16.5, due on December 9

1. I don't remember how to make sure the message is a point on the curve, and this section didn't explain it (that I could understand).

2. It was good to read this after already talking about it in class a couple times. It made sense. :)

16.4, due on December 7

1. I don't really understand what is going on when we add three points only to then add two points to find something that adds to infinity. I'm also having a hard time grasping the concept of working over a finite field GF(2^n).

2. This is random and unimportant, but I liked learning that a singularity is a point "where the partial derivatives with respect to x and y simultaneously vanish." I don't know if I've ever heard of that, but I think it sounds cool and interesting.

16.3, due on December 4

1. I didn't really understand the algorithm for factoring using elliptic curves. You just choose random curves until one of them works for you? How long do you wait before you deem one ineffective?

2. This section kind of felt like a lot of random things about the elliptic curve factoring algorithm without actually doing a good job of explaining how the algorithm itself works. But it could have been there and just went all over my head.

16.2, due on December 2

1. I don't really understand how to map a message to a point and how to get the message back from a point on an elliptic curve.

2. These elliptic curve things are kind of weird.

16.1, due on November 30

1. This is all a little more abstract than I'm able to grasp. Also, why do you take the negative of the y? I'm sure it's just how we choose to define addition, but why?

2. I'm interested to see how this applies to cryptography . . .

18.1-18.2, due on November 24

1. I didn't understand most of the section about error correcting codes, specifically how a code C can detect a certain number of errors.

2. I thought the part about ISBN numbers was really interesting. I had heard about it before, but never really got it. It helped to learn about it in the context of other error correcting codes.

2.12, due on November 23

1. I don't understand the set-up of Enigma, which I think makes it hard for me to understand the method of breaking it, although that seems to be hard to understand regardless.

2. I remember cyclic groups from 371, which I thought was cool. I also enjoyed the movie The Imitation Game, although I know that it is not completely accurate. 

19.1-19.2, due on November 18

1. I don't really understand how quantum bits work at all.

2. This is a cool overlap of math and physics. It's exciting that developments on this are happening right now!

14.1-14.2, due on November 14

1. I didn't understand the part about I and H(I||j) and why it works. I also had difficulty understanding why the Feige-Fiat-Shamir Identification Scheme actually works.

2. This seems really applicable and useful. That's something I like about this class. While it is just a lot of number theory, I like that it has context in the real world.

Exam Review, due on November 13

• Which topics and ideas do you think are the most important out of those we have studied?
• I think the most important topics are the different cryptosystems that we learned about and specific attacks on those systems. Most of the other stuff that we learned seemed to lead up to them.
• What kinds of questions do you expect to see on the exam?
• It's hard for me to know what to expect, because I feel like we used computers an awful lot in this unit. Even the concepts behind them seem to require bigger numbers, which we aren't really able to work with without a strong computer.
• What do you need to work on understanding better before the exam?
• I need to work on understanding Fermat's Little Theorem and Euler's theorem and how to apply them.
• I need to review all the different primality tests as well as attacks on the different cryptosystems.

12.1-12.2, due on November 11

1. I had a hard time following the notation and set-up of the Shamir threshold scheme.

2. This is all so cool to me. It just makes me feel like a spy, which I love. :)

9.1-9.4, due on November 7

1. It was hard for me to understand fully the set up of the RSA and ElGamal signatures and how they work between two people to produce the desired outcome.

2. I finally see how a hash function is useful. I know you said multiple times in class that signatures was one way that they were used, but I had a hard time understanding how until now.

8.4-8.5 and 8.7, due on November 6

1. I still don't really understand hash functions, so all of this was a little bit unclear. But I especially didn't understand that part about multicollisions.

2. I thought the birthday paradox was so cool!! I think it's awesome.

8.1-8.2, due on November 4

1. I didn't understand the Proposition on page 221 or the set-up of hash functions in general.

2. I'm so tired right now, that every time it says hash, I think of delicious chopped up and cooked potatoes. I'm hoping it means more tomorrow in class.

7.3-7.5, due on November 2

1. I didn't understand the part about the machines that do Diffie-Hellman problems/ElGamal decryption.

2. It's nice to see that the things we've learned in the past couple of classes lead up to something.

7.2, due on October 30

1. I don't understand the part in the Pohlig-Hellman Algorithm where you break x into x0 + x1q1 . . . and why it works. I also didn't really understand the Computing Discrete Logs Mod 4.

