Section 16.5, due on December 11

1. I had a difficult time understand the last two large paragraphs of 16.5. I don't get why integers mod p are replaced with the curve and why the number p-1 becomes n. I also don't get why "the use of the x-coordinate in the elliptic version is somewhat arbitrary", but at the same time "using the x-coordinate is an easy choice." Where does the x-coordinate come from?

2. I was really interested to read the last part of the section that shows that k*k^-1 was not 1 but was an integer congruent to 1 mod n. This proved that k*k^-1*A = A, even though it was not in the way I was originally thinking. I also was interested to see how elliptic curves could be used for encryption, key exchange, and digital signatures. While I know there has been a lot of study of cryptography, I would be interested to see if there are any other aspects of mathematics that could be used for cryptography.

I did the course evaluation.

Section 16.4

1. Why are we only working in the GF(4)? I understand that mod 2 is too small, but I don't understand why we don't go higher. At the end, it says we use GF(2^n). Were we just limiting it for this section?

2. I am really interested to learn and understand how the laws of GF fit together with elliptic curves and cryptography. I hope to be able to do that in class today.

Section 16.3, due December 6

1. This is silly, but I don't really understand what a singular curve is. Is it where the determinant is zero? If the p-1 method and trial division are included in factorization with singular curves, why do we need to learn specifically about it? It seems like it's not any more efficient than just trying to factor n by itself.

2. I'm amazed that elliptic curves can be used for factorization. It's interesting all of the ways that we can try to factor n, but a lot of them include and even end in the same step, such as finding an inverse mod p using the Euclidean algorithm. If there's not an inverse, then we've found a factor.

Section 16.2, due on December 4

1. Why are elliptic curves not used more if the traditional attacks, such as the Pohlig Hellman and the Baby Step-Giant Step, don't break them?

It seems like for a large enough K, there could be many potential values for j to test until you find a square. Is there an easier way to do this? Do people actually use elliptic curves for cryptography even though it seems complex and technical?

2. I think it's interesting that there's a way to estimate how many points an elliptic curve will have. I'm not sure I understand Hasse's theorem, but the application of it is very useful and neat. Finding individual points for an elliptic curve that has many points could take too long to be feasible.

Section 16.1, due December 2

1. I understand the idea of looking at an ellipse mod an integer, but I do not really understand how it ties to the addition law or how it helps in cryptography.

2. I haven't looked at ellipses for a few years, so it was interesting to see how they are used in the context of cryptography.  I had never thought about looking at an elliptical equation or graph mod an integer. The use of ellipses expands when we do this.

Sections 18.1-18.2, due November 26

1. I'm having a hard time understanding error correcting codes. One of the aspects I don't get is how we get d(C) using the Hamming distance. How do we pick u,v?
On a later note, what does the code rate mean? I don't understand the definition in the book.

2. I really like the idea of parity checks and other codes to figure out if the message was sent with an error. It seems like it would be hard to get a perfect message sent every time, so they seem very useful. I've done homework exercises with ISBN codes in other classes, and it's interesting to learn that such a common number that everyone sees uses a parity bit.

Section 2.12, due on November 25

1. In theory I understand the Enigma, but when I looked at figure 2.2, I realized that I don't understand it at all. I don't understand the different roles of the rotors, reversing drum, and the plugboard very well. The outcome of the Enigma makes sense, but how it gets there is a mystery.

2. I think it's really interesting how they attacked the Enigma by finding similar plaintexts to corresponding ciphertexts. I'm surprised that the original system of writing worked all the way until 1938, when a different method of transmitting keys was developed.

Section 19.3 and the blog, due November 22

1. I understand the basic idea of the Fourier transform, but I really don't understand Shor's algorithm. What is its goal? How does it work?

2. I really enjoyed reading the blog because it used fairly simple concepts. It helped me to understand that we can find the period for Shor's algorithm through his use of the clock analogy. However, I still don't really understand Shor's algorithm.

Section 19.1-19.2, due on November 20

1. How do Alice and Bob establish the bases they are working with? I didn't understand that from the reading.

2. I haven't taken a physics class is four years, so it was fun to revisit that area. I also knew what polarization looked like because 3D movie glasses are often polarized. When you put one of those lenses perpendicular to another lense, you can't see through them. I was really interested to find out that this concept can be used to establish a key.

Pre-Test Questions, due November 15

I think we have covered many aspects related to RSA, so I think really understanding how to encrypt, decrypt, and attack encrypted messages using RSA is an important topic. I also think that signatures are an important topic and are relevant in many different fields. They are used in a variety of situations, so knowing how to correctly use them and being aware of forged signatures is a useful skill.

