16.5, Due on December 8

This section was interesting to read because it was neat to see an alternate way of encrypting messages, signing documents, and exchanging keys using the elliptic curves. Finally I see the application of this whole chapter on elliptic curves! I also found the fact that this method protects against values of n with small prime factors to be interesting and poignant.
From section 16.5.1 I am wondering how a message is represented as a point on a curve. Apparently it was described in section 16.2 but I don't recall how to do it. One other aspect that I'm a little shaky on still is adding points to get a third point on an elliptic curve. [This is probably because I am writing this quite early, and we have two more class periods of practice before we get to this lesson.] I bring this up because adding points on an elliptic curve seems to at the heart of these crypto-systems.

16.4, Due on December 6

This section is really just a more specific portion of the other sections. I found it interesting that all the tangent lines are vertical to the curves of this modified elliptic curve. This type of crypto-analysis incorporates partial derivatives and lots of Calculus which I can appreciate.
I understand that elliptic curves are much easier to work with in mod 2, but I don't see why they are easier to work with in mod 2^n. That appears to be a big advantage but I don't see why. I am also confused about the end of the example on page 362. I am not sure how the authors found -(w,w^2).

16.3, Due on December 3

Frankly, I've found this whole chapter to be interesting. I love Algebra and this is a type of Abstract Algebra that is interesting. I find it fascinating that is can be used to factor large numbers. I don't understand everything that's going on but the main ideas are really intriguing. One of the things I don't understand is how it's easy to find B!P. That seems like a lot of work.
I also have two lingering questions about elliptic curves:

1. Doesn't a curve technically have an infinite amount of points? When we worked on the elliptic curve Z(E) mod 5 we only came up with 9 points. Are those the only solutions because our solutions must be integers mod 5?
2. When we are dealing with a finite number of points, such as Z mod 5, how can infinity be a point? It's hard for me to grapple that a curve with a finite number of points and finite values for coefficients can have infinity as a point. Am I missing something or is that just a difficult, abstract concept?

16.2, Due on December 1

The first portion of the section where the authors describe how we are going to use elliptical curves to factor large numbers was very interesting. "The situation where gcd(b,n)=1 fails... form the key to using elliptical curves for factorization" (pg. 353). I like that little teaser and it's making me moderately excited to read section 16.3.
I also found the first portion to be a review of what we went over on class on Monday. It's nice to approach the reading having already seen an example in class.
The hardest section for me was the last one about representing plaintext. This confuses me. I thought we were using the elliptical curves to factor n, not to encrypt anything. I also had a question about the example on page 353. Where did the equations for what x3 and y3 are congruent to come from? I see where the values came from but not the equations.

16.1, Due on November 29

I kept waiting for the cyrptography application in this section but it never came. Hopefully in the later sections I will see how it relates. I enjoyed reading the historical point on page 349, and it cleared up some confusion about the naming of elliptical curves for me. I also found the very last sentence of the section to be interesting, that infinity is the identity element of the abelian group. I will be very interested in hearing your explanation of that. It reminds me of complex analysis.
The hardest part for me to understand was the addition of elements in the group. I didn't quite follow the group what see what the point was. I struggled with this section a little bit just because I didn't see the motivation for the math. It seemed like the authors were just babbling on and I wasn't sure why. Are all the points on an 'elliptical' curve an abelian group with infinity as the identity?

2.12, Due on November 23

I always enjoy reading about the history and application of the different cryptosystems we have learn about. I liked reading about how the British sold Enigma machines to other countries without telling them it had been cracked. I also liked reading about the different aspects of the machine that made the different combinations plentiful. It's interesting to learn about early, physical cryptography machines, especially in light of the Quantum Computing we read about earlier.
One question I have is that it seems this is just a substitution cipher? I thought that at first, then it sounded like there was more going on, but then on the last page I read about Rejewski and his colleagues creating a code book for the different combinations, and therefore "the effect of the plugboard was then merely a substitution cipher"(pg. 55). What is going on here that makes this different from a substitution cipher?

Shor Reading and 19.3, Due on November 22

I enjoyed the reading on the web about Shor's Algorithm. It was written in a very lighthearted manner tailored to the layman. His analogies were very good and he brought up some great points. It's interesting to me to think about modulo exponentiation and factoring large values of n.
The part about the reading that confused me was the the clock example. I understood the 'experiment' but I didn't understand what we can deduce from the results.
The book reading was not very easy to follow. If I were to ask one question about it I would ask where the Fourier Transform came from? It seems pretty crucial to the Shor Algorithm but I found its explanation to be less than satisfactory. I am also confused about how the graph relates to the algorithm.

19.1 and 19.2, Due on November 19

One big question I have is regarding the strange physics notation. I have never seen vectors represented that way and I'm a little confused as to what they mean. Another question I have is: What does this have to do with cryptography? I was so confused by the reading that I didn't even see it's application to cryptography.
The thing I'm most looking forward to in this unit is the actual experiments in class. I get that we are representing light as a vector but I think it will make more sense in class when we actually do the experiment. I also like the idea of using light as a medium for sending hidden messages, even if I don't understand why or how.

