Chapter 8.4, Problem 87AE

Cryptography One method of encryption is to use a matrix to encrypt the message and then use the corresponding inverse matrix to decode the message. The encrypted matrix, $E$, is obtained by multiplying the message matrix, $M$, by a key matrix, $K$. The original message can be retrieved by multiplying the encrypted matrix by the inverse of the key matrix. That is, $E\text{ = }M\cdot K$ and $M\text{ = }E\cdot K^{-1}$.

a. Given the key matrix $K\text{ = }\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}1\\ 1\\ 1\end{array}\begin{array}{c}1\\ 0\\ 1\end{array}\right]$, find its inverse, $K^{-1}$. [Note: This key matrix is known as the $Q_2^3$ Fibonacci encryption matrix.]

b. Use your result from part (a) to decode the encrypted matrix $E\text{ = }\left[\begin{array}{c}47\\ 44\\ 47\end{array}\begin{array}{c}34\\ 36\\ 41\end{array}\begin{array}{c}33\\ 27\\ 20\end{array}\right]$.

c. Each entry in your result for part (b) represents the position of a letter in the English alphabet. What is the original message?

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