Section 16.5, due Wednesday December 8

1. (Difficult) The descriptions in this section were relatively easy to understand because they were analogous to other things that we have studied. In the El Gamal Digital System, when computing s, they use the x coordinate. Would it matter if the y coordinate was used? Was this was they were referring to at the very end of the section?

2. (Reflective) This was a pretty straightforward section because it was similar to what we have seen before. I find it interesting that there are similar signatures using elliptical curves. When El Gamal Signatures started to be used, was it very soon after that someone found the analogous signature using elliptical curves? I am curious how long it took someone to figure that out. This has been an interesting chapter overall. I am glad that I finally know a little bit about elliptical curves and how they are used.

Section 16.4, due Monday December 6

1. (Difficult) Why are elliptical curves over finite fields better to use for cryptography purposes? Can you explain more about the example over GF(4)? I was confused on how the points on the elliptical curve were found.

2. (Reflective) Because we have already learned about finite fields it was interesting to see them come up again. Elliptical curves are really interesting to me. The properties and the things that they can be used for are interesting and intriguing. I can only imagine that they just get more and more complicated, but also more fun to study.

Section 16.3, due Friday December 3

1. (Difficult) Section 16.3.1 and the example in that section were very confusing to me. I was confused on where the values were coming from and why certain computations were completed. I am excited to see an example in class, so hopefully the fogginess of that example can dissipate. Can you explain more about how smooth numbers relate to Hasse's Theorem?

2. (Reflective) I would have never guessed that Elliptical curve factorization relates to p-1 factorization, but I find that very interesting. The p-1 factorization method was very useful in factorization, and throughout the section it mentioned that Elliptical curve factorization was even more successful, which is a great thing. Various topics that one would not expect to relate often due in mathematics. I do find that interesting. When I was watching The Proof about Fermat's Last Theorem, I thought the same thing. I guess it is probably because of my ignorance about various subjects, but while I was watching it, it seemed like Andrew Wiles connected two very different mathematical subjects and related them to each other to proof the theorem. That is a fascinating thing.

Section 16.2, due Wednesday December 1

1.(Difficult) I don't understand perfectly how the discrete logarithm for Elliptical curves are the same as discrete logarithms. Is it similar to if you take the log of both side and the exponent comes down as a constant? More examples for how to turn plain text into cipher text would also be helpful.

2.(Reflective) Square roots mod p seem to come up a lot through many sections and it shows how important they are in cryptography. I would never have guessed that, but square roots mod p have come of often. Finding square roots mod p are difficult though, so what will happen when the numbers get too large to calculate? We only saw for small examples, but we may have large square roots to find.

Section 16.1, due Monday November 29

1. (Difficult) Can you explain the use of infinity in the addition of points of elliptical curves? They said that infinity played the role of the additive identity, and I just want to understand that more clearly. Why can't an elliptical curve have multiple roots?

2. (Reflective) I was excited to learn a little about elliptical curves. I understand that this section was very basic, but I enjoyed reading it. I found it interesting where the name for elliptical curves came from. I never knew that it was from elliptical integrals, which I also did not know about. I enjoyed this section as well because it made sense for the most part. I think it will be interesting to find out more about what elliptical curves are used for.

Section 2.12, due Tuesday, November 23

1. (Difficult) The actual encoding process was the most confusing part of the section. I don't understand the purpose of the set up of the machines, so that would be helpful to go over in class. I understood more about how to break the code because it used things that I have seen before.

2.(Reflective) This section deals with permutations, which we learned in Math 371. This permutations are all disjoint as well, so we know that they can be switched and it will not affect the outcome. This seems like an interesting coding process, but also inconvenient to have these large machines needed to send the code. I would imagine at the time it did not seem too bad, but not I imagine people would not enjoy using these machines very much.

Section 19.3 and explanation online, Due Monday, November 22

1. (Difficult) What does it mean by superposition in the Shor's Algorithm explanation online? I was really confused by this section. Even the example was not very clear, and hopefully class will help clear somethings up. I'm not sure if it was more the notation or just the concepts, but I was pretty lost throughout all of 19.3.

