The Hill cipher was the first cipher purely based on mathematics (linear algebra). To encipher a message, first the plaintext is broken into blocks of n letters which are converted to numbers, where A=0, B=1, C=2. Y=24, Z=25 (so each character is assigned to a number which is usually from the range of 00-25 for the characters A-Z.
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Upper case and lower case characters are treated equally). Then the encryption is done by multiplying the numbers with an n x n key matrix modulo 26 (if we have A-Z as our alphabet). The result is converted back to text producing the ciphertext. Let’s assume that we want to encode the message 'ACT' with the key 'GYBNQKURP'.¹ Since G=6, Y= 24, B=1 etc. Guitar serial numbers. We get the following matrix for the chosen key: The message is thus encoded by this vector: Key and message are multiplied with each other and apply modulo 26 to the result: This result (15, 14, 7) can be decoded by 'POH' which would be the output of the Hill cipher for the chosen message and the used key.
To decode the message, one would have to multiply the ciphertext with the inverse matrix of the key and apply modulo 26 to the result. Details: The key has to be chosen in such a way that there exists an inverse matrix for the key matrix because it would be impossible to decode the message otherwise. Therefore the determinant of the key matrix modulo 26 has to be co-prime to 26. Numbers co-prime to 26 are: 1,3,5,7,9,11,15,17,19,21,23,25. The determinant of the key matrix shown above is therefore calculated as such: Some implementations like only allow modulo values which are primes.
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This is better for security but no requirement of the original method. If a length like 26 is used, then this website complains e.g. 'Hill cipher won't work unless the alphabet length is prime.' This extra requirement can be achieved by adding e.g. An underscore as the first letter. Implementations without this additional restriction and with the possibility to choose matrix dimensions n other than 2 or 3 are: CrypTool 1, CrypTool 2, and SageMath.
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(1) This sample is taken from en.wikipedia.org/wiki/Hillcipher, 2017-06-05.
To perform matrix multiplication we 'combine' the top row of the key matrix with the column vector to get the top element of the resulting column vector. We then 'combine' the bottom row of the key matrix with the column vector to get the bottom element of the resulting column vector. The way we 'combine' the four numbers to get a single number is that we multiply the first element of the key matrix row by the top element of the column vector, and multiply the second element of the key matrix row by the bottom element of the column vector.
We then add together these two answers. Now we perform matrix multiplication, multiplying the key matrix by each column vector in turn. To perform matrix multiplication we 'combine' the top row of the key matrix with the column vector to get the top element of the resulting column vector. We then 'combine' the middle row of the key matrix with the column vector to get the middle element of the resulting column vector.
And similarly for the bottom row. The way we 'combine' the six numbers to get a single number is that we multiply the first element of the key matrix row by the top element of the column vector, multiply the second element of the key matrix row by the middle element of the column vector, and multiply the third element of the key matrix row by the bottom element of the column vector.
We then add together these three answers. Step 1 - Find the Multiplicative Inverse of the Determinant The is a number that relates directly to the entries of the matrix. For a 3 x 3 matrix it is found by multiplying the top left entry by the determinant of the 2 x 2 matrix formed by the entries that are not in the same row or column as that entry (that is the 2 x 2 matrix not including the top row or left column). Similar steps are done with the other two elements in the top row, and the middle value is subtracted from the sum of the other two. This is shown more clearly in the algebraic version below. Although this seems a bit of a random selection of letters to place in each of the discriminants, it is defined as the transpose of the cofactor matrix, which is much easier to remember how to work out. To find the cofactor matrix, we take the 2 x 2 determinant in each position such that the four values in that position are the four values not in the same row or column as the position in the original matrix.
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The adjugate is then formed by reflecting the cofactor matrix along the line from top left ot bottom right. Discussion The most important item that must be discussed regarding the use of the Hill Cipher is that not every possible matrix is a possible key matrix. This is because, in order to decrypt, we need to have an inverse key matrix, and not every matrix is invertible. Fortunately, we do not have to work out the entire inverse to find it is not possible, but simply consider the determinant. If the determinant is 0 or shares a factor, other than 1, with the length of the alphabet being used, then the matrix will not have an inverse. If this is the case, a different key must be chosen, since otherwise the ciphertext will not be able to be decrypted. In order to be a usable key, the matrix must have a non-zero determinant which is coprime to the length of the alphabet.
The security of a 2 x 2 Hill Cipher is similar (actually slightly weaker) than the Bifid or, and it is somewhat more laborious to implement by paper and pencil mmethods. However, as the key matrix size increases, so does the security, and also the complexity of operating the cipher. It is possible to increase the key size further than the 3 x 3 we have discussed here, but the mathematics involved gets rapidly more complex. Hill invented a machine that would mechanically implement a 6 x 6 version of the cipher, which was very secure. Unfortunately, the machine was unable to change the key setting, leaving it with limited use in the real world.