And assuming that the key agreement has been achieved, the second problem that is addressed by 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.

So, namely we want to come up with algorithms which should help me to do secure communication. And 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 m, m, m and so on.

Now, the way symmetric encryption works is as follows. So, imagine Sita has some message, it could be an email, it could be just a hi message, it could be anything, it could be her banking password. So, she has some message which is abstracted as a binary string, we call her message as plain text. We want to design an algorithm which we call as an encryption algorithm which takes a message m and the key k both of which are binary strings. And it should produce another binary string which we call a ciphertext.

Now, once Ram obtains this scrambled message, he will have a decryption algorithm, he will have in a sense, he will know that Sita has used an encryption algorithm whose details are publicly known and the corresponding matching decryption algorithm also will be publicly known. So, Ram will use the corresponding decryption algorithm.

So, 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?

So, 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.

So, 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, whatever Ram’s mixture that Sita has received, she takes that and to that she adds whatever components she has added to prepare her secret mixture. And same task is done by Ram. He takes Sita’s mixture and to that he will add his own contribution.

So now you can see that how exactly the concept that we have seen in the context of cyclic groups are useful to come up with a very important practical solution for a practical problem namely that of key agreement.