2. (Reflective) This was a good explanation for the Shor's Algorithm that was simple and easy to read. I have heard of Discrete Fourier Transforms before, but I have a very limited understanding of them. I used them in a couple of problems that I did during my summer of IMPACT. In my research last year, I read some basics about Fourier Transforms and on wavelets.

Sections 19.1 and 19.2, due Friday November 19

1. (Difficult) The notation of arrows is what confused me the most. I understood the way that Alice and Bob create the key stream, but I don't understand why the notation is the way it is or exactly how the method corresponds to quantum mechanics. Can you go over how Alice and Bob would detect an eavesdropper?
2.(Reflective) I am pretty sure I have done this experiment in my high school physics class. It is certainly an intriguing phenomenon. The use of quantum mechanics in cryptography seems odd to me. I would never have guessed that a physic application, such as quantum mechanics would be used to set up key streams. It shows how important research is because the applications of discoveries can be extended to many different fields.

Sections 14.1 and 14.2, due Wednesday November 17

1. (Difficult) Can you explain the Feige-Fiat-Shamir Identification Scheme? I guess what gets confusing is exactly how the pieces (x, y, b_i, and v_i) all fit together to create a zero-knowledge technique. I think just the technicality in the book is hard to follow, so a slower, simpler explanation would be helpful.

2.(Reflective) I have never heard from the zero-knowledge techniques before. The explanation in 14.1 made a lot of sense though and is an interesting concept. It is interesting that in fact Victor can never be fully convinced that Peggy knows how to get through the door, but after several tries, he can be pretty well convinced. It is interesting that in the book, it is called a zero-knowledge "proof" because that goes against what we actually consider a proof.

Sections 12.1 and 12.2, due Monday November 15

1. (Difficult) After the example of the Shamar threshold scheme, there was an explanation of why t-1 people could not get the secret. That explanation was confusing. Also, could you explain the Lagrange interpolation polynomial?

2. (Reflective) This is an interesting problem. I had never thought about distributing parts of a key, so that a certain number of people are needed to find the secret. It makes a lot of sense though because it spreads responsibility among various people and guarantees that not just one person can find the secret. What are other applications of the threshold scheme that weren't in the book? It kind of reminds me how when you take out money from a bank for a club or an organization, two signatures are often required or two people must be present to do so.

Midterm Questions, due Friday, November 12

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

Understanding the Chinese Remainder Theorem is important for this class and for many other math classes. RSA was the largest section that we studied since the last midterm, and so we should really know and understand RSA and the factoring methods. Square roots were also very important to understand and implement for factorization.

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

Chinese Remainder Theorem problems, Probability for Birthday Attack, The Pohlig-Hellman Algorithm (Possibly), the Concepts relating to RSA and factoring large n.

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

Factoring methods are things that I certainly have to go over and understand. I need to work on discrete logarithms and know to find them using the Pohlig-Hellman Algorithm. I need to make sure that I understand and can apply the concepts from ElGamal, hash functions, and digital signatures.

• Are there topics you are especially interested in studying during the rest of the semester? What are they?

I have heard of Elliptical Curves, and I'm glad that we will be going over them briefly so I at least have a general understanding of what they are and what they possibly are used for.

Sections 8.3 and 9.5, due Wednesday November 10

1. (Difficult) Section 8.3 was pretty confusing, but we were given a fair warning at the beginning of the section. I was confused mostly about the SHA-1 Algorithm. Some of the steps seemed very "random" because I had no idea where they were coming from, including the constant values K_t. It mentioned that the core of the algorithm was step 3, which was probably the most confusing part of the algorithm.

1. (Reflective) The Digital Signature Algorithm made more sense to me, so it was more interesting. It makes sense that why we need these secure ways to sign documents. If this wasn't secure, then people would be accused of signing contracts and other documents that they did not. This is really important in many transactions. The DSA is very similar to the ElGamal Signature Scheme. Are they basically the same other than the difference in the integer r?

