🧮 Affine Cipher — Step-by-Step Example

Welcome, codebusters! The Affine cipher uses a math formula to encrypt letters. Don't worry — the math is simpler than it sounds! Let's walk through it together.

🤔 What is the Affine Cipher?

The Affine cipher replaces each letter using this formula:

E(x) = (a × x + b) mod  26

Where:

• x = the number of the letter you want to encrypt (A=0, B=1, C=2, ... Z=25)
• a = the first key number (must be "coprime" with 26 — more on that below!)
• b = the second key number (can be any number 0–25)
• mod 26 = means "divide by 26 and keep the remainder" (like a clock that resets after 26)

What does "mod" mean? 🕐

Think of mod like a clock! If a clock has 26 hours and you're at hour 30, where are you? 30 - 26 = 4. That's mod!

• 28 mod 26 = 2 (28 - 26 = 2)
• 52 mod 26 = 0 (52 ÷ 26 = exactly 2, remainder 0)
• 7 mod 26 = 7 (7 is less than 26, so it stays the same)

What does "coprime" mean?

a must be coprime with 26, meaning a and 26 don't share any common factors (other than 1).

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

Invalid values: 2, 4, 6, 8, 10, 12, 13, 14... (these share a factor with 26)

Letter-to-Number Table

📜 The Problem

Encrypt the message: PLANET Using keys: a = 3, b = 5 Formula: E(x) = (3x + 5) mod  26

🕵️ Step-by-Step Encryption

Step 1: Convert each letter to a number

Step 2: Apply the formula to each number

P (x = 15): E(15) = (3 × 15 + 5) mod  26 = (45 + 5) mod  26 = 50 mod  26 = 24 50 ÷ 26 = 1 remainder 24 → Letter Y

L (x = 11): E(11) = (3 × 11 + 5) mod  26 = (33 + 5) mod  26 = 38 mod  26 = 12 38 ÷ 26 = 1 remainder 12 → Letter M

A (x = 0): E(0) = (3 × 0 + 5) mod  26 = (0 + 5) mod  26 = 5 mod  26 = 5 5 is less than 26, so it stays 5 → Letter F

N (x = 13): E(13) = (3 × 13 + 5) mod  26 = (39 + 5) mod  26 = 44 mod  26 = 18 44 ÷ 26 = 1 remainder 18 → Letter S

E (x = 4): E(4) = (3 × 4 + 5) mod  26 = (12 + 5) mod  26 = 17 mod  26 = 17 17 is less than 26, so it stays 17 → Letter R

T (x = 19): E(19) = (3 × 19 + 5) mod  26 = (57 + 5) mod  26 = 62 mod  26 = 10 62 ÷ 26 = 2 remainder 10 → Letter K

Step 3: Write the result!

✅ Ciphertext: YMFSRK

🔓 Now Let's Decrypt It!

You received the ciphertext: YMFSRK You know the keys are a = 3, b = 5.

Step 1: Find the "modular inverse" of a

To decrypt, we need to undo the multiplication by 3. We need a special number called the modular inverse of a.

The modular inverse of a is a number a⁻¹ such that:

a × a⁻¹ ≡ 1 (mod  26)

For a = 3, we need: 3 × ? = 1 (mod  26)

Let's test:

• 3 × 9 = 27 = 26 + 1 → 27 mod  26 = 1 ✅

So a⁻¹ = 9

💡 Tip: Here's a handy cheat sheet of modular inverses (mod 26):

(Memorize this table — it shows up a LOT in competition!)

Step 2: The decryption formula

D(y) = a⁻¹ × (y − b) mod  26

With our values:

D(y) = 9 × (y − 5) mod  26

Step 3: Decrypt each letter

Y (y = 24): D(24) = 9 × (24 − 5) mod  26 = 9 × 19 mod  26 = 171 mod  26 171 ÷ 26 = 6 remainder 15 → Letter P ✓

M (y = 12): D(12) = 9 × (12 − 5) mod  26 = 9 × 7 mod  26 = 63 mod  26 63 ÷ 26 = 2 remainder 11 → Letter L ✓

F (y = 5): D(5) = 9 × (5 − 5) mod  26 = 9 × 0 mod  26 = 0 mod  26 0 → Letter A ✓

S (y = 18): D(18) = 9 × (18 − 5) mod  26 = 9 × 13 mod  26 = 117 mod  26 117 ÷ 26 = 4 remainder 13 → Letter N ✓

R (y = 17): D(17) = 9 × (17 − 5) mod  26 = 9 × 12 mod  26 = 108 mod  26 108 ÷ 26 = 4 remainder 4 → Letter E ✓

K (y = 10): D(10) = 9 × (10 − 5) mod  26 = 9 × 5 mod  26 = 45 mod  26 45 ÷ 26 = 1 remainder 19 → Letter T ✓

Step 4: Read the answer!

✅ Plaintext: PLANET 🪐🎉

⚠️ What About Negative Numbers?

Sometimes when you subtract b, you get a negative number. Don't panic!

Example: If y = 2 and b = 5:

• y − b = 2 − 5 = −3

To handle negatives with mod 26, just add 26 until the number is positive:

• −3 + 26 = 23

Then continue with 23.

Another example: −10 mod  26 = −10 + 26 = 16

🧠 Key Things to Remember

1. Encryption: E(x) = (ax + b) mod  26
2. Decryption: D(y) = a⁻¹(y − b) mod  26
3. a must be coprime with 26 — valid values: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25
4. b can be any number from 0 to 25
5. Memorize the inverse table! It saves tons of time in competition
6. Negative mod: Just add 26 until positive

🎯 Practice Problem

Decrypt this message using a = 7, b = 2:

ZEBBW

Hint: Start by finding the modular inverse of 7 from the table above!

Happy calculating! 🔍🏆