ECC Example

We start by picking a secure elliptic curve.  This one is known as the 2013 Aranha–Barreto–Pereira–Ricardini Elliptic curve. It is given by $y^2 = x^3 + 117050x^2 + x$ defined over $\mathbb{Z}/(2^{221}-3)\mathbb{Z}$. 

We now choose a point $P$. 

The curve $E$ and point $P$ are public information for anyone to use.  

Now Alice chooses her value $n_{A}$ and Bob chooses his value $n_{B}$.  They do not share these with each other or anyone else; $n_A$ is known only to Alice and $n_{B}$ is known only to Bob. 

Alice computes $n_A P$ where $n_A P$ means $P$ added to itself $n_A$ times with elliptic curve addition. 

Alice now sends the point $Q_A$ to Bob over a public channel.  She can e-mail it, rent a billboard, run a commercial on TV, it doesn't matter.  It does not need to remain secret because the elliptic curve discrete logarithm problem says even if people know $P$ and $Q_A$, they cannot figure out $n_A$ from that information. 

Bob computes $Q_B = n_B P$ and sends that to Alice. 

Once Bob receives the point $Q_A$, he computes $n_B Q_A$.  Alice receives the point $Q_B$ and computes $n_A Q_B$. 

Bob and Alice now share the secret $2571462759950963338623653560522804306295245557988914858290462099836$.  They can use this as a key for AES to set up secure communication.