Assignment 42

1. I like how this section has examples for each specific point with discrete logs. I still don't feel confident doing them on my own, but having those examples help a little.

2. I wonder how people figured that we could use elliptic curves along with the idea of discrete logs in the first place. I am always amazed at how bright some people can be--far beyond me.

Assignment 41

1. I guess for me the hardest part is really that I'm behind because I haven't made it to class-food poisoning is not fun. So, I really need to buckle down and make sure I understand elliptic curves in general better. Since I won't make it today either, I need to study hard on this even moreso.

2. I definitely recognize the setup to GF(4) from abstract algebra, and I find it interesting that it has such a direct correlations to what we are doing here now. It makes sense to this it would be more useful than a curve mod 2.

Assignment 40

1. I am so glad that we have already done a lot with factoring. Otherwise, I would be totally lost right now. I'm still trying to figure out the very first example--how it was explained kind of lost me.

2. It's interesting that we begin by finding several curves--that is definitely different from how we did things before! There are a lot of things that we do differently with elliptic curves, even though other things remain the same.

Assignment 39

1. I feel like the points in this section weren't too difficult. What I still struggle with is the idea of elliptic curves themselves. There are just a few points about it--like coming up with the points at times--that still haven't clicked. It seems like once this is solidified in my mind, I'll be okay with everything else.

2. It is interesting how things like the P-H algorithm can be thwarted in the case of elliptic curves. It definitely does make it seem as though elliptic curve cryptography is more secure. I wonder what attacks we might be able to make on elliptic curves that we can't on our other systems though.

Assignment 38

1. Well this was an interesting read. Isn't addition supposed to be easy? I still have a hard time wrapping my head around the whole concept. For example, it really throws me off to see the graph of an elliptic curve where the ordered pairs don't really seem to make any sense on a normal Cartesian coordinate.

2. One thing that stood out to me reading this was at the beginning. It's cool that we are now moving towards methods that will reduce the workload, so that we can send things more easily and efficiently. If we can get things like this working--and later on, find new things that take even less to work--we should be able to do even more with cryptographical systems.

Assignment 34

1. I think going over how we can add these quantum vectors together could be helpful in order to ensure that I am thinking about it correctly. Other than this, I think I understand it pretty well.

2. This was a pretty fun reading, taking me back to my days in Chemical Engineering. My question on it, though, is how did we come up with it exactly? Why do we use it? It makes sense that it can be secure and all, but it feels like it's coming out of left field. :)

Assignment 33

1. Oh boy, 14.2 brings out the big numbers again. I liked the easy conceptual case. Oh well. Since I understand the concept alright I feel it won't be too bad, but a review on Sage might be nice in order to speed things up a bit for me. I feel like I'm not nearly as good at using Sage as others and that slows me down.

2. I really like this section of reading. It's definitely nice to have something like this to protect us, but it's also cool that it's all based on probability instead of strictly rigid proof. Overall, that is something neat about cryptography--it's (so far) mostly based on attackers being incredibly unlikely to crack the code, whereas other forms of math adhere only to proof.

Assignment 32

1. I feel the most important topic that we have studied so far is the RSA algorithm, as that is what is most commonly used today. It is also good to know the conceptual ideas behind why it works so well and how it is a secure system, as knowing these concepts will help us to be better crypto-thinkers.

2. I expect to see a good deal of computational questions along with questions dealing with systems like RSA and ElGamal, along with discrete logarithms potentially. Really, though, there is just so much to choose from!

3. Really, I feel that I just need to have a good review of everything since there is so much to remember. In particular, I think I need to focus on the Chinese Remainder theorem, primality and all theorems having to do with that, and all of the algorithms.

Assignment 31

1. I feel like I have a pretty good handle on all of the math in these sections, along with the idea. I guess I feel that I'll be good to go after doing a couple of examples and once I'm sure I can do it well with all of the ways presented.

2. This section seems pretty cool to me. I like how it takes us back to things like high school algebra too, since other systems we've been dealing with bring in things like discrete logarithms and whatnot. It's fun to be able to go back to the basics and still do cool cryptosystems.

Assignment 30

1. This was a pretty fun reading. Most everything made good sense to me here too, so I guess my question for today would be why use El Gamal when it seems that everything we've studied generally points to RSA being stronger? Is it because of one or two saving graces it has, or are there other reasons? I guess that's what I struggle with on this reading.

