16.1. Discrete Mathematics

Hello, everyone, welcome to this lecture. So, in this lecture we will continue our discussion on cryptography. And we will see some more applications of the concept from number theory and abstract algebra in the context of cryptography. Namely, we will see the definition of public key cryptosystem. And we will see 2 very popular instantiations of public key cryptosystem, namely that of ElGamal encryption scheme and RSA encryption scheme.

This introduction sets the stage for discussing cryptography, focusing on public key cryptosystems. Public key cryptography is essential as it allows secure communications over public channels without the need to share a secret key in advance. The lecture promises to delve into both the definition and practical examples of public key systems.

Imagine sending a postcard with your address on it—everyone can see it's yours, but only you can read the message inside because it's in a sealed envelope. Public key cryptography works similarly, where one key is public and visible to everyone, while the private key remains confidential.

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...

The Diffie-Hellman key exchange protocol enables two parties to establish a shared secret over an insecure channel. By leveraging mathematical concepts where solving discrete logarithm problems is complex, the protocol ensures that even eavesdroppers have a challenging time deciphering the agreed-upon key.

Think of Sita and Ram as two friends using walkie-talkies. They can set a shared channel number (the key) to talk privately. If someone else hears their conversations, they won't know what the channel number is, making their communications secure despite being in plain sight.

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 that is why to get around this problem, Diffie and Hellman proposed an architecture for a new type of cryptosystem...

While Diffie-Hellman provided a groundbreaking method for key exchange, it requires both users to be online simultaneously, posing challenges for asynchronous communications like email. This limitation led to the idea of public key cryptosystems, where one key can be public and used while the sender and receiver are not interacting in real time.

Imagine two friends planning to meet in a café. If both have to be there at the same time to agree on where to meet, things might not work out. Instead, if one writes down the meeting spot and leaves it somewhere public, the other can find it anytime, making the process more flexible.

So, let us see the architecture of public key cryptosystem. So, 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, there are two keys: a public key that anyone can access and a secret key that only the intended receiver possesses. This allows users to send encrypted messages to one another without prior arrangements for a shared key, automating secure communication processes.

Imagine you have a mailbox that anyone can drop letters in (the public key), but only you have the key to open it (the private key). This ensures that while everyone can send you messages, only you can read them.

And now 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...

The core security requirement for public key cryptography is that even if an attacker knows the public key and the encryption method, they should not be able to deduce the secret message without access to the private key. This non-reversible encryption is what maintains the integrity of secure communications.

Returning to the mailbox analogy, someone can see who sends you letters (public key), but they cannot access the letters themselves without your unique key. Hence, even if they intercept the letters, they can't understand the messages inside.

So, even though Diffie and Hellman thought about this architecture, this new system, they failed to give a concrete instantiation... But the race for coming up with the first instantiation of public key cryptosystem was won by another Turing Award winner triplet namely, RSA...

While Diffie-Hellman laid the groundwork for public key cryptography, it was the RSA algorithm that fully realized the concept, providing a mechanism for both public key generation and secure communication. The RSA system is based on number theory's complexity, particularly concerning prime factorization.

Think of it like a safe that only you can open. Diffie-Hellman designs the safe while RSA is the actual lock and key that make it usable in real life, ensuring that only the intended recipient can access the contents.

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