16.5 due December 11

1. While I understand how to send a message through an elliptic curve, I am not comprehending how our message can be a point x on our elliptic curve. Wouldn't the message then just be some (x,y)? How is that a meaningful message at all? I can understand key exchanges and signatures well enough, but the whole "Send a message as a point x" does not seem useful, even though it is said to be a lot easier to use this method with smaller primes.
2. After our last hw, and what is said in this chapter, it is really cool to see how we can use smaller primes for an elliptic curve equation! I think it is a pretty good step forward, because I'm sure a lot of the primes we are using are so crazily big when they don't need to be. It's been really fun this semester to see how cryptography works, and to be able to experience it first hand!

16.4 due December 9

1. This section was pretty straightforward. I think the only thing I would really like for clarification in class is the additions for GF(4), like why (0,w^2) + (w,w^2) = (w,1). It almost makes sense, but just a little more in depth as to why this, and the other examples work that that would be great!
2. When they mentioned in this chapter that we can work things out pretty well in mod2, I am wondering how this will work? I think they could mean in binary, like we used before, which would make sense. I hope that I am right!

16.3 due December 6

1. Why does it work that we can factor n using elliptic curves? I am trying to follow the logic that when we find 2P we find the inverse and if it is a number not 1, then it is our factor? How does that work? I feel like they didn't really explain it in the book very well.
2. I really hope that we didn't save a super easy method for last when finding factors! I don't think this will work for all cases, but I think it would be really unfair if it turns out that it is so much easier to factor this way! It definitely seemed easier in the example they gave us!

16.2 due December 4

1. Why are the number of points mod p going to be p? I thought in the previous section we decided that there would be 3 points, sometimes including infinity? And is Hasse's Theorem just a proof that there are going to be at least p points, or is that showing that there can be less than p points? Also, is it possible for there to be less than p points? What about significantly less than p?
2. I think this chapter is going to be a bit rough for me. What I am thinking right now is still how are we going to put a message into an elliptic curve equation? I feel that it seems a bit impractical, even with a fast working computer. Although I am not very knowledgeable with how this works still, I think that it would take more time to encode, and possibly be easier to cipher then an RSA or something along those lines.

16.1 due December 2

1. I don't think they really got into this, but I was a little confused at what our message is going to be in our elliptic curve equation. Is it going to be whatever x is? Will it be the a, b, and c? I feel that they didn't really go in depth on why this is going to be important for us as cryptographers.
2. Honestly, this section was not very interesting. I am a bit nervous on how we will be using these elliptic curve equations in the future. It seems to me like they will either be super easy, or super hard. I really cannot see an inbetween on this! I really hope somewhere in this chapter there is some sort of history lesson! I think those are the most interesting parts for me, and they motivate me to actually use what we learned!

18.1 and 18.2 due November 26

1. I was a little confused about Error Correcting Codes. About to the Notation on page 401 it starts talking about code lengths and codewords... what is the difference? I think I am confused too on what distance is too. Equivalence was also throwing me off a bit too. How are they equivalent when we Permute the symbols? It's all really new and I have to remember sets and stuff again..
2. It's interesting how cryptography and coding theory differ. One is sending messages over nonsecure channels, and one is sending messages over noisy channels. I wonder if there are times when both are used at the same time? I could see someone failing to send a clear message with the examples we learned about with one time pads, where the random integers were made from computers or other random-ish things. In circumstances like that though, it seems like a ciphertext would be even easier to break since you might need to send the key more then once, but could also have potential to be harder to break! I wonder which it really is.

2.12 due November 25

1. I got a little lost during the explanations of solving the keys to find the rotation used for that day. They talk about using the first and the fourth letters since they would be the same plaintext, but I wasn't sure if it was for just one key at a time, or for all the different keys we were given at once. Then they talk about how the permutation A sends k to d and D sends x to v. Where are they getting those from? Then it feels like out of nowhere they know what x is! So confusing..
2. I've always mentioned in my posts that I love history. This was no exception. What a cool concept the Enigma was! It seemed like a really creative and "out-there" type idea, I was surprised that the British were able to crack the codes and keep them secret for 30 years. It was really lucky for them, it seems like something like that where it was essentially one of the first electronic computers would be a lot harder to crack, but somehow they did it! Kudos to them for sure!

