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I have a plaintext "monday" and ciphertext "IKTIWM" and $m=2$. I want to find the key of the Hill cipher.
I made a matrix $$ \begin{bmatrix} a_1 & a_2 \\ a_3 & a_4 \end{bmatrix}\begin{bmatrix} m \\ o \end{bmatrix} = \begin{bmatrix} I \\ K \end{bmatrix} \pmod{26}$$
$X=\{\{m,o\},\{n,d\}\}$, $Y=\{\{I,K\},\{T,I\} \}$, I want to find $X \times K=Y$.
I will multiply this equation with inverse($X$).
But for the modulo inverse you need $gcd$(determinant($X), 26) =1$ . Which is not happening here.
These modular equations are not uniquely solvable:
$$\begin{bmatrix}7&2\\ 10& 20\end{bmatrix}, \begin{bmatrix}7&2\\ 23& 7\end{bmatrix}, \begin{bmatrix}20&15\\ 10& 20\end{bmatrix}, \begin{bmatrix}20&15\\ 23& 7\end{bmatrix}$$
are all the $2 \times 2$ matrices over $\mathbb{Z}_{26}$ would transform 'monday' to IKTIWM, the first and third have even determinant so are not invertible so the second or the fourth candidate encryption matrix is the correct one: invert them and check the rest of the text which is one is actually correct.
