Decryption
In order to decrypt, we turn the ciphertext back into a vector, then simply multiply by the inverse matrix of the key matrix (IFKVIVVMI in letters). (There are standard methods to calculate the inverse matrix; see matrix inversion for details.) We find that, modulo 26, the inverse of the matrix used in the previous example is:
Taking the previous example ciphertext of 'POH', we get:
which gets us back to 'ACT', just as we hoped.
We have not yet discussed one complication that exists in picking the encrypting matrix. Not all matrices have an inverse (see invertible matrix). The matrix will have an inverse if and only if its determinant is not zero, and does not have any common factors with the modular base. Thus, if we work modulo 26 as above, the determinant must be nonzero, and must not be divisible by 2 or 13. If the determinant is 0, or has common factors with the modular base, then the matrix cannot be used in the Hill cipher, and another matrix must be chosen (otherwise it will not be possible to decrypt). Fortunately, matrices which satisfy the conditions to be used in the Hill cipher are fairly common.
For our example key matrix:
So, modulo 26, the determinant is 25. Since this has no common factors with 26, this matrix can be used for the Hill cipher.
The risk of the determinant having common factors with the modulus can be eliminated by making the modulus prime. Consequently a useful variant of the Hill cipher adds 3 extra symbols (such as a space, a period and a question mark) to increase the modulus to 29.
Read more about this topic: Hill Cipher