19.3 due November 22

1. After looking at the website and then the book, I still have a little bit of confusion. At the beginning of the chapter, they talk about how we are trying to find an a and r such that a^r = 1(mod p) but in the example, we get to the point where 11^x-y = 1 (mod p). I think that's the point we were trying to get to, but wouldn't x-y=0? Maybe I am wrong, but I didn't think that this would work.
2. I was very surprised in the example when they said that a quantum computer couldn't use 21 like we could do in the example in our book! It seems like such a small number, and it shows we still have some progress to make in this field! It makes me think about how conspiracists believe that the NSA has a super quantum computer that could do anything! That would be interesting if they did, and would actually probably help public advancements in the future!

19.1-19.2 due November 20

1. Well, this whole section was confusing. I think the worst part was all the arrows. I really didn't understand what they meant. I know that they are bases or something, but I don't know what they represent at all, which made it hard to understand the rest of the section.
2. I definitely did not think we would talk about quantum mechanics in cryptography! It looks like in the introduction that it is still a young concept, and so we don't discuss it much, but I think that it has a lot of interesting potential for the future!

Review due November 15

• Which topics and ideas do you think are the most important out of those we have studied?
• I think that RSA and discrete logarithms are the most important topics that will be on this test.
• What kinds of questions do you expect to see on the exam?
• I am expecting a continued fractions question, a discrete logarithm question, a hash function, and maybe a RSA signature question that isn't too hard.
• What do you need to work on understanding better before the exam?
• I think what I need to work on the most is discrete logarithms. It played a big role, and I wasn't the best at solving discrete logs. I think most the time I got them wrong, though I do understand the concept and how to solve it.

12.1-12.2 due November 13

1. I definitely got a little lost in the middle of 12.2. They mention for the Shamir threshold scheme splitting the keys into subsets so that a smaller amount of people can get the message. But to me, it looked like they were still using everyone to get the message. Then they put it into a matrix and take the determinant for some reason. That's about where I got lost and felt like I was just reading without even understanding what I was reading! Clarification on this would be most helpful.
2. At the very beginning of 12.2 they mention having a mechanism where two out of three keys will control a weapon. I literally thought about a spy movie before I read the line that mentioned it. They use  a similar method in the White House Down movie that just came out. To launch a missile, there were three people that needed to give their authentication, including the president. It was a nice, cheesy thriller with Channing Tatum in it. It's cool that this concept is used in cryptography.

9.1-9.4 due November 11

1. These sections were pretty straightforward, but I feel like I might not understand quite fully what is happening. They mention in the introduction of chapter 9 that we are not trying to encrypt m at all, but in 9.1, Alice's signature is m^dA. Isn't this encrypting m then? I think there is something I am not quite getting.
2.  I was very interested in the birthday attack. It seems interesting to me to try and create two separate documents, and change them just slightly to try and make it so both have the same hash, then use their signature against them. At the end, they mention to do something small, like remove a comma, or any slight alteration to foil this method. I wonder how this would actually work in some cases. But I also am intrigued by the idea, especially after our last homework where we basically made two hashes similar in this way.

8.1-8.2 due November 5

1. I am not feeling good, so I was having a hard time understanding hash functions in general, but my main question I was wondering is why when h(m1) = h(m2), m1≠m2. Is it just the way that a hash function works that there are many ways to interpret it? I don't understand if we are encrypting and decrypting a function or not in this chapter.
2. I don't have much to reflect on. I am kind of out of it - I went into anaphylactic shock today and now I'm on some meds that are making me really tired. But I was trying to look at examples online about hash functions, and I noticed that they mentioned caches as an example. I always wondered how caches work, I know they do it for websites when you google it. But I couldn't quite understand how they work. But it's still interesting to me.

