Section 16.5; Due December 8

Difficult: The steps of the cryptosystems here are all rather difficult to follow, I look forward to seeing examples in class. I'm having trouble understanding how the repeated addition on the elliptic curve should be as difficult to find the coefficient "a" for as it is difficult to solve a discreet log problem. Is there a proof like with the Diffie Helman problem to show that this is the same as solving a discreet log?

Reflective: Wow, last section! This has been a great class, I've really enjoyed all that we have done and learned. It was great to get another practical use out of something as abstract as an elliptic curve. Is there also some RSA type algorithm that you can do with an elliptic curve? I would suppose not since we need to work mod a prime, as in section 16.2, but it seems there might be some other way. Maybe using two elliptic curves mod p_1 and q_1.

Section 16.4; Due December 6

Difficult: The discussion of addition with these elliptic curves was much more confusing than before. In the example, it seemed to say that (0,0)+(1,1) did not have a solution, then it added (0,0) again to get the identity (which I do not see), and then suddenly there was a solution because (0,0)+(1,1) had to be -(0,0)? It also says that elliptic curves with characteristic 2 are important to cryptography, but I didn't really see where it dicussed why these would be best.

Reflective: The mash up of finite fields with the new (to me) group structure on elliptic curves was more difficult to grasp than I thought it would be. It seems not too different that the curves mod p, but it is still difficult. You would think there would be more description in the book, this was a short section.

Section 16.3; Due December 3

Difficult: So we can just pick any elliptic curve and it will probably factor the number? How can we be sure within a good probability that we have a curve that will work? It is interesting that we can just keep picking curves and one may work though, not like the p-1 method. I also did not understand how the singular curves were worked with.

Reflective: So the p-1 and brute force algorithms are just special cases of the elliptic curve algorithm? Never thought of it that way before, of course. Or is it just that there are special cases of the elliptic curve algorithm that are the same as the p-1 and brute force algorithms? The moniker "smooth" for numbers with only small prime factors caught my eye as I have been working with "smooth" functions on manifolds a lot recently. Is there some relation in this terminology? I couldn't think of one.

Section 16.2; Due December 1

Difficult: I do not quite understand the idea of encrypting messages with an elliptic curve mod p. So, you pick your message and then attempt to build your curve based on using your message as a point? Or do you have the curve first and then try to guess a point on the curve based on your message by adding things to your message? If it is the second method, how would the person you are sending the message to know which part is junk? Or even if you did not have to add any junk?

Reflective: The elliptic curve mod p doesn't seem that much more difficult than the elliptic curve without modular arithmetic. The number of points on the curve being so unpredictable is interesting, but I suppose it is just another "finding square roots mod n" problem. If using a prime p, do we need to ensure that p=3 (mod 4) as we did before?

Section 16.1; Due November 29

Difficult: Why do they define addition of points on the curve as the reflection of the third point where the secant line hits the curve? It seems like it would be more natural to just define the sum of two points on the curve to be the point where the secant line hits the curve. Is there some sort of computational benefit to adding in the reflection that I just do not see? The formulas seem to follow easily from the definition given, but it would seem to me just as easy to come up with formulas for the nonreflected point.

Reflective: Putting a group structure on points of a curve is particularly interesting to me in light of the research I have done in topology. This structure does not seem to be quite as natural a structure as what I am used to, so it is more interesting to think of what it all means. For example, I spent some time thinking about the curve with two components (x(x-1)(x+1)) and what the definition of addition does with that group. It looks like the points on the circle would generate the full group, as you can obtain any point on the line from two points on the circle (including the point at infinity). Possibly they do not generate though, since I see no way you could get a point on the circle from one or two points on the circle.

Section 2.12, due November 23

Difficult: I do not fully understand the discussion of the permutations used to break Enigma. The text mentions that letters never mapped to themselves in the machine, but the 1 cycles in the examples seem to indicate that they could. Also, how exactly did deciding the cycles reduce the problem to a substitution cipher? The initial keys were still quite random, so it seems that there would be trouble figuring out the permutations you need. How many messages were needed to work out which permutations you were looking at for a day?

Reflective: This is just an extremely interesting section. I enjoyed learning more about the enigma machines. I had no idea that the British then used them to spy on former colonies, that is quite interesting. Good to see governments haven't changed quite so much as we might think. I was disappointed that the links you posted did not work for me, so I could not read the extra information from the NSA.

Sections 19.1 & 19.2; Due November 19

Difficult: I found it difficult to understand what physicists (and for that matter chemists) believe about photons. Do they really think that a photon does not have a state until it is observed? When did science become a religion? Anyway, that's not really answerable, I just get frustrated when people try to explain something with illogical conclusions. Also, there wasn't much to be confused about in this section really. The theoretical quantum bit key exchange makes sense. They talk about methods to use to transmit these keys (fiber optic etc.) and the distance they can send, does that mean that the method is not really theoretical any more? Also, I remember in my undergraduate chemistry discussing using the chirality of a molecule to encode messages, is there any more information on this method?

Reflective: It seems that just message-sending type cryptography isn't the meat of the quantum computing idea. Isn't the way computers communicate with their internal parts a form of cryptography? They must encode electronic signals and decipher what they mean. Maybe it would be better to be focusing on efficient ways to encode these messages rather than just secure message sending that could be accomplished with a flashlight and some polaroid filters.