Affine Cipher

Definition, operation, and interactive tool.

• Reading time: ~6 min

What is the Affine Cipher?

The Affine cipher is a monoalphabetic substitution that generalizes Caesar. Each letter of the alphabet (mapped to an index 0..25) is transformed by a linear function mod 26 with two parameters: a (slope) and b (shift).

Conceptually: map letters → numbers, apply the affine transform, then map numbers → letters. If a and 26 are not coprime, the function is not invertible and you cannot decrypt.

Formulas

Encryption:   C = (a·P + b) mod 26
Decryption:   P = a⁻¹·(C - b) mod 26

Condition:    a must be coprime with 26 (gcd(a,26)=1) so that a⁻¹ exists.
Valid a:      {1,3,5,7,9,11,15,17,19,21,23,25} ;  b ∈ {0..25}

Here a⁻¹ is the modular inverse of a in ℤ26, i.e., a·a⁻¹ ≡ 1 (mod 26).

How it works

1) Choose a coprime with 26 and a b in 0–25. 2) Normalize the text (accents → base). You may choose to preserve ñ/Ñ without encryption for compatibility—just document it. 3) Convert each letter to its index (A=0, …, Z=25), apply C = (a·P + b) mod 26, then map back to letters.

For decryption, compute a⁻¹ (via the extended Euclidean algorithm) and use P = a⁻¹·(C - b) mod 26.

Example

Quick example

Parameters: a=5, b=8 ⇒ E(x) = (5x + 8) mod 26
Plaintext: ATAQUE → indices A=0, T=19, A=0, Q=16, U=20, E=4

Computation: A: (5·0+8)=8 → I  |  T: (5·19+8)=103≡25 → Z  |  A: 8 → I  |  Q: 88≡10 → K  |  U: 108≡4 → E  |  E: 28≡2 → C

Ciphertext: IZIKEC

For decryption with a=5, the inverse is a⁻¹=21 (since 5·21=105≡1 mod 26).

History

  • Formalizes and generalizes classical substitutions (like Caesar) using modular algebra.
  • Widely used in teaching to introduce modular inverses and linear functions over ℤn.

Classic attacks

  • Frequency + known pairs: two plaintext/ciphertext pairs (P₁,C₁), (P₂,C₂) solve for a and b.
  • Brute force: the key space is small (12 valid a × 26 values of b = 312 combinations).
  • Language cues: guessing likely mappings (e.g., plaintext E to a frequent ciphertext letter) speeds up recovery.

It is still monoalphabetic: overall letter frequencies are preserved, which is why it’s vulnerable.

Pros and cons

Pros

  • Generalizes Caesar; ideal for teaching modular arithmetic and inverses.
  • Flexible parameters (a, b) and simple implementation.
  • Great for interactive tools and classroom exercises.

Cons

  • Monoalphabetic: vulnerable to frequency analysis and to brute force (312-key space).
  • If a is not coprime with 26, no inverse exists and decryption is impossible.
  • Provides no integrity or authenticity; it only hides letters via a linear function.

Affine Encryption & Decryption Tool

Options

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

b accepts negatives and large values; normalized mod 26 (e.g., 27→1, −1→25).