Sections 9.1-9.4, for Monday November 8

1. (Difficult) Can you explain in class the difference between a signature with appendix and a message recovery scheme? Actually, if I understand correctly, it is just that in the latter you actually can find the message m, where in the signature with appendix it is not easy to actually recover m. Is that correct? What are more examples where a signature would be used?

2. (Reflective) The most interesting section was the Birthday Attack on Signatures. It is a very clever way to trick people, but yet very easily avoided if one is weary of the attack. I find it interesting that some many of these ideas have been discovered. It fascinates me that people have come up ways to attack seemingly secure things, such as the Birthday Attack. Hashing itself is a good idea and seems like a practical thing to use, but as the Birthday Attack shows, it will sometimes fail to be secure.

Sections 8.4-8.5 and 8.7, due Friday November 5

1. (Difficult) Section 8.5 on Multicollisions was confusing to me. I understand that the hash of a message M is a compression of blocks of M into a fixed value, but I don't understand the concatenation part of finding multicollisions. Also, the time estimated did not make sense. Overall, this section was the most confusing and hard to follow.

2. (Reflective) I think the birthday paradox is very interesting. I have heard it several times and have had it explained to me, but I think I understand it better now. It is interesting because it seems intuitively wrong that the probability would be so high for two birthdays to match, but then I think of my birthday and how many people I know with the same birthday. I have met many people with the same birthday and expect that I will continue to do so.

Sections 8.1-8.2, due Wednesday November 3

1. (Difficult) The proof of the proposition on page 221 is a little fuzzy to me. I follow most of it, but another explanation in class would be helpful. Can a hash function be anything that satisfies those three conditions on pages 218-219? Was the simple example in 8.2 really a hash function because it was not strongly collision-free?

2. (Reflective) The simple hash example seemed very easy to compute, but did not seem like it would be very useful. XORing the columns would get rid of all the information in the message. Would you just use this type of has function if you were given (m, h(m)) and were checking to see if the message m, actually matched h(m)? The only other time I have heard of something similar to hash functions is when we were asked to read about one-way functions. One-way functions are turning out to be very useful functions to use in cryptography.

Sections 7.3-7.5, due Monday November 1

1. (Difficult) I was confused about the prepositions in Section 7.5.1. Those would be great to go over in class. I understood the Diffie-Hellman problems, but the propositions and proofs were confusing to me. Is the Diffie-Hellman Key Exchange as secure as RSA? Is it more secure?

2. (Reflective) It initially surprised me that the discrete logarithm problem was so hard, because initially it looks so simple. From these section, we see that this method of encryption and decryption is secure as far as we know. I wonder how it compares to RSA though, as I asked above. Is discrete logs as secure or more secure than RSA as far as we know currently?
The bit commitment made a lot of sense in real-world application. There are sometimes that predictions need to be made, but without a secure way to ensure that the prediction was not altered could present problems in the future.

Section 7.2 , due Friday October 29

1. (Difficult) I had a very difficult time with this section. I kind of understood the examples, but was lost for the most part. I understand that the discrete log is a hard problem, and the explanation of how to solve for it was very difficult to understand. The Baby Step, Giant Step section was also confusing to me.

2. (Reflective) I have never heard of these before in any other math class. Normally, I have at least heard of the topics we have gone over, but this is brand new to me. From the explanations, this seems like a very difficult topic, one that takes a lot of time and patience. I am hoping that with class and working out some solid example this new material will become less foreign.

Section 6.4, due Friday October 22 up to just before section 6.4.1

1. (Difficult) Is the p-1 factoring algorithm called that because at the end of the algorithm we look at the gcd(b-1,n)=d? Can you explain why you if you use the method at the top of pg. 183 to ensure that p-1 has at least one larger prime factor, that will make it hard to factor n=pq using the p-1 method?
2. (Reflective) This section seemed pretty straight-forward. I had never heard of this algorithm, but it is an interesting way to factor n. I am excited to learn about elliptical curve factorization because I have heard of elliptical curves before from "The Proof," the movie about Andrew Wiles proofing Fermat's Little Theorem. Obviously, the elliptical curves that he was using are much more complicated that we will be dealing with, but I think it will be fun to learn about.

