Section 16.5, due Wednesday December 8

1. (Difficult) The descriptions in this section were relatively easy to understand because they were analogous to other things that we have studied. In the El Gamal Digital System, when computing s, they use the x coordinate. Would it matter if the y coordinate was used? Was this was they were referring to at the very end of the section?

2. (Reflective) This was a pretty straightforward section because it was similar to what we have seen before. I find it interesting that there are similar signatures using elliptical curves. When El Gamal Signatures started to be used, was it very soon after that someone found the analogous signature using elliptical curves? I am curious how long it took someone to figure that out. This has been an interesting chapter overall. I am glad that I finally know a little bit about elliptical curves and how they are used.

Section 16.4, due Monday December 6

1. (Difficult) Why are elliptical curves over finite fields better to use for cryptography purposes? Can you explain more about the example over GF(4)? I was confused on how the points on the elliptical curve were found.

2. (Reflective) Because we have already learned about finite fields it was interesting to see them come up again. Elliptical curves are really interesting to me. The properties and the things that they can be used for are interesting and intriguing. I can only imagine that they just get more and more complicated, but also more fun to study.

Section 16.3, due Friday December 3

1. (Difficult) Section 16.3.1 and the example in that section were very confusing to me. I was confused on where the values were coming from and why certain computations were completed. I am excited to see an example in class, so hopefully the fogginess of that example can dissipate. Can you explain more about how smooth numbers relate to Hasse's Theorem?

2. (Reflective) I would have never guessed that Elliptical curve factorization relates to p-1 factorization, but I find that very interesting. The p-1 factorization method was very useful in factorization, and throughout the section it mentioned that Elliptical curve factorization was even more successful, which is a great thing. Various topics that one would not expect to relate often due in mathematics. I do find that interesting. When I was watching The Proof about Fermat's Last Theorem, I thought the same thing. I guess it is probably because of my ignorance about various subjects, but while I was watching it, it seemed like Andrew Wiles connected two very different mathematical subjects and related them to each other to proof the theorem. That is a fascinating thing.

Section 16.2, due Wednesday December 1

1.(Difficult) I don't understand perfectly how the discrete logarithm for Elliptical curves are the same as discrete logarithms. Is it similar to if you take the log of both side and the exponent comes down as a constant? More examples for how to turn plain text into cipher text would also be helpful.

2.(Reflective) Square roots mod p seem to come up a lot through many sections and it shows how important they are in cryptography. I would never have guessed that, but square roots mod p have come of often. Finding square roots mod p are difficult though, so what will happen when the numbers get too large to calculate? We only saw for small examples, but we may have large square roots to find.

Section 16.1, due Monday November 29

1. (Difficult) Can you explain the use of infinity in the addition of points of elliptical curves? They said that infinity played the role of the additive identity, and I just want to understand that more clearly. Why can't an elliptical curve have multiple roots?

2. (Reflective) I was excited to learn a little about elliptical curves. I understand that this section was very basic, but I enjoyed reading it. I found it interesting where the name for elliptical curves came from. I never knew that it was from elliptical integrals, which I also did not know about. I enjoyed this section as well because it made sense for the most part. I think it will be interesting to find out more about what elliptical curves are used for.

Section 2.12, due Tuesday, November 23

1. (Difficult) The actual encoding process was the most confusing part of the section. I don't understand the purpose of the set up of the machines, so that would be helpful to go over in class. I understood more about how to break the code because it used things that I have seen before.

2.(Reflective) This section deals with permutations, which we learned in Math 371. This permutations are all disjoint as well, so we know that they can be switched and it will not affect the outcome. This seems like an interesting coding process, but also inconvenient to have these large machines needed to send the code. I would imagine at the time it did not seem too bad, but not I imagine people would not enjoy using these machines very much.

Section 19.3 and explanation online, Due Monday, November 22

1. (Difficult) What does it mean by superposition in the Shor's Algorithm explanation online? I was really confused by this section. Even the example was not very clear, and hopefully class will help clear somethings up. I'm not sure if it was more the notation or just the concepts, but I was pretty lost throughout all of 19.3.

2. (Reflective) This was a good explanation for the Shor's Algorithm that was simple and easy to read. I have heard of Discrete Fourier Transforms before, but I have a very limited understanding of them. I used them in a couple of problems that I did during my summer of IMPACT. In my research last year, I read some basics about Fourier Transforms and on wavelets.