I expect to see a lot of encrypting and decrypting questions, especially with RSA. I also expect to see a couple of signature questions, where we need to sign the message and verify the signature.

I need to study everything a lot. I've missed a week and a half of school for fly-out job interviews since the last exam, so I really need to spend the weekend studying everything we've learned, so that I'm prepared to take the exam on Monday.

Sections 8.1-8.2, due November 6

1. I don't really understand the third requirement of a cryptographic hash function. I realize the importance of it, but I don't get the difference between strongly collision free and weakly collision free and how they are related.

2. I'm excited to be learning about hash functions. I understand that the example in 8.2 is very simple, but I am glad to be getting into even more complex cryptosystems. I am interested to learn what makes hashes more secure beyond just rotating.

Sections 6.5-6.7, 7.1 due on October 30

1. I had a couple questions from this reading. I have learned about public keys before, but what I've learned and what I read don't seem to completely mesh. Hopefully you will go over this in class, and I'll be able to see the connection.

In 7.1, I had a hard time grasping discrete logarithms. We've dealt with exponents before, so I don't understand how this is a new topic. I also don't really know where alpha comes from, and I'm guessing we'll be learning this in the future, but what's the application of discrete logarithms?

2. I was really interested to read about the application of RSA to treaty verification. I can see how that would be very valuable, so that the treaty stays secure and the same as what it was when it was agreed upon. The more I learn, the more it seems that cryptography is applicable in many areas of life.

I also enjoyed learning about the RSA challenge. I don't know if I could have come up with it myself, but I like learning how people break puzzles or use ingenuity to solve a problem.

Sections 6.4.1-6.4.2, due October 28

1.  Why is it important to understand theoretical methods when they can't actually be used in practice? I realize it's important to pay attention to the assumptions we make when we use these tests, such as the Miller-Rabin, but I have a hard time seeing the application of an unusable test. Maybe it's because I come from such an application-based major such as information systems.

2. The quadratic sieve seems very systematic, which I like. It's interesting that we can make a matrix out of the primes and look at factoring that way. This still seems like it would take awhile, which means that there still is not a good system for factorization.

Section 6.4, due October 25

1. The p-1 factoring method seems like it would work well, but I don't understand how or if it's any quicker than Fermat factorization. I don't think I understand it fully.

2. While inefficient for large numbers, the Fermat factorization seems more easy to use. It's a lot of repeated calculations, but it's interesting to me that it would work well, especially on a computer, up until a certain size of number.

Section 6.3, due October 23

1. I was doing okay with the use of the different tests until I got to the Soloway-Strassen Primality Test. I understand that its use is similar to the other tests, but it seems more difficult to use. Also, why do we use primality tests when they can only conclusively tell us if a number is composite and not if it's prime?

2. It's really interesting to me that we can find out if a number is composite much faster than we can factor it. These primality tests are surprisingly helpful, assuming that knowing a number is composite is helpful.

Section 3.9, due on October 18

1. For a composite modulus, we divide a more complex number into two of its factors.  Does it matter which factors we use, or do we just keep trying until there's a root for y mod p for some factor p?

2. I was really excited to learn about this topic because I have been wondering about it, amd I hadn't looked ahead in the book. I especially found the last theorem interesting that compared having a square root mod n to factoring. It doesn't solve our factoring problem, but it will be useful to use.

*Sorry this is late. I am on an interview fly-out trip.

Section 6.2, due on October 16

1. I am having a hard time understanding low exponent attacks. I understand how they are useful, but I am not understanding the theorem that explains how to use them. More specifically, how do you choose d? There are guidelines, but it seems like there would be a lot of choices.

2. I have never heard about a timing attack, and it seems like that greatly jeopardizes the security of RSA. The use of derivatives to find the times also is interesting and is another area of math that I get to be refreshed on. I would be really interested to know what preventative features, specifically, can be added to a physical implementation to prevent the timing attack.

Section 3.12, due October 14

1. I don't really understand the theorem in this section. I get the idea of continued fractions, but I don't really understand how to use the theorem, and more specifically where r and s come from.

2. I think continued fractions are really interesting. I'm most interested in the applications of this idea. I can see how it would be useful to be able to approximate a real number with a rational number. This offers a fairly easy way to do this.

Section 6.1, due October 11

1. I do not understand how the two claims given at the end of the section fit together. I understand them individually, and that their purpose is to show that finding the decryption exponent d is essentially as hard as factoring n. Are they supposed to work together or build on each other? Or are they just two separate claims that when put together prove a point?

