Affine Cipher Explanation

The Affine Cipher is a type of monoalphabetic substitution cipher that uses a simple mathematical function to encrypt and decrypt messages. Here’s a detailed breakdown:

How It Works

1. Numeric Representation:

   Each letter in the plaintext is first converted into a number. For the English alphabet, a common assignment is A = 0, B = 1, C = 2, ..., Z = 25.

2. Encryption Function:

   The cipher uses a linear function to transform each number. The general encryption formula is:
   E(x) = (a * x + b) mod m

   • x: The numerical equivalent of the plaintext letter.
   • a: The multiplicative key. It must be chosen such that it is coprime with m (typically 26 for English) to ensure a unique decryption is possible.
   • b: The additive key.
   • m: The size of the alphabet (e.g., 26 for English).
3. Decryption Function:

   To reverse the encryption, you use the inverse of the multiplicative key (denoted as a⁻¹, which exists because a and m are coprime). The decryption formula is:
   D(y) = a⁻¹ * (y - b) mod m

   • y: The numerical equivalent of the ciphertext letter.

Example

Suppose we choose:

• a = 5 (which is coprime with 26),
• b = 8,
• m = 26.

Encryption:
For a plaintext letter represented by x, the ciphertext is calculated as:
E(x) = (5x + 8) mod 26

For example, if x = 2 (which corresponds to the letter C):

E(2) = (5 × 2 + 8) mod 26
     = (10 + 8) mod 26
     = 18 mod 26
  

This corresponds to the letter S (if A = 0, then S = 18).

Decryption:
To decrypt, first find the modular inverse of 5 modulo 26. The inverse, a⁻¹, is the number such that (5 × a⁻¹) mod 26 = 1. In this case, a⁻¹ = 21 because (5 × 21 = 105, and 105 mod 26 = 1).

Then, for a ciphertext letter represented by y:

D(y) = 21 * (y - 8) mod 26
  

If y = 18 (for the letter S):

D(18) = 21 * (18 - 8) mod 26
      = 21 * 10 mod 26
      = 210 mod 26
      = 2
  

This brings us back to the plaintext letter C.

Security Considerations

• Simplicity: While the Affine Cipher introduces two keys (a and b), it is still a form of monoalphabetic substitution. This means each letter in the plaintext always maps to the same letter in the ciphertext, making it vulnerable to frequency analysis.
• Key Space: The requirement that a be coprime with 26 limits the possible values for a. For the English alphabet, there are only 12 valid choices for a (1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25), reducing the overall key space and making the cipher easier to break with modern methods.

In summary, the Affine Cipher is an elegant, though historically basic, encryption method that illustrates key concepts in modular arithmetic and cryptography, but it lacks the security needed for modern communications.