Section 16.5, due December 5

The hardest part of the section was the explanation of the El Gamal digital signature algorithm for elliptic curve cryptosystems.

It's interesting how much simpler certain cryptographic algorithms can be when they are implemented using elliptic curves.  This is because the operations that are analogous to modular multiplication and modular exponentiation on elliptic curves are much easier to perform, but are just as strong cryptographically as systems implemented using large prime numbers.  It's interesting to see alternate ways of implementing familiar cryptosystems.

Section 16.4, due December 3

This was a pretty confusing section.  I'm having a hard time understanding elliptic curves over finite fields.  An longer explanation of the example in the section would really help.

Elliptic curves are interesting but they just seem to become less and less and intuitive the more I learn about them.  It is interesting that NIST recommended 15 specific elliptic curves (mod 2) for use in elliptic curve cryptosystems.  I might be wrong, but it seems that ECC is easier to implement than RSA since you choose from fewer curves, and key generation is easier than RSA.  But it's still a strong system.  That's just my initial thought.

Dr. Cristian Tomasetti, Colloquium, Extra Credit

I didn't fully understand everything that Dr. Tomasetti talked about.  I was unfamiliar with some of the math that he talked about, so probably the most difficult part of the talk was just when he was going over some of the formulas he derived.

I really enjoyed Dr. Tomasetti's presentation.  I've been interested in cancer research for a long time, and I would definitely like to go into mathematical biology to have a chance to do the kind of work that Dr. Tomasetti does.  It was very fascinating to learn of the correlation between the age of cancer patients upon diagnosis and the number of pre-cancer phase mutations in those same patients.  I would definitely be interested in learning more about what researchers are doing to find ways to differentiate between which mutations lead to proliferation of passengers and which lead to proliferation of drivers.

Section 16.2, due November 28

The hardest part of the reading was the part about using elliptic curves to represent plaintext. 

Elliptic curves are definitely strange and not one of the most intuitive mathematical concepts that I've encountered.  But I definitely do want to know more about how they can be used in a cryptosystem.  Looking at elliptic curves mod p seems like it is definitely be a lot simpler than just looking at elliptic curves on the real numbers. 

Dr. Dave Richeson, Focus on Math, Extra Credit

The most difficult part of Dr. Richeson's talk to understand was some of the explanations of how certain constructions were done.  A few of them were just a little hard to follow.

I enjoyed Dr. Richeson's talk.  I'm taking a class on the history of math this semester, so a lot of his talk dovetailed nicely with what I've learned about geometry and geometric constructions.  They do seem like very elementary problems (even though they're impossible), and they might not have important applications to the real world, but it is still very interesting to see what mathematics is (and is not) capable of.

Section 16.1, due November 26

The most difficult part of the section was the part about the addition law for elliptic curves.  It wasn't super difficult to understand, but an example would be very helpful.

I think the most important part of the section, at least as it pertains to cryptography, is the fact that elliptic curve systems can do with 313 bit keys what certain conventional systems can only do with 4096 bit keys.  The is vast improvement in efficiency, made possible by a creative cryptosystem built on powerful mathematics.

Section 2.12, due November 20

This wasn't a very difficult section, but a more in depth explanation of how the enigma worked would be helpful, though I think I basically understand it.

But I have always been pretty interesting in the Enigma system.  I read The Code Book a long time ago and had the chance to read about the history of the Enigma and its effect on World War II.  Although cryptography has many non-civilian applications, it seems that it will always be an important part of warfare, for good or ill.