How to break the code?

The coding and decoding techniques that we discussed, used invertible matrices which represent linear transformation. The purpose of cryptography is to find a secure ways of trsmitting information taht prevents unautorized entities from learning content of the message. So for each specific way of coding one of the main quaetions needed to ba answered is the following:

How much information needed for someone to break the code?

Since we are using linear transformations for coding and decoding when we use matrices, we need to learn about their propreties. Recall that any linear transformation $L: V \rightarrow W$ is completely detemined by the image of a basis for $V$. So if $A$ is an $n \times n$ matrix, we need to know n-plaintext vectors $P_1, P_2, \dots , P_n$, and the ciphertext ( coded) vectors $AP_1, AP_2, \dots , AP_n$ to break the code. Breakng the code means obtining the matrix $A^{-1}$.

To do this we may form a matrix $P$

$$P = [P_1\vert P_2\vert\dots \vert P_n]$$

whose columns are plaintext vectors

$$P_1, P_2, \dots , P_n$$

and let

$$Q = [AP_1\vert AP_2\vert\dots \vert AP_n]$$

Hence, $Q = AP$ and $A^{-1} = PQ^{-1}$. This will give us the tool to decode the message. To use row operation to find $A^{-1}$, you may want to write $A^{-1} = PQ^{-1}$ as $A^{-1} Q = P$ or $Q^T \left( A^{-1} \right) ^T = P^T$. To find $A^{-1}$ you need first solve for $\left( A^{-1} \right) ^T$, by row reducing ( using Gaussian Elimination ) $[Q^T\vert P^T]$ to $[I\vert\left( A^{-1} \right) ^T ]$.