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Affine Cipher

Tool to decrypt/encrypt with Affine automatically. The Affine cipher uses a encrypting function with additions and multiplication (as the mathematical affine function) which convert a letter (of value x in a 26 letters alphabet) into another letter with value (ax + b) modulo 26.

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Affine Cipher -

Tag(s) : Cryptography, Substitution Cipher

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Affine Cipher

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Tool to decrypt/encrypt with Affine automatically. The Affine cipher uses a encrypting function with additions and multiplication (as the mathematical affine function) which convert a letter (of value x in a 26 letters alphabet) into another letter with value (ax + b) modulo 26.

Answers to Questions

How to encrypt using Affine cipher

Encryption uses a classic alphabet, and two integers, called coefficients or keys A and B.

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

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

Example: A=0, B=1,..., Z=25. It is possible to use A=1, ..., Y=25, Z=0 with the alphabet ZABCDEFGHIJKLMNOPQRSTUVWXY (not recommended).

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 alphabethref, it is the ciphered letter.

Example:

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

The Affine cipher text is the replacement of the letters by the new ones.

Example: DCODE is crypted SNVSX

How to decrypt Affine cipher

Affine decryption requires to know the two keys A and B and the 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 alphabethref.

Example: The alphabet ABCDEFGHIJKLMNOPQRSTUVWXYZ, starting at 0 gives A=0, B=1, ..., Z=25. There is also A=1, ..., Y=25 and Z=0 with the alphabet ZABCDEFGHIJKLMNOPQRSTUVWXY.

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 a modular inversehref.

Example: A coefficient \( A' \) for \( A = 5 \) with an alphabet size of \( 26 \) is \( 21 \) because \( 5 \times 21 = 105 \equiv 1 \mod 26 \).

Example: For S (\(y=18\)), \( x = A' \times (18-B) = 21*(18-3) \equiv 315 \mod 26 = 3 \)

For each value \( x \), corresponds a letter with the same position in the alphabethref: the coded letter.

Example: For S (\(x=3\)) corresponds the letter at position 3: D.

The plain text is the replacement of all characters with calculated new letters.

Example: The original plain text is DCODE.

How to recognize an Affine ciphertext?

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

How to decipher Affine without coefficient?

To crack Affine, it is possible to bruteforce all A and B coefficients. Use the Brute-force attack option.

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 A' value?

Calculate the modular inversehref of A, modulo the length of the alphabet.

How to compute B' value?

B' has the same value as 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

When Affine have been invented ?

No date of known author.

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 coprimehref. This limits A values to 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23 et 25 (if the alphabet is 26 characters long)

Can I 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.

Can I use a negative value for A?

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.

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Source code

dCode retains ownership of the source code of the script Affine Cipher. 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, etc.) which dCode owns rights can be transferred after sales quote. So if you need to download the Affine Cipher script for offline use, for you, your company or association, see you on contact page !

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