16.3, due December 3

1. I thought I understood elliptic curves until I read this section. I understood the first few paragraphs and that's about it. In the first example, why are we trying to compute 10!P? It's probably a bad sign that I don't even see how this relates to the goal of factoring n. Also, I don't quite understand the significance of smoothness in this factorization method. Singular curves were also confusing.

2. I like the concept of utilizing differences in p and q in order to factor n=pq. Assuming I follow what it is saying, the very last statement in this section was interesting. Does it mean that the p-1 method and trial division are a part of the algorithm? Or that performing the algorithm is equivalent to doing both methods?

16.3, due December 1

1. This section was surprisingly understandable. One thing I'm a little unclear about is the subsection on discrete logarithms on elliptic curves. How exactly is this the same as the discrete logarithm problem? Also, are we going to learn how this problem can be solved using Pohlig-Hellman or Baby Step Giant Step attacks?

2. It was cool to begin to see how this seemingly obscure and unrelated concept may be used to factor large numbers. I still don't know exactly why elliptic curves mod a composite number could help in factoring the large number, but hopefully we will find that out soon. I also liked the method for manipulating a message so that it lies along an elliptic curve. It's nice.

16.1, due on November 29

1. I don't understand the example given on page 350. I understand the general idea, but I don't understand how knowing two points on the curve can give you a third. Also, we get the intersection point Q. Why not make that our third point instead of it's reflection across the x axis, P_3?

2. I am intrigued by this. I really can't see, just from reading this section, how elliptic curves will be useful in cryptography. Hopefully I'm not supposed to be able know this at this point. The Addition Law given on page 352 makes things nice. Even though I don't understand why it's true yet, I like it.

2.12, due on November 23

1. Near the end of the section, when they start talking about permutations and how to find the three letter daily key, I start to get confused. We just started talking about cycles in abstract algebra, and I'm still not quite comfortable with them. Once we know AD, BE and CF, how do we find the key? Also, how does this key necessarily add to the security? It seems to me like it detracts from it. Can't the cryptanalysts simply start frequency analysis on the 7th character?

2. It's really cool to be learning about a system that played such a big role in World War II. I also liked that it was cracked. Go England. This is the first cryptosystem we've talked about which conceptually actually requires a machine, and the fact that it requires a machine makes me wonder if, with such strides in technology the world is making, there are newer variations of enigma which are harder to break.

19.3, due on November 22

1. There are some days when I understand why the general public hates math. Today is one of those days. I did not understand a word of the reading in the book. The explanation given online mostly made since, but as soon as we started talking about Fourier transformations I was lost. Maybe it's the fact that I'm computer retarded and don't even understand how classical computing works. Let alone quantum computing. Sure hope this is one heck of a lecture or I'll be a lost cause on the homework.

2. That clock analogy was really interesting. What a clever way to describe the concept behind such a monstrously ugly mathematical result. However, I'm afraid it didn't really work on me. I still don't know what the heck is going on. I suppose it is cool, however, that there in theory a quantum computer can do things so quickly.

19.1-19.2, due on November 19

1. Maybe I kind of just freaked out as soon as I read the word quantum. One thing I don't understand is how we know that photons will pass through certain polarized films around 1/2 of the time. I know quantum mechanics is heavily based on probability, but I didn't understand the book's explanation of this. I also don't really get the vectors involved.

2. Wow, I never would have related quantum theory to cryptography. It's amazing that the relationship exists, but I don't understand why we would need to use the quantum key distribution over something like the Diffie-Helman key exchange. Doesn't the fact that eavesdropping causes communication errors make this method not incredibly valuable?

14.1-14.2, due on November 17

1. As soon as we started applying hash functions to this seemingly simple and understandable concept, things went downhill. I didn't understand the second half of page 320 on. It seems like this confusing part is somewhat of an extension of the previous concepts of section 14.2. Also, I'm getting all of the sequences of numbers confused. The s's, v's, and j's all get jumbled together.

2. The broad concepts discussed are really interesting. I like the fact that something can be verified with a very high probability without any useful information being released to Victor, and therefore, by Eve. I'm curious to know if the Feige-Fiat-Shamir Identification Scheme is really what is used in ATM and other machines which must verify the identity of a person.

12.1-12.2, due on November 15

1. I had a hard time following the Vandermonde matrix, and struggled to understand how a subset of the people can find the secret. Also, is there some kind of theorem which says the minimum number of people who need to get together in order to find the secret for each of these methods? If the matrix is not invertible, it won't work right?

