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Hill Cipher

Tool to decrypt/encrypt with Hill cipher, a ciphering system similar to affine cipher but using a matrix for the gradient.

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Hill Cipher -

Tag(s) : Cryptography, Substitution Cipher

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# Hill Cipher

## Matrix Inversion

Tool to decrypt/encrypt with Hill cipher, a ciphering system similar to affine cipher but using a matrix for the gradient.

### How to encrypt using Hill cipher?

Hill cipher encryption uses a matrix $$M$$ (and an alphabet).

Example: Consider the plain text DCODE and the matrix $$M$$ (size 2): $$M = \begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix}$$ and the latin alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ

Split the text into n-grams with $$n$$ the matrix size, if needed, complete the last n-gram with any letter.

Example: The matrix $$M$$ is a 2x2 matrix, DCODE, split in 2-grams, becomes DC,OD,EZ (Z letter has been added to complete the last bigram)

Substitute the letters of the plain message with a value: their rank in the alphabet starting from 0.

Example: For ABCDEFGHIJKLMNOPQRSTUVWXYZ, it leads to A=0,B=1,...,Z=25. Sometimes ZABCDEFGHIJKLMNOPQRSTUVWXY is used in order to get A=1,B=2,...Y=25,Z=0.
Groups of letters DC, OD, EZ become the groups of values (3,2), (14,3), (4,25)

For each group of values P of the plain text (equivalent to a vector of size $$n$$), compute the matrix product: $$M.P \equiv C \mod 26$$ where $$C$$ is a vector (a group) of ciphered values and $$26$$ the alphabet length.

Example: $$\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix} \begin{pmatrix} 3 \\ 2 \end{pmatrix} \equiv \begin{pmatrix} 12 \\ 3 \end{pmatrix} \mod 26$$

From cipher values C, one can find the cipher letters with their rank in the alphabet.

Example: 12 is equal to M and 3 is equal to D.
So, DCODEZ is encrypted MDLNFN.

### How to decrypt Hill cipher?

Hill cipher decryption needs the matrix and the alphabet used. Decryption involves matrix computations such as matrix inversion, and arithmetic calculations such as modular inverse.

To decrypt hill ciphertext, compute the matrix inverse modulo 26 (where 26 is the alphabet length), requiring the matrix to be invertible.

Example: Using the example matrix, compute the inverse matrix (modulo 26) : $$\begin{pmatrix} 2 & 3 \\ 5 & 7 \end{pmatrix}^{-1} \equiv \begin{pmatrix} -7 & 3 \\ 5 & -2 \end{pmatrix} \equiv \begin{pmatrix} 19 & 3 \\ 5 & 24 \end{pmatrix} \mod 26$$

Decryption consists in encrypting the ciphertext with the inverse matrix.

Note that not all matrices can be adapted to hill cipher. The determinant of the matrix has to be coprime with 26. For a 2x2 matrix, the 4 numbers $$\{ a,b,c,d \}$$ must satisfy the condition that $$ad-bc$$ is coprime with 26.

### How to recognize Hill ciphertext?

The ciphered message has a small index of coincidence and similar ngrams can be coded using the same letters.

### How to decipher Hill without matrix?

dCode proposes to bruteforce test around 8000 combinations of 2x2 matrices and alphabets.

### What are the variants of the Hill cipher?

Hill is already a variant of Affine cipher. Few variants, except the use of large size matrices.

### When Hill cipher have been invented?

In 1929 by Lester S. Hill