16.5, Due on December 8

This section was interesting to read because it was neat to see an alternate way of encrypting messages, signing documents, and exchanging keys using the elliptic curves. Finally I see the application of this whole chapter on elliptic curves! I also found the fact that this method protects against values of n with small prime factors to be interesting and poignant.
From section 16.5.1 I am wondering how a message is represented as a point on a curve. Apparently it was described in section 16.2 but I don't recall how to do it. One other aspect that I'm a little shaky on still is adding points to get a third point on an elliptic curve. [This is probably because I am writing this quite early, and we have two more class periods of practice before we get to this lesson.] I bring this up because adding points on an elliptic curve seems to at the heart of these crypto-systems.

16.4, Due on December 6

This section is really just a more specific portion of the other sections. I found it interesting that all the tangent lines are vertical to the curves of this modified elliptic curve. This type of crypto-analysis incorporates partial derivatives and lots of Calculus which I can appreciate.
I understand that elliptic curves are much easier to work with in mod 2, but I don't see why they are easier to work with in mod 2^n. That appears to be a big advantage but I don't see why. I am also confused about the end of the example on page 362. I am not sure how the authors found -(w,w^2).

16.3, Due on December 3

Frankly, I've found this whole chapter to be interesting. I love Algebra and this is a type of Abstract Algebra that is interesting. I find it fascinating that is can be used to factor large numbers. I don't understand everything that's going on but the main ideas are really intriguing. One of the things I don't understand is how it's easy to find B!P. That seems like a lot of work.
I also have two lingering questions about elliptic curves:

1. Doesn't a curve technically have an infinite amount of points? When we worked on the elliptic curve Z(E) mod 5 we only came up with 9 points. Are those the only solutions because our solutions must be integers mod 5?
2. When we are dealing with a finite number of points, such as Z mod 5, how can infinity be a point? It's hard for me to grapple that a curve with a finite number of points and finite values for coefficients can have infinity as a point. Am I missing something or is that just a difficult, abstract concept?