2. The idea of digital signatures is fun to read about, as it has become such a key part of society today. The examples--particularly the one in 9.4--were fun to read about too. Are we going to do some signatures in class too? Because I feel that could be a fun project actually--making sure it can't be forged or something.

Assignment 29

1. So, I guess I still have a hard time seeing why we use hash functions since there can be so much collision and whatnot. I guess using the stronger algorithms take out the times where two completely different items can get the same hash, but maybe we could cover in class a bit more on why they are useful? I know that it's a fast way to verify and can be a sort of signature, but how sure can we really be?

2. I do like the birthday problem, just since it would always seem like "No, there's no way it's that likely" when you don't look at it mathematically. It is pretty interesting that it can be used to attack discrete log problems though--that's a pretty good idea!

Assignment 28

1. So, it seems like every example of a hash given was too slow or not collision resistant enough. What's the point of it then? You can't really use it for signatures if it isn't feasible. I know that the book presents a working method in 8.3, but since we don't read 8.3 for today I assume that there is something more valuable to what we read than what I got out of it, and so now I just wonder what that was. It must have gone over my head.

2. So, it was cool to see that if we know some m that isn't equal to m' where h(m)=h(m'), we can find the discrete logarithm. That definitely piqued my interest since we've spent so much time talking about how we assume it is too hard to find it. Too bad that method is too slow to use!

Assignment 27

1. I guess with these sections I don't quite see, mathematically, how it will be just as hard to compute the discrete log as it would be to do the computational Diffie-Hellman problem. I guess I just don't understand it enough just yet to see how the mathematics of these two problems weave together. It is nice to go over the ElGamal system now too since we have been discussing it a lot in class recently. A few examples of this system would be nice to see too.

2. I like the example of the football game. Does this method get used a lot in the real world these days? What other instances might we use for it?

Assignment 26

1. I feel like there is just so much to remember at this point. It's going to be hard for me to keep track of it all. I guess I just really need to practice all of it over and over. The first one seems really similar to what we've been doing lately, but then the rest all seem to differ so much from each other.

2. It's interesting that there is no really good way to solve discrete logarithms, but there are so many methods here to try to solve it. Looks like a lot of people have tried to figure it out. Like we said in class, if any of us could solve this issue, we'd definitely have a job many places!

Assignment 25

1. So, discrete logarithms. I haven't really done much with logarithms in a while, so I guess it's time to brush up on the basics. Overall, the concept doesn't seem too bad though--it seems to follow the general rules so far at least.

2. I really liked reading about the public key. It's just really cool that RSA can have a signature. We previously talked about how this has facilitated things like online shopping, and it's really nifty to me that this is all possible because of cryptography.

Assignment 24

1. The quadratic sieve method seemed pretty interesting. We had already covered the methods from 6.4.2, so I follow those pretty well, but I don't see all the logic behind the quadratic sieve as well as I would like to. I think part of it is that I need to review some of the more basic operations of modular arithmetic and how they can relate to such methods. The matrix this method finds is also interesting, but I guess it doesn't yet make full sense.

2. I look forward to using the quadratic sieve method (once I understand it a bit better). It seems like it'll be pretty fun to use and I like how it uses the matrix to let us "look up" more information regarding our number of interest.

Assignment 23

1. Well, I definitely understand the inefficient methods--it figures that the one we'll really focus on is the one that I really don't understand. Haha, such is life right? I really like how the book generally gives and example to work through, so maybe we could do one together in class?

2. I do like how we use RSA because of how difficult it is to factor, and then we now move to studying factoring. So, I haven't heard of anything beyond RSA so far, so does that mean we are going to see that factoring is still difficult overall?

Assignment 22

1. Okay, so there are so many parts to the Miller-Rabin Primality Test. It is going to be difficult for me to keep them all straight! It's interesting also that we have ways of saying that n is composite, but otherwise we can only say that n is probably prime for the most part.

2. This was a pretty interesting read. I like the idea of figuring out whether some large number may or may not be prime. Obviously, it doesn't give you whether something IS prime or not, or else systems like RSA would be woefully inadequate. However, it does make me think about whether it would be possible to one day do so.

Assignment 21

1. This is some pretty interesting stuff. I feel like I have a decent handle on the usage of the Legendre symbol, but I would like a bit more practice with the Jacobi symbol. It just appears to be a little more involved to me (in terms of the important properties and whatnot that it relies on).