7.3-7.5 due November 4

1. I was a bit confused about the instructions in 7.5. I don't see hoe bob computing tr^-a mod p will give him the message m. They give us the reason why, but it wasn't really helping me. The whole process seemed pretty confusing to me.
2. Along with the process seeming confusing, it also doesn't feel like it's very secure. All eve needs to figure out is a. The only thing preventing her is the fact thy it is hard to compute discreet logs is very hard. But I feel like it could be done on a very good computer. I personally would not use this method.

7.2 due November 1

1. I was a little confused about the equation we were given where b = a^x, and a^(p-1)=1(mod p) implies that a^(p-1)/2 = +- 1(mod p). why is this true? Was it a rule we learned already? I can't remember if it was or not.
2. Reflectively, I was wondering to myself how they come up with these names for different methods. At the beginning of the chapter, it talks about the important birthday attack. Really? Who comes up with this? Before we had Rijndael and LUCIFER. I know they usually mean something specific, but I am pretty sure in some cases, like LUCIFER, they will change around words so that it will become that anagram.

6.4.1 and 6.4.2 due October 28

1. In 6.4.1, they start showing us a relation 17078^2 = 2^6 * 3^2 * 11 ( mod 3837523) gives the row 6,2,0,0,1,0,0,0. Where did this "row" come from? I know in the book it says each base gives a row in a matrix where the entries are the exponents of the primes, but it still feels like they just got these out of thin air.
2. A couple homeworks ago, we had to find x^2 = y^2(mod n) and x ≠ y(mod n). I had a hard time with that homework, and i think I actually just gave up on that specific question. This section really helps explain a little more to me at least how to come up with those numbers. I was just so confused before and couldn't even think of how to do that problem.

6.4 due October 25

1. I guess I am still a little confused when it comes to this kind of stuff because a lot of it all the sudden is "often we use this - choose random this - go until you decide to stop" I know that it is because of how difficult everything is, but I am just so used to math being exact always. How will I decide when I should stop?
2. It's a little crazy how hard it really is to factor large primes. In this age of computers, it surprising th think that it really is an issue. I feel like I just grew up thinking computers could do anything you wanted. The book mentions a quantum computer being used to help factor primes quickly. I am pretty sure that is what everyone thinks they have done at the NSA facility here in Utah. That would definitely be interesting if they had.

6.3 due October 23

1. I was a little confused with the Miller-Rabin Primality Test. Near the end of the test, we are told to continue finding new b's until stopping or reaching b k-1. I don't know where the k came from. I am pretty sure they are just saying to stop after a while, but maybe I'm wrong. It just feels like they came up with a k out of nowhere.
2. We learned about pseudoprimes a long time ago in my number theory class, but I definitely liked how this book described it more. It was way more obvious what they were in this text, while in my other text I was pretty confused. It's funny how some stuff is easier in this book, while other material makes sense in the other book.

3.10 due October 20

1. I already learned about this in my number theory class. I think the only thing was was confusing me when I first learned about it was when we were going from a big number to a small number, like for example (9|24). When you flip it, sometimes the examples given were (24|9) and others were -(24|9). How do we know when it is negative and positive? I kept messing it up sometimes.
2. I really like taking this class and number theory at the same time because the classes are very similar. It is nice because number theory shows more just the basic theory and proofs of what we are given, and cryptography uses them for more specific and "real life" examples. I T.A. for a college algebra class, and that's the question I get over and over, "Will we use this in real life?" I just tell them I'm still in college and not real life yet. But I am happy I am taking a class that I will hopefully use in real life.

3.9 due October 18

1. I wasn't too confused with this section because I already went over this in number theory. But, I was wondering why we had to use p = 3(mod 4) for our prime in the proposition. Is it because of something specific? I've seen it a couple times, but I don't know why it works.
2. It's interesting to see after going back and forth from chapter 3 to chapter 6 how they are building us up for RSA. They mention in this section about how we choose n=pq, and I think about how we are using that already! I think this means stuff is going to get a little harder in RSA!

