RSA Algorithm

 RSA Algorithm

RSA (Rivest–Shamir–Adleman) is a widely used public-key cryptosystem, meaning it utilizes a pair of mathematically linked keys for encryption and decryption. Here's a deeper look into RSA:

Core Concept:

• RSA leverages the mathematical difficulty of factoring large prime numbers.
• It's computationally easy to multiply large prime numbers together, but extremely hard to reverse the process and find the original primes.

Key Generation:

1. Prime Numbers: Two large, distinct prime numbers (p and q) are chosen as the foundation.
2. Modulus (n): The product of p and q (n = p * q) becomes the public key's modulus.
3. Euler's Totient (φ(n)): A mathematical function (Euler's totient) is applied to n, resulting in φ(n).
4. Public Exponent (e): A public exponent (e) is chosen such that 1 < e < φ(n) and has no common factors with φ(n).
5. Private Exponent (d): Using a mathematical equation (modular multiplicative inverse), a private exponent (d) is derived that satisfies the equation (e * d) ≡ 1 (mod φ(n)).

Key Distribution:

• The public key (e, n) is freely distributed. Anyone can encrypt messages using this key.
• The private key (d, n) is kept confidential. It's used for decryption.

Encryption Process:

1. Plaintext Conversion: The message (plaintext) is converted into numerical blocks smaller than the modulus (n).
2. Encryption: Each plaintext block (M) is raised to the power of the public exponent (e) and then modulo n. This creates the ciphertext block (C) using the formula: C = M^e (mod n).

Decryption Process:

1. Ciphertext Received: The receiver gets the ciphertext block (C).
2. Decryption: The private exponent (d) is used to decrypt the message. The original block (M) is recovered using the formula: M = C^d (mod n).

Security and Applications:

• The security of RSA relies on the difficulty of factoring large n. As key sizes increase, so does the difficulty of breaking the encryption.
• RSA is commonly used for:
  • Secure communication (HTTPS, digital signatures)
  • Secure key exchange for other encryption algorithms
  • Digital signing of documents to ensure authenticity

Limitations:

• RSA is computationally slower than symmetric algorithms like AES.
• It's not ideal for encrypting large amounts of data directly. It's often used to encrypt smaller chunks like keys for symmetric encryption.

Overall, RSA remains a cornerstone of public-key cryptography, providing secure communication and digital signatures for various applications.