Affine Cipher

What is the Affine Cipher?

The Affine cipher is a type of monoalphabetic substitution cipher that uses a mathematical formula to encrypt each letter. It combines multiplication and addition modulo 26, making it more secure than simple substitution ciphers like Caesar.

The encryption formula is: E(x) = (ax + b) mod 26

Where 'a' and 'b' are the keys, and 'x' is the letter's position (A=0, B=1, etc.).

How the Affine Cipher Works

Encryption

1. Convert each letter to its numerical value (A=0, B=1, ..., Z=25)
2. Apply the formula: (a × x + b) mod 26
3. Convert the result back to a letter

Decryption

1. Find the modular multiplicative inverse of 'a' (called a⁻¹)
2. Apply: a⁻¹ × (y - b) mod 26
3. Convert the result back to a letter

Key Requirements

The key 'a' must be coprime with 26 (their greatest common divisor is 1). This ensures that every letter maps to a unique ciphertext letter and decryption is possible.

Valid values for 'a': 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25

The key 'b' can be any number from 0 to 25 (the shift amount).

Special Cases

• a=1: Reduces to Caesar cipher (just a shift)
• a=1, b=0: Identity (no change)
• a=25, b=25: Equivalent to Atbash cipher

Affine Cipher in Geocaching

The Affine cipher appears in geocaching puzzles because:

• Mathematical: Appeals to puzzle-makers who like math
• Two keys: More complex than simple shift
• Brute-forceable: Only 312 possible key combinations (12 × 26)
• Educational: Teaches modular arithmetic concepts

Breaking the Affine Cipher

Without the key, you can try:

• Brute force: Only 312 combinations to try
• Frequency analysis: Most common letters are E, T, A, O
• Known plaintext: If you know any word, solve for a and b
• Pattern analysis: Look for common words like THE, AND

Mathematical Background

The Affine cipher uses modular arithmetic. The modular multiplicative inverse a⁻¹ satisfies: (a × a⁻¹) mod 26 = 1