Remember, we begin by doing this cell, which enables us to turn words into numbers. But really, what we care about are the numbers, because there are lots of other ways to turn words into numbers. So just use this if you want a quick number for a message.

There is a quite useful application of D-H called key exchange. Here is the basic concept.

Two people trying to pass information (universally called Alice and Bob) want to decide on a secret key for using some encryption routine. Unfortunately, they know that someone may be listening in on their decision. Instead of trying to send a secret key one has chosen, they try to create a secret key using (essentially) public means. Here's how it works.

• First, Alice and Bob jointly pick a big prime \(p\) and a base for exponentiation \(g\), presumably with \(1<g<p\).
• Now, they each secretly choose an exponent; maybe Alice chooses \(m\) and Bob chooses \(n\).
• Then each of them exponentiates \(g\) to their secret power!
• Then they pass off these numbers to each other.

Thus, now they have a secret key (\(g^{mn}=g^{nm}\)) they can easily compute but which a spy in the middle cannot. Feel free to try this with your own numbers you pick!

This gives an encryption key useful to both Alice and Bob, to protect from any potential Eve who might be listening in. (That's Eve for eavesdropping, believe it or not – also a universal person in these stories.)

On the down side, if Eve not only is listening, but actually has access to Alice and Bob's transmissions and can change them, she can actually add her own exponent \(\ell\) to the game, so that she pretends to have secret key \(g^{m\ell}\) with Alice and secret key \(g^{n\ell}\) with Bob. Both of their keys' security is now compromised. Such a situation is known as a "Man in the Middle" attack. There is no obvious way to stop this if Eve has that much power.