2. I really think the idea and use of a present key cryptosystem is interesting. I can definitely see its merits in that you don't have to send a key to the other person, and it's not incredibly difficult to use. I am enjoying learning about the algorithms that go along with the cryptosystems I've only previously heard about from a technology standpoint.

Sections 3.6- 3.7, due October 9

1. I've always had a hard time understanding Euler's Theorem, even though I've learned about it before. I'm especially having hard time keeping all of the symbols and their meanings straight and remembering how the different theorems used in the proof applies to Euler's Theorem.

2. I really liked learning about the three pass protocol. Because of the example given, I understood what was going on when they put it in mathematical terms. Although it is vulnerable to attacks and requires multiple communications, it's useful because both Bob and Alice have to lock/unlock the message.

Sections 3.4-3.5, due on October 7

1. I followed the examples for the Chinese Remainder Theorem, but I got lost at the Lemma and its proof in Section 3.4. I understand its usefulness in understanding the Chinese Remainder Theorem, but I don't get the proof, or why they start the proof where they did.

2. I really thought Section 3.5 was interesting. It combines what I know about binary with what I know about congruences to find answers to congruences, even when working with large numbers. This seems like it will be very useful going forward. The only part I didn't understand was the sentence "If we want to compute a^b (mod n), we can do it with at most 2 log2(b) multiplications.

Exam Prep Questions, due October 4

1. I think the most important topics we have covered are the areas of math we have covered, such as modulus and fields. These repeatedly come up in different cryptosystems, so I think they are the most important in getting an overall understanding of cryptography.
2. I expect to see two questions for each cryptosystem we've covered: one for encryption and one for decryption. For those that we haven't memorized or completely learned,  I expect to see questions about the processes of encryption, if not the specific algorithms,
3. I need to work on understanding decrypting. I can encrypt in any of the systems we've talked about, but I have often had trouble on the homework with decryption.

Class Progress Questions, due on September 30

1. I have usually spent about three to four hours on the homework, although I'm not understanding everything. The lecture and the reading help with the homework, but not to the point that I know what I'm doing the first or second times I look at it. I have to really dive into it to attempt it.

2. I think when I'm doing the homework correctly and thoroughly, that contributes to my overall learning in the class. I have also learned a lot by working with other people on the homework. When I try to keep up with the lectures, that also helps me to learn.

3. I think I would learn more effectively if I had taken a math class more recently, but that's not something that I can fix now. Going forward, I really need to try to keep with the lectures. I often feel like the topics covered in class are covered very quickly. I need to stay up with them, so I can ask questions when the topic arises instead of being clueless later.

Section 3.11, due on September 27

1. The main thing that I didn't quite understand from the reading was all of the conditions for a field. Initially it was the three conditions given on page 95. However, on page 97, there are another three conditions that are said to be used to construct a finite field. What is the difference between those two sets of conditions?

2. I really enjoyed reading about polynomials and fields. I have always been able to understand polynomials better than the modulus function, so hopefully that will help me with this section. I was also intrigued to be introduced to the idea of fields, but they make sense.They potentially seem very powerful.

Sections 4.5-4.8 due on September 25

1. I'm not sure I really understand salt. I understand its aim, but I don't understand the process or computation behind salt.

2. I liked learning about the computations behind a meet-in-the-middle attack because I had learned about them in a few of my classes, but I didn't know exactly what was going on. It's interesting that even though you can double and triple encrypt the plaintext, the level of security still isn't that high and definitely not high enough that you would trust your message to be completely secure.

Sections 4.1, 4.2, 4.4 due on September 23

1. How is/was DES secure when you use the same Initial Permutation and Expansion Permutation tables every time? I realize that computing capabilities were limited in 1974, but as they increased, I feel like the tables should have changed, especially since DES is not a group.

2. One thing I found interesting was the lack of trust people had of the NSA. It seems like that's a recurring theme. I also was intrigued by expander functions. I had never seen those before, but they are definitely effective. As I read about both DES and the simplified DES-type algorithm, I realized how deep into computers both security and hacking has gotten. We are learning about how to encrypt a set of bits of information, which seems so tiny, when in reality, it could be hacked.

Sections 2.9-2.11, due September 20

1. For the LFSR, and for other cryptosystems we've learned about, it seems like there's a certain amount of educated guessing. For the LFSR, it's guessing the length of the recurrence. Is there a systematic way to guessing what the length of the recurrence is, or should I just start from 1 and work my way up?

2. My first thought was "Blum-Blum-Shub? Who names anything, let alone a pseudo-random bit generator, something like "Blum-Blum-Shub?" Anyway, I had never heard of a linear feedback shift register, so I found it really interesting. The use of binary is easier and harder to understand in some ways than using letters. For example, I'm glad to be working with mod 2 now instead of mod 26, but these ciphers are more difficult to decrypt because of the limited characters that could be repeating.

