Diffie-Hellman key exchange

Let's say you're in the middle of your classroom and you want to send a message to your friend located at the back (by shouting it out). However you don't want your other classmates to understand the message.

Well you've learned in your computer security class that if you and your friend both have a shared key, you can encrypt the message using the key, send it to him, and then he can then decrypt the message it using the same key.

However in this case both of you don't have any shared key.
What can you do?

In fact this seems to be impossible, but it isn't. You can use the Diffie-Hellman key exchange method.

This is explained beautifully in this image taken from Wikipedia:

You are Alice and your friend is Bob.

The dark orange and dark blue paint are some random secrets decided by each of you.

Everyone in your classroom knows the yellow paint. They also know the "light orange" paint and the "light blue" paint (since this is exchanged publicly by Alice and Bob)

After applying the above method, both Alice and Bob will end up with a common secret: the dark brown paint which they can subsequently use as a shared key to exchange secret messages.

Let's see where the difficulty arises for your classmates to obtain the dark brown paint (the shared key)

They know the yellow, light orange and light blue paints.

But from the yellow and light orange paint, they cannot recover the dark orange paint.

And from the yellow and light blue paint, they cannot recover the dark blue paint.

Thus they cannot obtain the dark brown paint.

In fact, the above paint operations can be defined by some mathematical functions which are easy to compute (mixing paints) but computationally difficult to reverse (unmixing paints) where it would take millions or billions of years for even the fastest supercomputers to find out the answer.

The original implementation of the protocol uses the function g^A mod p (known as Modular Exponentiation function)

Someone knowing g & p (the yellow paint) and g^A mod p (the light orange or light blue paint) cannot find the value of A (the dark orange or dark blue paint) in a feasible amount of time for large values (known as Discrete Logarithm function).

More details here.

You can use this method to avoid man-in-the-middle (MITM) attacks, even if someone is listening to your communications from the beginning before you've agreed on a common secret key.