6.2 due October 15

1. One thing that was taking me a lot of time to understand was the Low Exponent Attacks. They were trying to show if d<(1/3)N^(1/4), then d could be calculated quickly. They then go on to prove this, but it was super confusing with lots of different integers, roots, and powers. I think it was just looking really abstract to me, so it wasn't making much sense.
2. They mention in the book that these attacks to RSA are more because of mistakes. It made me think of what happened to Adobe recently. They were hacked, and the hackers got a bunch of account and credit card information. I don't know if they use RSA or something else, but I was really surprised to hear that a high up company like that getting hacked. I wonder if they just made a small mistake that allowed hackers to exploit, or if they were super super lucky.

6.1 due October 11

1. RSA seems pretty straightforward compared to stuff we've learned previously like DES. I think the only thing I wasn't following was why we had to make sure de = 1(mod(p-1)(d-1)) and 1 = gcd (e, (p-1)(q-1). But I'm sure its just so that it's actually easy do decrypt for Bob.
2. Like I said before, I am just happy this isn't quite as complex as DES. That section killed me. This seems easy, which makes it seem like it should be easy to attack, but I'm sure it's not. I was blown away  by how big the primes are though, 100 digits! That's crazy! I just hope we don't have to use super large primes for homework!

3.6-3.7 due October 9

1.  The Three-Pass Protocol was a little complex to me. I understood the nonmathematical example that they give to demonstrate how the Three-Pass Protocol worked, but when we got to the real math, I was barely grasping it. I think it was mostly just the different K1, K2 and K3. Is that a new K each time? How does that give us the real K? I think that maybe it just doesn't make too much sense because we don't have plaintext or ciphertext, just K.
2. Everything but the Three-Pass Protocol is stuff we just went over in my Number Theory class. I actually feel that this book does a better job at explaining it then my Number Theory book or professor does, which is kind of sad. But I like that these sections have been a little similar to each other so that I can understand everything a lot better. It's like taking 2 hours to understand 1 hour of stuff!

Review due Oct 4

• Which topics and ideas do you think are the most important out of those we have studied?
• I think the most important topics were the DES and chapters beyond that, since they seem the most applicable to "real life". It is something I expect to see more of if I get a job that uses cryptography.
• What kinds of questions do you expect to see on the exam?
• I am expecting that we will have to solve basic cyphertexts that can be done by hand. Probably coding and decoding for the different types of systems we learned.
• What do you need to work on understanding better before the exam?
• During the last part of this unit I was super busy with some family issues, so I think that I need to review DES and the last two units we went over before the test. I just struggled trying to do the homework and didn't have time to get help so I think reviewing the last couple chapters and the harder concepts will do me well.

5.1-5.4 due October 2

1. Well first off, this whole section was confusing from the beginning to the end. The algorithm in 5.1 made sense for the most part, but the in depth part in 5.2 was pretty cryptic to me(haha). I think it would be amazing if we went over the different layers in depth(with examples!) to show how they actually work!
2. I noticed in 5.1  that the first transformation - ByteSub - that it is a non-linear layer that is used for resistance to differential and linear cryptanalysis attacks. It kind of caught my eye, we haven't really used any sort of differential attacks(to my knowledge at least). I'm sure I'll regret saying it later, but it sounds interesting and like a good challenge to try.

Feedback due September 30

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I spend about 1 to 2 hours on the homework, depending on how difficult it is. I feel that the reading prepares me the most for them, but that if there is a hard concept that learning about it more in depth in class helps me out more then the reading.
• What has contributed most to your learning in this class thus far?
• Honestly, I think that Google has helped me the most in learning in the class. If there's something I can't understand very well, I usually go to Google because I usually don't have time during office hours to talk to the professor. I'm a pretty visual learner, and there will usually be at least one website that helps me to understand in the most effective way.
• 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.)
• I think that what would and already helps me the most in the class is examples. The more, the better. If I can follow along to an example, it helps me to understand the homework so much more, especially if I get stuck on something I maybe don't understand very well. I think classtime is used the most effectively when we go over the material to help clarify for the first half, then spend the second half going over examples to help understand even better.