Sections 19.1 and 19.2, due Friday November 19

1. (Difficult) The notation of arrows is what confused me the most. I understood the way that Alice and Bob create the key stream, but I don't understand why the notation is the way it is or exactly how the method corresponds to quantum mechanics. Can you go over how Alice and Bob would detect an eavesdropper?
2.(Reflective) I am pretty sure I have done this experiment in my high school physics class. It is certainly an intriguing phenomenon. The use of quantum mechanics in cryptography seems odd to me. I would never have guessed that a physic application, such as quantum mechanics would be used to set up key streams. It shows how important research is because the applications of discoveries can be extended to many different fields.

Sections 14.1 and 14.2, due Wednesday November 17

1. (Difficult) Can you explain the Feige-Fiat-Shamir Identification Scheme? I guess what gets confusing is exactly how the pieces (x, y, b_i, and v_i) all fit together to create a zero-knowledge technique. I think just the technicality in the book is hard to follow, so a slower, simpler explanation would be helpful.

2.(Reflective) I have never heard from the zero-knowledge techniques before. The explanation in 14.1 made a lot of sense though and is an interesting concept. It is interesting that in fact Victor can never be fully convinced that Peggy knows how to get through the door, but after several tries, he can be pretty well convinced. It is interesting that in the book, it is called a zero-knowledge "proof" because that goes against what we actually consider a proof.

Sections 12.1 and 12.2, due Monday November 15

1. (Difficult) After the example of the Shamar threshold scheme, there was an explanation of why t-1 people could not get the secret. That explanation was confusing. Also, could you explain the Lagrange interpolation polynomial?

2. (Reflective) This is an interesting problem. I had never thought about distributing parts of a key, so that a certain number of people are needed to find the secret. It makes a lot of sense though because it spreads responsibility among various people and guarantees that not just one person can find the secret. What are other applications of the threshold scheme that weren't in the book? It kind of reminds me how when you take out money from a bank for a club or an organization, two signatures are often required or two people must be present to do so.

Midterm Questions, due Friday, November 12

• Which topics and ideas do you think are the most important out of those we have studied?

Understanding the Chinese Remainder Theorem is important for this class and for many other math classes. RSA was the largest section that we studied since the last midterm, and so we should really know and understand RSA and the factoring methods. Square roots were also very important to understand and implement for factorization.

• What kinds of questions do you expect to see on the exam?

Chinese Remainder Theorem problems, Probability for Birthday Attack, The Pohlig-Hellman Algorithm (Possibly), the Concepts relating to RSA and factoring large n.

• What do you need to work on understanding better before the exam?

Factoring methods are things that I certainly have to go over and understand. I need to work on discrete logarithms and know to find them using the Pohlig-Hellman Algorithm. I need to make sure that I understand and can apply the concepts from ElGamal, hash functions, and digital signatures.

• Are there topics you are especially interested in studying during the rest of the semester? What are they?

I have heard of Elliptical Curves, and I'm glad that we will be going over them briefly so I at least have a general understanding of what they are and what they possibly are used for.

Sections 8.3 and 9.5, due Wednesday November 10

1. (Difficult) Section 8.3 was pretty confusing, but we were given a fair warning at the beginning of the section. I was confused mostly about the SHA-1 Algorithm. Some of the steps seemed very "random" because I had no idea where they were coming from, including the constant values K_t. It mentioned that the core of the algorithm was step 3, which was probably the most confusing part of the algorithm.

1. (Reflective) The Digital Signature Algorithm made more sense to me, so it was more interesting. It makes sense that why we need these secure ways to sign documents. If this wasn't secure, then people would be accused of signing contracts and other documents that they did not. This is really important in many transactions. The DSA is very similar to the ElGamal Signature Scheme. Are they basically the same other than the difference in the integer r?

Sections 9.1-9.4, for Monday November 8

1. (Difficult) Can you explain in class the difference between a signature with appendix and a message recovery scheme? Actually, if I understand correctly, it is just that in the latter you actually can find the message m, where in the signature with appendix it is not easy to actually recover m. Is that correct? What are more examples where a signature would be used?

2. (Reflective) The most interesting section was the Birthday Attack on Signatures. It is a very clever way to trick people, but yet very easily avoided if one is weary of the attack. I find it interesting that some many of these ideas have been discovered. It fascinates me that people have come up ways to attack seemingly secure things, such as the Birthday Attack. Hashing itself is a good idea and seems like a practical thing to use, but as the Birthday Attack shows, it will sometimes fail to be secure.