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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.

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  • $\begingroup$ I am making a matrix X={ {m,o}, {nd} },Y={ {I,K} ,{T,I} },I want to find X*K=Y; $\endgroup$ Commented Feb 1, 2019 at 11:20
  • $\begingroup$ I edited it.I don't know how to write a matrix here. $\endgroup$ Commented Feb 1, 2019 at 11:23
  • $\begingroup$ Hint: not all systems of 6 equations with 4 unknowns have a unique solution. Find them all. $\endgroup$
    – fgrieu
    Commented Feb 1, 2019 at 11:52
  • $\begingroup$ these are the equations. 12a + 14b = 8 , 12c + 14d = 10 ,13a + 3d = 19 ,13c + 3d = 8 , 24b=22 , 24d= 12. I have replaced a1 with a , a2 with b , a3 with c , a4 with d. Can we solve them? $\endgroup$ Commented Feb 1, 2019 at 12:33
  • $\begingroup$ you are right. But can you help me solve it? $\endgroup$ Commented Feb 1, 2019 at 12:55

1 Answer 1

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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.

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