16.1.1. Lecture - 64

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.

In this chunk, the lecture introduces the concept of public key cryptography, which is a critical aspect of modern cryptography. Public key cryptography allows secure communication over the internet without the need for both parties to be online simultaneously. This solves a major problem faced by earlier cryptography methods that required both parties to share a secret key.

Think of public key cryptography like sending a locked box via mail. The sender can lock the box with a padlock that anyone can use to secure their message, but only the receiver has the key to open it. This allows anyone to send secure messages without needing to communicate a special key first.

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.

The Diffie-Hellman key exchange protocol enables two parties to establish a shared key through a public channel. It leverages mathematical problems that are hard to solve, ensuring that even if someone intercepts their communication, they won't be able to deduce the shared key without significant effort.

Imagine Sita and Ram want to negotiate a secret code while shouting across a park. They agree on a code way in advance, and while shouting out their contributions, they cleverly ensure that no one else can figure out the final code even if they hear part of the conversation.

So, remember, the symmetric key cryptosystem, the same key is used both for encryption and decryption.

In symmetric key systems, both parties use the same key to encrypt and decrypt messages. This means that both individuals need to find a secure way of sharing the key right from the beginning, which can be a logistical issue, especially in situations where the parties are not always in direct communication.

Think of symmetric key encryption like two friends who create a secret password. They need to meet in person to share this password securely. If one friend moves to another city, continuing to use the same password becomes problematic.

So, in this system, the receiver will have 2 keys, a key which we call as 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.

Public key cryptosystems use two keys: the public key, which anyone can know and use to encrypt messages, and the private (or secret) key, which is kept confidential by the recipient and is used to decrypt messages. This architecture allows for secure communication without the need for both parties being online simultaneously.

Imagine a mailbox where anyone can drop letters in (the public key), but only the person who owns the mailbox can open it and read the messages inside (the private key). This setup allows for secure communication without revealing the mailbox key.

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... should not be able to figure out what exactly in the underline message.

The security of a public key cryptosystem relies on the assumption that while the public key and the encryption method are known, it is computationally infeasible for an attacker to derive the secret key or the original message, given the ciphertext. This ensures that even if the mechanism is public, the sensitive information remains protected.

It's like having a Fort Knox style vault where you can see the giant door (the public key) and know how it's constructed, but without the specific combination (the private key), you can't get in. The detailed design is available for all, yet it remains secure due to the complexity of unlocking it without the correct access.

If I am a receiver, and if I am a amazon, for instance, I do not have to worry, who is the potential sender... Anyone who wants to communicate to me has just to use that public key.

Public key cryptography simplifies the process of secure communication by eliminating the need for prior arrangements of shared keys. Any sender can encrypt a message using the receiver's public key without needing to personally exchange any secret information first.

Much like a public phone number: anyone can call you without having to coordinate that share of secret beforehand. They only require your number to connect—just as senders need the public key to encrypt messages for the receiver.

So, this encryption scheme is called as ElGamal encryption scheme attributed to Tahir Elgamal...

The ElGamal encryption scheme is a specific implementation of public key cryptography. It builds upon the Diffie-Hellman protocol and introduces the concept of using a combination of random values, making the encryption process secure while ensuring that each message encrypted with the same key appears differently.

Envision sending a message wrapped in different colored paper each time, even if the core message is the same. This way, onlookers cannot tell that the same message is being sent repeatedly.

Now, let me give you briefly a description of the RSA public key cryptosystem...

RSA is another popular public key cryptographic system that relies on the difficulty of factoring large numbers. It utilizes two large prime numbers to generate keys and allows secure data transmission. The encryption and decryption methods are straightforward, yet effective due to the underlying mathematical principles.

Think of RSA like a restaurant with a lock box: only a certain number of chefs (prime numbers) can create keys to lock and unlock the box, making it exceedingly difficult for outsiders to gain access to the recipes (or messages inside), hence keeping them secure for restaurant operations.

But this forms the basis of the RSA public key cryptosystem, a very interesting public key cryptosystem.

Although RSA is widely used, it has vulnerabilities, such as being deterministic in nature—encrypting the same message multiple times results in the same ciphertext. This can expose patterns and potential information to attackers, which is a significant security flaw. Solutions exist to add randomness, preventing such detections.

It's similar to sending identical messages wrapped in identical gift boxes multiple times; the same appearance may signal to snoopy friends that it's the same gift, potentially leading them to guess what’s inside.

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