Section 6.3, due Wednesday October 20

1. (Difficulty) The Miller-Rabin Primiality Test was the most confusing test that was discussed in the section. I understood the steps, but how does the result conclude if the number is prime or not?More examples in class would be wonderful. Also what is the difference between strong pseudoprimes and just pseudoprimes?

2. (Reflective) It is interesting that there are many tests that guarantee that a number is composite, but for a long time there were not tests that could guarantee primality. It still takes some work, it seems, with the new algorithm by Agrawal, Kayal, and Saxena to discover a number is a prime. Are there any other known tests that guarantee primality? It seems like there will/would be more tests soon.

Section 3.10, due Monday October 18

1. (Difficulty) I don't understand exactly what the theorems are used for. The theorems and the examples make sense, but I do not see the link to cryptography right away. How are the squares of x used in cryptography? Also, why in Jacobi Symbols, when we are working with composite numbers does n have to be odd?

2. (Reflective) I have not seen any of these things before in my classes. I understood the examples and the various steps that were taken. It is interesting that some problems that seem simple still to now have a quick algorithm to solve them. It was mentioned right at the end of the section that there is still no quick algorithm to solve the quadratic residuosity problem. I thought this section was interesting and enjoyed reading it because the propositions and examples were relatively easy to follow.

Sections 6.5-6.7 and Section 7.1, due Wednesday October 27

1. (Difficult) Is a trapdoor in RSA simply the decryption exponent, d? I would like to understand more about the connection with primitive roots and discrete logarithms. I still feel like I don't exactly understand how to find primitive roots and how that would be connected to discrete logs. Will you explain more fully why it is difficult to solve for x in a discrete logarithm?

2. (Reflective) I found it clever the way that Bob is able to know that a message comes from Alice. It makes a lot of sense to have Alice apply her decryption exponent and then Bob's public encryption, so that Bob is able to undo the message and know it was from Alice. I find it interesting how important one-way functions are. Often in other math classes we want to be able to easily compute (or find) the inverse of a function, but the inverse of one way functions are not supposed to be easily computed. I can certainly see the application and the need for one way functions for security. I liked the message for the RSA challenge. It was certainly not a message that someone could try and get and encrypt, using the public encryption exponent.

Section 6.4.1, due Monday October 25

1. (Difficult) I understand the method of finding the numbers 9398, 19095, etc., but why does that method guarantee that the squares of that number are factored into small primes? Why are small primes important in this factorization method as well?

2. (Reflective) It is amazing how factorization has improved over the last 50 years. They showed the table on page 185 that shows the factorization methods. We have certainly come a long way. I still feel like factoring should not be so hard, but when the numbers get large it makes sense that it is difficult to find the prime factors. Is the reason why small prime factors are important in the numbers 9398, 19095, etc. because we can quickly check those and find the prime factorization? We don't have to go all the way up to the sqrt(n) to factor if we know that the numbers are a product of small prime numbers.

A Brief History on Primes

1. (Difficulty) I understood most of the information because I have sen it before, but I did not get all of the proof that 2^n-1 being prime implies that n is prime. I understand how it was used, but the proof was just done quickly, so I didn't understand it all. Other than that I followed most of the talk because it was a review from previous math classes, including cryptography.

2. (Reflective) I liked that fact that the talk was very simple and made sense. You didn't need to know a lot, or anything specifically, about number theory to understand the lecture. I have heard about several of those theorems many times, and it just shows how important they really are. I have seen Fermat's little theorem in three class, and this was the fourth exposure to it. It is amazing how useful primes are, and yet we still do not know how to prove several of the conjectures mentioned in the talk. The conjectures seem so simple, but the proofs are far from that. I think that is really interesting. Many of the conjectures seem so simple, but yet we do not know of their authenticity. Number theory must be a fascinating branch of mathematics, which I would like to more about.

