The Diffie-Hellman Key Exchange Protocol

Nikita and Michael agree on a prime number $p$ and an integer $g$ that has order $p-1$ modulo $p$. (So $g^{p-1}\equiv 1\pmod{p}$, but $g^{n} \not\equiv 1\pmod{p}$ for any positive $n<p-1$.) Nikita chooses a random number $n<p$, and Michael chooses a random number $m<p$. Nikita sends $g^n\pmod{p}$ to Michael, and Michael sends $g^m\pmod{p}$ to Nikita. Nikita can now compute the secret key:

$$s = g^{mn} = (g^m)^n \pmod{p}.$$

Likewise, Michael computes the secret key:

$$s = g^{mn} = (g^n)^m \pmod{p}.$$

Now Nikita uses the secret key $s$ to send Michael an encrypted version of her critical message. Michael, who also knows $s$, is able to decode the message.

Meanwhile, hackers in The Collective see both $g^n\pmod{p}$ and $g^m\pmod{p}$, but they aren't able to use this information to deduce either $m$, $n$, or $g^{mn}\pmod{p}$ quickly enough to stop Michael from thwarting their plans. Yeah!

The Diffie-Hellman key exchange is the first public-key cryptosystem every published (1976). The system was discovered by GCHQ (British intelligence) a few years before Diffie and Hellman found it, but they couldn't tell anyone about their work; perhaps it was discovered by others before. That this system was discovered independently more than once shouldn't surprise you, given how simple it is!