Public key tools, such as Diffie-Hellman and RSA, can be used for key exchange, encryption, and digital signatures. We will discuss the applications of these scenarios later, but first, we need to cover the main concept of public key cryptography.
What problems public key solved?
In symmetric key cryptography, one of the most challenging aspects is securely distributing and managing the shared secret key. If an attacker intercepts this key during transmission, the entire encryption system is compromised.
Public key cryptography solves this problem by using two keys: a public key (which can be freely shared) and a private key (which is kept secret). The public key is used to encrypt data, and only the corresponding private key can decrypt it. This means the secret key never needs to be transmitted over the network, thus eliminating the key distribution problem.
This approach effectively solves the problem of securely delivering and managing encryption keys, as the private key does not need to be shared, and the public key can be freely distributed without compromising security. The strength of public key cryptography lies in the fact that even if someone intercepts the public key, they cannot decrypt the data without the corresponding private key, which is computationally infeasible to derive.
How does the public key work?
To make the public key mechanism work, we need to implement a trapdoor function scheme. But what is a "trapdoor"? Before we discuss what a trapdoor is, we need to understand why we need it.
In order to encrypt and decrypt without sharing the secret key, we need a function that can encrypt messages but is difficult to reverse (invert) without the secret information. This secret information, which is not shared over the network, is called the trapdoor. A trapdoor function is a function that allows for easy encryption, but is hard to decrypt without the trapdoor. Only with the trapdoor can the ciphertext be easily decrypted.
How does the trapdoor function scheme work?
A trapdoor function scheme T, defined over X and Y, consists of a triple of algorithms (G, F, I).
G is a probabilistic algorithm that generates a public key (pk) and a secret key (sk).
F is a deterministic algorithm that takes the public key (pk) and a message x (where x∈X) and produces ciphertext y. This is like the encryption process.
I is a deterministic algorithm that takes the secret key (sk) and ciphertext y and outputs the original message x. This is like the decryption process.
How can a trapdoor function scheme be used in key exchange?
To explain how the trapdoor function can be used in key exchange, let’s consider an example where Alice and Bob exchange keys.
First, Alice generates a public key and a secret key using algorithm GGG and sends the public key to Bob. When Bob receives Alice's public key, he generates ciphertext yyy using the message xxx and the public key. Bob then sends the ciphertext to Alice. Alice can decrypt the ciphertext using her secret key and algorithm III.
In this example, the secret key is never transmitted over the network, and only with the secret key can Alice decrypt the ciphertext.