Section 3.9, due Friday October 15

1. (Difficult) Can you explain why -1 cannot be a square root of mod p when p = 3 mod 4, on pages 86-87? Does that have something to do with the fact that if p = 3 mod 4, then p =-1 mod 4? Also, I assume you are going to go over an example in class, which would be helpful. I follow the example in the book pretty well, but some of the steps are confusing. Can you explain the step at the end, where the gcd(a-b, n) = p?

2. (Reflective) It amazes me some of the propositions and theorems are discovered. I wonder how someone discovered that we need p = 3 (mod 4) for this to work. It is certainly a useful theorem for factoring, it just seems like the needed conditions are somewhat arbitrary. This material is also new to me, so that may be why that seems rather random, but possibly should be intuitive. This proposition certainly will be helpful in RSA, since this proposition is computationally equivalent to factoring n. I assume that this has been used many times in attacks of RSA.

Section 6.2, due Wednesday October 13

1. (Difficult) The Theorem on page 170 was the most confusing part of the section. I understood the beginning inequalities, but it was just difficult to follow all the way through. Also, the explanation on the short plaintext and how to pad it correctly was a little confusing. I understand the attack on a short plaintext, but I want to understand the padding of the plaintext portion of that section. I find it interesting that Bob can discover if it is a valid encryption by the amount of zeros on the end of the ciphertext, but I want to understand exactly why more clearly.
2. (Reflective) I find it interesting that these attacks are discovered, especially the last one by a young student at Stanford. RSA seems so secure to me because the magnitude of the numbers used is so large. I am not as fluent in computer's abilities, so to me RSA seems very secure. The last attack was the more interesting to me. The fact that Eve can attack simply using the time it takes for Bob to decrypt is not only impressive, but scary that so much information can be found that way.

Section 3.12, due Monday October 11

1. (Difficult) I understand the theorem, but I am not exactly sure why it is needed in the discussion of continued fractions. Does it have to do with the face that p_k/q_k gives a better approximation to x than any of the preceding rational numbers? Also, is the recurrence relationships related to, or actually, the Euclidean Algorithm? The example demonstrates that the quotients from the Euclidean algorithms are the floored functions after ever step.

2. (Reflective) I have never learned about continued fractions, but this is a really good method. The approximations are surprisingly accurate once you have gone out enough. I remember doing various approximations for numbers, specifically pi, but I never have used this method. I imagine that continued fractions not only could be used in application for good approximation, but could be easily be a research. I think that would be an interesting thing to study and learn more about.

Section 6.1, due Friday October 8

1. (Difficult) When would be a situation when Claim 2 happened? When would you know both e and d? From the example, it seems like you would either only know n and e if you were the public, or you would know p, q, d already and thus would have no need to factor n because you already have the factors. Could you talk about the Pretty Good Privacy. I was confused by that paragraph.
2. (Reflective) RSA seems like a pretty secure way to encrypt a message, but I wonder how anyone thought of this method. I learned about it briefly in Math 190, but it makes more sense to me now. I remember doing simple examples, but for very small primes. It makes sense that the prime numbers have to be so large, or else n, could be factored easily using a computer.

Sections 3.6-3.7, due Wednesday October 6

1. (Difficult) I have heard of primitive roots, but I have not used them in practice. The proof of the proposition makes sense, but I would like to see more examples of how it is applied. The Three-Pass Protocol would be nice to go over as well. If Bob has K^(aba-1b-1), how do we know that the exponents are communative and that K^(aba-1b-1) = K (mod p)? It says it uses the Basic Principle, but I don't understand exactly how it was applied.
2. (Reflective) A lot of the terms in these two sections I have seen in Math 190 and Math 371. They are certainly important concepts because they keep coming up. I have not worked a lot with the phi-function, so that is more new to me than Fermat's Little Theorem, but it makes sense. I really enjoy these sections and understanding Fermat's Little Theorem and it's applications better. The Basic Principle was new to me in these sections as well, but has interesting applications as was seen in the Three Pass Protocol.

