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# 数学代写|密码学代写Cryptography代考|CSE599 Levine and Hill

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## 数学代写|密码学代写CRYPTOGRAPHY代考|Levine and Hill

Lester Hill (Figure 6.1) is almost always given credit for discovering matrix encryption, but as David Kahn pointed out in his definitive history of the subject, Jack Levine’s work in this area preceded Hill’s. Levine (Figure 6.2) also pointed this out in a 1958 paper. ${ }^1$ Nevertheless, matrix encryption is typically referred to as the Hill cipher. Levine explained how this happened: ${ }^2$
Sometime during 1923-1924 while I was a high-school student I managed to construct a cipher system which would encipher two completely independent messages so that one message could be deciphered without disclosing that a second message was still hidden (I also did for three messages). Not long thereafter (end of 1924) Flynn’s Weekly magazine started a cipher column conducted by M. E. Ohaver (Sanyam of A.C.A.), a most excellent series appearing every few weeks. Readers were encouraged to submit their systems which Ohaver then explained in a later issue. So I sent him my system mentioned above and it appeared in the Oct. 22, 1926 issue with its explanation in the Nov. 13, 1926 issue. Ohaver gave a very good explanation and some other interesting remarks […] In 1929 Hill’s first article appeared in the Math. Association Monthly Journal recognized he had a very general system which could encipher plaintext units of any length very easily, but the basic principle was the same as my 2-message system (or 3-message). I had some correspondence with Hill and I think told him of my youthful efforts. All this to explain why I believe my system was the precursor to his very general mathematical formulation (I also used equations).

## 数学代写|密码学代写CRYPTOGRAPHY代考|How Matrix Encryption Works

Matrix encryption can most easily be explained by example. Consider Oscar Wilde’s quote “THE BEST WAY TO DEAL WITH TEMPTATION IS TO YIELD TO IT.”We first replace each letter with its numerical equivalent: $\begin{array}{lllllllllllllllllllllllllllllll}0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 & 25\end{array}$
This gives
$\begin{array}{rlllllllllllllllllllllllllllllllllll}19 & 7 & 4 & 1 & 4 & 18 & 19 & 22 & 0 & 24 & 19 & 14 & 3 & 4 & 0 & 11 & 22 & 8 & 19 & 7 & 19 & 4 & 12 & 15 \ 19 & 0 & 19 & 8 & 14 & 13 & 8 & 18 & 19 & 14 & 24 & 8 & 4 & 11 & 3 & 19 & 14 & 8 & 19 & & & \end{array}$
The key we select in this system is an invertible matrix (modulo 26$)$ such as $\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)$. To encipher, we simply multiply this matrix by the numerical version of the plaintext in pieces of length two and reduce the result modulo 26 ; for example,
$$\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 19 \ 7 \end{array}\right)=\left(\begin{array}{c} 6 \cdot 19+11 \cdot 7 \ 3 \cdot 19+5 \cdot 7 \end{array}\right)=\left(\begin{array}{c} 191 \ 92 \end{array}\right)=\left(\begin{array}{c} 9 \ 14 \end{array}\right)(\text { modulo } 26)$$

gives the first two ciphertext values as 9 and 14 or, in alphabetic form, J O. Continuing in this manner we have
$$\left(\begin{array}{lc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 4 \ 1 \end{array}\right)=\left(\begin{array}{c} 9 \ 17 \end{array}\right) \quad\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 4 \ 18 \end{array}\right)=\left(\begin{array}{l} 14 \ 24 \end{array}\right) \quad\left(\begin{array}{lc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 19 \ 22 \end{array}\right)=\left(\begin{array}{l} 18 \ 11 \end{array}\right)$$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}0 \ 24\end{array}\right)=\left(\begin{array}{c}4 \ 16\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}19 \ 14\end{array}\right)=\left(\begin{array}{c}8 \ 23\end{array}\right) \quad\left(\begin{array}{lc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}3 \ 4\end{array}\right)=\left(\begin{array}{c}10 \ 3\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}0 \ 11\end{array}\right)=\left(\begin{array}{c}17 \ 3\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}22 \ 8\end{array}\right)=\left(\begin{array}{c}12 \ 2\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 7\end{array}\right)=\left(\begin{array}{c}9 \ 14\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 4\end{array}\right)=\left(\begin{array}{c}2 \ 25\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}12 \ 15\end{array}\right)=\left(\begin{array}{l}3 \ 7\end{array}\right) \quad\left(\begin{array}{lc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 0\end{array}\right)=\left(\begin{array}{c}10 \ 5\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 8\end{array}\right)=\left(\begin{array}{l}20 \ 19\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}14 \ 13\end{array}\right)=\left(\begin{array}{c}19 \ 3\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}8 \ 18\end{array}\right)=\left(\begin{array}{c}12 \ 10\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}19 \ 14\end{array}\right)=\left(\begin{array}{c}8 \ 23\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}24 \ 8\end{array}\right)=\left(\begin{array}{c}24 \ 8\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}4 \ 11\end{array}\right)=\left(\begin{array}{l}15 \ 15\end{array}\right)$
$$\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 3 \ 19 \end{array}\right)=\left(\begin{array}{c} 19 \ 0 \end{array}\right) \quad\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 14 \ 8 \end{array}\right)=\left(\begin{array}{c} 16 \ 4 \end{array}\right) \quad\left(\begin{array}{ll} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 19 \ 23 \end{array}\right)=\left(\begin{array}{c} 3 \ 16 \end{array}\right)$$

