RSA Encryption Project (macOS)

This project demonstrates a simple RSA encryption and decryption example using C++ on macOS. It includes two versions of an encryption implementation: one using unsigned long long int for calculations, and a more optimized version using mpz_class from the GMP (GNU Multiple Precision Arithmetic) library.

Note: This project is designed specifically for macOS. The SecRandomCopyBytes function used for secure random number generation is part of Apple's Security framework and will not work on other platforms like Linux or Windows.

Table of Contents

• Project Overview
• Files in Project
• Dependencies
• Installation
• Compiling and Running
• Usage
• Keywords

Project Overview

The project provides two versions of RSA encryption:

1. encryption.cpp: This file uses unsigned long long int for modular exponentiation and a brute-force modular inverse (find_d) implementation. It is less efficient and should be used for educational purposes or small values only.

2. encryption_optimized.cpp: This file uses GMP's mpz_class type for arbitrary-precision arithmetic, which allows for much larger numbers, making it suitable for actual cryptographic use cases. It includes optimized modular exponentiation and modular inverse calculations, leveraging the Extended Euclidean Algorithm.

Files in Project

• encryption.cpp: Basic RSA encryption with unsigned long long int and brute-force modular inverse.
• encryption_optimized.cpp: Optimized RSA encryption with GMP's mpz_class for large number support.

Dependencies

• macOS: This project relies on macOS's SecRandomCopyBytes for secure random byte generation.
• GMP Library (GNU Multiple Precision Arithmetic): Required only for encryption_optimized.cpp to handle large numbers and provide optimized arithmetic functions.

Installation

To install GMP on macOS, use Homebrew:

brew install gmp

installation is only needed to run the encryption_optimized.cpp version

Compiling and Running

1. Basic Implementation (encryption.cpp)

This version uses unsigned long long int for calculations and is compiled as follows:

clang++ -std=c++17 -framework Security -o enc encryption.cpp

2. Optimized GMP Implementation (encryption_optimized.cpp)

This version uses GMP for high-precision arithmetic. Compile with the following command:

clang++ -std=c++17 -framework Security -o enc encryption_optimized.cpp -lgmp -lgmpxx -I/opt/homebrew/include -L/opt/homebrew/lib

• Note: Adjust include and library paths (-I/opt/homebrew/include and -L/opt/homebrew/lib) if GMP is installed in a different location.

Usage

Both versions accept a plaintext message as a command-line argument. The message will be encrypted using RSA encryption and then decrypted to show the original message.

Run the program with:

./enc YourMessageHere

Example

./enc Yousef

How It Works

1. encryption.cpp

• Uses unsigned long long int for calculations, which can only handle smaller values, making it suitable for basic or educational use.
• Implements find_d (modular inverse) with a brute-force algorithm, which is slower and less efficient for larger values.
• Provides basic RSA functionality: it generates random prime numbers, computes the public and private keys, and performs encryption and decryption.

2. encryption_optimized.cpp

• Uses GMP's mpz_class for arbitrary-precision arithmetic, enabling it to handle large RSA key sizes securely.
• Includes optimized functions for modular exponentiation and modular inverse (find_d) using the Extended Euclidean Algorithm, which is faster and more suitable for cryptographic use cases.
• Suitable for larger messages and realistic RSA encryption due to its support for very large numbers.

Keywords

• RSA Encryption: Public key cryptography algorithm used for secure data transmission.
• macOS: Platform dependency, uses macOS-specific SecRandomCopyBytes.
• GMP: GNU Multiple Precision Arithmetic Library, necessary for high-precision calculations.
• Modular Arithmetic: Key concept in cryptography, used in both modular exponentiation and modular inverse.
• Extended Euclidean Algorithm: Algorithm for finding the modular inverse, used in optimized version.
• Secure Random Number Generation: Used in cryptographic applications, specifically for generating keys.
• Public Key Cryptography: Encryption system where encryption and decryption use different keys.
• High-precision Arithmetic: Important for cryptography to handle large numbers securely.

License

This project is open for educational purposes.