Exam 1 Prep

◦Which topics and theorems do you think are the most important out of those we have studied?
I think those that deal with congruence and facts learned about Z_p would be the most important. This would include subjects such as the number of integers relatively prime to some given n, whether a given number is quadratic residue, etc...

◦What kinds of questions do you expect to see on the exam?
Some application, maybe say solving Jacobi numbers, or solving for x in some large polynomial in Z_[some composite]. Also to write out proofs that are similar to the ones we have done in class/homework. Maybe more proofs of the infinitude of primes.


◦ What do you need to work on understanding better before the exam?
I just need more practice seeing and working the more important proofs. I understand them, but I am not sure I could reproduce them on demand.

Section 3.1.1

Difficult: I wasn't able to really follow the proof of Lemma 3.1.1. I don't see how we transfer from the set of F9k) for all k element of the Naturals, then limit it to a single f(k) and then re-apply it to all f(k).

Reflective. I think that it is interesting that there are so many proofs of the infinitude of primes, and that they all show us something a little different about the primes.

Jacobi Numbers

Difficult: The reading was really short and I already knew about Jacobi numbers so I didn't really find anything hard. The results shown follow pretty straightforward from Legrange numbers.

Reflective: I read a little more and saw about Euler psuedo-primes. I know there are systems that lack carmichle type numbers, and I was interested to see another. I find probalistic methods very interesting.

Section 2.6

Difficult: I had a hard time following the proofs of the two Lemmas given by Gauss. They were long, and I got lost early, which didn't help.

Reflective: I find it interesting that we can easily determine if a number is a quadratic residue, but determining what x is, is computationally hard. I find it so interesting that it can be so easy to prove existence without proving what an item is.

Section 2.5.2

Difficult: Following the initial description of the Chinese Remainder Theorem was also hard. I am having a hard time correctly following all of its pieces.

Reflective: I like how the fact that we can reduce polynomials comes into play for the finite field math of AES. I also like how the quadratic formula can be used in Z_p.

Section 2.5.1

Difficult: I found the description of the Chinese remainder theorem hard to follow. Also there examples I felt jumped around. It was hard to follow what they were doing at each individual step.

Reflective: It is interesting to see how we can expand what we have been learning to solve systems not only in Z_p, but also in in Z_n, where n is composite.

Section 2.4.4

Oops, last time I blogged about the wrong section (2.4.5, today's section). Here is what I should have written for last time.

Difficult: I still have a hard time conceptualizing what is going to be a primitive root and what is not. I feel that it should be a little more obvious, but maybe it isn't.

Reflective: The existance of psuedo-primes is interesting, especially how there are even a set of numbers where they will always appear prime using Fermat's Little Theorem. It makes it clear that we can't extrapolate too much from what we have.

Section 2.4.5

Difficult: I did have a little bit of a problem following the proof of Lemma 2.4.5.2. I understand it now, but it was a little harder to follow because it switched back to using the division algorithm, which I wasn't expecting.

Reflective: I liked how we offed an alternative proof of the theorem 2.4.5.2. I really like it when I can see multiple proofs for the same concept, it helps me see how everything is connected.

Section 2.4.3

Difficult: Once again not too much was difficult because this is basic in the areas of cryptography that I study. The only difficult thing was following all the proofs in the book. I just don't feel they are written as clearly as they could be.

Reflective: Once again I find this information to be extremely interesting. It amazes me how much is later derived from these very concepts. It is also interesting to see how it is so trivial to determine much of this material with congruences for computers.

Section 2.4.2

Difficult: This section was mostly review from my 2 cryptography classes and my abstract algebra class. As such the only difficult part was Theorem 2.4.2.4, but just because I had never seen it before. It makes sense, though, and it is a nice contradiction proof.

Reflective: I am constantly amazed by what we pull out of congruences and modular arithmatic. Just today I had to re-explain finite field math for AES to someone, and it is crazy how well it works. I am interested to see what we will pull out of it this semester.

Section 2.4

Difficult: The most difficult part at first was reviewing the concept of a UFD, and Euclidean domains, though this second time through I think I understand it without and problem. The congruence stuff made sense because I have done it several times before.

Reflective: I find it interesting how much can be learned about number systems by mapping them to others, as shown in the relationship between Euclidean domains and UFDs.