Sections 3.4-3.5 due October 4, 2010

1. (Difficult) I have seen this before but some of the methods for using the Chinese Remainder Theorem were different than I have seen before. I was a little confused about the example with x = 7 (mod 12345) and x= 3 (mod 11111). If you could go over that one in class, it would be helpful. The binary example was also interesting, but a little confusing, and I would love to see in class.
2. (Reflective) I have seen the Chinese Remainder Theorem in at least two other math classes, but it alwasy interesting how I understand it more every time that I relearn it. When I first learned it in Math 190, it made little sense to me, but now after learning it in Math 371 and rereading about it in this class it has become clearer. The fact that it has come up in several classes shows that it is an important concept in mathematics.

Exam Questions

1. Which topics and ideas do you think are the most important out of those we have studied?
I think that understanding the Euclidean Algorithm is very important as well as modular arithmetic. Also, understanding how different modes of operations will be important. The finite fields section is crucial to understanding AES and other types of cryptosystems.
2. What kinds of questions do you expect to see on the exam?
Computationally, I expect to see the Euclidean Algorithm on the exam, the Hill cipher, and recursion. I expect that there will also be theoretical problems, which will test the understanding of the different modes of operation and the different types of ciphers. I expect that an affine map and substitution cipher will be incorporated into the exam.
3. What do you need to work on understanding better before the exam?
I need to make sure that I understand the steps in various cryptosystems and how to encrypt/decrypt. I also need to work on finite fields. I just do not feel completely ready for the exam because I need to understand the cryptosystems more clearly, including the modes of operation, (CTR, CFB, etc.)

Sections 5.1-5.4 for September 29

1. (Difficult) The key schedule was the most confusing part. I read it through twice, and I think that is makes more sense, but I don't understand why some of the steps were taken or used to create the key schedule. Also, a little more explanation about why MC was not included in the last round would be helpful. The reading seemed to go a lot more smoothly though because we had already started learning about it in class.

2. (Reflective) I liked this section. It was an interesting section that I felt that I understood for the most part. I find it interesting that the number of rounds was chosen simply to have a pad for safety. I understand that there needed to be more than seven rounds, but ten still seems arbitrary to me. Some of these cryptosystems make me wonder who ever thought of such ideas. How were they discovered and implemented for the first time? I guess in math there is a lot of things like that. Some ideas and theorems proved seem so abstract or even random, but yet they are applicable and work in the world.

Questions for September 27

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

It depends on the assignment, but this last week probably took me a total of 3-3.5 hours. The first 9 problems were actually relatively quick for me, but the one time pad probably took at least half of the total time that I spend on the homework. I think that together both the lecture and the reading prepare me well for the homework. When I read, I often feel confused and am not exactly sure how some things work, but the lecture helps me understand. Plus in lecture, we often do an example or two similar to the homework, so that helps a lot.

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

The homework has been the most helpful for my learning. The reading is a good preparation for lecture, and lecture is a good preparation for the homework, but the homework is where I actually use the tools that we mentioned and start to understand the material better.

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

I think that the class is going well, and I cannot think of any suggestions right now.

Section 3.11 for September 24

1. (Difficult) I understood most of the information about fields, but why does x+1 = x^2 in the group Z_2[X]? I know that it was a law in the first example, but I do not understand where it comes from. The last page was a little fuzzy for me once it went into recurrence. I understand how to set up the matrix and most of the steps and explanation, but I don't know what Lagrange's theorem in group theory is, so I didn't understand exactly how the value 2^m -1 comes into play.

