What do you mean by Encryption? Describe RSA algorithm with examples.

Encryption is a process of converting plaintext or readable data into ciphertext or unintelligible form using cryptographic algorithms and keys. The purpose of encryption is to protect sensitive information, such as personal data, financial transactions, and communications, from unauthorized access, interception, and tampering by unauthorized parties. Encryption ensures confidentiality, integrity, and privacy of data by rendering it unreadable to anyone without the corresponding decryption key.

One of the most widely used encryption algorithms is the RSA algorithm, named after its inventors, Ron Rivest, Adi Shamir, and Leonard Adleman. RSA is an asymmetric encryption algorithm, meaning it uses two different keys for encryption and decryption: a public key and a private key.

Here's how the RSA algorithm works:

1. Key Generation:

   • Choose two large prime numbers, p and q.
   • Calculate their product, n = p * q.
   • Calculate Euler's totient function, φ(n) = (p-1) * (q-1).
   • Choose a public exponent, e, such that 1 < e < φ(n) and gcd(e, φ(n)) = 1 (e and φ(n) are coprime).
   • Calculate the private exponent, d, such that (d * e) ≡ 1 (mod φ(n)).
   • The public key is (n, e), and the private key is (n, d).

2. Encryption:

   • Sender Alice encrypts plaintext message M using Bob's public key (n, e) to produce ciphertext C.
   • Ciphertext C is computed as: C = M^e mod n.

3. Decryption:

   • Receiver Bob decrypts the ciphertext C using his private key (n, d) to obtain the original plaintext message M.
   • Plaintext message M is computed as: M = C^d mod n.

Example of RSA Algorithm:

Let's demonstrate the RSA algorithm with smaller numbers for illustration purposes.

1. Key Generation:

   • Choose two prime numbers: p = 11, q = 17.
   • Compute n = p q = 11 17 = 187.
   • Compute φ(n) = (p-1) (q-1) = (11-1) (17-1) = 160.
   • Choose public exponent e = 7 (e is relatively prime to φ(n)).
   • Compute private exponent d such that (d * e) ≡ 1 (mod 160). For e = 7, d = 23.
   • Public key (n, e) = (187, 7), Private key (n, d) = (187, 23).

2. Encryption:

   • Sender Alice wants to send message M = 88 to Bob.
   • Encrypt M using Bob's public key: C = 88^7 mod 187 = 11.

3. Decryption:

   • Receiver Bob receives ciphertext C = 11.
   • Decrypt C using Bob's private key: M = 11^23 mod 187 = 88.

Thus, Bob successfully decrypts the ciphertext to obtain the original plaintext message, which is 88.

In summary, the RSA algorithm provides a secure method for encryption and decryption, leveraging the mathematical properties of prime numbers and modular arithmetic to protect sensitive data and ensure secure communication over insecure channels.