And assuming that the key agreement has been achieved, the second problem that is addressed by the cryptography, the second core problem, I should stress here, it is not the case that secure communication is the only problem, the second core problem addressed by cryptography startup secure communication. So, the setting here is the following, we will assume that Sita and Ram has already executed the key agreement protocol over the internet, and they have agreed upon a common key.

And now using this common key, we would require Sita and Ram to come, we would require some algorithms which are publicly known, according to which Sita can convert or encrypt her message into some garbled text into some garbage and communicate to Ram and Ram should be able to convert back those garbage or scrambled text back to the original contents using the same key, k which Sita has.

So that is the second problem addressed by cryptography. By secure communication here I mean that, if there is a third party or Ravana, who knows the public description of your algorithm but does not know the value of key then even after observing the communication happening between Sita and Ram and even after knowing the full protocol description according to which these messages have been computed, the Ravana should not be able to come up with the values of m1, m2, m3 and so on.

So, it turns out that there are two kinds of, two classes of cryptographic algorithms which we use. The first category is that of private key or symmetric key encryption. In symmetric key encryption, the setting is the following. It will be ensured that a common key is already shared between Sita and Ram by some mechanism, say, by running a key agreement protocol and no one else apart from Sita and Ram knows the value of that key.

So, the analogy could be that, assume Sita and Ram have already exchanged a key for a physical lock. If Sita has a message, what she can do is, she can take a box, keep her message written in a paper inside the box and close the box with a lock and using the key that she has. Now she can send this locked box by a courier or anything. So, if there is a third person who does not have the key for opening the lock of the box, he would not be able to do that.

How at the first place they can do that? Because everything will be now happening over a public channel because it is not the case that Sita and Ram knew beforehand in advance. It is like saying the following, if I want to do a transaction over the internet; Amazon may not be knowing well in advance that a person called Ashish Chowdhury, would like to do a transaction with Amazon. So, I will be doing my transaction at a run time, how at the first-place key agreement has taken place?

It was a folklore belief that it is not possible to agree upon a common key by interacting over a public channel. But the Turing Award winner, Diffie and Hellman, proved this belief to be incorrect, by coming up with their seminal key exchange protocol.

The main idea used in their key exchange protocol is the following. They observed that there are plenty of tasks in this universe which are asymmetric, they are asymmetric in the sense, they are very easy to compute in one direction. That means, it is very easy to go from one state to another state but extremely difficult to reverse back the effect of that action.

So, to begin with, both Sita and Ram will be starting with some common publicly known color. And now, what they will be doing is the following. They will prepare independently some secret mixtures. So, Sita will prepare her secret mixture independently and Ram will be preparing his secret mixture independently, by adding a secret color, individually and then they will publicly exchange their mixtures.

Now, what is the goal? The goal for Sita and Ram is to come up or agree upon a common mixture which should be known only to them. So, what they can do is, they can individually add the secret component that they have added, to the copy of the mixture that they are receiving from the other party.

For the adversary or for the third person to compute ⟨k1⋅k2⟩, he should know α or k2. Because if any of these 2 values is learned by the attacker, he can easily compute ⟨k1⋅k2⟩. But for learning α or k2 he has to basically solve discrete log. That means, if I ensure that computing discrete log is extremely time consuming for this attacker and by time consuming means, at least it takes say 10 years or 15 years then, I can say that this protocol is safe.

So, it turns out that, if we instantiate this protocol with my group being ℤ that means, if I ensure that my group G is the set ℤ where, p is some 2048 bit prime number then, using current best computing speed machines for solving a random instance of discrete log, it will take order of several years.

So, now you might be wondering that what should be the choice of the group, how big it should be and so on. So, it turns out that, if we instantiate this protocol with my group being ℤ that means, if I ensure that my group G is the set ℤ where, p is some 2048 bit prime number then, using current best computing speed machines for solving a random instance of discrete log, it will take order of several years and hence, an adversary who tries to attack the scheme will fail to do that.