2. Although I did understand the theory behind these concepts, I think it's really interesting that they exist. It seems counterintuitive that any subset of t people can find the message. Are there instances where threshold schemes are used other than military and business? I know these two areas practically run the world, but it would be cool to hear about more situations where this is used.

Exam 2, due on November 12

1. I feel like the most important topics we studied are the RSA and ElGamal crypto systems. Both of these fall under the concept of public key systems. If you don't understand the public key concept, there is no way you can understand RSA or ElGamal. Also, the math behind these systems is important: factoring large numbers, discrete logarithms, etc.

2. Since we won't have maple on the test, I am expecting conceptual questions. For example, if a system is set up in such and such way, how can eve break in and find the message. Or, why is it infeasable for eve to be able to find this parameter? I also expect some number theory problems which don't require maple. Like solving a discrete log problem by hand.

3. I need to understand hash functions better. I guess I still just don't understand the point. Signatures will also be good for me to study a lot. Most of all, however, I need to work on my understanding of the theory behind the algorithms and such. I know that they work, but a lot of times I don't know why they work.

4. I would be interested in studying a bit of coding theory, but I'm not sure that fits in the curriculum. Chapter 13, "games" looks like fun. But maybe the title of the chapter is deceiving, and the material not as fun as it sounds.

8.3, 9.5, due on November 10

1. It would be helpful in class for you to go over each step of the SHA-1 Algorithm. I suppose I understand the main idea of the hash function, but how would one be designed? Why is each step in the SHA-1 Algorithm, for example, necessary? My other question is when will the digital signature algorithm be used?

2. Hash functions amaze me. Not that they are particularly intriguing in and of themselves, but HOW on Earth do people sit down and come up with them? Looking over the SHA-1 Algorithm, all I see is a bunch of seemingly random operations which take an input and produce a smaller output. It's really cool to me that there is an method to this madness.

9.1-9.4, due on November 8

1. I was a little bit confused about the ElGamal signature scheme. Also, I'm really wondering how to escape "blind signatures." They seem necessary, and I'm interested in knowing how to safeguard against the dangers involved in them. One other thing I didn't understand was on page 246 when it explains why s/k is the signed message. After the therefore, it says k^(ed)m^(d)/k=m^d (mod n). I don't see how this is justified.

2. I quite enjoyed reading about birthday attacks on signatures. What a clever way to use probability to get Alice to sign a fraudulent document. I was even more excited about Alice's decision to add a comma. Poor Fred thought he had her fooled, but the joke's on him. He should have paid more attention in English class. RSA signatures were also interesting.

8.4-8.5, 8.7, due on November 5

1. What didn't I understand? The question is what did I understand. I guess I'm still just unsure about hash functions. Also, why would we use the Birthday Attack if baby step giant step is "superior," as the book says? When it's talking about multicollisions, how do we know that we can find the two blocks m_0 and m'_0 in 2^(n/2) steps?

2. I think the birthday paradox is fascinating. It really doesn't make sense intuitively, but I liked the way they explained it as not looking for a match with "your" birthday necessarily, but a match between any two students. I kind of understood the birthday attack on discrete logarithms, and I'm excited to learn more about that in class.

7.3-7.5, due on Novemeber 1

1. The subsection 7.5.1 talking about the security of ElGamal ciphertexts was hard to follow. I understand what each of the propositions is saying, but I don't quite understand either of the proofs. Also, in section 7.3, I had a hard time connecting the football analogy to the math behind it.

2. I liked this section because I understood it; this is something that is becoming more and more rare. It was interesting to learn a method by which Alice and Bob can both know a key without being in the same room. Also, the ElGamal system was interesting. I was wondering if it is computationally more difficult to solve discrete logarithms or to factor large primes.

7.2, due on October 29

1. I understand the concept of discrete logs, but I had a hard time following any of the methods to compute them. Under the "computing discrete logs mod 4" section, I don't understand the conclusion which says "if we believe that finding discrete logs for p congruent to 3 (mod 4) is hard, then so is computing such discrete logs (mod 4). Nor did I understand the lemma presented in this section. I was able to follow the Pohlig-Hellman Algorithm for about the first half. When it starts saying x=x_1+x_2q+..., I don't really get it.

