Part C: Application of linearity
1. How can Bob decrypt the message?
Bob can get the message by multiplying the Message by the Secret message then decoding the numbers into letters.
2. Can Eve decrypt the message without finding the inverse of the secret matrix?
Yes she can. She can right a system of equations and solve for each component of the secret message.
- $ \begin{pmatrix} 1 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix}\begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 3 \end{pmatrix} $
- Multiply out
- $ a+4g=2, b+4h=0, c+4i=0 \, $
- $ d=0, e=1, f=0 \, $
- $ a+g=0, b+h=0, c+i=0 \, $
Solving These Equations yields the Secret Matrix
- $ \begin{pmatrix} 2/3 & 0 & 4 \\ 0 & 1 & 0 \\ 1 & 0 & 1 \end{pmatrix} $