2. This stuff just seems really cool to me. I've always like equations, and I look forward to working with this stuff. I'm a bit behind on what's been going on in class because I've missed the last two since I got sick, but I look forward to seeing how we are going to use this with the systems we use.

Assignment 20

1. This section makes sense more or less to me. I suppose if anything is difficult, it would be getting used to the compositional method that was used. Does it work no matter how you factor n? I guess, in the end, I'm just not used to looking for it too. So, if I can get used to it then I think I'll be okay.

2. I enjoyed this reading. It really seems to make sense after we have already learned about exponential powers being used modularly. It also makes sense as we are using it in terms of n=pq, and how we can either find the square root or the factorization for n. Overall, it just makes sense to learn about this here as I feel it ties in perfectly with everything we've been discussing right now.

Assignment 18

1. Now, this is was an interesting read. I remember doing thing with repeating fractions when I took History of Mathematics. None of it really seems to be too difficult for me on it since I am somewhat familiar with it. It is also nice that they gave us a general form for a repeating fraction at the very end of the reading in order to improve the efficiency of it all. I know that we are supposed to have something here that was difficult for us to understand, but I felt that everything made sense while I was reading through it. I guess the only thing that kind of surprised me in the reading was how they came up with the theorem out of nowhere--did they originally decide on 1/2s^2 because it fit a pattern or what?

2. I wonder how we are going to use this in our different cryptic systems. It is interesting and all, but we have been working with really big numbers lately. In a similar way, I guess we could work with irrationals? In any case, I really don't see yet what the application to our systems will be, but I'm sure it will be interesting. Or are we possibly moving on to a new system already?

Assignment 17

1. Ok, so I think I understand the idea of it all, but it all just seems to be so gross! I definitely wouldn't want to do this by hand--are we going to use computers when we deal with the RSA? If we are, I'll probably need some pointers on how to write it into SAGE. I can pretty easily see how this relates to what we've recently covered, but the numbers are much larger now, so it seems unlikely to me that we'll solve these by hand. If we do plan to solve these by hand, I suppose I'd be able to understand it well enough to do it for smaller primes p and q.

2. How do people even come up with this? It really is pretty brilliant. I especially like how they explain what is happening in a non-mathematical way first as it helps me get a better context for conceptual understanding of the system's efforts. I guess I really wonder what we plan to do with it now though. Are we going to play with RSA on the computer to send or receive secure messages, encoding them and decoding them? Are we going to do it by hand with smaller primes? It's definitely important to know about since it has so many applications today, but what can I consciously do with it?

Assignment 16

1. I feel like I could still understand section 3.6 better. It seems as though Fermat's and Euler's finding will come to be very important in the near future, so I'd like to understand them through and through. I feel like I mostly get what they are working towards, but I just need that little bit extra that the lecture in class usually gives me before I feel like I really know what we are looking for.

2. Ok, so I found that it was really cool how we could use Fermat's Theorem to find that 2^53 is congruent to 8 (mod 11) so quickly. That's pretty classy that. It was nice to have the explanation of the three-pass protocol--are we going to do anything with that? That would be interesting to work with.

Assignment 15

1. I feel I'm going to struggle with getting the hang of finding a (mod m*n) and also with modular exponentiation. It'll just take a bit of practice, and I feel that going over an example in class will help me out immensely. The concept makes sense to me though.

2. I guess what this reading makes me think about is how we are going to be using values of mod that require an m*n conversion. I know we are eventually going to work up to primes of that magnitude, so I wonder how exactly we'll fit this into stuff like that. How are we going to use it in practice?

Assignment 14

1. I feel the most important topics we have discussed are the actual encryption systems themselves and how to decrypt them. In particular, the last couple of ones we've gone over seem to be even more important as they are much stronger overall.

2. On the exam, I expect to see question that test our basic understanding of the different systems and how to encrypt and decrypt using the various methods we've seen in class.

3. I feel like I can definitely use some extra time understanding the last two weeks worth of material a bit better. I guess I just feel like I'm not fully understanding everything I should be in order to be ready for the exam.

Assignment 13

1. I find that these systems are so much easier to understand when we go over them in class. I think having somebody who really knows how it works explaining it while simultaneously writing it down helps me so much. I guess that's been the difficulty for me on this reading--complete comprehension.

