Multiplicative Cipher

The Multiplicative Cipher (or Decimation Cipher) is an Affine cipher (ax+b) with the value b null (equal to 0), so a multiplication by $ a $.

Multiplicative encryption uses a key $ k $ (an integer) and an alphabet.

Each letter is associated with its rank $ c $ in the alphabet (starting from 0).

For each character of the plain message, apply the following calculation:

$$ c \times k \mod 26 $$

($ 26 $ being the number of letters in the alphabet)

The number obtained indicates the rank in the alphabet of the corresponding numbered letter.

Decryption can be done in 2 ways:

— Mathematically, calculate the modular inverse $ k^{-1} $ of the key modulo 26 and apply the calculation for each letter:

$$ c \times k^{-1} \mod 26 $$

— By substitution, in fact, during encryption each letter is associated with only one other, by calculating all the possible associations (by encrypting the 26 letters of the alphabet) then it is possible to deduce an alphabet substitution that will serve as a decryption table.

For the encryption to be reversible (so that the message can be decrypted), the key must be a coprime number with 26 (where 26 is the number of letters of the alphabet).

So there is an infinite number of possible keys, but many will give identical messages, because for a $ k $ key, then the $ k + 26 $ key gives an identical cipher.

The message is an alphabetical substitution, the frequency analysis should make it possible to find the most common letters.

The index of coincidence is unchanged from plain text.

The letter A remains unchanged ans id always encoded A

For a given alphabet, there are only a few possible keys.

The 26-letter Latin alphabet allows only 11 keys: 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 (these are coprime numbers with 26).

The multiplicative cipher is a simplification of the Affine cipher.

The multiplicative cipher has little interest, but it is often used for learning computer science and ciphers.