Cryptography One method of encryption is to use a matrix to...

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, $E=M \cdot K$ and $M=E \cdot K^{-1}$. (a) Given the key matrix $K=\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{array}\right]$, 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 $$ \text { matrix } E=\left[\begin{array}{lll} 47 & 34 & 33 \\ 44 & 36 & 27 \\ 47 & 41 & 20 \end{array}\right] \text {. } $$ (c) Each entry in your result for part (b) represents the position of a letter in the English alphabet $(A=1, B=2$, $C=3$, and so on). What is the original message?

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, $E=M \cdot K$ and $M=E \cdot K^{-1}$.
(a) Given the key matrix $K=\left[\begin{array}{lll}2 & 1 & 1 \\ 1 & 1 & 0 \\ 1 & 1 & 1\end{array}\right]$, 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
$$
\text { matrix } E=\left[\begin{array}{lll}
47 & 34 & 33 \\
44 & 36 & 27 \\
47 & 41 & 20
\end{array}\right] \text {. }
$$
(c) Each entry in your result for part (b) represents the position of a letter in the English alphabet $(A=1, B=2$, $C=3$, and so on). What is the original message?

Key Concepts

Numerical Encoding of Text

Numerical encoding of text is a method where each letter of an alphabet is assigned a corresponding number (for example, A=1, B=2, etc.). In cryptographic schemes, these numbers can be organized in matrices, so that mathematical operations, such as matrix multiplication, can be applied directly to encoded messages. This representation bridges the gap between textual data and mathematical processing, enabling encryption and decryption techniques.

Matrix Inversion

The matrix inverse is a fundamental concept in linear algebra that provides a means to reverse the effects of matrix multiplication. For an encryption key matrix, if the inverse exists, multiplying the encrypted matrix by the inverse key matrix retrieves the original message matrix. Calculating the inverse is a crucial step in decryption processes that rely on linear transformations.

Matrix Multiplication in Cryptography

Matrix multiplication is used in cryptography to encode and decode messages by transforming the original message represented as a matrix. In such schemes, the message matrix is multiplied by a key matrix to produce an encrypted matrix. This operation is central to many cryptographic systems because it can conceal the original data, and its invertible properties allow for later recovery of the original message.

Transcript

00:01 So in this problem, we're working with some cryptography, and it makes the note in here that the original message could be retrieved by multiplying the cryptid matrix by the inverse of the key matrix.

00:16 Okay, so the first part, they give us the key matrix k.

00:20 So we need to find k inverse.

00:23 Now, the easy way to do this is to go to the matrix calculator.

00:27 So i'll go over here to desmos .com.

00:32 And then the math tools menu pull up that matrix calculator right there, and you get this matrix calculator.

00:39 So i'll add a new matrix here, and our matrix is a 3 by 3.

00:47 And so the entries are 2.

00:51 1, 1, 1, 1 ,0.

00:57 Let's see 1, 1, 1 ,0, and then 1 -1 -1.

01:04 1, 1, 1.

01:11 There's our matrix.

01:13 So then all i have to do is go a and a inverse with that key right there.