Assignment 26

1. I feel like there is just so much to remember at this point. It's going to be hard for me to keep track of it all. I guess I just really need to practice all of it over and over. The first one seems really similar to what we've been doing lately, but then the rest all seem to differ so much from each other.

2. It's interesting that there is no really good way to solve discrete logarithms, but there are so many methods here to try to solve it. Looks like a lot of people have tried to figure it out. Like we said in class, if any of us could solve this issue, we'd definitely have a job many places!

Assignment 25

1. So, discrete logarithms. I haven't really done much with logarithms in a while, so I guess it's time to brush up on the basics. Overall, the concept doesn't seem too bad though--it seems to follow the general rules so far at least.

2. I really liked reading about the public key. It's just really cool that RSA can have a signature. We previously talked about how this has facilitated things like online shopping, and it's really nifty to me that this is all possible because of cryptography.

Assignment 24

1. The quadratic sieve method seemed pretty interesting. We had already covered the methods from 6.4.2, so I follow those pretty well, but I don't see all the logic behind the quadratic sieve as well as I would like to. I think part of it is that I need to review some of the more basic operations of modular arithmetic and how they can relate to such methods. The matrix this method finds is also interesting, but I guess it doesn't yet make full sense.

2. I look forward to using the quadratic sieve method (once I understand it a bit better). It seems like it'll be pretty fun to use and I like how it uses the matrix to let us "look up" more information regarding our number of interest.

Assignment 23

1. Well, I definitely understand the inefficient methods--it figures that the one we'll really focus on is the one that I really don't understand. Haha, such is life right? I really like how the book generally gives and example to work through, so maybe we could do one together in class?

2. I do like how we use RSA because of how difficult it is to factor, and then we now move to studying factoring. So, I haven't heard of anything beyond RSA so far, so does that mean we are going to see that factoring is still difficult overall?

Assignment 22

1. Okay, so there are so many parts to the Miller-Rabin Primality Test. It is going to be difficult for me to keep them all straight! It's interesting also that we have ways of saying that n is composite, but otherwise we can only say that n is probably prime for the most part.

2. This was a pretty interesting read. I like the idea of figuring out whether some large number may or may not be prime. Obviously, it doesn't give you whether something IS prime or not, or else systems like RSA would be woefully inadequate. However, it does make me think about whether it would be possible to one day do so.

Assignment 21

1. This is some pretty interesting stuff. I feel like I have a decent handle on the usage of the Legendre symbol, but I would like a bit more practice with the Jacobi symbol. It just appears to be a little more involved to me (in terms of the important properties and whatnot that it relies on).

2. This stuff just seems really cool to me. I've always like equations, and I look forward to working with this stuff. I'm a bit behind on what's been going on in class because I've missed the last two since I got sick, but I look forward to seeing how we are going to use this with the systems we use.

Assignment 20

1. This section makes sense more or less to me. I suppose if anything is difficult, it would be getting used to the compositional method that was used. Does it work no matter how you factor n? I guess, in the end, I'm just not used to looking for it too. So, if I can get used to it then I think I'll be okay.

2. I enjoyed this reading. It really seems to make sense after we have already learned about exponential powers being used modularly. It also makes sense as we are using it in terms of n=pq, and how we can either find the square root or the factorization for n. Overall, it just makes sense to learn about this here as I feel it ties in perfectly with everything we've been discussing right now.

Assignment 18

1. Now, this is was an interesting read. I remember doing thing with repeating fractions when I took History of Mathematics. None of it really seems to be too difficult for me on it since I am somewhat familiar with it. It is also nice that they gave us a general form for a repeating fraction at the very end of the reading in order to improve the efficiency of it all. I know that we are supposed to have something here that was difficult for us to understand, but I felt that everything made sense while I was reading through it. I guess the only thing that kind of surprised me in the reading was how they came up with the theorem out of nowhere--did they originally decide on 1/2s^2 because it fit a pattern or what?

2. I wonder how we are going to use this in our different cryptic systems. It is interesting and all, but we have been working with really big numbers lately. In a similar way, I guess we could work with irrationals? In any case, I really don't see yet what the application to our systems will be, but I'm sure it will be interesting. Or are we possibly moving on to a new system already?

Assignment 17

1. Ok, so I think I understand the idea of it all, but it all just seems to be so gross! I definitely wouldn't want to do this by hand--are we going to use computers when we deal with the RSA? If we are, I'll probably need some pointers on how to write it into SAGE. I can pretty easily see how this relates to what we've recently covered, but the numbers are much larger now, so it seems unlikely to me that we'll solve these by hand. If we do plan to solve these by hand, I suppose I'd be able to understand it well enough to do it for smaller primes p and q.

2. How do people even come up with this? It really is pretty brilliant. I especially like how they explain what is happening in a non-mathematical way first as it helps me get a better context for conceptual understanding of the system's efforts. I guess I really wonder what we plan to do with it now though. Are we going to play with RSA on the computer to send or receive secure messages, encoding them and decoding them? Are we going to do it by hand with smaller primes? It's definitely important to know about since it has so many applications today, but what can I consciously do with it?

Assignment 16

1. I feel like I could still understand section 3.6 better. It seems as though Fermat's and Euler's finding will come to be very important in the near future, so I'd like to understand them through and through. I feel like I mostly get what they are working towards, but I just need that little bit extra that the lecture in class usually gives me before I feel like I really know what we are looking for.

2. Ok, so I found that it was really cool how we could use Fermat's Theorem to find that 2^53 is congruent to 8 (mod 11) so quickly. That's pretty classy that. It was nice to have the explanation of the three-pass protocol--are we going to do anything with that? That would be interesting to work with.

Assignment 15

1. I feel I'm going to struggle with getting the hang of finding a (mod m*n) and also with modular exponentiation. It'll just take a bit of practice, and I feel that going over an example in class will help me out immensely. The concept makes sense to me though.

2. I guess what this reading makes me think about is how we are going to be using values of mod that require an m*n conversion. I know we are eventually going to work up to primes of that magnitude, so I wonder how exactly we'll fit this into stuff like that. How are we going to use it in practice?

Assignment 14

1. I feel the most important topics we have discussed are the actual encryption systems themselves and how to decrypt them. In particular, the last couple of ones we've gone over seem to be even more important as they are much stronger overall.

2. On the exam, I expect to see question that test our basic understanding of the different systems and how to encrypt and decrypt using the various methods we've seen in class.

3. I feel like I can definitely use some extra time understanding the last two weeks worth of material a bit better. I guess I just feel like I'm not fully understanding everything I should be in order to be ready for the exam.

Assignment 13

1. I find that these systems are so much easier to understand when we go over them in class. I think having somebody who really knows how it works explaining it while simultaneously writing it down helps me so much. I guess that's been the difficulty for me on this reading--complete comprehension.

2. What I wonder about with this system--and others like it--is how practical is it to use overall. For example, what exactly is it used for, and why? How well does it work compared to the others we've seen so far. I can make the obvious deductions (I'm pretty sure it's better than substitution for example), but I wonder how this system holds up against other ones.