14.1 and 14.2, Due on November 17

The first sections was the most easily understood by me. I enjoyed reading about the Zero-Knowledge Protocol and the Tunnel diagram was very helpful. It's interesting to me how useful different mathematical ideas are in real contexts. The notion of finding square roots as a tool to secure information is really cool to me.
The second section was less easy for me to follow. I didn't catch all the steps, such as what Peggy and Victor need to check. I am wondering if this Feige-Fiat-Shamir identification scheme is meant to be a cryptosystem. I think it is not. I am also curious to know where else I might see Zero-Knowledge Techniques in place. It seems like an interesting idea and a quick Wikipedia search taught me a little more about it. I found it not so ironic that the example on Wikipedia used Victor and Peggy as well. Those must be real mathematical terms.

12.1 and 12.2, Due on November 15

One confusing part for me was in the example at the bottom of page 299-300. By using the Lagrange interpolating polynomial they came up with some numbers but I don't understand where the numbers came from. I'm not sure why they divided some of the large integers by 5 either. I understood most of the example except for that part. I also didn't understand the Vandermonde Matrix. That probably led to me being confused about the Lagranfe interpolating polynomial.
The interesting part of these sections for me was the different approaches to solving the same problem. As a mathematics education major we are always taught to represent things in multiple ways. The book did a nice job of explaining how to break this secret in three different ways. I also liked the first section because it was explained well and not too difficult.

Exam II Review

I will admit Dr. Jenkins, I am pretty nervous for this second exam. To try and alleviate that stress I'm doing my best to be well prepared and studying early. As I have been studying the topics I feel the least sure about are:

• The Pollard Rho Factorization Method
• Diffie-Hellman Key Exchange
• Encryption with Hash Functions
• Digital Signatures

Those are the concepts that make me scared. As far as the problems we should be able to know, I have no idea what to expect in regards to:

• Signing documents using RSA or ElGamal
• Describe strengths, weaknesses, and attacks for algorithms we have studied in class.

As I have been studying I am consistently pleased with how much more sense the mathematics makes the second time around. There are many ideas that I understand now that I didn't know when I first read and learned about them. That is making me more confident in my ability to do well on the test and I'm hoping that as I keep studying they will continue to make more sense as opposed to me just memorizing what to do.
As far as questions that I expect to see on the exam, I am guessing there will be a problem regarding finding a square root, a problem regarding calculating exponentials modulo n, and calculating Jacobi or Legendre symbols. I bet there will also be a problem where we are asked to describe one of the primality tests or factorization tests, and one where we are given a cryptosystem ans asked to describe a weakness or strength of it.

8.3 and 9.5, Due on November 10

I did not enjoy reading section 8.3. Hash functions have been a struggle for me from the start and this section only added to my woes. I don't understand the SHA-0 process, why letters such as A and B equal 1010, 1011 respectively, or what the 'W' variables are. My brain just does not think like a computer.
I enjoyed reading section 9.5 much more. You discussed it briefly on Monday and it's always easier to read something when the ideas have already been introduced. I like the idea of having alpha^q =1 mod p instead of using alpha as a primitive root.
I have one small suggestion regarding the homework. It would be nice if there was a key to selected problems posted online. I can usually solve all ten problems but I get the feeling there is often a better way than the method I chose. If it's too much work don't worry about it, and the grader is pretty good about catching and marking individual errors, but if you already have a key for the grader it would help my test preparation to see other ways of solving the problems.

9.1-9.4, Due on November 8

I enjoyed reading about the Birthday Attacks on Signatures. I believe you touched on it in class on Friday because it sounded very familiar. I also liked reading about the ElGamal Signature Scheme because it sounded familiar from what we have already learned studied.
Now, there are a few things I am still confused about. One big item is hash functions. I know that hash functions do not compose a cryptosystem but they are obviously used in cryptosystems. Once something has been sent through a hash function how do you get the message out again? I am very confused about them and their place in cryptosystems. Another question I have is about digital signatures. Are digital signatures anything special besides a specific message? It seems to me that they are essentially just messages that are unique to a sender. I'm not sure why the entirety of chapter nine is dedicated to digital signatures if they are just certain types of messages.

8,4-8.5, 8.7, Due on November 5

The section on the Birthday Paradox (and Birthday Attack) was the most interesting to me. I have learned that paradox before, and even put the mathematical probability to the test. I was at a party and counted 22 people so I went around and asked everyone for their birthday. Unfortunately there were no matches, making me look pretty foolish, but I told them if one more person showed up they would have the same birthday as someone already there. However, I realize now even with 23 people the probability is still only slightly better than 50%. Maybe next time I'll try it with 60 people. With the Birthday Attack I am wondering why anyone would use it instead of the Baby Step Giant Step method. The book claims that it doesn't provide a guarantee of a match, and that the BSGS method is generally faster anyway.
The last section on using hash functions to encrypt was hard for me. It looked a lot like the example you did on Wednesday but I'm still a little confused. I keep thinking hash functions are going to look like nice algebra functions but it's becoming increasingly more obvious that they are a whole different breed of functions.