2. I had the thought as I was reading that cryptography is just a lot of number theory. I also had the thought, "this is much funner than just plain number theory." It's because this points to something and has cool applications. It's not just proofs! :)

6.5-6.7 and 7.1, due on September 28

1. I was somewhat confused by the discrete logarithms section. I don't really understand how they work.

2. I think the thing with decrypting your message with your key and then encrypting it with their key, so that they decrypt with their key and then encrypt with your key to get the message is really clever.

6.4.1-6.4.2, due on October 26

1. I didn't understand the part about putting the prime factor powers in a matrix and finding linear dependencies (I don't fully remember this), and how the numbers that they got were 0 (mod 2).

2. I found the quadratic sieve method of factoring interesting. I'm impressed someone came up with that.

6.4, due on October 23

1. I didn't really understand how The p-1 Factoring Algorithm works or how it gives us a factor of n.

2. This is just a fun fact: I'm giving a presentation on Fermat tomorrow in my History of Math class, and it's just kind of cool to see his influence in so many different places, specifically here in number theory that relates to Cryptography.

6.3, due on October 21

1. I didn't understand the part about how to pick a prime number.

2. Is this used to make sure we pick good p's and q's?

3.10, due on October 19

1. I had a hard time understanding 4 and 5 of the Jacobi Proposition. How can n be congruent to anything other than 0 (mod n) (in 4.)?

2. I know I've learned the Jacobian before . . . I don't remember how to do it, so is this the same thing, by the same guy, or completely different altogether?

3.9, due on October 16

1. This section made little sense to me. It seemed like they were pulling random numbers out of a hat and putting them together. I didn't understand the Proposition at the beginning, which probably contributing to not understanding the rest of the section.

2. I'm interested to see how this is used in terms of RSA. I figure it breaks down somewhere, or RSA wouldn't be secure. Or I didn't understand the section and that's not even applicable.

6.2, due on October 14

1. I have no idea how the Timing Attacks works.

2. It's nice to know all these rules, but I still don't feel very confident in being able to pick p,q,e that work well.

3.12, due on October 12

1. I didn't understand the theorem or how it applied. The rest of the material was kind of overall confusing too.

2. I'm interested to see how this applies to cryptography. It seems very random.

6.1, Due on October 9

1. I was slightly confused by the claims, especially the second one. They seem like they contradict the fact that RSA is secure, but maybe I'm not reading them correctly. Either way, I had difficulty understanding this part.

2. This made me really excited because I think it's a really cool idea. It also makes me want to discover a way to quickly factor super large primes, even though I know it won't happen. It does make me wonder if someone will in the future . . .

3.6-3.7, due on October 7

1. I didn't understand Euler's phi-function at all. That made the rest of the reading somewhat confusing.

2. The discussion in 3.6.1 was nice to see how this stuff is applicable to cryptography. It made me more motivated to understand it.

3.4-3.5, due on October 5

1. I had some difficulty understanding the General Form of the Chinese Remainder Theorem. I just got a little lost in the different variables.

2. Modular Exponentiation reminded me of the Egyptian method of doubling (to find products and quotients) that I learned about in MTHED 300. It's a similar idea of doubling and then adding the ones that you want. This was a cool connection to make. :)

Study Prep, due on October 2

1. Which topics and ideas do you think are the most important out of 

those we have studied?

I think the more modern ciphers are important as well as the ideas of the other ciphers we've learned.

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

I expect to see similar problems to the ones on the homework (processes relating to different cipher systems), possibly with the addition of some definition/term type questions.

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

I had a hard time with the DES homework, so I need to work on understanding that better. I also need to just solidify my understanding of each of the other topics before the exam.

5.1-5.4, due on September 30

1. I had difficulty understanding how the key gets changed for each round. I was also confused by the decryption process, especially finding IARK.

2. I find it fascinating that someone came up with this. Are there logically reasons behind each algorithm structure, or is it largely creativity? How do people come up with these? I wonder.

Reflection, due on September 28

1. How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
I usually spend about two hours on a homework assignment. Sometimes it's more or less. 
For the most part, both the lectures and readings have been good about preparing me for the homework. Occasionally, though, something will seem to pop out of no where. 

2. What has contributed most to your learning in this class thus far?
I think the most helpful thing for me has been reading before class (and thinking about it enough to write a reflection) to give me context and a first look and then having the lecture reiterate what I read.

3. What do you think would help you learn more effectively or make the class better for you? (This can be feedback for me, or goals for yourself.)
Sometimes the lectures seem to spend a lot of time on less important things and then gloss over important or harder to understand things. I think I can do a better job of fully doing the homework so that I can practice and let learning happen there.

