Section 6.1, due December 2

I found it a little confusing why single points were bad in our scenario. It seems fine to add a single point to itself. It may not be a very interesting curve, but I could not see why it would be banned. Also, I did not quite understand what characteristic 2 meant.

However, I did find it interesting that elliptic curves are all abelian groups. We know abelian groups are really just products of additive groups in Zn, so it seems that we could just classify elliptic curves as such. It may just be too hard to map the points on the curve to their corresponding points in the products of Zn. However, if this was done, it would seem to open up all sorts of opportunities.

Section 5.4.2 due November 30

In reading about the public key systems, it was a little confusing at first all the notation associated with RSA. After they gave an example, it became much clearer. I think the part that threw me at first was the different bases for the numbering system. I thought it was going to do it all in base 26 or so, but it makes sense that it is easiest to convert to base 10 first.

I did find it fascinating that this area seems to be changing. It is like those trying to encrypt are always on the lookout for better algorithms, and they hope to find them before that other party can figure out a method to decrypt too fast. Another part that was interesting was how this depends on a problem taking a very long time to solve relative to the creation of the problem. After all, to encrypt something, it needs to be quick, but those trying to break the code have much more time to tear it apart.

Section 5.4 due November 28

I found it difficult at first to understand the matrix cipher, but after seeing the example it made quite a bit more sense.

I found all the various methods for encryption to be interesting. There are just so many ways, but some of them seem to be rather vulnerable. I wondered why they would include the simple examples like the Caesar cipher, since it is easily broken, but I guess like most things, it is important to start with the basics before getting down to the tough stuff.

Section 5.3.2 due November 22

The proof for the Ludqw Lehmer test was a little difficult for me to understand. I did not follow the proof very well. It was bouncing around in all sorts of areas, and in general just hard to see where it was going. However, it was rather interesting how it brought in so many different concepts.

Another part that was interesting was how it revolves around the Mersenne primes. They seem to be a particularly useful breed of primes. Also, it seems that they would be 3 mod 4 which we discussed in cryptography to be a desirable trait in a prime number.

Section 5.1, 5.2 due November 15

I found it a little confusing how Legendre's Formula worked. I could not quite follow what it was doing or how exactly it worked. From what I did understand, it was there to find numbers relatively prime to m. I figure if one multiplied a bunch of primes together, and then used this method, they could find a fair number of primes.

Something that I found interesting was the general idea of testing for primes. The methods that were introduced seemed mostly like those that I learned in grade school. However, I imagine that it will get more sophisticated. Another side note, showing that the sum of twin primes converges was rather interesting as well.

Exam Review, due November 11

I feel that the major portion of this section was spent handling how there are an infinite number of primes, and also Dirishlet's theorem (some of the simple cases). Also, quite a bit of time was spent discussing arithmetic and multiplicative functions as well.

I would expect some proofs for some general concepts that we have covered (similar to the homework problems). I would also expect some computational problems such as continued fractions or Fibonacci/golden ratio questions.

I would like to go over a little bit of the concept behind sums of squares problems. Just section 3.2 in general.

I would like to go more into how to find primes in general.

Section 4.2, due November 9

I found it a little tough to follow Bertrand's theorem. The logic was a little difficult to follow and all of the simplifications sort of got jumbled in my mind. I could not quite follow where it was going or how it was arriving at certain conclusions.

However, I did find it interesting how we can set so many bounds on the number of primes, how big the primes are, and some sums of primes as well. It seems that we can say quite a bit about the distribution of primes whereas I figured it was more or less random.