2. I liked the Index Calculus. I basically understood this method. My question is, what makes one of these methods more useful than the others? Also, when will a public key where the trapdoor is finding discrete logs be more useful than RSA?

6.5-6.7, 7.1, due on October 27

1. Treaty verification was a little bit hard to understand. The sentence which says
"...but then x would probably not be a meaningful message, so A would realize that something had been changed" was confusing to me. I guess I don't understand how this is a reverse RSA like the text says. Also, section 6.7 used the word "non-repudiation" a couple of times. What does this mean?

2. I hadn't really thought about the possibility of more public key systems. Reading about this makes me curious as to what other kinds of one-way functions exist and are used in cryptography. It was also cool that the RSA challenge was completed in such a relatively short time. This reminds me of when I was trying to figure out number 10 of last week's homework.

6.4.1, due on October 25

1. This may be ridiculous, but on page 184 at the bottom where it lists the equivalent squares mod 3837523, I don't really understand why the numbers on either side of the equality aren't the same. Because aren't the primes just the factorization of the number on the left? I guess the matrix and linear dependencies mod 2 just threw me off and I'm not sure why we go about it that way.\

2. For one thing, I thought table 6.1 was pretty cool. It's amazing how far we've come in 50 years. In 50 more years will we have a fast method for factoring any large number without any special conditions? It is also cool that there was a jump in factoring capabilities because of the increased usefulness of factoring in cryptography.

6.4, due on October 22

1. Just under the statement of the p-1 factoring algorithm, there is a sentence which says "By Fermat's theorem, ..., so p will occur in the greatest common divisor of b-1 and n. What does it mean for a number to occur in the greatest common divisor of two numbers? This doesn't make sense to me. Also, in class, maybe go over the explanation of this factoring algorithm a little slow. There are a lot of weird steps that I have a hard time following.

2. I liked the Fermat factorization method, perhaps because I actually understand it. Too bad it can't really be applied these days. It's cool that there are other factoring methods being developed. It makes me wonder if anybody will ever find a quick way to factor any given large integer.

6.3, due on October 20

1. I didn't understand the concept of pseudoprime or strong pseudoprime. It would be nice if you could explain that in class for a minute. Also, I didn't quite grasp the part where it talks about why the Miller-Rabin primality test works.

2. Even though I didn't understand why it worked, I think the Miller-Rabin primality test is really cool. I also think it's interesting that we can say that a number is "probably prime." I didn't catch in the reading how "probable" it actually is.

3.10, due on October 18

1. This was quite the difficult section for me to wrap my head around. I guess I need to just warm up to the notations and not think of (a/b) as a normal rational number. Also, unless I'm completely confused, it looks like there may be a typo on page 91. At the bottom where it says (a/n)=(-1)^((n-1)/2), I think instead of -1 it should be "a". But I'm not sure.

2. I suppose it was pretty cool to be able to see if numbers are squares of other numbers mod another number using this method. I'm still not quite sure how it will apply to RSA, but I guess I will find out soon enough!

3.9, due on October 15

1. I don't understand why they picked 3 and 4 in the proposition. Is that just one pair of numbers that will work? Also, it took me a minute to understand how they got the solutions + or - 15 and + or - 29, but I got it.

2. This was exciting because I can see how it might apply to RSA. Also that very last statement in italics, saying how finding the square roots is computationally equivalent to factoring, was nice.

6.2, due on October 10

1. Just about every attack confused me. I don't know what it is with me and attacks, but I never understand them. Maybe it's because I'm such a peacemaker;). Anyways, the attack I understood the most was the timing attack. For the low exponent attack, I could barely follow the proof of the theorem presented. I was able to understand the example, but as far as the theory behind the attack goes I was lost.

2. It's interesting that the timing attack was discovered by an undergraduate. It really makes me wonder what other attacks are out there. Something else that I thought was cool is that these attacks can be prevented a long as the algorithm is implemented with them in mind.

3.12, due on October 11

1. I had a hard time understanding the very last part where it describes the "faster method." Other than that, the section was pretty straight forward.

2. I have never thought this concept before. It's really cool. It's cool that if the number you are approximating is irrational, then you will end up with an infinite series with an upper bound as the number you are approximating. If the number you are approximating is rational, eventually you will get an exact representation of it using this algorithm.

6.1, due on October 8

1. I am still a little bit confused about Euler's theorem. The idea of RSA makes sense to me, but some of the math is hard for me to follow. Maybe once I am forced to use Euler's theorem on the homework I'll finally start to warm up to it.

2. It's pretty fascinating that Alice can send a message to Bob and not even know the key! It's intuitively unsound, but the process makes perfect sense. It was also interesting to read about the application with banks.

3.6-3.7, due on October 6

1. These were some difficult concepts to grasp. First of all, in section 3.5, the book used a giant pi to notate something. I don't know what that something is so it was hard to follow any results or proofs of results that used it. I don't think I'd have any problem using the theorems shown, but I don't feel completely confident in my understanding of the proofs of them.

2. I was just thinking to myself the other day, "self, I wonder if there is a way to count the number of integers which are relatively prime to a given number." Research in number theory would be fun and pretty interesting because it would be exciting to come up with questions like the one I was thinking about and try to work with them.

3.4-3.5, due on October 4

1. The Chinese Remainder Theorem is a bit difficult for me to put to use. I understand the concept, but I know if I were asked to actually use it, it would take me a minute to do it on my own.

2. I really liked the example where it found the solution to x^2=1 mod 35 using the chinese remainder theorem. The theorem provides a really nice way to solve problems like these which I would otherwise have no idea how to solve. The method the book showed for finding exponents mod n was also pretty interesting.

Exam 1, due on October 1

• Which topics and ideas do you think are the most important out of those we have studied?
• All of chapter 3
• DES
• AES
• What kinds of questions do you expect to see on the exam
• Decipher using any of the methods learned
• Use of the mathematical tools learned 
• Questions about attacking systems 

• What do you need to work on understanding better before the exam?
• Methods of Attack
• Modes of Operation
• DES

5.1-5.4, due on September 29

1. It was a little bit hard to remember which step was which as I was reading along. I'd have to keep looking back a page or two to remind myself what BS, SR, MC and ARK meant. Also, in section 5.4, it's hard for me to see why the design used escapes the attacks mentioned.

2. It is interesting how fast diffusion occurs in Rijndael; that in two rounds, all 128 output bits depend on all 128 input bits. Another thing I like about this system is the simplicity of the s boxes. These s boxes seem more practical than those in DES.

Questions, due September 27

1. I have probably spent an average of 3 hours on each assignment. I feel that they were very fair and kind of fun. The lectures and reading did prepare me for them.

2. The assignments and lecture have contributed most to my learning. The book is okay in general, but sometimes I feel completely lost while reading it. When I go to class the next day I understand most of the material without much of a problem.

3. I feel like I need to spend more time and concentration on the readings. If I really plugged through it and tried harder to understand what is going on, I think I'd learn better.

3.11, due on September 24

1. I don't really understand the concept of generating polynomials. There wasn't really a definition in the text unless I missed it. Also, I had a hard time making the connection between these polynomial fields with LFSR sequences. I understand when it talks about GF(2^m) as a vector space over Z_2, but as soon as it introduces the matrix M_x I get lost.

2. My favorite part of this class so far is the math. Like I've said in the past, I'm a math major and I don't know any computer science, so it's refreshing to read a chapter that's written in my language. It is interesting how much F[x] has in common with the integers. I have never thought about the polynomials in that way.

4.5-4.8, due on September 22

1. I thought this entire reading assignment was difficult. I don't know a think about computer science and I feel like this text makes the assumption that the reader is familiar with cs. Also, I wasn't able to follow the way most concepts were explained. Hopefully things will make more sense in class.

2. I suppose it is interesting that methods were developed which can attack DES so quickly. I haven't the faintest idea what those methods are or how they work, but it seems incredible that such an intricate and complex system.

4.1-4.2, 4.4, due on September 20

1. The last part of section 4.4 was hard for me to follow. It may be because I haven't completed a course in Abstract Algebra and am therefore unfamiliar with group theory. Besides that, the trickiest part of this reading assignment was keeping track of all of the steps involved in DES. There are several parameters to keep track of within each step of the algorithm.

2. I loved reading about DES! I imagine that the guys who came up with it had a great time inventing it. It is so complex and intricate that I can't imagine how it can ever be broken. But I guess computers these days are a bit smarter than me. My favorite part was that if I went slowly enough, I was able to follow every step of the algorithm. I feel confident that if given enough time (days), I could successfully encrypt a short message using DES.

2.9-2.11, due on September 15

1. I really struggled with this reading assignment. I don't know a thing about computer science or programming so a lot of the terms and language that the book used went way over my head. I especially didn't understand most of section 2.11. Specifically, the proof at the end of 2.11 was really hard for me to understand.

2. It is really interesting to me that so much information can be communicated using just the numbers 0 and 1. Intuitively it doesn't make sense. I also liked how using certain random bit generators are able to reduce the number of bits needed to create much bigger keys.

2.5-2.8, 3.8, due on September 15

1. Basically this whole reading assignment was difficult for me. I didn't really have a hard time understanding how to encrypt or decrypt any of the ciphers, but I struggled understanding why the attack methods would work. This was especially true for the Playfair and ADFGX Ciphers.

2. I really love linear algebra. It has been my favorite subject since I've been studying math. So it is exciting to see it used in Cryptography. Not only is it used, but it seems to be really valuable.

2.3, due on September 13

1. I struggled a little bit with the part where the book explains why the method they showed for finding the key length works. It kind of makes sense to me, but if I were asked to explain it to another person, I'm not sure how well I'd be able to do it.

2. I loved learning the ciphertext only attack of this method. As I read the first couple pages explaining the method, I couldn't think of any way to possibly break the code. After reading the whole chapter, however, it was amazing to see how simple it is to break. Even though I'm still not totally clear on how it works, it's really cool to see probabilities working to the cryptanalyst's advantage.

2.1-2.2 and 2.4, due September 10

1. I had a hard time understanding the chosen plaintext and chosen ciphertext methods of attack for the affine cipher. The rest of 2.2 was pretty understandable, but I didn't quite understand the methods of attack, especially under the chosen plaintext when it says "The first character of the ciphertext will be alpha*0+beta=beta..." Maybe I just need to try to better understand the general ideas of chosen plaintext and chosen ciphertext.

2. My favorite part of the reading was in section 2.4 when it decrypts the declaration of independence. It was really cool to see letter frequency and pair frequency actually used to decipher such a seemingly difficult ciphertext. I'm going to use the letter frequencies given in the book to my advantage in hangman.

Guest Speaker, due on September 10

1. I struggled with the concept of the Deseret Alphabet. If it was something the Saints deemed important enough to develop, why is it not used anymore? It seems to me that it would still be valuable for children to learn in addition to the English Alphabet.

2. The whole presentation really held my attention. Cryptography isn't something I'd readily connect with church history. The speaker really opened my eyes to the importance of ciphers in the history and development in our Church. My favorite cipher discussed was the pig pen cipher, mostly because it looked almost like some middle eastern language, but is actually constructed very simply.

3.2-3.3, Due on September 3

1. I haven't completed abstract algebra, and therefore haven't worked with modular arithmetic very much. The most difficult part of this reading was shifting my way of thinking from real number arithmetic to modular arithmetic.

2. The most interesting part of the reading for me was at the end of 3.3 when it talks about fractions and dividing. The whole idea of multiplicative inverses is brand new to me and I find it pretty cool.

1.1-1.2 and 3.1, due on September 1

1. The part of this reading which was most difficult for me to understand was the proof of the theorem which states that if a and b are two integers with at least one of them nonzero, and d=gcd(a,b), then there exist integers x, y such that ax+by=d. I understood up until "Start with j=1."

2. The most interesting part of this material for me was section 1.1, where it talked about the history and importance of Cryptography. It really got me excited for this course, because I've been spending a lot of time doing research in pure math and will be fun to be doing something with apparent applications.

Introduction, due on September 1

1. I am a junior this year, and my major is Mathematics (possibly a double major with Economics).
2. 290, 313, 314, 334, 341. I am currently enrolled in 371, 362, and 485.
3. I am taking this course to determine my interest in this area of mathematics as a possible career option. I also plan on applying for an NSA summer program.
4. I have some experience in Maple.
5. I have little to no programming experience, but I feel semi-comfortable using Maple to complete homework assignments.
6. The most effective teacher I've had is Dr. Barrett. My favorite part of his teaching style is how reasonable he was; he was able to put himself in the position of the student and teach according to our needs. One thing he did was try to ensure that homework assignments didn't take over 3 hours. He also made study guides which gave us an idea of what would be on exams while still offering exam questions which required a solid understanding of the material.
3. Something unique about me: I am blonde and a mathematician.
4. Your scheduled office hours work for me.