A gentle introduction to elliptic curve cryptography

Abstract: Let's say you order some snacks from Amazon.  To do this, you need to give Amazon your credit card number.  This information is sent on a public channel, which means that it's accessible to outside eavesdroppers.  Naturally you don't want a hacker using your card to order his own snacks, so to keep your card number safe, you need a way to encrypt it so that Amazon can figure out your card number but the hacker can't.  

The solution to this problem comes from something called public key cryptography.  One of the best known methods of encryption is the elliptic curve Diffie Hellman key exchange protocol, which I will explain in this talk.  I'll also give a brief overview of some other cryptosystems and discuss the benefits and drawbacks of each.  Here is a motivating question: if I tell you that 5^n is equivalent to 

6879587787378509891410449896487199349357858646119270556081924724964524986206921232974707347355011349015062428 

mod 

10513733234846849736873637829838635104309714688896631127438692162131857778044158273164093838937083421380041997,

can you tell me what n is?  If you can solve such problems, please come collect your Fields medal.  If not, then I'll explain on Friday why you (probably) can't do this on your laptop and why that fact is important to the privacy of all our online communications.