Math 485

Sunday, December 6, 2009

In this section, I had a hard time going through the application of ElGamal digital signatures on an elliptic curve. The outline seems very similar to the signature we already learned (outside of elliptic curves), and maybe that's why its tripping me up. It is so similar that I'm having a hard time making the distinctions.
I really loved this section as well. It helps me to apply a new concept to an older one that I already understand. That helps me to understand both subjects better. The only thing I would've liked to have seen more of in this section is examples. I followed the outlines pretty well, except that of the signature. Elliptic curves are still pretty new and are still taking some getting used to. It would have been wonderful for me to see more examples.

The only thing I found a little bit difficult was the addition of points in a field on an elliptic curve. I followed what was said in the book, but it is still a little bit unclear how to do it.
This section of reading was really neat because it brought back fields and presented a new approach to using them. I didn't understand fields at first, but after receiving some help and explanation I really feel like I've got them. It was nice to be able to take a relatively new concept and apply it to a relatively old one. Putting the two together helped my understanding of fields as well as my understanding of elliptic curves.

I'm still a little bit unclear as to how elliptic curves work. I follow along in lecture and understand the examples, but I'm still a little cloudy. The fact that elliptic curves can be singular is also a stumbling block for me.
This section of reading referred back to two other methods of factoring n. It referred back to the p-1 method as well as the Quadratic Sieve. I was intrigued at how similar this "elliptic curve" factorization was to the P-1 method.

The most difficult part of this reading for me was the plaintext section. I couldn't really follow how to encode plaintext and how to get it back. Hopefully this will be cleared up in lecture on Wednesday.
The best part about this reading is that it brought back discrete logs. Not that I love them terribly, but they are doable. It didn't go too far into detail about it, but I can do discrete logs. And, a modified Pohlig-Helman algorithm wouldn't be too hard to work through. I also enjoyed the first part of the reading, where we learned about elliptic curves mod p. I've also got a feeling that we can use this method to help us factor n. Not that I want to, but I think it may lead to that.

The hardest thing to understand about this section was the "infinity" concept. I understand where they're saying infinity is on the graph, but I'm having trouble understanding the addition with infinity. Its really close, but not quite. Lecture should help clear this one up. I also wasn't sure what "abelian group" meant. I learned it in 371 last year, but I can't remember what an abelian group is...
This section of reading really reminded me of differential equations. They took implicit derivatives, they found the slope of the tangent line to a curve, and so on. I also really liked that they clarified the origin of an elliptic curve. At first, I thought it had something to do with ellipses, which it does. But it didn't deal directly with ellipses like I originally thought.

I'm not really into machinery very much. Maybe that's why I am not quite visualizing what's going on with Enigma. Although I'm not really understanding the process yet, I love the concept. I'll definitely need a review of cycles and a run-through of how the whole process works.
Its amazing that Enigma was broken 30 years before anyone knew it was broken! And, how sneaky of those Englishmen to sell broken parts to people who didn't know they were broken... Enigma seems to be a fairly complicated system with rotors and a keyboard and a plugboard. But it is nice to have something physical to look at instead of having "quantum" theories floating around. It's nice to have something to touch and feel and visualize. I also really love that Enigma uses cycles. That was one of my favorite parts of Math 371. It was one of the things I understood the best. :)

Okay, so if I'm understanding this right, Shor's Algorithm is another way to factor a number n, which has two large prime factors, p and q. So, this is basically a quantum way to solve an RSA system? It was great to read the article online, and very nice of Scott Aaronson to explain it in a way that was understandable. Reading through section 19.3, I felt that I already understood what it was saying, because I had had an introduction to it already. Although Aaronson did bring the discussion full-circle, to come back and relate how this "parallel universe" thing works in Shor's Algorithm, he didn't give any examples or anything. I imagine this would be hard to do, but it was a little bit hard for me to understand the full circle without seeing an example.
I completely understood the thumbtack analogy. It was an interesting way to find a period. I had to read that section about 3 times before it started to make any sense. And then I wondered: "If I didn't have school or church or meetings at all, what would be my typical day? How long would my period be?" It got me thinking that my typical would probably be shorter than 24 hours. If I were locked in room without a clock and without a window, it might be different, but when I'm bored I try to sleep. But, that could also have something to do with being pregnant. :)