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I'm trying to create a hill cipher utility. One feature I want is to be able to compute the key if you have the plaintext and ciphertext.

$C$ = ciphertext matrix ($2\times 2$), $P$ = plaintext matrix $\left(2\times\frac{N}{2}\right)$, $K$ = key ($2\times 2$).

$PK = C$

So if I have $C$ and $P$, then $K = P^{-1}C$. How do I best find $P^{-1}$?

EDIT: It is important to note that this is all mod $26$. Sorry.

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    See also [this](http://stackoverflow.com/questions/960190) related question on StackOverflow.2012-10-05

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You could use the direct formula for $2 \times 2$:

$ \pmatrix{a & b\\ c & d}^{-1} = \dfrac{1}{ad-bc}\pmatrix{d & -b\\-c & a}$

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    Yes! Also, consider that $2$ and $13$ and any multiples (basically just even numbers and the number $13$) do not have modular inverses.2012-10-05