Application of Linearity
1. Bob knows the encryption matrix
The letters that Alice is sending will be sent in groups of three, then multiplied by her encryption matrix which we will call [A]. We will call her letter message x, and the encrypted output y. The dimension calculation is:
$ (3\times 3)*(3\times 1)=(3\times 1)\, $
$ [A]* \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} = \begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix} $
Now for bob to decrypt the message, he must multiply the inverse of [A], which is also a 3X3 matrix, to the left side of both sides of the equation. This will isolate the x values, which are the original message. Bob can find the inverse of the matrix using the determinants, some Gaussian methods, but we will assume he has a calculator or matlab to do it for him.
$ [A]^{-1}*[A]* \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} =[A]^{-1}* \begin{bmatrix} y_1\\ y_2\\ y_3 \end{bmatrix} = \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix} $
He knows all the y values, A, and the inverse of A, so he can find the x's, and thus the decoded message.
