Hybrid Cryptography

It turns out the only ways we currently know to do public-key crypto take a long time do calculate (as compared to symmetric-key crypto)
Using public-key to encrypt large messages would be too slow!
We take a hybrid approach:
Pick a random 128-bit key for a symmtric-key cryptosystem
Encrypt the large message with that symmetric key (use, e.g. AES)
Encrypt the 128-bit key with a public-key cryptosystem
Send the symmetric-encrypted message and the public-encrypted key to Bob.

This hybrid approach is used for almost every cryptography application on the Internet today.