ElGamal Cryptosystem

Overview

This repository contains a Jupyter Notebook implementing:

1. ElGamal Cryptosystem: Key generation, encryption, and decryption.
2. Prime Number Computations: Finding large prime numbers using the Sieve of Eratosthenes.

Each section includes detailed explanations and Python implementations to support theoretical concepts with concrete examples.

Contents

• elgamal.ipynb: Implementation of the ElGamal cryptosystem.
• number_theory.ipynb: Analysis of cyclic groups and even-order elements.
• prime_numbers.ipynb: Prime number computations with the Sieve of Eratosthenes.
• README.md: This documentation file.

Installation & Usage

Prerequisites

Ensure you have Python and Jupyter Notebook installed:

pip install jupyter numpy sympy

Running the Notebook

1. Clone the repository:

git clone https://github.com/chrishounwnu/rsa_Elgmal_ecosystem_Pohlig_Hellman_Cryptography.git

2. Navigate to the project directory:

cd rsa_Elgmal_ecosystem_Pohlig_Hellman_Cryptography

3. Start Jupyter Notebook:

jupyter notebook

4. Open the relevant .ipynb file and run the cells.

Project Details

1. ElGamal Cryptosystem

• Generates a large prime p of at least 700 digits.
• Computes private and public keys.
• Encrypts and decrypts messages.
• Uses Python for number generation and modular arithmetic.

2. Prime Number Computations

• Finds the (5000000 + n)-th prime using the Sieve of Eratosthenes.
• Counts prime numbers between 2^25 and 2^26.
• Uses an optimized approach for large-scale prime computations.

Example Outputs

• ElGamal Encryption & Decryption:
  • Public Key: (g, p, pk)
  • Private Key: sk
  • Ciphertext: (c1, c2)
  • Decrypted Message: Original text
• Number Theory:
  • Number of even-order elements for different p, n values.
• Prime Computations:
  • 5000000 + n-th prime found.
  • Prime count in the given range.

License

This project is open-source under the MIT License.

Author

Christophe HOUNWANOU - GitHub Profile