Part 1
Since Bob does not know a specific output given a specific input, he must use the inverse of the secret matrix.
Part 2
I don't believe that Eve will be able to decrypt the message because I don't think that the system is linear. If the system was linear she would be able to decrypt the message without the inverse of the secret matrix, however, for a system to be linear:
The system is obviously time invariant because it doesn't matter what time you evaluate the encrypted code with the secret matrix, it will always give you the same output. I'm not exactly sure how she would do it, I need some time to think about it. Does anyone have any ideas? Or is this completly wrong?
