Section 16.5, due on December 11

1. I had a difficult time understand the last two large paragraphs of 16.5. I don't get why integers mod p are replaced with the curve and why the number p-1 becomes n. I also don't get why "the use of the x-coordinate in the elliptic version is somewhat arbitrary", but at the same time "using the x-coordinate is an easy choice." Where does the x-coordinate come from?

2. I was really interested to read the last part of the section that shows that k*k^-1 was not 1 but was an integer congruent to 1 mod n. This proved that k*k^-1*A = A, even though it was not in the way I was originally thinking. I also was interested to see how elliptic curves could be used for encryption, key exchange, and digital signatures. While I know there has been a lot of study of cryptography, I would be interested to see if there are any other aspects of mathematics that could be used for cryptography.

I did the course evaluation.

Section 16.4

1. Why are we only working in the GF(4)? I understand that mod 2 is too small, but I don't understand why we don't go higher. At the end, it says we use GF(2^n). Were we just limiting it for this section?

2. I am really interested to learn and understand how the laws of GF fit together with elliptic curves and cryptography. I hope to be able to do that in class today.

Section 16.3, due December 6

1. This is silly, but I don't really understand what a singular curve is. Is it where the determinant is zero? If the p-1 method and trial division are included in factorization with singular curves, why do we need to learn specifically about it? It seems like it's not any more efficient than just trying to factor n by itself.

2. I'm amazed that elliptic curves can be used for factorization. It's interesting all of the ways that we can try to factor n, but a lot of them include and even end in the same step, such as finding an inverse mod p using the Euclidean algorithm. If there's not an inverse, then we've found a factor.

Section 16.2, due on December 4

1. Why are elliptic curves not used more if the traditional attacks, such as the Pohlig Hellman and the Baby Step-Giant Step, don't break them?

It seems like for a large enough K, there could be many potential values for j to test until you find a square. Is there an easier way to do this? Do people actually use elliptic curves for cryptography even though it seems complex and technical?

2. I think it's interesting that there's a way to estimate how many points an elliptic curve will have. I'm not sure I understand Hasse's theorem, but the application of it is very useful and neat. Finding individual points for an elliptic curve that has many points could take too long to be feasible.

Section 16.1, due December 2

1. I understand the idea of looking at an ellipse mod an integer, but I do not really understand how it ties to the addition law or how it helps in cryptography.

2. I haven't looked at ellipses for a few years, so it was interesting to see how they are used in the context of cryptography.  I had never thought about looking at an elliptical equation or graph mod an integer. The use of ellipses expands when we do this.