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?

16.2, Due on December 1

The first portion of the section where the authors describe how we are going to use elliptical curves to factor large numbers was very interesting. "The situation where gcd(b,n)=1 fails... form the key to using elliptical curves for factorization" (pg. 353). I like that little teaser and it's making me moderately excited to read section 16.3.
I also found the first portion to be a review of what we went over on class on Monday. It's nice to approach the reading having already seen an example in class.
The hardest section for me was the last one about representing plaintext. This confuses me. I thought we were using the elliptical curves to factor n, not to encrypt anything. I also had a question about the example on page 353. Where did the equations for what x3 and y3 are congruent to come from? I see where the values came from but not the equations.

16.1, Due on November 29

I kept waiting for the cyrptography application in this section but it never came. Hopefully in the later sections I will see how it relates. I enjoyed reading the historical point on page 349, and it cleared up some confusion about the naming of elliptical curves for me. I also found the very last sentence of the section to be interesting, that infinity is the identity element of the abelian group. I will be very interested in hearing your explanation of that. It reminds me of complex analysis.
The hardest part for me to understand was the addition of elements in the group. I didn't quite follow the group what see what the point was. I struggled with this section a little bit just because I didn't see the motivation for the math. It seemed like the authors were just babbling on and I wasn't sure why. Are all the points on an 'elliptical' curve an abelian group with infinity as the identity?

2.12, Due on November 23

I always enjoy reading about the history and application of the different cryptosystems we have learn about. I liked reading about how the British sold Enigma machines to other countries without telling them it had been cracked. I also liked reading about the different aspects of the machine that made the different combinations plentiful. It's interesting to learn about early, physical cryptography machines, especially in light of the Quantum Computing we read about earlier.
One question I have is that it seems this is just a substitution cipher? I thought that at first, then it sounded like there was more going on, but then on the last page I read about Rejewski and his colleagues creating a code book for the different combinations, and therefore "the effect of the plugboard was then merely a substitution cipher"(pg. 55). What is going on here that makes this different from a substitution cipher?

Shor Reading and 19.3, Due on November 22

I enjoyed the reading on the web about Shor's Algorithm. It was written in a very lighthearted manner tailored to the layman. His analogies were very good and he brought up some great points. It's interesting to me to think about modulo exponentiation and factoring large values of n.
The part about the reading that confused me was the the clock example. I understood the 'experiment' but I didn't understand what we can deduce from the results.
The book reading was not very easy to follow. If I were to ask one question about it I would ask where the Fourier Transform came from? It seems pretty crucial to the Shor Algorithm but I found its explanation to be less than satisfactory. I am also confused about how the graph relates to the algorithm.