2. (Reflective) I have seen most of this stuff in Abstract Algebra. I remember doing a short lesson and homework assignment on irreducible polynomials and feeling really confused, but I think that I understand it better now. It makes more sense now that we have been doing some homework problems using binary numbers, and the example in the book used Z_2[X]. I still do not understand all of the theory behind groups and fields, but it is becoming clearer every time I study it a little more. It is definitely a different kind of mindset to think of certain polynomials being irreducible or not irreducible mod 2, when that is not always the case when working with real numbers.

Sections 4.5-4.8 for September 22

1. (Difficult) Section 4.5 was pretty difficult to understand. I didn't understand the differences in many of the five modes of operation. I understood the first one, ECB, and the way it is attacked, but the Cipher Feedback (CFB) and Output Feedback (OFB) were confusing, even with the diagrams. I did not fully understand the Meet-in-the-Middle Attack either. I understand that this attack shows that the composition of two keys does not make it any more secure, but the process is not clear to me.

2. (Reflective) I find it interesting the way that DES was broken. I took a seminar on parallel computing, and the use of several computers to break one code makes a lot of sense. Using multiple computers at one time is much more efficient than using one computer. Parallel computing makes many seemingly "impossible" problems possible.

I really like the concept of "interesting" decrypted messages. When I first read that, I was wondering what it meant, but then the explanation explained it. It shows that even though the DES system can be broken by using parallel computing and checking the keys, that a lot of work is being done that is somewhat useless. Many of the keys are simply discarded.

Sections 4.1, 4.2, and 4.4 for September 20

1. (Difficult) I was very confused about the DES method and the simplified example provided in section 4.2. I think it was mainly the notation and the use of bits that made it confusing for me. I understood some of the simpler steps, such as using the S-boxes and XOR, but it was the process and steps that confused me. Using the Initial Permutation table was clear, but then I did not understand how to use several of the other tables. I understand that R is the last 32-bits of a sequence of 64-bits, but what is the function E(R)?

2. (Reflective) I find it interesting the history of many of the different methods of cryptosystems. It is amazing how something that seems secure 30 years ago can now be relatively easily broken. Also, the fact that DES was used and then replaced shows the importance of continuing to study cryptography and a lot of other subjects. Different methods are meant to be broken eventually and replaced by a better one, and then a another one, and another. There was a psychologist that believed that good theories and ideas have to be falsifiable. That is, that ideas need to have the option to be false, so that better ideas can come about in an effort to proof current ideas false. It is an interesting thing to think about. We come up with pretty good ideas, and then we try to disprove their security so that better ideas will emerge.

Sections 2.9-2.11 due September 17

1. (Difficult) I understood the proposition on Pg. 48, but the proof was difficult to follow. I understood the first part when it is shown that the det(M_n) = 0 for all n>N, but it was contradiction portion that I was getting confused. I also didn't understand how the period of a sequence is at most 2^m -1.

2. (Reflective) Using the bits for the key is something that I understand better now because the group we switched projects with used that as their cryptosystem. It is certainly an very good idea, because the key can be very long and hard to break. It makes sense that a one-time pad is unbreakable. The key is just as long as the plaintext, so really the only way to decrypt the message is with the key. Blum-Blum-Shub pseudo-random generator seemed like a good idea, but like it said it the book the numbers get large and it takes time to calculate. I had never thought about how important the randomness of the key would be, but it must be random or else someone could find a pattern or find a way to discover the key.

Section 3.8 and Sections 2.5-2.8 for September 15

1. (Difficult) I understand how many of the messages are encrypted, but it is the attacks that often do not make complete sense to me. In the ADFGX cipher, I understand how they encode a message, but the attack does not make sense. Similarly, the attack for the block ciphers seem very complicated. I assume that the block cipher attacks are more complicated due to the discussed definitions of diffusion and confusion, where one change in the plain text can alter several letters in the cipher text.

2. (Reflective) I have never found the inverse of a matrix, mod (n), but it makes sense how to find the inverse. I think that algebra, especially matrices, can be very useful in cryptography and that was shown in the block ciphers. Matrices are very useful tools in many areas of mathematics, especially applied mathematics. I did research using matrices, and they are very useful in many areas of mathematics.

