Hill Cipher
The Hill Cipher is a polygraphic substitution cipher based on linear algebra. It uses matrix multiplication to transform blocks of letters (vectors) into ciphertext vectors. The key is an invertible matrix, and decoding requires the inverse of that matrix modulo 26.
What You'll Learn
Introduction to Hill Cipher
The Hill Cipher is a polygraphic substitution cipher based on linear algebra. It uses matrix multiplication to transform blocks of letters (vectors) into ciphertext vectors. The key is an invertible matrix, and decoding requires the inverse of that matrix modulo 26.
Prerequisites
How to Solve Hill Cipher Problems
Step 1: Convert plaintext letters to numbers (A=0 to Z=25)
Step 2: Group letters into vectors of size n (n×1 matrices)
Step 3: Multiply each vector by the n×n key matrix (mod 26) to get ciphertext vector
Step 4: Convert resulting numbers back to letters
Step 5: For decoding, multiply ciphertext vectors by the inverse of the key matrix (mod 26)
Example Problem
Example: Decode 'DH' using key matrix [[3,3],[2,5]] (mod 26). Solution: Step 1: D=3, H=7 Step 2: Find inverse of matrix [[3,3],[2,5]] mod 26 Determinant = (3×5 - 3×2) = 15-6=9 mod 26 Inverse of 9 mod 26 is 3 (since 9×3=27≡1 mod 26) Inverse matrix = 3 × [[5,-3],[-2,3]] = [[15,-9],[-6,9]] mod 26 = [[15,17],[20,9]] Step 3: Multiply: [3,7] × [[15,17],[20,9]]? Wait, row vector times matrix: [3,7] * [[15,17],[20,9]] = [3*15+7*20, 3*17+7*9] = [45+140, 51+63] = [185,114] mod 26 = [185-26*7=185-182=3, 114-26*4=114-104=10] → [3,10] → D(3), K(10) Answer: DK
Pro Tips & Tricks
- The key matrix must be invertible modulo 26
- Determinant must be odd and not divisible by 13
- For 2×2 matrix [[a,b],[c,d]], inverse = (det^-1) × [[d,-b],[-c,a]] mod 26
- Common key sizes: 2×2, 3×3 (for digraphs and trigraphs)
- The cipher is polygraphic (encrypts blocks of letters at once)
- Hill cipher is resistant to frequency analysis
Shortcut Methods to Solve Faster
Common Mistakes to Avoid
Practice Worksheets
Practice makes perfect! Work through these worksheets to master Hill Cipher. Each worksheet contains 20 questions with detailed explanations. Start from Worksheet 1 and progress through increasing difficulty levels.
Exam Importance
Hill Cipher is an important topic for various competitive exams. Here's how frequently it appears:
Ready to Master Hill Cipher?
Start with Worksheet 1 and work your way up to expert level! Each worksheet includes: