section 16.5

Difficult: I'm a little confused about the El Gamal with elliptic curves.

Interesting: It's interesting how fairly simple it is to convert these systems into elliptic curves

section 16.4

Difficult: I'm still confused about what omega is in GF(4).

Interesting: The fact that finding -P from P in mod 2 is different than other cases is interesting to know.

section 16.3

Difficult: I don't think I really understood how to find the factors using elliptic curves very well.

Interesting: I thought the idea of smooth and b-smooth numbers was interesting since I've never really learned about them before by name. I didn't know there was a name for those numbers! I think it's pretty awesome that someone cared about them enough to name them, though I do wonder why they chose those names.

section 16.2

Difficult: Discrete logarithms sound like they would be difficult to do with elliptic curves.

Interesting: It's interesting that there's a way to use elliptic curves to encode/decode messages. In all honesty, before that last bit of the section, I was like, what the heck does this have to do with cryptography. And now I am beginning to see the possibilities. Pretty cool how people can come up with these ideas for cryptosystems.

section 16.1

Difficult: I'm not sure I understood the adding points part very well.

Interesting: the whole concept was interesting since I've never learned about it before that I can remember, but I don't really get what this has to do with Cryptography.

sections 18.1 and 18.2

Difficult: I didn't understand the Hamming distance very well.

Interesting: It's interesting that they've come up with a way to correct errors in code words, etc.

section 2.12

Difficult: I didn't understand the process very well for attacking it

Interesting: It's funny that this is what our reading is for tomorrow since I was actually just talking to someone about Enigma earlier today when I told them I was in a Cryptography class. I think it's interesting (though I guess unsurprising) that the British sold captured machines to former colonies and kept the fact that they had broken it a secret for so long.

Shor's algorithm and section 19.3

Difficult: Shor's Algorithm and the quantum Fourier transformation was kind of confusing

Interesting: The whole idea of a quantum computer was interesting. Especially since the authors said that they aren't yet a reality and yet the ideas are out there on what the computer would be able to do even though it isn't a reality. Also, the internet reading will really interesting to read. Definitely sounded like something you would have written just because of the guy's personality...

sections 19.1 and 19.2

Difficult: It doesn't seem that difficult as far as the key distribution part goes, but I'm still confused about quantum mechanics itself

Interesting: It's really weird/interesting to find out that quantum mechanics has applications in cryptography. The filter/overview of quantum mechanics discussion was interesting

sections 14.1 and 14.2

Difficult: I don't think I understood very well what the heck was going on with the tunnel analogy. how can Peggy always come up the correct side of the tunnel if she doesn't know which way the door opens? I'm confused.

Interesting: the story at the beginning was interesting. It's weird that someone could manage to put a fake atm anywhere period without security/mall officials or whoever asking questions to validate that the machine was legitimate.

Midterm 2 Review

• Which topics and ideas do you think are the most important out of those we have studied?
• What kinds of questions do you expect to see on the exam?
• What do you need to work on understanding better before the exam?
Thinking about the answers to these questions can help guide your study.

I don't know how to solve any of these problems by hand very well at all!! Please go over that in class!

sections 12.1 and 12.2

Difficult: I'm not sure I understand the Shamir threshold scheme very well, but I'm sure with pracetice, I'll be able to understand it better

Interesting: I like the whole you need two people to reconstruct idea. It's a lot like how you need multiple keys to set off those crazy bombs the military uses, etc. I like that it makes it harder for people to find since the pieces are divied up.

sections 9.1-9.4

Difficult: Are we going to have to break any of these? Because I'm not sure I'd be able to until possible after reading later sections that probably go into methods to break them. Though I'm sure they are similar to how we break RSA and ElGamal. I also thought the ElGamal Signature Scheme was kind of complicated, but still kind of like the normal ElGamal encryption method, so maybe it's not as bad as it looks? Could you do examples of these methods in class?

Interesting: I thought it was interesting how the different encryption methods could be modified to create personalized signatures and also that this idea hasn't (explicitly) come up before these sections really. I also thought the difference between the signature with appendix and message recover scheme was interesting.

sections 8.4-8.5 and 8.7

Difficult: I'm kind of confused about how to use the birthday idea to attack different ciphers

Interesting: I found it interesting that they spent so much time talking about the probability of repeating birthdays when almost everyone already knows about the birthday paradox.

sections 8.1-8.2

Difficult: I thought the second hash method would be difficult to do--for long messages, especially.

Interesting: I thought the hash function in 8.2 was interesting because it was more effective than the first one described, I wonder what other hash methods there are that would be more effective.

sections 7.3-7.5

Difficult: I think the Diffie Hellman Key Exchange would to interesting and difficult to encrypt/decrypt by hand for sure, but I also think the Computational Diffie-Hellman Problem would be difficult to compute period. Could you show an example in class?

Interesting: The bit commitment section was really interesting. I had never really thought of it before, but it makes since in some situations that someone would make a bet/prediction or something like that and would not want the other person to know what they sent until a certain day/time. It's kind of like sending a gift before someone's birthday and not wanting them to open it until their birthday, even though it was sent early. I think it's a really cool way to accomplish this, because before, you just had an honor system so to speak, but this way, you can't change the bet/whatever, and Bob can't open it until you give him the 'key' so to speak.

section 7.2

Difficult: I still don't think I understand their notation very well, the whole x = L sub alpha (beta) thing is hard for me to keep track of which part means what,but I'm sure time/hw problems will help with that so I can keep them all straight. Are we going to be expected to factor all these bigger numbers and mod them in our heads/using only a basic calculator on the next exam? Because I'm pretty sure that would extend the amount I need to set aside for the test substantially.

Interesting: The Baby Step, Giant Step Attack was pretty interesting. I also thought the fact that they named the attack that was pretty funny. But it is cool that/interesting to think about since the two lists are interconnected and there's a match, but since it only works well for primes up to 10^20, it makes me wonder what a better method would be.

Sections 6.4.1 & 6.4.2

Difficult: I didn't understand the part about linear dependencies very well in 6.4.1. I'm also not sure I understood the matrix connection very clearly.

Interesting: it is really interesting how far we've come with factoring algorithms in the last 30 or so years and how big of an impact computers have had on our mathematical and other advances. Who knew such a relatively small thing would have such far reaching impacts?! It's pretty cool, not going to lie.

PS, as you can see below, I accidentally posted the entry on my personal blog instead of my Cryptography blog--sorry!!

sections 6.5-6.7 and section 7.1

Difficult: The discrete logarithm problem looked difficult, I didn't understand very well why n had to be the smallest positive integer such that alpha^n is congruent to 1 mod p for us to get that x=(L_alpha)(beta) though. Could you explain why in class?

Interesting: It's crazy how many people and computers it took to find the congruence relations of the desired type for this problem! and then how relatively they were able to reduce the matrix and determine the dependencies. I also thought the message was pretty interesting... mostly weird, but I guess it makes sense since they didn't want anyone to just arbitrarily guess what it was and get the moneys without doing the work.

6.4 up to just before section 6.4.1

Interesting: I think it's crazy how many awesome ideas Fermat came up with. Seriously, were all of these ideas actually his or did he just steal it from people who didn't care to publish or is some of his work actually someone else's that historians decided to attach his name to? But I guess when you think about Euler, Fermat's work pales in comparison as far as breadth goes. Still, Fermat had some pretty crazy ideas and theorems. I do like that there are other methods of factoring large numbers other than checking every prime up to and including the square root of the number, because for large numbers, it's impractical to do this.

Difficult: I'm not sure that I understood the p-1 factoring algorithm very well or at least I don't know how we would know if we picked a "good enough" a and bound B for the algorithm. Also, how would we know when to stop computing?

Section 6.3

Difficult: I feel like the Solovay-Strassen Primality Test could be very difficult, because translating fractions into modular has always been time consuming and confusing for me and I have a tendency of accidentally messing up somewhere along the way. So maybe if we could go over some examples and really solidify how to translate fractions into mod, that would really help me.

Interesting: Whenever I've learned/talked about how to check if something is prime or not, I've always been pointed to using the square root, but like the readings said, this isn't practical for very large numbers. I've never really had any experience with other methods of determining whether or not something was prime, so I really liked that they presented many different kinds of thinking about and finding whether or not a number is prime. I especially thought the Miller-Rabin Primality Test was really interesting and it kind of made me wonder why I've never really worked like anything like this in previous math classes and why everyone always talks about using the square root and then using all those primes when it's not exactly practical for very large numbers.

Section 3.10

Interesting: I have to admit that the proposition at the beginning was really interesting to me because it seems on the surface to be unrelated principles, and yet it's provable, and the proof isn't super complicated either. I don't know why, but it just amazes me sometimes how people can connect so many seemingly unrelated ideas to create theorems, etc. that really help advance our understanding of mathematics. I also thought the properties of the Jacobi symbol were interesting in the way that the authors applied them. I don't think I knew that Jacobi's ideas could really relate to reducing fractions, and yet it does! It also doesn't help that I don't remember how I really used the symbol before too though--maybe I knew and just forgot.

Difficult: I think it will definitely take practice to get used to using Legendre's and Jacobi's symbols in this way since I don't remember using it this way before (in fact, I really don't remember what I used it for before at all). So it will definitely take time for that to process.

Section 3.12

difficult: I've already done work with continued fractions before, so it's not super hard for me. But I could see how some people could have a hard time with it since it's a weird concept and not something that we're really work with that often if at all.

interesting: I liked how they talked about the approximation for pi and how you can approximate for different fractions too and that they actually went through and showed you the process in numbers instead of just symbols.

Section 6.1

Difficult: I don't think I really understand why this algorithm works. Could you explain the why more in class? I can see that it does work, but why?

Interesting: I thought it was interesting that they were able to come out with a way to send a key in a secure method. I also thought it was interesting that it wasn't until 1997 that it finally came out that someone else has come up with a method before 1977. But seriously, how do these people come up with these things!?!? I also thought PGP was a funny acronym!

10/4/13

Which topics and ideas do you think are the most important out of those we have studied?

how to "crack" different ciphertexts and various encryption and decryption methods... so pretty much everything.

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

definitions, probably algorithm type stuff--especially with rings, since we spent two days on that. things like that--hopefully things that don't take a really long time though, but since it's an exam, the really long problems probably won't be exam though.

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

keeping the different encryption/decryption methods straight--it's really hard for me to remember them all!

Sections 5.1-5.4

Interesting: I felt like this encryption method itself was really interesting--how do people come up with so many cool and crazy methods of encryption/decryption?!? But I though it was really cool how it covered a broad range of mathematics--binary, matrices, GF(2^8), modular arithmetic, etc.

Difficult: I feel like reading these processes is not very easy to understand, because it's not the same as applying and it's hard for me to visualize what I would need to do when pretty much everything they talk about is very abstract. But the homework and lecture will definitely help me better make the connection and understand how to tackle this kind of encryption/decryption.

Q&A entry

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?

it depends--usually about 2 hours, but sometimes more and sometimes/rarely much less than that

• What has contributed most to your learning in this class thus far?

working on homework with a group instead of by myself and being able to communicate my ideas verbally and if I get the question first, explaining what I did to others

• What do you think would help you learn more effectively or make the class better for you?

I'm not sure. I like the class. I'm just forgetful about the blog entries. But that's all me, not you. I think you're doing a good job.

Section 3.11 to 3.11.2

Difficult: I thought 3.11.1 could be difficult, mostly because I don't remember it very well from Abstract Algebra, but I feel like their explanation is very good, so that should help me as I practice this procedure in homework problems and during lecture.

Interesting: I thought it was interesting that only Linear Algebra is required to take this class and yet there is a discussion of fields, which I feel like is in the realm of Math 371. Most of the students in the class have probably already taken Abstract Algebra, but I wonder how confusing this section is for those who do not really have a very clear and concise definition of what a field is. However, I feel like the section was a good overview of ideas addressed in Math 371, so maybe it wouldn't be as big of a problem as I think it could be.

4.1, 4.2, and 4.4

Interesting: I think it's interesting that someone named their algorithm Lucifer... I liked that they gave a simplified DES encryption method before they went into a deeper discussion in a later section.

Difficult: I think the DES encryption method is very complex and that all those steps would make it easy to mess something up or forget about something.

Sections 2.9-2.11

Difficult: cracking the linear feedback shift register sequence method looks very time consuming to decrypt for long keys.

Interesting: I thought the one time pad method was interesting, because it seems to be the first that they author's have noted to be truly unbreakable for a ciphertext only attack (which I'm guessing means you need some complex computer program to crack it).

Section 3.8 and Sections 2.5-2.8

Difficult: I thought the inverted matrices using modular arithmetic was difficult because I've never used matrices in the context of modular arithmetic before and modular arithmetic is already time-consuming and sometimes confusing to me still anyway.

Interesting: I loved the Sherlock Holmes story and that they included it in the textbook!! It does make me wonder if Elsie knew how to decrypt it, but had just forgotten or was embarrassed about it. I also thought it was interesting that the 2.6 ciphers used matrices and that they were used during WWI

Section 2.3

Difficult: the part about the dot product and vectors was difficult--mostly because I haven't done anything with vectors or dot products in at least a year. And finding the key using dot products would be difficult too.

Interesting: I really like how easy this textbook is to understand and follow compared to other math textbooks that I've used. Anyway, I thought the vigenere cipher was really interesting and that it's a good way to "shake up" so to speak the substitution method to make it seemingly more random. I also like how you can use a word or a set of numbers as your key interchangeably.

Sections 2.1-2.2 and 2.4

Difficult: I think I'd have a lot of problems with the affine ciphers. I feel like I don't understand how they work very well and decrypting them would be hard for me.

Reflection: I thought it was cool that they used "we hold these truths to be self evident..." as the cipher text they decrypted in 2.4. I also liked that they discussed how to crack the different methods they presented.

Guest Lecture

Difficult: I didn't find any of the presentation difficult, though I do think that the deseret alphabet would be a difficult one for me to transition to and I also think that decoding the secret word different alphabet for each letter one would be hard for me to crack.

Reflection: it was cool to learn how cryptography has been used in the church. It would make sense that it appeared, but I especially liked how she talked about that one guy who did tons of different ciphers and sending letter methods. I also like that she talked about how difficult (and probably annoying) it was for some of the receivers to decrypt the messages.

Sections 3.2 and 3.3

Difficulty:
I found the extended Euclidean algorithm to be the most difficult. Again, when I learned the Euclidean algorithm from Dr. Cannon, he taught us a method to do both simultaneously (we had a 4 column chart with x, y, q, and r as the column headings and then kept going until we got the correct answer). I think I should be able to figure it out soon, but it will take a lot of effort to remember well enough for time-efficient computations.

Reflective:
When I saw about the multiplication, etc. tables for (mod n), they reminded me a lot of the tables we made in Abstract Algebra. I do have to say that I like the proofs in this book way better than the "proofs" in my Spring Term Survey of Geometry textbook (those proofs were all over the place and usually incomplete/not very rigorous... unsurprisingly, that was the only semester that that textbook will be used).

Sections 1.1-1.2 and 3.1

I didn't notice that this reading was due the same day as the get-to-know you questionnaire!! Sorry it's late!! I also didn't realize/forgot soon after lecture that the blog posts are due the midnight BEFORE lecture and thought they were due by 5 pm day of lecture... but at least I realized my mistake early! I will definitely do them on time in the future!

Difficult:
The Euclidean algorithm! (Still!) Dr. Cannon did the Euclidean algorithm different than any of the other professors or even any of the textbooks that I have come in contact with, but I think I either lost, misplaced, or threw away my notes on it, because I really can't remember how we did it, though I remember the general ideas. The "normal" way is super difficult for me to use, because that's not how I learned it. I seriously have this problem at the beginning of every school year.

