# Public Key Cryptography

Public Key Cryptography, also known as asymmetric cryptography, is a cryptographic system that uses a pair of keys: a public key and a private key. Each key can be used to encrypt or decrypt data, but the keys are not interchangeable.

The Public Key Cryptography system works by using a pair of keys, one public and one private. The public key is known to everyone, and can be freely distributed. The private key, on the other hand, is kept secret and known only to the owner of the key pair.

To send an encrypted message to someone using Public Key Cryptography, you would use their public key to encrypt the message. Once the message is encrypted, it can only be decrypted by the owner of the private key.

Similarly, if someone wants to digitally sign a message, they would use their private key to sign the message. Anyone with access to the public key can then verify that the message was indeed signed by the owner of the private key.

The security of Public Key Cryptography is based on the fact that it is computationally infeasible to determine the private key based on the public key. This is known as the “computational hardness assumption”.

The algorithm used in Public Key Cryptography is typically based on mathematical problems that are easy to solve in one direction, but difficult to solve in the opposite direction. For example, the RSA algorithm is based on the fact that it is easy to multiply two large prime numbers together to obtain a composite number, but it is much harder to factor that composite number back into its original prime factors.

In summary, Public Key Cryptography provides a secure way to exchange messages and digital signatures over an insecure network. It allows users to communicate securely without having to share a secret key.

RSA (Rivest-Shamir-Adleman) is a public-key cryptosystem that is widely used for secure data transmission over the internet. It was first proposed in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman. RSA encryption is based on the fact that it is easy to multiply two large prime numbers together, but it is difficult to factor the product of two large prime numbers back into the original primes. This mathematical problem forms the basis of RSA encryption and decryption.

The RSA algorithm involves three main steps: key generation, encryption, and decryption.

Key Generation: In RSA, each user has two keys: a public key and a private key. The public key is shared with others and used for encryption, while the private key is kept secret and used for decryption. The key generation process involves the following steps:

1.1 Select two large prime numbers p and q.

1.2 Compute n = p * q.

1.3 Compute Φ(n) = (p - 1) * (q - 1), where Φ is the Euler’s totient function.

1.4 Select a random integer e such that 1 < e < Φ(n) and e is co-prime to Φ(n).

1.5 Compute d such that (d * e) mod Φ(n) = 1.

The public key is (n, e), while the private key is (n, d).

- Encryption: To encrypt a message M, the sender uses the receiver’s public key (n, e) to compute the ciphertext C as follows: C = M^e mod n

Where ^ denotes exponentiation and mod is the modular operation. The ciphertext C can be sent to the receiver over an insecure channel.

- Decryption: To decrypt the ciphertext C, the receiver uses their private key (n, d) to compute the plaintext message M as follows: M = C^d mod n

Where ^ denotes exponentiation and mod is the modular operation. The plaintext message M can then be read by the receiver.

The security of RSA is based on the fact that it is difficult to factor the product of two large prime numbers. An attacker would need to factor n into its prime factors in order to compute d from e, but the factorization problem is believed to be computationally infeasible for large enough values of p and q. Therefore, the security of RSA depends on the size of the key, which is typically 2048 or 4096 bits long.

RSA has become one of the most widely used public-key cryptosystems in the world, used in applications such as secure email, online banking, and digital signatures. However, it is important to note that RSA is vulnerable to attacks if the keys are not generated properly or if the implementation is flawed. Therefore, it is important to use secure key generation techniques and to ensure the proper implementation of RSA.