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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 Encoder

### 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 D 3 $5 \times 3 + 3 = 18$ S O 14 $5 \times 14 + 3 = 73 = 21 \mod 26$ 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 = 1 A' = 1 A = 3 A' = 9 A = 5 A' = 21 A = 7 A' = 15 A = 9 A' = 3 A = 11 A' = 19 A = 15 A' = 7 A = 17 A' = 23 A = 19 A' = 11 A = 21 A' = 5 A = 23 A' = 17 A = 25 A' = 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.

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