Affine Cipher
Affine Cipher

Affine Cipher



Create an implementation of the affine cipher, an ancient encryption system created in the Middle East.

The affine cipher is a type of monoalphabetic substitution cipher. Each character is mapped to its numeric equivalent, encrypted with a mathematical function and then converted to the letter relating to its new numeric value. Although all monoalphabetic ciphers are weak, the affine cypher is much stronger than the atbash cipher, because it has many more keys.

The encryption function is:

E(x) = (ax + b) mod m

  • where x is the letter's index from 0 - length of alphabet - 1
  • m is the length of the alphabet. For the roman alphabet m == 26.
  • and a and b make the key

The decryption function is:

D(y) = a^-1(y - b) mod m

  • where y is the numeric value of an encrypted letter, ie. y = E(x)
  • it is important to note that a^-1 is the modular multiplicative inverse of a mod m
  • the modular multiplicative inverse of a only exists if a and m are coprime.

To find the MMI of a:

an mod m = 1

  • where n is the modular multiplicative inverse of a mod m

More information regarding how to find a Modular Multiplicative Inverse and what it means can be found here.

Because automatic decryption fails if a is not coprime to m your program should return status 1 and "Error: a and m must be coprime." if they are not. Otherwise it should encode or decode with the provided key.

The Caesar (shift) cipher is a simple affine cipher where a is 1 and b as the magnitude results in a static displacement of the letters. This is much less secure than a full implementation of the affine cipher.

Ciphertext is written out in groups of fixed length, the traditional group size being 5 letters, and punctuation is excluded. This is to make it harder to guess things based on word boundaries.

General Examples

  • Encoding test gives ybty with the key a=5 b=7
  • Decoding ybty gives test with the key a=5 b=7
  • Decoding ybty gives lqul with the wrong key a=11 b=7
  • Decoding kqlfd jzvgy tpaet icdhm rtwly kqlon ubstx
    • gives thequickbrownfoxjumpsoverthelazydog with the key a=19 b=13
  • Encoding test with the key a=18 b=13
    • gives Error: a and m must be coprime.
    • because a and m are not relatively prime

Examples of finding a Modular Multiplicative Inverse (MMI)

  • simple example:
    • 9 mod 26 = 9
    • 9 * 3 mod 26 = 27 mod 26 = 1
    • 3 is the MMI of 9 mod 26
  • a more complicated example:
    • 15 mod 26 = 15
    • 15 * 7 mod 26 = 105 mod 26 = 1
    • 7 is the MMI of 15 mod 26


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Last updated 10 August 2022
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