2. What I wonder about with this system--and others like it--is how practical is it to use overall. For example, what exactly is it used for, and why? How well does it work compared to the others we've seen so far. I can make the obvious deductions (I'm pretty sure it's better than substitution for example), but I wonder how this system holds up against other ones.

Assignment 12

1. I'd say I average a couple of hours each assignment up to now, although the last ones have been much longer. Lecture and reading do give me a good basic understanding, but sometimes I still feel a little lost on the last couple of assignments.

2. For me, working with the problems has been most beneficial for me so far. I really enjoy having the chance to decrypt things on my own and try to better understand how the different encryptions actually work.

3. If homework assignments continue at this level, I'm going to need to find more time to work on them. I feel like I'd like to say something else here too, but I'm not sure what it may be. I feel like class time is used pretty well after all. I guess if I get too stuck I'll try to make it to office hours more often, though it can be hard with my schedule at times. It has been really helpful when I can make it though, so thanks for that!

Assignment 11

1. Oh fields. We meet again abstract algebra! Taking it over the spring was hard enough, so I'm here thinking that a review of it in class would be most excellent to have. In the end, I feel that a review would really help my full understanding of what we are preparing to cover. I mean, I remember the basics of fields and whatnot, but (especially since I took it in the spring term and thus at an accelerated rate) a lot of it is eluding me as I read through it. So, I'll definitely be reviewing on my own, but any additional help would be more than welcome.

2. One thing that I find interesting through these readings were when we had equations suddenly pop out, seemingly out of nowhere. For example, when we multiply in GF(2^8), we are dealing with X to the exponents of 8, 4, 3, 1, and 0. Before that, though, we were representing all elements as polynomials of degree 7 or less. I don't quite get yet how this works out, even reading through it again and again. I wonder how they got this, and how it helps us out in the end.

Assignment 7

1. The most difficult parts of this reading was probably the section on block ciphers. I found it to be difficult mostly because of all the terms in there. This says to me that I still need to review all of the ciphers that we have covered before, and learn all of the ciphers that we have not yet covered. I feel that a good review of these ciphers, along with a better understanding of how the other ciphers work, will clear up and misunderstandings I have about this section.

2. I really enjoyed reading through the examples of cryptography being used, whether it was fictional (as in Sherlock Holmes' case) or historical (as with the WWII codes). It still amazes me how people could think of such things for codes. I feel like coming up with this on my own would be out of my league. That's probably why it's so much fun to read about it all.

Assignment 6

1. I feel that the most difficult thing for me from the reading was thinking in terms of matrices and vectors and not seeing them the way I want to think about them there in the text. Once I went off to the side in order to see it in the way I imagined it, it became easier for me to visualize what was being described. I would have like more visuals in the actual text, but it did describe it more than enough for me to figure it out in a way that made sense to me.

2. I found the methods for cracking this cipher to be very interesting. How does somebody come up with that? It makes sense to me to compare expected values against the frequency at which a given character generally appears, but I would not have thought to compare dot products in order to do this. It would be really nice to try using this method once or twice in class!

Assignment 5

1. I thought that the hardest part of this reading was the section on affine ciphers. I am at least somewhat familiar with the shift and substitution ciphers, and this type was just that one step further in terms of sophistication--to me at least. Luckily, we had just gone over modular division, so that part didn't really phase me at all. It just gets a lot more difficult when you suddenly have to keep track of more than one characteristic. Really, if it were up to me, I'd only try to crack codes if they were based on one characteristic unless the code was really important. If only those trying to read my codes thought that way too.

2. Reading through this really makes me think about my project, especially when it comes to determining what a letter may represent based on how often it show up. I hadn't really been thinking of this fully when we were brainstorming how to make a good code, and so ours may fall short of the mark we set for it if Alice were to discover the method by which the code is made. After all, even if each character is represented by three new characters, it can still be blocked off in a similar manner. So, overall, this was a good reading at this point in time in order to hit home how a code needs to avoid such an inspection through a stronger set-up.

Assignment 4

1. I didn't know a lot of the history behind what we went over in class. It was all very interesting, and I feel that she did a very good job at relating it to us, but I do always like to have a good understanding of what is being discussed. For me, the more I know about something, the better able I am to relate to it and make it meaningful. So, while it was very interesting, I kept finding myself wondering more about the stories and histories instead of always focusing on the coding methods.

