This was an interesting question Professor Boutin
Part 1
How can Bob decrypt the message?
Bob can decrypt the message by multiplying the inverse of the 3-by-3 secret matrix with the coded message.
Part 2
Can Eve decrypt the message without finding the inverse of the secret matrix?
The asnwer is "yes." She can find the inverse of the secret matrix from the intercepted message. Or she can apply the intercepted information linearly to a wanted set of data.
The coded and decrypted message can be arranged in a 3-by-3 matrix form.
- $ Coded = \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix} \ $
- $ Decrypted = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \\ \end{bmatrix} \ $
Thus
$ Coded * A\ = Decrypted $
Or (A is the Secret Matrix)
$ \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix}\ * A = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \\ \end{bmatrix} $
- $ A = \begin{bmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \\ \end{bmatrix} * \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \\\end{bmatrix}^{-1} = \begin{bmatrix} 1/2 & 0 & 4/3 \\ 0 & 1 & 0 \\ 1/2 & 0 & 1/3 \\\end{bmatrix} $
Applying the secret matrix A to the message (2, 23, 3), we get,
$ Message\ = \begin{bmatrix} 2 & 23 & 3 \\\end{bmatrix} * \begin{bmatrix} 1/2 & 0 & 4/3 \\ 0 & 1 & 0 \\ 1/2 & 0 & 1/3 \\\end{bmatrix} = $
Part3
Method 1.
Previously, we observe that (1,0,4,0,1,0,1,0,1) yields (2,0,0,0,1,0,0,0,3).
(1, 0, 4) -----> Matrix Encryption -----> (2, 0, 12) ......................... (1)
(1, 0, 1) -----> Matrix Encryption -----> (2, 0, 3) ......................... (2)
Because the system is linear, (1) - (2) results in the subtraction of the result of (2) from that of (1).
(0, 0, 3) -----> Matrix Encryption -----> (0, 0, 9) ......................... (3)
(0, 1, 0) -----> Matrix Encryption -----> (0, 1, 0) ......................... (4)
Or
(0, 23, 0) -----> Matrix Encryption -----> (0, 23, 0) ......................... (4)'
Multiply (2) by 4 yields
(4, 0, 4) -----> Matrix Encryption -----> (8, 0, 12) ......................... (5)
Subtract (1) from (5)
(3, 0, 0) -----> Matrix Encryption -----> (6, 0, 0) ......................... (6)
Now multiply (6) by 2/3 to get (2, 0, 0).
(2, 0, 0) -----> Matrix Encryption -----> (9, 0, 0) ......................... (7)
Now add (7), (4)', and (3) together to get (2, 23, 3).
(2, 0, 0) + (0, 23, 0) + (0, 0, 3) ---------> Matrix Encryption ---------> (9, 0, 0) + (0, 23, 0) + (0, 0, 9)
(2, 23, 3) -----> Matrix Encryption -----> (9, 23, 9).
Method 2.
Use the inverse of the secret matrix.