## 数学代写|密码学代写CRYPTOGRAPHY代考|矩阵加密如何工作

$\begin{array}{rlllllllllllllllllllllllllllllllllll}19 & 7 & 4 & 1 & 4 & 18 & 19 & 22 & 0 & 24 & 19 & 14 & 3 & 4 & 0 & 11 & 22 & 8 & 19 & 7 & 19 & 4 & 12 & 15 \ 19 & 0 & 19 & 8 & 14 & 13 & 8 & 18 & 19 & 14 & 24 & 8 & 4 & 11 & 3 & 19 & 14 & 8 & 19 & & & \end{array}$我们在这个系统中选择的键是一个可逆矩阵(模26$)$ 例如 $\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)$。要进行加密，我们只需将这个矩阵乘以长度为2的明文的数值版本，并对结果取模26;例如，
$$\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 19 \ 7 \end{array}\right)=\left(\begin{array}{c} 6 \cdot 19+11 \cdot 7 \ 3 \cdot 19+5 \cdot 7 \end{array}\right)=\left(\begin{array}{c} 191 \ 92 \end{array}\right)=\left(\begin{array}{c} 9 \ 14 \end{array}\right)(\text { modulo } 26)$$

$$\left(\begin{array}{lc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 4 \ 1 \end{array}\right)=\left(\begin{array}{c} 9 \ 17 \end{array}\right) \quad\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 4 \ 18 \end{array}\right)=\left(\begin{array}{l} 14 \ 24 \end{array}\right) \quad\left(\begin{array}{lc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 19 \ 22 \end{array}\right)=\left(\begin{array}{l} 18 \ 11 \end{array}\right)$$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}0 \ 24\end{array}\right)=\left(\begin{array}{c}4 \ 16\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}19 \ 14\end{array}\right)=\left(\begin{array}{c}8 \ 23\end{array}\right) \quad\left(\begin{array}{lc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}3 \ 4\end{array}\right)=\left(\begin{array}{c}10 \ 3\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}0 \ 11\end{array}\right)=\left(\begin{array}{c}17 \ 3\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}22 \ 8\end{array}\right)=\left(\begin{array}{c}12 \ 2\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 7\end{array}\right)=\left(\begin{array}{c}9 \ 14\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 4\end{array}\right)=\left(\begin{array}{c}2 \ 25\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}12 \ 15\end{array}\right)=\left(\begin{array}{l}3 \ 7\end{array}\right) \quad\left(\begin{array}{lc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 0\end{array}\right)=\left(\begin{array}{c}10 \ 5\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}19 \ 8\end{array}\right)=\left(\begin{array}{l}20 \ 19\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}14 \ 13\end{array}\right)=\left(\begin{array}{c}19 \ 3\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}8 \ 18\end{array}\right)=\left(\begin{array}{c}12 \ 10\end{array}\right)$
$\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}19 \ 14\end{array}\right)=\left(\begin{array}{c}8 \ 23\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{c}24 \ 8\end{array}\right)=\left(\begin{array}{c}24 \ 8\end{array}\right) \quad\left(\begin{array}{cc}6 & 11 \ 3 & 5\end{array}\right)\left(\begin{array}{l}4 \ 11\end{array}\right)=\left(\begin{array}{l}15 \ 15\end{array}\right)$
$$\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 3 \ 19 \end{array}\right)=\left(\begin{array}{c} 19 \ 0 \end{array}\right) \quad\left(\begin{array}{cc} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{c} 14 \ 8 \end{array}\right)=\left(\begin{array}{c} 16 \ 4 \end{array}\right) \quad\left(\begin{array}{ll} 6 & 11 \ 3 & 5 \end{array}\right)\left(\begin{array}{l} 19 \ 23 \end{array}\right)=\left(\begin{array}{c} 3 \ 16 \end{array}\right)$$

## 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。