Tool to decrypt/encrypt with Hill cipher, a ciphering system similar to affine cipher but using a coefficient matrix instead of 2 affine coefficients (gradient).

Hill Cipher - dCode

Tag(s) : Poly-Alphabetic Cipher

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See also: Inverse of a Matrix

Tool to decrypt/encrypt with Hill cipher, a ciphering system similar to affine cipher but using a coefficient matrix instead of 2 affine coefficients (gradient).

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

Example: The plain text is DCODE and the matrix is $ 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 multiplication">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 $, find back 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.

**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.

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

Any reference to an actual **hill** or mountain is a clue.

dCode proposes to bruteforce test around 6000 combinations of 2x2 matrices (with digits between 1 and 9) and alphabets.

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

In 1929 by Lester S. **Hill**

dCode retains ownership of the source code of the script Hill Cipher online. Except explicit open source licence (indicated Creative Commons / free), any algorithm, applet, snippet, software (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt, encrypt, decipher, cipher, decode, code, translate) written in any informatic langauge (PHP, Java, C#, Python, Javascript, Matlab, etc.) which dCode owns rights will not be released for free. To download the online Hill Cipher script for offline use on PC, iPhone or Android, ask for price quote on contact page !

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Source : https://www.dcode.fr/hill-cipher

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