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Affine Cipher

Tool to decrypt/encrypt with Affine cipher, an encryption function with additions and multiplication that code a letter into another with value (ax + b) modulo 26.

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Affine Cipher -

Tag(s) : Substitution Cipher

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Affine Cipher

Affine Decoder





Manual Parameters and Options







Affine Encoder










Answers to Questions (FAQ)

What is the Affine cipher? (Definition)

Affine encryption is the name given to a substitution cipher whose correspondence is given by an affine function endowed with 2 coefficients A and B.

How to encrypt using Affine cipher

Encryption uses a classic alphabet, and two integers, called coefficients or keys A and B, these are the parameters of the affine function Ax+B.

Example: Encrypt DCODE with the keys A=5, B=3 and the English/latin alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ.

For each letter of the alphabet is associated to the value of its position in the alphabet (starting at 0).

Example: By default, A=0, B=1, …, Z=25, but it is possible (but not recommended) to use A=1, …, Y=25, Z=0 using the alphabet ZABCDEFGHIJKLMNOPQRSTUVWXY.

For each letter of value $ x $ of the plain text, is associated a value $ y $, resulting of the affine function $ y = A \times x + B \mod 26 $ (with $ 26 $ the alphabet size). For each value $ y $, corresponds a letter with the same position in the alphabet, it is the ciphered letter. The Affine ciphertext is the replacement of all the letters by the new ones.

Example: DCODE is crypted SNVSX

Plain letter$ x $$ y $Cipher letter
D3$ 5 \times 3 + 3 = 18 $S
O14$ 5 \times 14 + 3 = 73 = 21 \mod 26 $V

How to decrypt Affine cipher

Affine decryption requires to know the two keys A and B (the one from encryption) and the used alphabet.

Example: Decrypt the ciphered message SNVSX with keys A=5 and B=3

For each letter of the alphabet corresponds the value of its position in the alphabet.

Example: The alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ, starting at 0 gives A=0, B=1, …, Z=25.

For each letter of value $ y $ of the message, corresponds a value $ x $, result of the inverse function $ x = A' \times (y-B) \mod 26 $ (with $ 26 $ the alphabet size)

The value $ A' $ is an integer such as $ A \times A' = 1 \mod 26 $ (with $ 26 $ the alphabet size). To find $ A' $, calculate its modular inverse.

Example: A coefficient $ A' $ for $ A = 5 $ with an alphabet size of $ 26 $ is $ 21 $ because $ 5 \times 21 = 105 \equiv 1 \mod 26 $.
For S ( $ y = 18 $ ), $ x = A' \times (18-B) = 21 \times (18-3) \equiv 315 \mod 26 = 3 $

For each value $ x $, corresponds a letter with the same position in the alphabet: the coded letter. The plain text is the replacement of all characters with calculated new letters.

Example: For S ( $ x = 3 $ ) corresponds the letter at position 3: D, etc. The original plain text is DCODE.

How to recognize an Affine ciphertext?

A message encrypted by Affine has a coincidence index close to the plain text language's one.

Any reference to an affine function (in a straight line), a graph, an abscissa or an ordinate is a clue (the function $ f(x) = ax + b $ can be represented in an orthonormal coordinate system like a classical affine function, it is therefore possible from a graph to find the slope coefficient $ a $ and the y-intercept $ b $).

The Caesar cipher is a special case of the Affine cipher where A is 1 and B is the shift/offset.

How to decipher Affine without coefficient A and B?

To crack Affine, it is possible to bruteforce/test all values for A and B coefficients. Use the Brute-force attack button.

If the alphabet is 26 characters long, then A coefficient has only 12 possible values, and B has 26 values, so there are only 312 test to try.

How to compute the decryption function?

Pour an affine encryption with the function $ y = A x + B $, then the reciproqual decryption function is expressed $ y' = A' x + B $

How to compute A' value?

Calculate the modular inverse of A, modulo the length of the alphabet (see below for pre-calculated values).

How to compute B' value?

B' has the same value as B, for this reason, this variable should not be called B' but B.

What are the A' values?

The value of A' depends on A but also on the alphabet's length, if it is a classic one, it is 26 characters long. The values of A' are then:

A = 1A' = 1
A = 3A' = 9
A = 5A' = 21
A = 7A' = 15
A = 9A' = 3
A = 11A' = 19
A = 15A' = 7
A = 17A' = 23
A = 19A' = 11
A = 21A' = 5
A = 23A' = 17
A = 25A' = 25

Why is there a constraint on the value of A?

The Bezout's theorem indicates that A' only exists if A and 26 (alphabet length) are coprime. This limits A values to 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 and 25 (if the alphabet is 26 characters long)

Is it possible to use a key A not coprime with 26?

Yes, but an automatic decryption process becomes impossible, a single ciphered letter will have multiple plain letters possible.

Does a negative value for A exists?

Yes, but it exists a positive corresponding value, a value of A = -1 is equals to a value of A = 25 (because 25 = -1 mod 26).

Is there a limitation on B value?

No, B can take any value.

All the values of B modulo 26 (length of the alphabet) are equivalent. So if B is negative, there is an equivalent positive value of B.

Example: 'B = -1' is equivalent to 'B = 25' (modulo 26)

Why is this encryption so called affine?

In mathematics, an affine function is defined by addition and multiplication of the variable (often $ x $) and written $ f(x) = ax + b $. The affine cipher is similar to the $ f $ function as it uses the values $ a $ and $ b $ as a coefficient and the variable $ x $ is the letter to be encrypted.

When was Affine invented?

No date nor known author for affine cipher.

Source code

dCode retains ownership of the online 'Affine Cipher' tool source code. Except explicit open source licence (indicated CC / Creative Commons / free), any 'Affine Cipher' algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any 'Affine Cipher' function (calculate, convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (Python, Java, PHP, C#, Javascript, Matlab, etc.) and no data download, script, copy-paste, or API access for 'Affine Cipher' will be for free, same for offline use on PC, tablet, iPhone or Android ! dCode is free and online.

Need Help ?

Please, check our dCode Discord community for help requests!
NB: for encrypted messages, test our automatic cipher identifier!

Questions / Comments

Thanks to your feedback and relevant comments, dCode has developed the best 'Affine Cipher' tool, so feel free to write! Thank you!


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