16.5, due on December 8

1. (Difficult) This section didn't get too much into deriving things, and was, in  way, a review of some previous concepts, so it wasn't too bad.  I still find it difficult to think intuitively about elliptic curves.  I don't know why adding and multiplying behave like they do (or even how they behave).  Doing some examples on the computer the other day though leads me to believe doing the types of calculations required for these crypto systems is possible without having full understanding of what's going on.  You can perform the necessary operations without considering all of the details.  It seems like some intuition, like thinking about what points at infinity mean, would be helpful to better understanding these cryptographic applications of elliptic curves.
2. (Reflective) I am curious how well elliptic curve cryptography performs.  I understand you can use smaller keys (I think I've heard that), and I think we've learned we can use smaller primes.  Do these savings outweigh the more complicated basic operations.  It seems like adding and multiplying with elliptic curves would be more computationally expensive than multiplying and performing exponentiation. Could be wrong.
   Also, I wasn't aware all of these elliptic curves analogues of other systems existed.  Perhaps I've heard of elliptic curve Diffie Hellman being used?  I've also heard of elliptic curves being used with Identity-Based Encryption.  Are elliptic curves used in other real-world security situations?