8.1-8.2, Due on November 3

One connection I can draw from these sections is that the hash functions feel a lot like the AES and DES algorithms. Maybe it's the XORing or the three letter acronyms (MD4,MD5) but something about has functions reminds me of those earlier computer algorithms. However, I think the difference comes in the fact that the has functions are one way. I recall finding inverse operations for the different modes of operation in an earlier homework and I don't believe that is possible with has functions.
While there were many confusing parts for me, the one I am most intrigued and eager to learn more about is the proposition on page 221. I don't see why such a vague supposition can lead to the answer to a discrete logarithm problem which we just learned is very hard to solve. It seems that if we have a function that is not surjective we shouldn't be able to solve a discrete logarithm problem. [I think that's what m not equal to m' but h(m)=h(m') implies, although I'll admit it's written differently than I normally see it and I'm not totally convinced myself that it does imply it's not surjective].

7.3-7.5, Due on November 1

I am interested in how cryptography answers a variety of problems that go beyond just disguising messages. For example, I was intrigued by section 7.3 in regards to making sure Alice doesn't change her message and Bob doesn't read the message until after an event as occurred. In section 7.4 the authors claim that there is a situation involving elliptical curves where the decision Diffie-Hellman problem is solvable quickly which I am interested in. I recall getting a teaser in an earlier section about elliptical curves as well.
The hardest part for me was understanding the ElGamel cryptosystem. I understand the big idea because I understand public key cryptography (not completely, but mostly) but I don't see how the public and and private keys are derived in regards to the discrete logs problem. The RSA was easy because the private key was just two large primes, but I'm not sure what the private keys are in the ElGamel system.
P.S. My high school German teacher was named Mrs. El Gamel and she told me 'El Gamel' was Arabic for 'The Camel.'

7.2, Due on October 29

I enyoed reading about Baby Step, Giant Step (section 7.2.2). Not only did it remind me of Buzz Aldrin's famous quote, but it also reminded me of the meet-in-the-middle attacks from the earlier sections. I also liked reading this section in light of the material about factoring. Knowing that the past two weeks have been spent learning how to factor in order to break weak RSA systems helped me understand why we are learning all this information about discrete log problems. I'm guessing that the discrete logarithm problems are going to be used in a public key cryptosystem and all this information we are learning is going to help us crack weak versions of that system.
The two hardest subsections for me were 7.2.1 about the Pohlig-Hellman Algorithm and 7.2.3 about the Index Calculus. The examples were quite long and despite finally understanding the log notation I had difficulty understanding it in light of the examples.

6.5-6.7 and 7.1, Due on October 27

The most interesting portion of these sections for me was the idea of trapdoors. I liked reading about the variety of trapdoors available and how they differ according to the public key system. I also enjoyed reading about other types of public key cryptosystems besides the RSA. It's easy to avoid thinking outside the box but this section helped me see some other alternatives to the RSA.
One thing I didn't really understand was the discrete logs in section 7.1. I am familiar with the idea of logarithms but I don't understand the x=L[subalpha](Beta) notation. I also don't really see how the discrete logs apply to anything so far. I imagine their application will become more apparent in the ensuing sections. The other hard section for me was 6.6- An Application to Treaty Verification. I don't quite know what all is going on there.

6.4.1, Due on October 24

I guess the thing I am most confused about is how we factor the numbers into smaller primes. The example of the basic principle on page 183 starts by saying 9398^2 = 5^5 * 19 (mod 3837523) but I don't really see how they got that or how I would factor an even bigger number. I believe, upon finishing the chapter, that they give the explanation on page 185 but I didn't quite see how it all related. I'm not sure where this equation of in+sqrt(in) + j^2 came from or how it helps us factor n into smaller primes.
The most interesting part of this section for me was the idea of using matrices. I enjoy linear algebra and I loved my Matrix Analysis class so I'm pretty eager to see additional applications of matrices. This matrix formed in this section must serve a purpose, although I have not successfully identified it yet. I understand how the matrix is formed, but I don't see where the linear independence comes in, or why we move to mod 2.

6.4, Part I, Due on October 22

Fermat's Factorization was the most interesting portion of this quick section. It is so simple but very powerful for small numbers and it still took a genius to come up with it. I like it because I can understand it and it uses the idea of working backwards to factor (relatively) large numbers.
The p-1 Factoring Algorithm was the only other topic discussed, and it wasn't crystal clear to me. One question I have is how B is chosen. It's only description is that it is a bound but I'm not sure what it is bounding. This method looks a lot like the Miller-Rabin Primality Test  except it actually gives us factors instead of just determining if a number is prime or composite.