Section 16.5

The first little bit of this section went over what we talked about in class yesterday, going over applying different cryptosystems to elliptic curves and how you might translate some of the same ideas into elliptic curve math.
The diffie hellman key exchange follows suit, in that we are translating an idea we have covered before and putting it into elliptic curves and this part seems pretty straightforward.
The signature part again seems like a concept that I should be able to understand better but i was a little confused when it starting talking about how k and k^-1 dont multiply to get one but sometimes you could get out infinity.  If k is just an integer shouldnt it just come out as one?  k can't be a point because that would make k^-1 make even less sense...

Section 16.4

It makes sense that for computer usage that having elliptic curves mod 2 would be worthwhile, what i hadnt thought of is that the equations would need to be modified to not make them = infinity everywhere.  If we are finding the points on a line though mod 2 would there ever be more than 5 points total?  It seems like no matter the curve that they would always only be able to have the same 5 points.  I guess that that is probably why it says that usually elliptic curves mod 2 are not generally big enough.

Section 16.3

Its interesting how using a random curve that we can can factor prime numbers.  Probably factor numbers as large as we have ever worked with just by trying one random curve... Also it is still intriguing to me, that the thought process is "try to find 3P, oh it didnt work, well that helped us to factor n"  which seems like a much trickier problem than finding 3P.
What I am confused about, is what is the doing this by hand method? How could we apply this on a the final, or what kinds of questions could be asked about it that I could solve by hand?
What are those other times besides RSA when I would want to factor a really large number...?

Section 16.2

Curves mod p are what we started to talk a little bit about yesterday, which was good because I liked Math 371 so talking about rings and groups feels more familiar than some of the other material that we have covered this semester.  Also we talked about "addition" so that concept makes sense.
Is the method for finding out the number of points beyond anything that we will cover?  It seems like even if there was some kind of basic algorithm it could be nicer than counting everything up.  unless its a small graph.
Nit-picky I know but I dislike how the book says "this might not look like a log problem, but it is clearly the analog of the log problem...." if it doesn't look like it, how is it clearly it?
So is it possible that when you are choosing a ciphertext or making one I guess that it doesnt work and doesnt code? Or what is it exactly that is failing?