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Affine Cipher

# Affine Cipher

Medium

## Instructions

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`

### Running and testing your solutions

#### From the command line

Simply type `make chez` if you're using ChezScheme or `make guile` if you're using GNU Guile. Sometimes the name for the scheme binary on your system will differ from the defaults. When this is the case, you'll need to tell make by running `make chez chez=your-chez-binary` or `make guile guile=your-guile-binary`.

#### From a REPL

• Enter `(load "test.scm")` at the repl prompt.
• Develop your solution in `affine-cipher.scm` reloading as you go.
• Run `(test)` to check your solution.

#### Failed Test Cases

If some of the test cases fail, you should see the failing input and the expected output. The failing input is presented as a list because the tests call your solution by `(apply affine-cipher input-list)`. To learn more about `apply` see The Scheme Programming Language -- Chapter 5

### Source

Wikipedia
Last updated 7 December 2022
Edit via GitHub

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