The RSA (Rivest-Shamir-Adleman) algorithm is one of the most widely used public key cryptosystems. It was invented by Ron Rivest, Adi Shamir, and Leonard Adleman in 1977. RSA is named after their surnames and is based on the mathematical complexity of the integer factorization problem.

Key Generation:

1. Prime Number Generation: Choose two large prime numbers, p and q. These primes should be kept secret.
2. Modulus Calculation: Calculate the modulus, n, by multiplying p and q: n = p * q. The modulus represents the key size and is made public.
3. Euler’s Totient Function: Calculate Euler’s totient function, φ(n), where φ(n) = (p-1) * (q-1). This value is kept secret.
4. Public Key Generation: Choose a public exponent, e, which is a positive integer less than φ(n) and relatively prime to φ(n). The public key is the pair (e, n).
5. Private Key Calculation: Compute the private exponent, d, such that (d * e) mod φ(n) = 1. The private key is the pair (d, n).

Encryption: To encrypt a plaintext message, M, using the public key (e, n):

1. Convert the plaintext message into a numerical representation, typically using a specific encoding scheme.
2. Apply the encryption formula: C = M^e mod n, where C is the ciphertext.
3. The resulting ciphertext, C, is the encrypted form of the plaintext message.

Decryption: To decrypt the ciphertext, C, using the private key (d, n):

1. Apply the decryption formula: M = C^d mod n, where M is the decrypted plaintext.
2. The resulting value, M, is the original plaintext message.

Security: The security of RSA is based on the difficulty of factoring large composite numbers into their prime factors. Breaking RSA encryption requires factoring the modulus, n, into its prime factors, p and q. As long as the primes are large enough, factoring becomes computationally infeasible and provides the security of the RSA algorithm.

Key Length: The security of RSA depends on the key length. Longer keys offer stronger security against brute-force attacks and advancements in computational power. Common key lengths for RSA range from 1024 bits to 4096 bits, with 2048 bits being widely used.

Applications: RSA has numerous applications in secure communication, digital signatures, key exchange, and secure data transmission. It is widely used in protocols like SSL/TLS for secure web browsing, secure email, secure file transfer, secure remote access, and many other cryptographic systems.

It’s important to note that while RSA is a widely adopted algorithm, its performance in terms of encryption and decryption speed is slower compared to symmetric key algorithms. As a result, RSA is often used in conjunction with symmetric key cryptography in hybrid cryptosystems to combine the benefits of both approaches.

RSA Encryption Algorithm

RSA encryption algorithm is a type of public-key encryption algorithm. To better understand RSA, lets first understand what is public-key encryption algorithm.

Public key encryption algorithm:

Public Key encryption algorithm is also called the Asymmetric algorithm. Asymmetric algorithms are those algorithms in which sender and receiver use different keys for encryption and decryption. Each sender is assigned a pair of keys:

• Public key
• Private key

The Public key is used for encryption, and the Private Key is used for decryption. Decryption cannot be done using a public key. The two keys are linked, but the private key cannot be derived from the public key. The public key is well known, but the private key is secret and it is known only to the user who owns the key. It means that everybody can send a message to the user using user’s public key. But only the user can decrypt the message using his private key.