This paper introduces a public-key cryptosystem, also known as RSA, developed by Rivest, Shamir, and Adleman. The system allows for secure communication without the need for a shared secret key. It enables encryption and decryption using a pair of keys: a public encryption key and a private decryption key. The encryption process involves raising a message to a public power and taking the remainder when divided by a public product of two large primes. Decryption uses a secret power that is the modular inverse of the encryption power. The security of the system relies on the difficulty of factoring large numbers. The system has two important applications: secure communication and digital signatures. For secure communication, a message can be encrypted using the recipient's public key, which can only be decrypted by the recipient's private key. For digital signatures, a message can be signed using a private key, and anyone can verify the signature using the corresponding public key. This ensures the authenticity of the message and prevents forgery. The system is based on the concept of a "trap-door one-way function," which is easy to compute in one direction but difficult to reverse without specific knowledge. The security of the system is based on the difficulty of factoring large numbers, which is a well-known problem in mathematics. The paper also discusses the mathematical foundations of the system, including the use of Euler's totient function and modular arithmetic. The paper also addresses the challenge of securely distributing keys in a public-key system. Unlike traditional symmetric key systems, public-key systems do not require a secure channel for key distribution. Instead, the public key can be freely shared, while the private key remains secret. The paper also discusses the practical implementation of the system, including efficient algorithms for encryption and decryption, and the selection of large prime numbers for key generation. The paper concludes that the proposed system is secure and has the potential to revolutionize secure communication and digital signatures. It also acknowledges the importance of further research into the security of the system and the need for continued efforts to find efficient factoring algorithms.This paper introduces a public-key cryptosystem, also known as RSA, developed by Rivest, Shamir, and Adleman. The system allows for secure communication without the need for a shared secret key. It enables encryption and decryption using a pair of keys: a public encryption key and a private decryption key. The encryption process involves raising a message to a public power and taking the remainder when divided by a public product of two large primes. Decryption uses a secret power that is the modular inverse of the encryption power. The security of the system relies on the difficulty of factoring large numbers. The system has two important applications: secure communication and digital signatures. For secure communication, a message can be encrypted using the recipient's public key, which can only be decrypted by the recipient's private key. For digital signatures, a message can be signed using a private key, and anyone can verify the signature using the corresponding public key. This ensures the authenticity of the message and prevents forgery. The system is based on the concept of a "trap-door one-way function," which is easy to compute in one direction but difficult to reverse without specific knowledge. The security of the system is based on the difficulty of factoring large numbers, which is a well-known problem in mathematics. The paper also discusses the mathematical foundations of the system, including the use of Euler's totient function and modular arithmetic. The paper also addresses the challenge of securely distributing keys in a public-key system. Unlike traditional symmetric key systems, public-key systems do not require a secure channel for key distribution. Instead, the public key can be freely shared, while the private key remains secret. The paper also discusses the practical implementation of the system, including efficient algorithms for encryption and decryption, and the selection of large prime numbers for key generation. The paper concludes that the proposed system is secure and has the potential to revolutionize secure communication and digital signatures. It also acknowledges the importance of further research into the security of the system and the need for continued efforts to find efficient factoring algorithms.