Reflective:
I thought it was really interesting how they said in the first chapter that "modern cryptography is a field that draws heavily upon mathematics,computer sciences, and cleverness" and that they made cleverness its own "field" so to speak. I feel like helping students develop cleverness in a constructive way in the classroom and help them recognize that being clever is a skill--even if they aren't the best at mathematics or computer sciences. And if we can find a way to bring opportunities into the classroom for students to extend their 'cleverness,' this could really help students better understand mathematics because if we can make that connection between the mathematics and cleverness, then they will see that they can do mathematics and be more confident in their mathematical ideas.

About Me

What is your year in school and major?
Senior, Mathematics Education

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

Calculus II

Elementary Linear Algebra

Multivariable Calculus

Differential Equations

Math 290- Fundamentals of Mathematics

Abstract Algebra

Theory of Analysis I

Survey of Geometry

History of Mathematics (not that the class could really be counted as a math class per se,but we did do some math problems)

Physics 121- Principles of Physics I (from back when I was a Math major)

(aka, this is my last math class for the math education major)

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

Because when I found out about this class when I was in Math 290, I decided that (if I had stayed a Math major) Cryptography would be something that I would be interested in pursuing as a career. I've always loved breaking the simple texts, etc. that they give you in grade school and Math 290 and I think learning to create well protected information and learn how to decrypt ciphered information will also be beneficial for future use if for some odd reason, I needed to send a secure(ish) message or needed to decrypt a random message (like if my kids think they are being so cool and smart by using a simple alphabet replacement technique to cipher letters, etc.). But really, I just think Cryptography is a super cool field... and this class is most likely not going to be as difficult for me as Number or Graph Theory (whichever is offered in the Fall semester) would have been for me, since I'm not that interested in those subjects, etc. And in this class I get to learn how to use a math solving software, which I haven't had the opportunity to do yet in my other classes.

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?

Wolframalpa is about all the experience I've got. No. I'll get comfortable, but I have no real experience, so we'll see. I'm excited to get to know how to use this software, though.

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?

My best math professor/class that I've had so far is Brother Hendrickson's Task Design and Assess Understanding (MthEd 277), which you could say isn't very math-y, but we actually worked on a lot of mathematics in that class. He had us work in groups and would walk around the room and sit down with each group and ask them what they were doing and why they were doing that, etc. and if they were going in the wrong direction, he would guide the group to a more effective on task way of solving the problem. He also had very out-of-the-box methods of teaching. For Trig functions, he had us figure out the height of a Ferris Wheel using what we know about right triangle trig (in particular, the 30-60-90 and 45-45-90 triangles), etc. He had a set of tasks that built on each other where we essentially were playing around for a while until we saw some pattern or something and went for it and, in the process, unknowingly learned some mathematical principle. It was pretty awesome.

My least effective math teacher taught the... I don't know, Super Honors? section of 7th grade math (we were using the 8th grade math textbook, whereas the other "lower" honors and normal math classes were using the 7th grade textbook). My understanding is that she was only really qualified to teach elementary level math, but since mid-middle school isn't that far from elementary school math, they hired her (but maybe that was just a rumor). It got to the point to where she would have us split into groups and teach the class, and she really didn't have good classroom management skills. For example, one day a fellow classmate poured GermX in her coffee... she only found out about it because someone eventually told her,though she did notice that there was something floating in it, she just didn't know who did it. Moral of the story is, no one really liked her as a teacher, and what I learned in that class, I learned from the textbook and from doing homework... during class.

Write something interesting or unique about yourself.

I have Celiac Disease (yay! ...obviously that was drenched in sarcasm). I have an 11 month old daughter, and baby number two is on the way (due April 4th-ish). I'm from Columbia, Tennessee (an hour south of Nashville).

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

Those times should work, but I'll have to bring Elsie with me.