Abstract: This is the talk that I gave last week at the Rose-Hulman Undergraduate Conference. It is directed at a level for bright undergraduates and is therefore an elementary introduction suitable for someone knowing nothing about modern cryptography.

Electronic commerce requires the transmission of private information over public channels. Mathematically-based public-key cryptosystems have been developed for this purpose, but they all share one potentially disturbing feature, namely that our peace of mind depends on an eavesdropper's inability to solve some hard math problem. I shall describe society's increasing dependency on number theory and whether we can sleep soundly in light of some recent advances.