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# 数学代写|离散数学代写Discrete Mathematics代考|Classical Cryptography

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## 数学代写|离散数学代写Discrete Mathematics代考|Classical Cryptography

Cryptography is about making secret communication to make messages secure in the presence of adversaries. By encryption, an original message, called plaintext, is transformed into a coded message, called ciphertext. This transformation is performed before the plaintext is transmitted or stored. The reverse process is called decryption and is performed after the ciphertext is received or retrieved. The algorithm used for encryption and decryption is often called a cipher, and the process of encryption and decryption requires a secret key. A key is a number (value) that the cipher operates on, without which the unauthorized parties must not be able to recover the original message.

In classical cryptography, symbols, characters, letters, and digits were directly manipulated with the sole goal to provide secrecy through obscurity. It appears that encrypted messages were first developed in ancient Egypt using disordered hieroglyphics, which consisted of visual symbols and characters. There were then other forms of message concealment, which were developed by the Greeks, namely stenography, through which a secret message was hidden within an ordinary nonsecret message, and by the Spartans, namely scytale, by which a narrow strip of parchment was wound on a rod and the message written across the adjoining edges.

A well-known classical cryptography technique, which was developed by Romans, was the Caesar cipher, a simple encryption method based on substitution. The Caesar cipher shifts each letter in the alphabet by three letters forward, for instance, the letter $G$ becomes J. It thus requires a letter three places further along, while wrapping the letters at the end of the alphabet around to the letters at the beginning of the alphabet that is $X$ wraps around to $\mathrm{A}, \mathrm{Y}$ to $\mathrm{B}$, and $\mathrm{Z}$ to $\mathrm{C}$.

Mathematically described, in the Caesar cipher, each letter is coded by its position relative to others. To this effect, an integer $i \in{0,1, \ldots, 24,25}$ replaces a letter whose position in the alphabet ${A, B, \ldots, Y, Z}$ is the $i$ th; for instance, $D$ is the fourth letter in the alphabet, that is $i=3, \mathrm{D}$ is thus replaced by 3 . Assuming the nonnegative integer $p \leq 25$, the functions providing the encrypted message and the decrypted message are $f(p)=(p+3) \bmod 26$ and $f^{-1}(p)=(p-3) \bmod 26$, respectively.

A slight generalization of the Caesar, cipher called the shift cipher or the additive cipher, is when 3 is replaced by the integer $b$, called a key. In other words, the numerical equivalent of each letter is shifted by $b$, thus yielding the following functions:
$$\left{\begin{array}{l} \text { Encryption } \rightarrow f(p)=(p+b) \bmod 26 \ \text { Decryption } \rightarrow f^{-1}(p)=(p-b) \bmod 26 \end{array}\right.$$

## 数学代写|离散数学代写Discrete Mathematics代考|Modern Cryptography

To encrypt a message, an encryption algorithm, an encryption key, and the plaintext are needed, and to decrypt a message, a decryption algorithm, a decryption key, and the ciphertext are required. The encryption and decryption algorithms are public (i.e., anyone can access them), but the keys are secret and thus need to be protected.

Number theory uniquely plays a pivotal role in modern cryptography. Cryptography has become increasingly complex and its applications more varied. The major requirements for a system employing cryptography are as follows:

• To provide an easy and inexpensive means of encryption and decryption to all authorized users in possession of the appropriate key.
• To ensure that the task of producing the plaintext without the key is made extremely difficult and time-consuming.
Relying on the processing power and speed of modern computers, original messages are no longer encoded in characters in a specified language, nor are they encoded one at a time. Modern cryptography operates on binary bit sequences and relies on publicly known algorithms for encoding the message. Secrecy is obtained through a secret key, which is used as the seed for the algorithms. In modern cryptography, encryption and decryption can be carried out rapidly using complicated functions that are designed to be resistant to attack. The computational difficulty of algorithms in conjunction with the fact that only the parties interested in secure communication possess the secret key makes it extremely difficult for anyone else to obtain the original information.

The underlying need for modern cryptography stems from the fact there are essential applications in today’s world that require sensitive information to be fully protected. Some of the widely popular applications requiring cryptography are electronic and mobile commerce transactions, email privacy, secure remote surveillance, file transfers of confidential data, secure e-voting, banking data, secure cloud computing, medical records, and secure remote access.

There are numerous threats that can arise in transmission and storage of data, the array of attacks is constantly widening, and network security is continually becoming more challenging. The notion of security is tied to computing power, as a coded message is only as safe as the amount of computing power needed to break it. In short, the goal is to make undecipherability by an adversary as difficult as possible. Protecting information in its storage and retrieval as well as in transactional and messaging services is always of paramount importance.

The primary reasons to make messages secure through cryptographic mechanisms are as follows:

• Confidentiality: ensuring the transmitted message containing confidential data is hidden from unauthorized parties.
• Authentication: verifying the communicating parties are those they claim to be.
• Integrity: confirming that the message content has not been tampered with.
• Nonrepudiation: not being able to deny the transmission between the two parties has taken place.
There are fundamentally two types of adversaries. Passive adversaries are a threat to confidentiality, as they do not interrupt, alter, or insert any data. Active adversaries additionally threaten integrity and authentication. In any event, potential adversaries may have powers and resources ranging from minimal to unlimited.

## 数学代写|离散数学代写Discrete Mathematics代考|Classical Cryptography

$\$ \$$Vleft { Encryption \rightarrow f(p)=(p+b) \bmod 26 Decryption \rightarrow f^{-1}(p)=(p-b) \bmod 26 正确的。 \ \$$

## 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。