3.11-3.11.2, due on September 25

1. I had difficulty understanding some of the facts at the end and the comparison with integers.

2. This brought back a lot of memories of Abstract Algebra. I recognized a lot of the things it talked about, even though I've forgotten some of the details.

4.5-4.8, due on September 23

1. I was confused by the idea of salt. I don't understand how it works or how it makes a password more secure.

2. I think it's interesting that there was a national standard encryption system. Is there still one today? I also find it interesting that they chose not to change DES even though they knew that it was weak. That just seems risky.

4.1-4.2 and 4.4, due on September 21

1. I had a hard time understanding how to get Ki from K. I think I followed the rest for the most part, but I don't feel like I have a good grasp of it; it would take some time to figure out how to apply it to a real problem.

2. This seems like such an elaborate scheme. It seems so fabricated with so many rules, and it seems that it would take so much longer to encrypt than was worth the security. However, I bet it's faster and pretty efficient with a computer. It's just hard for me to see.

2.9-2.11, due on September 18

1. I did not understand the linear feedback shift register sequences. I had a hard time understanding how it was even set up, so all of the subsequent information was hard to understand without that context.

2. Some of these ciphers seem too inconvenient to even think about using. I wonder why they came about and if they are actually used anywhere.

3.8 and 2.5-2.8, due on September 16

1. I had difficulty understanding how to decrypt a message encrypted using the ADFGX cipher. I also got a little lost in the explanation of the Hill cipher and got a little confused by the end.

2. I really enjoyed reading the Sherlock Holmes example. It was a nice break from the usual textbook reading while still being relevant.

2.3, due on September 14

1. The most difficult part for me was when they explained why the method for finding the key length and key. I got really lost and confused with all the different vectors and i and j.

2. I have loved reading about each of these new ciphers. I think they are clever and interesting to learn about. It's interesting to see how some build on earlier ciphers (like the Vigenere building on the shift cipher) while others are completely unique.

2.1-2.2 and 2.4, due on September 11

1. The hardest part for me to understand was the part about finding the decryption equation for an affine cipher while working in modular.

2. I found the Affine Ciphers interesting. I've never considered shifting by an equation as opposed to a number and I can see how it would help make the cryptosystem a little harder to break.

Guest Lecture, due on September 11

1. The most difficult thing for me was that the presentation sometimes jumped from one thing to another without much flow. I sometimes go lost in all the different names that I missed some of the actual information.

2. I really loved this presentation. I especially loved learning about the Pig Pen cipher and Larrabee's Cipher. I never thought of either as a possibility, and I found both so clever.

3.2 and 3.3, due on September 4

1. I had difficulty understanding multiplicative inverses (4th Proposition in 3.3) and how they are (or maybe not) related to fractions in mod. Those ideas were confusing to me.

2. I remember first learning about this in 290 and being really confused. Then we learned about it again in 371 more in depth and some of it started making a little more sense. It is nice to see it/use it for a third time.

1.1-1.2 and 1.3, due on September 2

1. I didn't understand the Theorem in 3.1.3 about the ax + by = d equation. I'm confused with the r2 = a(-q2)+b(1+q1q2) part. I can't see how they got that or the next couple of steps. I'm also not sure how the result proves the original proposition.

2. I know it was brief, but I liked how they mentioned the German Enigma. I recently watched The Imitation Game (about the guy who cracked it), which has made me really excited about this class. I liked how the reading helped me remember that. 

Introduction, due on September 2

Hello!

• What is your year in school and major?
• I am a Junior majoring in Math.
• Which post-calculus math courses have you taken?
• MATH 290, 313, 314, 334, 341, 371
• Why are you taking this class?
• This class is required for my major. It is also the subject within math that sounds the most interesting to me, and I would like to see if it is a career that I would want to pursue.
• Do you have experience with Maple, Mathematica, SAGE, or another computer algebra system?  Programming experience?  How comfortable are you with using one of these programs to complete homework assignments?
• I have no experience with a computer algebra system and very little programming experience (just CS 142). I am not comfortable using these programs due to my limited experience.
• Tell me about the math professor or teacher you have had who was the most and/or least effective.  What did s/he do that worked so well/poorly?
• Dr. Lawlor has been the most influential teacher I have had. He knew the best way (from years of experience, I presume) to explain each of the concepts in a way that made sense. He had stories and examples that helped illustrate the concepts and keep interest. He was also really open to helping outside of class even at times outside of his office hours. This was important for me because I had a conflict during his normal office hours.
• Write something interesting or unique about yourself.
• I play the organ.
• If you are unable to come to my scheduled office hours, what times would work for you?
• MWF before 11am and TTh before 10 or from 11-12.