Hill Cipher Overview

Study Notes

Hill Cipher Overview

The Hill cipher is a polygraphic substitution cipher.
It encrypts multiple letters at a time, unlike simpler substitution ciphers that encrypt one letter at a time.
This allows for a more complex and potentially stronger encryption compared to monoalphabetic ciphers.
The cipher uses linear algebra, specifically matrix multiplication, for encryption and decryption.

Key Generation

A key matrix is used for encryption and decryption.
The key matrix is a square matrix of size n x n, where n represents the number of letters to be encrypted simultaneously.
The entries of the key matrix are integers chosen from a predefined set (e.g., {0, 1, ..., 25}).
The key and its corresponding inverse (used for decryption) must be invertible.
Invertibility is crucial for ensuring a unique decryption is possible.

Encryption Process

The plaintext is divided into blocks of n letters.
Each block is treated as a vector of size n.
The corresponding key matrix is multiplied with the vector using matrix multiplication modulo 26, to create a ciphertext vector.
The ciphertext vector is converted into n ciphertext letters.
This process is repeated for each block of plaintext.

Decryption Process

The ciphertext is divided into blocks of n letters similar to the encryption process.
Each block is treated as a vector of size n similar to the encryption process.
The inverse of the key matrix is used to multiply the ciphertext vector.
This multiplication is performed modulo 26 again.
The resultant vector gives the corresponding plaintext block which is converted into n plaintext letters.

Example

Consider a 2x2 Hill Cipher with key matrix [2 5; 3 7].
Plaintext is "PAY".
The plaintext letters are represented as numbers (P=15, A=0, Y=24).
This forms the vector [15 0].
Key matrix is multiplied with the vector in modulo 26.
Encryption result would be [5 15].
These numbers translate back into ciphertext letters giving a ciphertext of "FC".

Matrix Invertibility

For the Hill cipher to be invertible, the determinant of the key matrix must be relatively prime to 26.
This ensures that the multiplicative inverse of the determinant, modulo 26, exists.
Without this condition, some plaintext blocks might not be uniquely decrypted.
This condition is essential to guarantee the existence of a mathematical inverse matrix.

Security Considerations

Although the Hill cipher is more complex than basic substitution ciphers, it is vulnerable to known plaintext attacks.
If enough plaintext-ciphertext pairs are known, the key matrix can be determined through linear algebra techniques.
The block size (n) plays a major role in security. A larger block size typically leads to greater strength against cryptanalysis. Larger key spaces also generally imply stronger ciphers.
The Hill cipher is considered weaker than modern symmetric-key ciphers in practice.
Modern cryptography emphasizes advanced techniques like symmetric and asymmetric key cryptography, which offer substantially better security.

Practical Applicability

The Hill cipher is generally not suitable for modern, real-world applications.
Though educational, it's not robust against modern cryptanalysis.
It has historical value in demonstrating principles of polygraphic substitution and matrix operations in cryptography.

Description

Explore the workings of the Hill cipher, a polygraphic substitution cipher that encrypts multiple letters using matrix multiplication. Learn about key matrix generation and the importance of invertibility for effective decryption. This quiz will enhance your understanding of encryption methods in cryptography.