16.5

1. This section wasn't extremely difficult, but I would like to investigate what properties inverses in some field have when they are considered in an enclosing structure, like the ring Z.

2. We've talked a couple times about how there is no real notion of smallness when it comes to points on the curve E, and how it is for that reason that we can't use the same methods to attack the discrete logarithm problem for elliptic curves. At first I was tempted to say that we picked the wrong analogy and that comparing it to a discrete logarithm was guiding our intuition in the wrong direction, but I suppose it is the same problem if we used multiplicative notation instead of additive. Anyway, my point is that perhaps there is someway to define smallness based on certain properties, etc.

16.4

1. The most difficult part for me was realizing that one of the more significant issues in GF(2) and GF(4) (maybe GF(2^n) in general?) is finding the additive inverse of an element and not adding two elements together, but I guess it turns out to be the same thing. It takes an extra step.

2. Even though fields in Z_2 are a more natural fit for computers, does that mean that fields Z_2[x] are also? (This next part is all "if I'm not mistaken".) In the text, only a few lines were dealt with: vertical, horizontal, and the line y=x. A line could have the slope of any element in the group. In GF(4), names were given to each element, so I suppose in, say, GF(256), even though most of the elements may have the ring element x in their representation, this is a different x than the one appearing in y=mx+b, so if considering the line through (0,1) and (1, x^2+x+1) on some hypothetical curve, we would obtain y=(x^2+x+1)x. Is that y=x^3+x^2+x? In other words, are the x's the same? I'm leaning towards yes.

16.3

1. I had a question about the p-1 analogue for factoring but in writing it, I saw the answer. However, why the process isolates the behavior of one of the factors is still a mystery to me. That and the entire second half of the reading.

2. It's interesting to me that choosing a random curve (mod n) results in essentially a random number of points, around n, on the curve. Since there is a practical limit to the size of numbers that can be factored using this method, I'm going to assume that the time complexity goes up exponentially with the number of digits (so essentially linearly with the value of the number). I have nothing of worth to say about elliptic curves themselves, unfortunately.

16.2

1. Pardon my ignorance, but how can we take a derivative of a function (implicit or not) when there is no (obvious) notion of continuity. Don't we use difference quotients or something? Is that the same thing in this case?

2. I am consistently amazed at the connections and applications that seemingly obscure branches of math have. First, the recognition that many of the results of, say, topology come not from a construct itself (say, the complex numbers), but from the properties that construct has (field properties). It then allows one to interpret things like fractions in finite fields where all members are representable by integers. More pertinently, the investigation of arc length of ellipses led to a more intense study of elliptic curves which turn out to be extremely fascinating. I must say apologetically that I have nothing of interest to say about elliptic curves over Z mod n. This is all new to me.