section 7.2

Difficult: I still don't think I understand their notation very well, the whole x = L sub alpha (beta) thing is hard for me to keep track of which part means what,but I'm sure time/hw problems will help with that so I can keep them all straight. Are we going to be expected to factor all these bigger numbers and mod them in our heads/using only a basic calculator on the next exam? Because I'm pretty sure that would extend the amount I need to set aside for the test substantially.

Interesting: The Baby Step, Giant Step Attack was pretty interesting. I also thought the fact that they named the attack that was pretty funny. But it is cool that/interesting to think about since the two lists are interconnected and there's a match, but since it only works well for primes up to 10^20, it makes me wonder what a better method would be.

Sections 6.4.1 & 6.4.2

Difficult: I didn't understand the part about linear dependencies very well in 6.4.1. I'm also not sure I understood the matrix connection very clearly.

Interesting: it is really interesting how far we've come with factoring algorithms in the last 30 or so years and how big of an impact computers have had on our mathematical and other advances. Who knew such a relatively small thing would have such far reaching impacts?! It's pretty cool, not going to lie.

PS, as you can see below, I accidentally posted the entry on my personal blog instead of my Cryptography blog--sorry!!

sections 6.5-6.7 and section 7.1

Difficult: The discrete logarithm problem looked difficult, I didn't understand very well why n had to be the smallest positive integer such that alpha^n is congruent to 1 mod p for us to get that x=(L_alpha)(beta) though. Could you explain why in class?

Interesting: It's crazy how many people and computers it took to find the congruence relations of the desired type for this problem! and then how relatively they were able to reduce the matrix and determine the dependencies. I also thought the message was pretty interesting... mostly weird, but I guess it makes sense since they didn't want anyone to just arbitrarily guess what it was and get the moneys without doing the work.

6.4 up to just before section 6.4.1

Interesting: I think it's crazy how many awesome ideas Fermat came up with. Seriously, were all of these ideas actually his or did he just steal it from people who didn't care to publish or is some of his work actually someone else's that historians decided to attach his name to? But I guess when you think about Euler, Fermat's work pales in comparison as far as breadth goes. Still, Fermat had some pretty crazy ideas and theorems. I do like that there are other methods of factoring large numbers other than checking every prime up to and including the square root of the number, because for large numbers, it's impractical to do this.

Difficult: I'm not sure that I understood the p-1 factoring algorithm very well or at least I don't know how we would know if we picked a "good enough" a and bound B for the algorithm. Also, how would we know when to stop computing?

Section 6.3

Difficult: I feel like the Solovay-Strassen Primality Test could be very difficult, because translating fractions into modular has always been time consuming and confusing for me and I have a tendency of accidentally messing up somewhere along the way. So maybe if we could go over some examples and really solidify how to translate fractions into mod, that would really help me.

Interesting: Whenever I've learned/talked about how to check if something is prime or not, I've always been pointed to using the square root, but like the readings said, this isn't practical for very large numbers. I've never really had any experience with other methods of determining whether or not something was prime, so I really liked that they presented many different kinds of thinking about and finding whether or not a number is prime. I especially thought the Miller-Rabin Primality Test was really interesting and it kind of made me wonder why I've never really worked like anything like this in previous math classes and why everyone always talks about using the square root and then using all those primes when it's not exactly practical for very large numbers.

Section 3.10

Interesting: I have to admit that the proposition at the beginning was really interesting to me because it seems on the surface to be unrelated principles, and yet it's provable, and the proof isn't super complicated either. I don't know why, but it just amazes me sometimes how people can connect so many seemingly unrelated ideas to create theorems, etc. that really help advance our understanding of mathematics. I also thought the properties of the Jacobi symbol were interesting in the way that the authors applied them. I don't think I knew that Jacobi's ideas could really relate to reducing fractions, and yet it does! It also doesn't help that I don't remember how I really used the symbol before too though--maybe I knew and just forgot.

Difficult: I think it will definitely take practice to get used to using Legendre's and Jacobi's symbols in this way since I don't remember using it this way before (in fact, I really don't remember what I used it for before at all). So it will definitely take time for that to process.

Section 3.12

difficult: I've already done work with continued fractions before, so it's not super hard for me. But I could see how some people could have a hard time with it since it's a weird concept and not something that we're really work with that often if at all.

interesting: I liked how they talked about the approximation for pi and how you can approximate for different fractions too and that they actually went through and showed you the process in numbers instead of just symbols.

Section 6.1

Difficult: I don't think I really understand why this algorithm works. Could you explain the why more in class? I can see that it does work, but why?

Interesting: I thought it was interesting that they were able to come out with a way to send a key in a secure method. I also thought it was interesting that it wasn't until 1997 that it finally came out that someone else has come up with a method before 1977. But seriously, how do these people come up with these things!?!? I also thought PGP was a funny acronym!

10/4/13

Which topics and ideas do you think are the most important out of those we have studied?

how to "crack" different ciphertexts and various encryption and decryption methods... so pretty much everything.

What kinds of questions do you expect to see on the exam?

definitions, probably algorithm type stuff--especially with rings, since we spent two days on that. things like that--hopefully things that don't take a really long time though, but since it's an exam, the really long problems probably won't be exam though.

What do you need to work on understanding better before the exam?

keeping the different encryption/decryption methods straight--it's really hard for me to remember them all!

Sections 5.1-5.4

Interesting: I felt like this encryption method itself was really interesting--how do people come up with so many cool and crazy methods of encryption/decryption?!? But I though it was really cool how it covered a broad range of mathematics--binary, matrices, GF(2^8), modular arithmetic, etc.

Difficult: I feel like reading these processes is not very easy to understand, because it's not the same as applying and it's hard for me to visualize what I would need to do when pretty much everything they talk about is very abstract. But the homework and lecture will definitely help me better make the connection and understand how to tackle this kind of encryption/decryption.