RSA

RSA (Rivest-Shamir-Adleman) is a widely used cryptographic algorithm that enables secure communication and data encryption. It was first introduced in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman. RSA is based on the mathematical properties of prime numbers and modular arithmetic.

How RSA Works ?

The RSA algorithm involves the use of two keys: a public key and a private key. The public key is used for encryption, while the private key is used for decryption. Here's a brief explanation of how RSA works:

 

  • -  Choose two large prime numbers, p and q.
  • -  Compute their product, n = p * q, which is used as the modulus.
  • -  Calculate Euler's totient function, φ(n) = (p - 1) * (q - 1), which represents the count of positive integers less than n that are coprime to n.
  • -  Select a public exponent, e, which is a small prime number and relatively prime to φ(n).
  • -  Compute the private exponent, d, such that (d * e) mod φ(n) = 1. In other words, d is the multiplicative inverse of e modulo φ(n).


  • -  To encrypt a message M, which is represented as a number, the sender uses the recipient's public key (e, n).
  • -  Compute the ciphertext C by raising M to the power of e and taking the remainder when divided by n. C = (M^e) mod n.

  • -  The recipient uses their private key (d, n) to decrypt the ciphertext.
  • -  Compute the original message M by raising the ciphertext C to the power of d and taking the remainder when divided by n. M = (C^d) mod n.

  • The security of RSA relies on the difficulty of factoring large numbers into their prime factors. Breaking RSA encryption would require factoring the modulus n, which is computationally infeasible for sufficiently large prime numbers.

  • RSA has found extensive use in secure communication protocols, such as HTTPS for secure web browsing and email encryption. It provides a way for individuals and organizations to transmit sensitive information securely over public networks.