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# 数学代写| Public Key Systems 离散数学代写

## 数学代写| Public Key Systems 代考

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## 离散数学代写

A public key cryptosystem (Fig. 10.7) is an asymmetric key system where there is a separate key $e_{k}$ for encryption and $d_{k}$ decryption with $e_{k} \neq d_{k}$. Martin Hellman and Whitfield Diffie invented it in 1976 . The fact that a person is able to encrypt a
\begin{tabular}{l|l|}
\hline Table. & $10.3$ DES encryption \
\hline Step & Description \
\hline 1 & Expansion of the 32 -bit half block to 48 bits (by duplicating half of the bits) \
\hline 2 & The 48 -bit result is combined with a 48 bit subkey of the secret key using an XOR operation \
\hline 3 & The 48 -bit result is broken in to $8 * 6$ bits and passed through 8 substitution boxes to yield $8 * 4=32$ bits (This is the core part of the encryption algorithm) \
\hline 4 & The 32 -bit output is re-arranged according to a fixed permutation \
\hline
\end{tabular}$10.5$ Public Key Systems
Table. $10.5$ Advantages and disadvantages of public key cryptosystems
\begin{tabular}{l|l|}
\hline Only the private key needs to be kept secret & Public keys must be authenticated \
\hline The distribution of keys for encryption is convenient, & It is slow and uses more computer \
as everyone publishes their public key and the & resources \
private key is kept private & \
It provides message authentication as it allows the & Security Compromise is possible (if \
use of digital signatures (which enables the recipient & private key is compromised) \
to verify that the message is really from the particular & \
sender) & The sender encodes with the private key that is known only to sender. The receiver decodes with the public key and therefore knows that the message is from the sender \
\hline Detection of tampering (digital signatures enable the receiver to detect whether message was altered in transit) & Loss of private key may be irreparable (unable to decrypt messages) \
\hline Provides for non-repudiation & \
\hline
\end{tabular}
compute problem for large $n$ since there is no efficient algorithm to factorize a large integer into its prime factors (integer factorization problem).
compute problem f a large integer into
(ii) The function $f_{g, N}: x \rightarrow g^{x}(\bmod \mathrm{N})$ is a one way function since it is easy to compute. However, the inverse function $f^{-1}$ is difficult to compute as there is no efficient method to determine $x$ from the knowledge of $g^{x}(\bmod \mathrm{N})$ and $g$ and $\mathrm{N}$ (the discrete logarithm problem).
(iii) The function $f_{k, N}: x \rightarrow x^{k}(\bmod \mathrm{N})$ (where $\mathrm{N}=p q$ and $p$ and $q$ are primes) and $k k^{\prime \prime} \equiv 1(\bmod \mathrm{N})$ is a trapdoor function. It is easy to compute but the inverse of $f$ (the $k^{\text {th }}$ root modulo $\mathrm{N}$ ) is difficult to compute. However, if the trapdoor $k^{\prime}$ is given then $f$ can easily be inverted as $\left(x^{k}\right)^{k^{\prime}} \equiv x(\bmod \mathrm{N})$
10.5.1 RSA Public Key Cryptosystem
Rivest, Shamir and Adleman proposed a practical public key cryptosystem (RSA) based on primality testing and integer factorization in the late $1970 \mathrm{~s}$. The RSA algorithm was filed as a patent (Patent No. 4,405, 829) at the U.S. Patent Office in December 1977. The RSA public key cryptosystem is based on the following assumptions:

• It is straightforward to find two large prime numbers.
• The integer factorization problem is infeasible for large numbers.

\开始{表格}{l|l|}
\hline 表。 & $10.3$ DES 加密 \
\hline 步骤和说明 \
\hline 1 & 将 32 位半块扩展为 48 位（通过复制一半位）\
\hline 2 & 使用 XOR 操作将 48 位结果与密钥的 48 位子密钥组合在一起
\hline 3 & 将 48 位结果分解为 $8 * 6$ 位并通过 8 个替换框得到 $8 * 4=32$ 位（这是加密算法的核心部分）\
\hline 4 & 32 位输出根据固定排列重新排列 \
\hline
\end{tabular}$10.5$ 公钥系统

\开始{表格}{l|l|}

\hline 只有私钥需要保密 & 公钥必须经过身份验证 \
\hline 用于加密的密钥分配方便，& 速度慢，使用较多电脑 \

\hline 篡改检测（数字签名使接收方能够检测消息在传输过程中是否被更改）& 私钥丢失可能无法弥补（无法解密消息）\
\hline 提供不可否认性 & \
\hline
\end{表格}

(ii) 函数 $f_{g, N}: x \rightarrow g^{x}(\bmod \mathrm{N})$ 是单向函数，因为它易于计算。然而，逆函数 $f^{-1}$ 很难计算，因为没有有效的方法来根据 $g^{x}(\bmod \mathrm{N})$ 和 $的知识确定$x$g$ 和 $\mathrm{N}$（离散对数问题）。
(iii) 函数 $f_{k, N}: x \rightarrow x^{k}(\bmod \mathrm{N})$ (其中 $\mathrm{N}=pq$ and $p$ and $q$是素数），$kk^{\prime \prime} \equiv 1(\bmod \mathrm{N})$ 是陷门函数。它很容易计算，但 $f$ 的倒数（$k^{\text {th }}$ 根模 $\mathrm{N}$ ）很难计算。但是，如果给定陷门 $k^{\prime}$，则 $f$ 可以很容易地反转为 $\left(x^{k}\right)^{k^{\prime}} \equiv x(\ bmod \mathrm{N})$
10.5.1 RSA公钥密码系统
Rivest、Shamir 和 Adleman 在 1970 年后期提出了一种实用的公钥密码系统 (RSA)，该系统基于素数测试和整数分解。 RSA 算法于 1977 年 12 月在美国专利局作为专利（专利号 4,405, 829）提交。RSA 公钥密码系统基于以下假设：

• 找到两个大素数很简单。
• 整数分解问题对于大数是不可行的。

## 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