Sections 2.5-2.8, 3.8 due September 18

1. The book mentions the concepts of confusion and diffusion and relates them to some of the systems we've learned about. However, the classical cryptosystems were not mentioned. My question is do all block ciphers have the properties of confusion and diffusion? Do the Playfair and ADFGX ciphers have those properties?

2. I am really enjoying how the book logically builds up to more complex cryptosystems. It's very easy to follow. I specifically appreciated the review of inverting a matrix and learning about the Playfair cipher, which I had heard referenced in movies, but I didn't really know what it was. I also enjoyed the Sherlock Holmes reference, which helped me realize how widespread and well-used encryption methods really are.

2.1-2.2, 2.4 due September 13

1. The part of this reading that I had the most trouble understanding was the affine cipher. I understand the encryption method, but the decryption got me confused. I understand congruence, but I didn't understand why we were finding a multiplicative inverse of x (mod 26). Could there be more than one desired inverse?

2. I have always been fascinated with substitution ciphers, so I was happy to read 2.4. I loved the reference to U.S. historical figures, and their uses of substitution ciphers. More specifically, I'm glad that the book went through an example of trying to break a cipher through frequency counts. Although knowing which letter e is replaced with is helpful, it still does not tell us the remainder of the cipher. Unless someone has access to a computer (which almost everyone does these days) or they really want to know the message, the substitution cipher is a decent deterrent to discovering the message.

Guest Lecturer, due on September 13

1. One topic that I didn't really understand was the message that was never delivered related to plural marriages not being stopped outside of the U.S. I can definitely see the negative ramifications if it had been sent, but I think that if that was actually the way it was divinely supposed to be, the message would have gotten through another medium. I suppose I don't really understand any kickback or excitement about that story.

2. I really enjoyed the guest lecturer, in part because I had heard of things like the Deseret alphabet, but I had never known what they were used for. The examples she gave of Parley P. Pratt writing to his wife and John Smith writing to Brigham Young were fascinating uses of the Deseret alphabet to learn about. The use of the mason code and simple substitution ciphers also made me think that there must be other, lesser known uses of codes by members of the Church. I had never heard of the secret names in the Doctrine and Covenants either, so I was excited to learn about that.

3.2-3.3, due on September 9

1. I honestly don't really understand the extended Euclidean algorithm. I can follow what the example is doing, but I'm not sure that I'd be able to replicate it. I specifically got lost after it says "an easy calculation shows" because I don't understand why we plug in x5 and y5 instead of continuing with the sequences. I'm getting lost in the details.

2. I was really interested in 3.3 because congruences have come up a couple of times in my studies - both in programming and in networking. We can figure out how many bits are leftover once a certain storage capacity is filled using the mod function. I was glad to learn more about something I had already learned about.

1.1-1.2 and 3.1, due on September 6

1. (Difficult) The most difficult part of this reading was getting used to reading theorems and proofs again. I haven't taken a real math class in over two years, and so 3.1 was a nice re-introduction. I could understand most of the theorems and proofs, but I had a hard time following the proof of the last corollary of the section. I understand what the corollary is saying, but I am barely understanding the proof.

2. (Reflective) I have learned a little bit about cryptography in an information security class I took as part of my program, so reading this section made me excited to approach cryptography from a mathematical standpoint instead of an information security standpoint. In programming basic business applications, we have tried to include some checks for data integrity and authentication, but being able to incorporate more cryptography will be beneficial.

Introduction, due on September 6

My name is Heather Dunnigan. I'm a graduate information systems student. I'm doing the integrated 5-year program, so I will be graduating this April with a bachelor's degree and a master's degree. Beyond calculus, I   have taken Math 290, Math 341, and Math 313. I am taking this class because I got permission to use this class as the last requirement for my math minor, and I saw a flyer for it four years ago when I was a freshman, and I have wanted to take it ever since. It complements my major nicely, and it sounds fun.

I don't have any experience with a computer algebra system, but I do have experience with programming. I have taken a few classes in Java, and I have gotten a little bit of exposure to other languages. I am completely comfortable using SAGE to complete homework assignments.

I had the same math professor for both 290 and 341, and I thought he was very effective. He allowed time for questions to be answered in class, and he was very easy to approach outside of class. He was willing to explain things over again, even if he had just explained it. I could tell he respected his students, and he truly wanted them to understand.

As for something interesting about me, I am from a suburb of Detroit, MI, and this past summer I interned at Deloitte in San Francisco.