Affine Cipher

What it is

• Letters A–Z are mapped to numbers 0–25. Encryption applies the formula: y = (a · x + b) mod 26, where x is the 0-based index of a letter (A=0, B=1, ..., Z=25). This turns decrypted letters into encrypted letters.
• Decryption inverts it: x = a⁻¹·(y − b) mod 26, where a⁻¹ is the modular inverse of a modulo 26.
• Example (a=3, b=11): E(4) → y=3·4+11=23→X; T(19) → y=3·19+11=68≡16→Q. To decode, multiply (y − b) by a⁻¹ mod 26.

Key facts you need

• Valid a (coprime with 26): 1,3,5,7,9,11,15,17,19,21,23,25. b is 0–25.
• Decrypt formula: x = a⁻¹·(y − b) mod 26. a⁻¹ exists only if gcd(a,26)=1.
• Only 312 keys total (12×26) → brute force feasible; test a=1 first (Caesar family).

Step-by-step solving workflow

1. Quick screen: try a=1 with all b (Caesar). If none fit, continue.
2. If you have two plaintext–ciphertext letter pairs (cribs), solve for a and b using modular equations; then verify on a few letters/words.
3. Otherwise brute force over valid a and b; pick the clean English result.

Solving (a, b) from two letter pairs

If x₁→y₁ and x₂→y₂ (0–25 numbering):

• a·x₁ + b ≡ y₁ (mod 26)
• a·x₂ + b ≡ y₂ (mod 26) Subtract: a·(x₁ − x₂) ≡ (y₁ − y₂) (mod 26). If d = (x₁ − x₂) has an inverse mod 26, then a ≡ (y₁ − y₂)·d⁻¹ (mod 26), b ≡ y₁ − a·x₁.

Worked example (full walk-through)

Ciphertext

ZGXY SBWX JN YBQ VLUYXNN, JQ JN YBQ SBWX.

Given key: a = 3, b = 11.

1. Find a⁻¹ mod 26

• 3·9 = 27 ≡ 1 (mod 26) → a⁻¹ = 9.

2. Decrypt a few letters to confirm

• Z G X Y → 25 6 23 24.
• x = 9·(y − 11) mod 26 → 25→(14)→9·14=126≡22→W; 6→(−5≡21)→9·21=189≡7→H; 23→(12)→9·12=108≡4→E; 24→(13)→9·13=117≡13→N → WHEN.
• Next word SBWX → LOVE. Looks good; proceed.

3. Full decryption

WHEN LOVE IS NOT MADNESS, IT IS NOT LOVE.

Practice

1. Try a=5, b=8; decrypt: ZEBBW
2. Suppose you think E→C and T→Z. Solve for (a, b) and verify.
3. Brute force: KQXK TFQE → find (a, b) and plaintext.

Answers

1. a=5 → a⁻¹=21; ZEBBW → TERRA.
2. E(4)→C(2), T(19)→Z(25): d=11; d⁻¹=19; a≡(2−25)·19≡5; b≡2−5·4≡8 → (a,b)=(5,8).
3. One valid result: (a,b)=(11,6) → GOOD LUCK.

Common mistakes

• Using a not coprime with 26 → no a⁻¹ → cannot decrypt.
• Mixing 1–26 with 0–25 indices; stick to 0–25.
• Forgetting to wrap y−b into 0..25 before multiplying by a⁻¹.

Quick reference

• Try Caesar first (a=1). If it fails, brute force valid a and b.
• Or use two letter pairs (cribs) to solve for (a, b) and verify.
• Decrypt: x = a⁻¹·(y − b) mod 26; keep spaces/punct.

See also

• Caesar (a=1 special case), Random Aristocrat, K1/K2/K3 Aristocrat.