3.11 - 3.11.2

1. I was pretty confused with the first section of this chapter. They give us an example with four elements, GF(4) = {0,1,w,w^2} and the laws
1. 0 + x = x
2. x + x = 0
3. 1 * x = x
4. w+1 = w^2
I guess I'm just a little confused. Is this in a mod2? mod4? Or is it its own mod? Also what is the notation GF(4)? Maybe it was said in a different chapter, but I don't remember reading about it. They use the same notation later in 3.11 as well, but I don't understand what it's supposed to mean.
2. I am using division with functions right now in my number theory class, as well as teaching it to my students for math 110. It's kind of funny how I'm doing it in three different places! It makes it a little easier to understand after doing it three times in one day. Hopefully this section won't be too hard.

4.5-4.8 due September 25

1. I had a hard time understanding DES in class last time, and reading about it again confused me still. I was the most confused with the Output Feedback, just reading it and looking at the diagram made me super confused. We should definitely go over it in class. We could go over the Counter and Cipher Feedback as well, but those two weren't quite as hard to understand.
2. I liked reading about how the RSA held a couple contests to break a DES cipher. I actually was really surprised that the first time it only took 5 months! I was honestly expecting at least a year. But it was even more surprising to see that the second contest a year later only took about 40 days! That's such a big difference in so little time. It really does show that the DES cipher is pretty dated with new technology coming out. I'm excited to see if there is anything even more hard(and confusing) to crack.

4.1, 4.2, 4.3 due September 23

1. This section was a handful! I was having quite the time trying to understand what was being talked about. This will be weird, but the hardest part for me to understand were actually the diagrams. I would think I had something figured out, then looking at the diagram, like Figure 4.1 on page 115, the Feistel System, I would be lost again. I don't think it was very explanatory, or at least comprehendible. I would like examples in class of basically everything, but mostly what was talked about in section 4.2.
2. First off, in the introduction, I thought it was pretty funny that the IBM algorithm was called LUCIFER. I think it would be pretty great to know if it actually just happened that way, or if the people who made the algorithm chose to call it that because it's a beast or something. I thought that it was pretty cool that it was so recent though, in 1974, because its a lot more relevant to what we have learned recently.

2.9-2.11 Due September 20

1. I was having a hard time understanding Linear Feedback Shift Register Sequences. It looks like it might just be a simple formula, but I guess whats confusing me is how you can get a lot of different keys from that. I feel that a lot of keys will be similar and there won't be as much variety. But I also think I might be misunderstanding what the method is all about. If there's anything I want an example of in class, it would be this.
2. I was pretty intrigued that one-time pads are basically unbreakable. I think with the modern age of computers, that this wasn't possible, but after reading I understood that it could be possible. I still think that it might be at least plausible if you had the worlds fastest computer to try to decrypt it, but even then, it is so simple, yet so hard to break. I liked the example that FIOWPSLQNTISJQL could be either wewillwinthewar or theduckwantsout in plaintext. It shows that there are so many ways the code could be decrypted and make sense, but still not be right.

3.8 and 2.5-2.8 due September 18

1. One thing that confused me were the ADFGX ciphers, as well as the block ciphers. It was a lot of work, and made it a little hard to understand.  I guess it was the fact that not only we have a keyword, but we're only putting part of the message into a matrix, then crypting the message seprately. At least that's what I think it meant. As with everything in this book, an example would be nice in class, probably for the Playfair, ADFGX, and Block ciphers.
2. I always say that I like the history of cryptography, and this section was just as good. I think my favorite part was learning about the ADFGX cipher, and how it was used by the British forces. The fact that it was deciphered by French cryptanalyst Georges Painvin was pretty great, and how they were able to decrypt a lot of messages. I wonder how long it took the British to realize they'd been compromised!

