Discuss RSA Algorithm

RSA (Rivest-Shamir-Adleman) is one of the first public-key cryptosystems and is widely used for secure data transmission. Here's a step-by-step explanation of how it works:

1. Key Generation:

   • Two large prime numbers, p and q, are randomly selected.
   • These two numbers are multiplied together to produce n, which is used as the modulus for both the public and private keys.
   • The totient, φ(n), is calculated. This is (p-1)(q-1).
   • A public key exponent, e, is chosen such that 1 < e < φ(n) and e and φ(n) are coprime.
   • A private key exponent, d, is calculated as the modular multiplicative inverse of e mod φ(n).

2. Encryption:

   • The sender takes the recipient's public key (n, e) and represents the plaintext message as a series of numbers less than n.
   • Each number is encrypted using the formula c = m^e mod n, where m is the plaintext message and c is the ciphertext.

3. Decryption:

   • The recipient uses their private key (n, d) to decrypt the ciphertext.
   • The original plaintext message is recovered using the formula m = c^d mod n.

The security of RSA relies on the fact that, while it is easy to multiply two large prime numbers together to produce a composite number, it is computationally infeasible to do the reverse operation – that is, to factorize a composite number into its constituent primes. This is known as the RSA problem.