2. The whole presentation was very interesting to me. I really enjoyed it, and I feel like it was definitely well prepared. I had never really even considered the usage of codes in the early church years. Also, if it was used so much then, I imagine it still is now? I laugh a bit too, because I see myself more-so as the first representative who thought that everyone would be a gentleman like he was: they wouldn't dare to read another's mail, because he would never do so himself! But this really is just naive. It's too bad it is that way and we need codes for anything other than just fun.

Assignment 3

1. I haven't really dealt with modular division much in the past. Instead, I would usually just think my way through the problem in order to find a suitable solution. Because of this, I was somewhat unfamiliar with the thinking process the reading took me through. I definitely prefer for things to feel right for me when I do them, so I had to read over this section a few times before I felt comfortable with it. Now that I've done so, I'm okay with it, but it was the most difficult part of the reading for me.

2. I find the prospect of working with powers in modular math to be pretty intriguing. Like division, it is something I haven't dealt with before. This reading didn't delve into it in depth just yet--which is why it was easier for me to deal with--but I wouldn't be surprised if it was more difficult in the end, especially since it's likely we'll be dealing with large numbers. However, I'm really interested to see how this applies to cryptography. While it's true that I don't really know anything about it just yet, I'm can't help but wonder what it'll be like when we get to that point.

Assignment 2

1.   The most difficult part of this reading was trying to make sense of the different methods and terms that were flung out in the reading that I have not understanding of, such as all of the different algorithms mentioned. This section will undoubtedly make a lot more sense to me once I do know those terms, and I'm sure we'll shortly find out more about most of them, but it was difficult for me now because I kept trying to figure out as much as I could about each of these terms.

2. I really liked how the reading went through the differences between symmetric and public keys. I feel that will be a very important distinction, especially because of their different uses and levels of practicality from case to case. I wonder though--how much data is required before a symmetric key is more efficient? From the next section, I really appreciated the distinction that was made between authentication and non-repudiation. In our discussion in class, they had still seemed pretty similar to me, so this additional reading on the matter was very helpful for me to clear up why we still distinguish between the two. Reading through 3.1, it is apparent that we'll need these basic methods of mathematics in order to really delve into cryptography, but I wonder exactly how we will need them in order to do so. The first chapter talked a lot about different algorithms that make use of factorization, so things like divisibility, primes, and the Euclidean algorithm will definitely be useful, but I wonder exactly how we will wind up using them. I guess, at this point, only time will tell.

Assignment 1

1.   What is your year in school and major?

            I am a senior studying Mathematics Education.

2.   Which post-calculus math courses have you taken?  (Use names or BYU course numbers.)

            I have taken Linear Algebra, Multivariable Calculus, Ordinary Differential Equations, Theory of Analysis, and Abstract Algebra.

3.   Why are you taking this class?  (Be specific.)

            Cryptography sounds like a it'll be a really fun subject to study and better understand. I look forward to learning the mathematics behind the algorithms we use to ensure data integrity.

4.   Do you have experience with Maple, Mathematica, SAGE, or another computer algebra system?  

            Sadly, no. 

5.   Programming experience?  How comfortable are you with using one of these programs to complete homework assignments?

            I have some limited experience with Visual Basic from years gone by. I don't know how well I might be able to complete assignments using this, but I do feel fairly competent as an amateur when it comes to the logic behind programming.

6.   Tell me about the math professor or teacher you have had who was the most and/or least effective.  

             One professor I quite enjoyed was Professor Williams from the Mathematics Education department. I felt he was able to teach the material very well and also was very interested in how well we learned it.

7.   What did s/he do that worked so well/poorly?

             I've had a lot of instructors that I really liked, and only a couple that could have helped me more. In each of these cases, though, I feel that the most prevalent reason was that the professor either did or did not show me that they cared about me learning the subject matter. If they would open themselves up to the class and made themselves approachable while requiring that the material was mastered, I did quite well. On the other hand, those who didn't really want to be there passed that sentiment onto me.

8.   Write something interesting or unique about yourself.

            I have a lot of interests. I really enjoy things like reading, cooking (especially the eating afterwards), and pretty much all sports. Hopefully this helps make me a more rounded person.

9.   If you are unable to come to my scheduled office hours, what times would work for you?

            I may not be able to make the given office hours. If this is the case, I could make it during the hour directly afterwards.

Just some stuff for fun to start...

This explains why Eve became the way she did. I guess here enhanced capabilities as a machine really helped her crack the code.