RSA

RSA Encryption

Overview

RSA (Rivest-Shamir-Adleman) is a widely used asymmetric encryption algorithm. It is based on the mathematical difficulty of factoring large prime numbers, making it secure and effective for encrypting sensitive data and digitally signing messages.

Key Concepts

1. Asymmetric Encryption:

   • RSA uses a pair of keys: a public key and a private key.
   • The public key is used to encrypt data, and the private key is used to decrypt it.
   • This allows for secure communication, as only the intended recipient with the private key can decrypt the message.

2. Public and Private Keys:

   • Public Key: Can be shared openly and is used for encrypting messages.
   • Private Key: Must be kept secret and is used for decrypting messages and signing data.

3. Prime Factorization:

   • RSA’s security relies on the difficulty of factoring the product of two large prime numbers.
   • While it’s easy to multiply two primes, it’s computationally infeasible to reverse the process and find the original primes from the product.

How RSA Encryption Works

1. Key Generation:

   • Select Two Large Primes (p and q): These primes are kept secret.
   • Calculate n = p * q: This forms part of both the public and private keys.
   • Compute φ(n) = (p-1) * (q-1): Euler’s totient function.
   • Choose Public Key Exponent (e): A number that is coprime with φ(n).
   • Determine Private Key (d): The modular inverse of e mod φ(n).

2. Encrypting a Message:

   • Message M: The plaintext message, which must be converted to a numeric value m (0 ≤ m < n).

   • Encryption:

     java

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     Ciphertext c = m^e mod n

   • The ciphertext c is sent to the recipient.

3. Decrypting a Message:

   • Decryption:

     java

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     Message m = c^d mod n

   • The recipient uses their private key (d, n) to decrypt the ciphertext and retrieve the original message.

How RSA is Used to “Sign” Messages

1. Digital Signatures:

   • RSA can also be used to sign messages, ensuring the authenticity and integrity of the message.
   • The sender uses their private key to sign the message, and the recipient uses the sender’s public key to verify the signature.

2. Signing a Message:

   • Hash the Message: The original message is hashed using a cryptographic hash function (e.g., SHA-256).

   • Sign the Hash:

     bash

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     Signature s = hash^d mod n

   • The sender transmits both the original message and the signature.

3. Verifying the Signature:

   • Verification:

     lua

     Copy code

     Hash = s^e mod n

   • The recipient hashes the received message and compares it to the decrypted hash from the signature.

   • If the hashes match, the signature is valid, confirming that the message was not tampered with and was indeed sent by the owner of the private key.

Applications of RSA

1. Secure Communication:

   • Used in SSL/TLS protocols to secure internet communications.
   • Ensures that data transmitted over the internet is encrypted and protected from eavesdroppers.

2. Digital Signatures:

   • Widely used in software distribution, to verify the authenticity of the software.
   • Ensures that the software hasn’t been altered after being signed by the developer.

3. Key Exchange:

   • RSA is used to securely exchange keys in encrypted communications, allowing symmetric keys to be securely shared between parties.

Security Considerations

• Key Size: Modern RSA implementations typically use key sizes of 2048 bits or higher to ensure security.
• Vulnerabilities: RSA can be vulnerable to various attacks if improperly implemented, such as timing attacks, chosen-ciphertext attacks, or inadequate key generation.