Section 16.2 Due 1 Dec 2010

1. The thing I don't quite understand is why we would want to mod an elliptic curve. It seems to me that gives us less points on the curve to work with. In cryptography, don't we want as many options as possible, leaving more room for error on the part of the attacker? In short, how does mod-ing an elliptic curve increase advantages? Perhaps this won't be made clear til we start examining the cryptography applications of elliptic curves.
2. I'm starting to guess that as far as cryptography applications go, elliptic curves may be useful once we DO mod a point because then it will be harder to undo the mod, thus increasing the security.  One other thing, I got lost on why they used 2773 as an example when it isn't prime. How important is it for p to be prime when modding? What goes wrong when p is not prime?

Section 16.1 Due 29 Nov 2010

1. The hardest part for me to understand was the first example they give us to examine. It was hard to follow where all the numbers were coming from and why they were substituting certain expressions for specific variables in various other expressions. I also didn't understand why they could say that the sum of the three roots is minus the coefficient of x^2. Then later in the reading it suggests that -4 is a double root. Yet earlier in the reading it suggested as a "technical point" that we wouldn't worry about multiple roots until 16.3.
2. When doing these readings, I like to predict how the new ideas and techniques being introduced are going to be used in cryptography. I'm guessing that somehow the author of a certain key is going to use a made up equation (made up according to certain guidelines). Then the author is going to pick two (possibly random?) points and use them to find the third point based on the line that goes through P1 and P2 as described in the text. The coordinates of the third point will then be used to create an encryption and decryption key. I don't know HOW the coordinates will be used (unless the original two points are chosen so that the third point is composed of two large random numbers and then we would be in the realm of RSA) but I'm pretty sure they will be used to create E[k] and D[k]. I'm guessing all this purely on the ideas we've discussed on public cryptosystems. The third point is easy to find only to the author of the equation of the elliptical curve. It would be nigh impossible for a random citizen to find the original two points based purely on the third point. This leaves me with the question, "what components will be made public? P3? P2 & P1?" My guess is P3. Then the author can use P1 and P2 to create a decryption key.

Shor algorithm and section 9.3 Due 22 Nov 2010

1. It was a very well written article. It didn't use hardly any math (except periods and mods). But there were still a few issues I had with it. Finding phi(n) corresponds to finding a multiple of the period mod some N. However, how does one choose N? I'm still having trouble figuring out how to choose N. Furthermore, once one DOES choose N, and finds the period then how can one be sure of the corresponding phi(n)? For example, in the article they let N=15 and found the period to be 6 and phi(n)=12 and explained that sure enough 6 divides 12. When I read this, I thought, "but 6 also divides 24 and 36 and 66, etc." How can I find phi(n) based solely on the period?
2. There was a part explaining something called repeated squaring. It seems to me that's the same as our exponentiating a number mod n that we've learned. On another note, or maybe this would best fit under the previous section, it seems like this article would fit better under our section that examined RSA rather than quantum cryptosystems?

Section 14.1, 14.2 Due 17 Nov 2010

1. I've read the sections twice over and am still having trouble understanding how exactly Peggy can use square roots of factors of x to show Victor that she has factored n. How is showing Victor x[1] or x[2] the same as showing Victor that she has factored n? Isn't she giving him enough information so that he can actually deduce what s is, or for that matter what the factors of n are? I think I'm getting lost in the labeling of variables somehow.
2. If Peggy were to mis-choose which numbers to send to Victor as proof, wouldn't there be worse consequences than Victor knowing that Peggy didn't find a factor of n? Wouldn't Victor know that for some reason Peggy was just shooting in the dark, hoping to find some secret information or get something out of Victor? How does this apply to the example of crooks trying to steal sensitive information?

Section 12.1, 12.2 Due 15 Nov 2010

1. I'm going to go back and re-read the sections. In general, the material is new enough that I'm still trying to sort it all out. As for things that were harder to understand than others, I don't really understand the algorithm given as an example of a threshold scheme. I think that, as usual, once I see it a second and third time (in class) and have a chance to sort it out as I'M writing it down, it will start to come to me. Specifically, I don't understand how the pairs/points (t,w) are combined in such a way as to reconstruct a message.
2. One thing that I'm becoming more and more interested in is applications. I'd like to see more applications of the threshold schemes. It gives some really good examples (banks, inheritances, etc), but I'm wondering if there are any other examples.

Due 12 Nov 2010

I think the most important ideas to understand are the basics behind RSA (since we spent so much time and RSA related problems can still be found on the homework). I also think that it will be important to know the ideas behind the different methods of factoring large numbers that are the product of two large primes. Cracking RSA messages is centered on factoring n=pq. In relation to this are the discrete log problems.

I need to have a better understanding of the most recent stuff that we've covered in class. That stuff is harder to understand because I haven't had as many opportunities to put them into practice. Toward the top of the list of things I'm having trouble remembering is the ElGamal cryptosystem.

Also, I'm wondering how exactly we're going to be tested on some of these things. I know we won't need a computer, but there are some things on the list of things to know that I can't see how we can be tested on without a computer. For example, continued fractions.

Section 8.3, 9.5 Due 10 Nov 2010

1. The SHA-1 Algorithm is unlike anything I've encountered in the course thus far. I thought DES was hard, but that was nothing! To get myself through the algorithm, I have to read it really slowly, often times reading and re-reading portions of it. I guess this means not so much that the algorithm is hard to understand, but more or less it is hard to follow. One thing that is on the back of my mind is the test and its relation to this algorithm. I'm just barely seeing it for the first time. I'm wondering how I'm going to be expected to show my understanding of the algorithm in a testing situation.
2. One point of interest are the values held by H[0], H[1], ... , H[4]. They all seem to be loosely related. Some are just the reverse of others, for example. Were these values chosen for a reason? Also, how do the specific values fit in to the actual algorithm?

Section 9.1-9.4 Due 8 Nov 2010

1. I don't quite understand how the added signature idea fits into RSA. That is to say, I understand RSA just fine, but I don't understand using RSA with the additional signature. I don't understand where all the different numbers are coming from (how they were computed, why they were computed, where they go, their significance). It seems to me that the signature is simply added onto the end or the beginning of the message before/after encryption. But I still don't understand exactly how this verifies the sender/recipient.
2. The end of the reading describes how someone could make slight changes to an electronic document and how this could foil either or both parties involved in the process. I'd be interested more in the details behind this. Why exactly does this change the legal status of the contract? Doesn't the document still say essentially the same thing except with a few spaces added (or whatever the case may be)? Also, if someone signs one document, how does changing the document mean that they essentially signed another one?

Section 8.1-8.2 Due 2 Nov 2010

1. Two things I don't understand. I don't understand the proposition about finding discrete logs only given two messages whose hash functions are equal. I got bogged down in some of those steps. Especially because it says "Suppose . . . " Whenever it says something like that, I think "I may suppose that, but does that make it true?" Also, the simple has example they give involves XORing columns in a matrix. I don't understand how this is going to make a unique h(m). It seems to me that working with a mod that is so small, there would be many different messages that would map to the same XORed function.
2. Really the only thing I found interesting was that hash functions are used a lot in digital signatures. I would like to know more about how this works or how hash functions fit into that idea.