What is a good way to demonstrate elliptic curve cryptography?
An encrypted chat application by itself will not really demonstrate anything, since from the users' point of view it will not (if done properly) be any different from an unencrypted one. To demonstrate the encryption, you would need to also use Wireshark or something similar to sniff packets on the network and see that they are encrypted. But if simply showing a ciphertext and saying "this is encrypted" is enough, you can just encrypt a file, which apparently you can already do.