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# 数学代写代考| Symmetric Key Systems 离散数学

## 数学代写| Symmetric Key Systems 代考

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## 离散数学代写

$10.4$ Symmetric Key Systems
A symmetric key cryptosystem (Fig. 10.6) uses the same secret key for encryption and decryption. The sender and the receiver first need to agree a shared key prior to communication. This needs to be done over a secure channel to ensure that the
$10.4$ Symmetric Key Systems
167
Fig. $10.6$ Symmetric key cryptosystem
shared key remains secret. Once this has been done they can begin to encrypt and decrypt messages using the secret key. Anyone who is able to encrypt a message has sufficient information to decrypt the message.

The encryption of a message is in effect a transformation from the space of messages $\mathscr{l} b$ to the space of cryptosystems $\mathbb{C}$. That is, the encryption of a message with key $k$ is an invertible transformation $f$ such that:
$$f: \mathscr{M} \rightarrow^{k} \mathbb{C}$$
The cipher text is given by $\mathrm{C}=\mathrm{E}{k}(\mathrm{M})$ where $\mathrm{M} \in \mathscr{l} b$ and $\mathrm{C} \in \mathbb{C}$. The legitimate receiver of the message knows the secret key $k$ (as it will have transmitted previously over a secure channel), and so the cipher text $\mathrm{C}$ can be decrypted by the inverse transformation $f^{-1}$ defined by: $$f^{-1}: \mathbb{C} \rightarrow^{k} \mathscr{M}$$ Therefore, we have that $\mathrm{D}{k}(\mathrm{C})=\mathrm{D}{k}\left(\mathrm{E}{k}(\mathrm{M})\right)=\mathrm{M}$ the original plaintext message.

There are advantages and disadvantages to symmetric key systems (Table 10.2), and these include:
Examples of Symmetric Key Systems
(i) Caesar Cipher
The Caesar cipher may be defined using modular arithmetic. It involves a shift of three places for each letter in the plaintext, and the alphabetic letters are represented by the numbers $0-25$. The encyption is carried out by addition (modula 26). The encryption of a plaintext letter $x$ to a cipher letter $c$ is given by $^{3}$ :
${ }^{3}$ Here $x$ and $c$ are variables rather than the alphabetic characters ‘ $x$ ‘ and ‘ $c$ ‘.Table. $10.2$ Advantages and disadvantages of symmetric key systems
\begin{tabular}{|l|l|}
\hline Advantages Encryption process is simple (as the same key is used for encryption and decryption) & Disadvantages A shared key must be agreed between two parties \
\hline It is faster than public key systems & Key exchange is difficult as there needs to be a secure channel between the two parties (to ensure that the key remains secret) \
\hline It uses less computer resources than public key systems & If a user has $n$ trading partners then $n$ secret keys must be maintained (one for each partner) \
\hline It uses a different key for communication with every different party & There are problems with the management and security of all of these keys (due to volume of keys that need to be maintained) \
\hline & Authenticity of origin or receipt cannot be \
& proved (as key is shared) \
\hline
\end{tabular}
$$c=x+3(\bmod 26)$$
Similarly, the decryption of a cipher letter $c$ is given by:
$$x=c-3(\bmod 26)$$
(ii) Generalized Caesar Cipher
This is a generalisation of the Caesar cipher to a shift of $k$ (the Caesar cipher involves a shift of three). This is given by:
$$\begin{array}{cl} f_{k}=\mathrm{E}{k}(x)=x+k(\bmod 26) & 0 \leq k \leq 25 \ f{k}^{-1}=\mathrm{D}{k}(c)=c-k(\bmod 26) & 0 \leq k \leq 25 \end{array}$$ (iii) Affine Transformation This is a more general transformation and is defined by: $$\begin{gathered} f{(a, b)}=\mathrm{E}{(a, b)}(x)=a x+b(\bmod 26) \quad 0 \leq a, b, x \leq 25 \operatorname{and} \operatorname{gcd}(a, 26)=1 \ f{(a, b)}^{-1}=\mathrm{D}_{(a, b)}(c)=a^{-1}(c-b)(\bmod 26) \quad a^{-1} \text { is the inverse of } a \bmod 26 \end{gathered}$$

$10.4$ 对称密钥系统

$10.4$ 对称密钥系统
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$$f: \mathscr{M} \rightarrow^{k} \mathbb{C}$$

(i) 凯撒密码

${ }^{3}$ 这里的 $x$ 和 $c$ 是变量，而不是字母字符’$x$’和’$c$’.Table。 $10.2$ 对称密钥系统的优缺点
\begin{表格}{|l|l|}
\hline 优点 加密过程简单（加密和解密使用相同的密钥） & 缺点 共享密钥必须由两方商定 \
\hline 它比公钥系统更快，并且密钥交换很困难，因为双方之间需要有一个安全通道（以确保密钥保持秘密）\
\hline 它比公钥系统使用更少的计算机资源 & 如果用户有 $n$ 贸易伙伴，则必须维护 $n$ 密钥（每个伙伴一个）\
\hline 它使用不同的密钥与每个不同的方进行通信&所有这些密钥的管理和安全性都存在问题（由于需要维护的密钥量）\
\hline & 原产地或收据的真实性不能是 \
& 证明（因为密钥是共享的）\
\hline
\end{表格}
$$c=x+3(\bmod 26)$$

$$x=c-3(\bmod 26)$$
(ii) 广义凯撒密码

$$\开始{数组}{cl} f_{k}=\mathrm{E}{k}(x)=x+k(\bmod 26) & 0 \leq k \leq 25 \ f{k}^{-1}=\mathrm{D}{k}(c)=c-k(\bmod 26) & 0 \leq k \leq 25 \结束{数组}$$ (iii) 仿射变换 这是一个更一般的转换，定义为： $$\开始{聚集} f{(a, b)}=\mathrm{E}{(a, b)}(x)=a x+b(\bmod 26) \quad 0 \leq a, b, x \leq 25 \operatorname {and} \operatorname{gcd}(a, 26)=1 \ f{(a, b)}^{-1}=\mathrm{D}_{(a, b)}(c)=a^{-1}(cb)(\bmod 26) \quad a^{ -1} \text { 是 } a \bmod 26 的倒数 \结束{聚集}$$

## 密码学代考

• Cryptosystem
• A system that describes how to encrypt or decrypt messages
• Plaintext
• Message in its original form
• Ciphertext
• Message in its encrypted form
• Cryptographer
• Invents encryption algorithms
• Cryptanalyst
• Breaks encryption algorithms or implementations

## 编码理论代写

1. 数据压缩（或信源编码
2. 前向错误更正（或信道编码
3. 加密编码
4. 线路码