2.3 Due September 16

1. Well, this whole section was pretty confusing. I think it was just a hard to understand the shifting then using the matrix with probabilities. I'm able to understand it more in class when it's done visually on the whiteboard step by step, so hopefully that will help!
2. For the project 4 we were supposed to do, I was originally thinking about using matrixes to code and decode a message. This process seems like it could have been used to crack my code, and I hadn't even thought of it that way before! I think the process isn't quite as complex as it seems right now, but I think that once I understand it I will probably like this specific process.

2.1-2.2 and 2.4 due September 13

1. The one thing I was having a hard time with was section 2.2. I understand how to use modules, but how they were using that to encrypt a function wasn't making sense much. I was also confused at the four 'sections' of "ciphertext only, known plaintext, chosen plaintext, and chosen ciphertext" and what they meant. In the last two it would say something small like choose the letter a as the plaintext, and how it gives the key. I guess it's probably simple, but it wasn't really making sense at how having one letter gives away the key.
2. The thing I liked about these sections were how they talked again about historical times encrypting messages was used. I thought the example of Julius Caesar was really cool because of how simple his ciphertext was, but it probably was something relatively unknown to most people so it made it advanced for their day. I also liked the example of Thomas Jefferson and Benjamin Franklin. It was basically the same concept, yet so many years apart. We're definitely part of a great generation, where we can use computers to make such advances in cryptography.

Guest Lecture, due September 13

1. During this lecture, the only thing that was hard for me to comprehend was the Deseret Alphabet. I guess every sound had one symbol only, and this to me just didn't seem like it would be possible to turn into an alphabet. There are just so many different sounds you can make phonetically, it seems like you couldn't manage to make a symbol for each sound. But I am a little biased I'm sure, since I only know English and it's alphabet.
2. One thing that I thought was really interesting was the crypted names in the Doctrine and Covenants. It was super cool that they substituted names with such unusual ones, but funny that they didn't keep their own key so there were some names that no one remembered. They had to piece things together years after to remember the names.

3.2 and 3.3 due on September 9

1. I think the difficult part of the text today was basically all of 3.2. We are trying to solve ax + by = d, which I understand, but this part of the text was really hard to follow, and the explanation left me more confused then before. It would be nice in class to get a better explanation with an example.
2. 3.3 reminded me of my proofs class I took 3 years ago. We kind of glossed over mods, but that class is the only other place I remember even learning about it. I just remember my professor telling us that it is very useful in cryptography, so I am excited to see the connections and how useful it really will be!

1.1-1.2 and 3.1, due on September 6

1.  I am learning the same thing about gcd's and Eucilds algorithm, so none of this is especially new. I think the only thing that looks little difficult is the Prime Number Theorem. I am just a little confused at how it works. It would be nice to have an example in class and seeing how it is important.
2.  So far, what I think is interesting is all the examples of cryptography from WWI and WWII. It's cool to see how compared to then, how complex it is to send a secret message. I especially liked the example of the Germans sending a message saying everything is normal, when it was actually encrypted with a new key each day.

Introduction Due on September 6

• What is your year in school and major?
• I am a senior graduating this semester in Mathematics
• Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)
• 290, 300, 313, 314, 334, 341, 342, 352, 360, and 371
• Why are you taking this class?  (Be specific.)
• I am planning on applying to the NSA after I graduate, and I feel like this class will be really helpful in gaining experience.
• 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 not had experience with Maple, Mathematica or SAGE. I have had programming experience in Java and R. I think I will be fine using any program given.
• 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?
• I think teaching straight from the textbook is what makes a math professor very hard to follow. What is helpful in my experience is when a professor gives examples that are not in the book. It gives you more experience in what you're learning and something else to refer to if you are confused.
• Write something interesting or unique about yourself.
• I like to draw with chalk, I am participating in the Chalk the Block in the Riverwoods this coming week.
• If you are unable to come to my scheduled office hours, what times would work for you?
• I will be ok.