16.2. Public Key Cryptography

So, let us start with the definition of public key cryptography. What exactly is public key cryptography? Why exactly we need that and so on? So, this is the Diffie Hellman key exchange protocol which allows 2 parties, Sita and Ram to talk over the internet publicly and agree upon a common key k. And if we perform all the operations over a sufficiently large group where a random instance of discrete log problem is very difficult and any third party in a reasonable amount of time will not be able to come up with the value of key that is Sita and Ram have agreed upon.

Public key cryptography is a method that allows two parties to communicate securely over an unsecured medium. In this method, the Diffie-Hellman key exchange protocol allows two parties (Sita and Ram) to publicly exchange information and agree on a secret key (k). This is made secure because the operations are done over a mathematical group where solving the discrete log problem is difficult for outsiders. Therefore, while others can see the communication, they cannot easily derive the key.

Think of public key cryptography like two people agreeing on a secret handshake. Even if others see them talking, they can't deduce the handshake unless they know precisely how it is formed, making it exclusive to those two.

But the downside of the Diffie Hellman key exchange protocol is that it requires both the parties to be online. So, imagine the 2 parties in different time zones then it hinders the spontaneity of applications like email. So, for instance, if Sita and Rama are in 2 different time zones and Sita wants to send an encrypted mail to Ram then Sita will initiate a Diffie Hellman key exchange protocol instance with Ram, she will send her message namely m to Ram.

The Diffie-Hellman key exchange requires both parties (Sita and Ram) to be online at the same time to agree on a shared key. This poses challenges in real-world scenarios, especially with different time zones. For instance, Sita might want to send an email to Ram, but if Ram is not available (as he is sleeping), Sita cannot encrypt her message until they complete the key exchange.

Imagine trying to set a secret combination with a friend but only over phone calls. If one of you is busy or unavailable, you can't finish setting up the secret combination to communicate securely. It’s like trying to agree on a new secret without both being available, leading to delays.

So, let us see the architecture of public key cryptosystem. In this system, the receiver will have 2 keys, a key which we call us public key, pk available in the public domain. And there will be another key, sk which will be secret key and available only with the receiver.

In a public key cryptosystem, each receiver has two keys: a public key (pk) that anyone can access and a secret key (sk) that only the receiver knows. Anyone who wants to send a confidential message to the receiver uses the public key to encrypt their message. The receiver then uses their secret key to decrypt the message back to plaintext, ensuring that only they can read it.

Think of it like a mailbox. The public key is like the mailbox number that anyone can see and use to send you letters, but only you have the key to open the mailbox and read those letters.

The security property that we require here is that if there is a third party an attacker, who knows the description of the public key, who knows the description of the encryption algorithm, who knows the description of the decryption algorithm and who also knows the description of the cipher text, should not be able to figure out what exactly in the underline message.

The critical security aspect of public key cryptography is that even if an attacker knows the public key, encryption algorithm, and ciphertext, they should not be able to deduce the original plaintext message. This is because the secret key, which is necessary for decryption, remains unknown to them, preserving the confidentiality of the communication.

Imagine if a thief could see your mailbox and the letters within it but has no key to open it. They can’t read your private letters, even though they can see the mailbox number and know there are letters inside.

Even though Diffie and Hellman thought about this architecture, this new system, they failed to give a concrete instantiation. Namely, a concrete encryption algorithm, concrete decryption algorithm, a concrete mechanism of coming up with a public key and a concrete mechanism of coming up with a secret key. And the race for coming up with the first instantiation of public key cryptosystem was won by RSA.

Although Diffie and Hellman conceptualized public key cryptography, they did not develop functional algorithms for encryption and decryption. This need was filled by RSA, which provided a practical way to create the keys and execute the encryption/decryption processes outlined by the public key system.

It's like having a great idea for a new device, but not being able to build it yourself. Then, someone else comes along and actually creates the device so that people can benefit from it. RSA turned the theoretical idea into a usable technology.

If I am a receiver, and if I am a amazon, for instance, I do not have to worry, who is the potential sender, he can be any entity from the world. Whoever wants to communicate with me, I just have to publish my public key for him which I can do once for all.

In a public key cryptosystem, the receiver only needs to share their public key once, and anyone can use it to send encrypted messages. This simplifies key management since there's no need to securely exchange keys with every potential sender.

Imagine a bank that freely hands out a form for people to fill out their account details. Anyone can fill it out to deposit money, making transactions easy without needing to meet with the bank each time.

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