography can be found in Menezes, van Oorschot, and Vanstone’s Handbook [9].

The RSA system is widely used in practice, for example in SSL (Secure Socket Layer), which in turn is used in https (secure http). Any time you visit a “secure site” on the internet (to read e-mail or to order merchandise), your computer generates a public key and a private key for you and uses them to make sure that your credit card number and other personal data remain secret. Interestingly, although one might think that the mathematics behind such a scheme is very advanced and complicated, this is not so. In fact, little more than the material of Section $7.1$ is needed. Therefore, in this section we are going to explain the basics of RSA.

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写|Correctness of The RSA System

We begin by proving the correctness of the inversion formula $(*)$. For this we need a classical result known as Fermat’s little theorem.

This result was first stated by Fermat in 1640 but apparently no proof was published at the time and the first known proof was given by Leibnitz (1646-1716). This is basically the proof suggested in Problem 7.7. A different proof was given by Ivory in 1806 and this is the proof that we give here. It has the advantage that it can be easily generalized to Euler’s version (1760) of Fermat’s little theorem.

Theorem 7.6. (Fermat’s Little Theorem) If $p$ is any prime number, then the following two equivalent properties hold.
(1) For every integer, $a \in \mathbb{Z}$, if $a$ is not divisible by $p$, then we have
$$a^{p-1} \equiv 1(\bmod p)$$
(2) For every integer, $a \in \mathbb{Z}$, we have
$$a^{p} \equiv a(\bmod p)$$
Proof. (1) Consider the integers
$$a, 2 a, 3 a, \ldots,(p-1) a$$
and let
$$r_{1}, r_{2}, r_{3}, \ldots, r_{p-1}$$

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写| Public Key Cryptography; The RSA System

RSA系统在实践中被广泛使用，例如在SSL (安全套接字层) 中，而SSL又在https (安全http）中使用。每当您访问 互联网上的“安全网站”（阅读电子邮件或订购商品）时，您的计算机都会为您生成公钥和私钥，并使用它们来确保您 的信用卡号和其他个人数据保持机密。有趣的是，尽管人们可能认为这种方案背后的数学非常先进和复杂，但事实并 非如此。事实上，只不过是部分的材料7.1是必需的。因此，在本节中，我们将解释 RSA 的基础知识。

## 数学代写|离散数学代写DISCRETE MATHEMATICS代写| Correctness of The RSA System

(1) 对于每个整数， $a \in \mathbb{Z}$ 如果 $a$ 不可被整除 $p$ ，然后我们有
$$a^{p-1} \equiv 1(\bmod p)$$
(2) 对于每个整数， $a \in \mathbb{Z}$ 我们有
$$a^{p} \equiv a(\bmod p)$$

$$a, 2 a, 3 a, \ldots,(p-1) a$$

$$r_{1}, r_{2}, r_{3}, \ldots, r_{p-1}$$

## 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。