I also enjoyed the story about Sherlock Holmes. It was an excellent example of how someone imitate someone else using cryptography.

2.3, due on September 13

1. (Difficult) I was confused about the explanation about how to find the key length. I understand that the dot product when A_i is multiplied by A_i is the highest, but I am still confused about how that relates to the matching up of the letters when displacing the message. Also, finding the key using the second method was more confusing than the displacement and frequency analysis, and still is a little unclear.

2. (Reflective) I had never heard of the Vigenere Cipher, but I think it is a good idea. The fact that all you need for the key is one word is impressive. It doesn't surprise me that it was thought to be secure for that length of time. It would be a hard message to simply look at and break; thus, it is even more impressive that people have come up with methods to break the code. Even though the reasoning confuses me a little, I find it really cool that you can find the length of the key simply by the coincidences in matching letters when using displacement.

2.1-2.2 and 2.4, due on September 10

1. (Difficult) In Section 2.4, when the book discusses substitution ciphers, it was a little confusing how they deducted the letters from the table. It took a little while to understand how one frequency of two successive letters, led to deducing another letter. The explanation was clear, but it just shows how tedious it can be to decipher the ciphertext would be. The affine ciphers were also more challenging than a simple shift cipher. The modular arithmetic would be more challenging using an affine cipher.

2. (Reflective) Although it was difficult to follow sometimes, I think it is cool that we are able to often decipher a substitution cipher by using the frequency of letters. It is interesting that we have the statistics on the frequency of letters in the English language, but it does make it easier and possibly to break the code. Like I mentioned in the difficulty section, the reasoning takes a little time to understand how one letter can deduce another letter, but it is very interesting. A long line of ciphertext can look intimidating, but using the frequency of letters, we can often figure it out without using any computer at all.

Guest Lecturer, due on September 10

1. (Difficult) I found that the most difficult part was initially trying to decrypt the pig pen code. It was very clever though and did not take too much work. Many of the codes were easy to understand because they were simple encryption methods, including substitution ciphers.

2. (Reflective) I found this lecture very interesting and wish we had been able to hear more. I never realized that the church had used so many different codes before. It makes sense that for any type of organization that codes would be important to protect information. It was interesting to hear about the Doctrine and Covenants and to find out that the names were still not changed even in 1982. I also thought that the pig pen code was very clever. I really enjoyed the lecture!

3.2 and 3.3, due on September 3

1. (Difficult) I have heard many of the definitions before, but the most difficult part was the division operation in modulus arithmetic. I have learned it previously, but I have not practiced it enough to be complete comfortable. Also working with fractions mod n is something I have never learned. It seems more confusing to write 1/2 mod n instead of the inverse of 2 mod n, but I could see how when using large numbers 1/2 mod n would be more convenient.

2. (Reflective) I have seen most of the material in previous math courses, including Math 290 and Math 371. The Euclidean algorithm and extended Euclidean algorithm still impress me in their simplicity, but ingenuity. The fact that we can find the gcd of two integers and then using our work find the linear combination of the two integers that equal the gcd is very impressive. Also, as I mentioned about, working with fractions in modular arithmetic seems more confusing to me because I have never learned it before, but it makes sense when the working with large numbers that fractions will simplify things.

1.1-1.2 and 3.1, due on September 1

1. (Difficult) The difference between code and cipher is still not very clear to me. If I understand correctly, in a code words are replaced by codewords, which depends on the structure of the message. In a cipher, the message is encrypted by an algorithm, which is not dependent on the structure of the message. The Prime Number Theorem was also new to me, so it took a little while to understand.

2. (Reflective) Section 3.1 was mostly review from Math 371, but I think that the concepts are important for review. Several proofs in the section I have seen several times, but needed to be familiarized with them again. The most interesting parts of the reading was the application of non-repudiation. I had never thought about how important that would be especially in purchases on the internet. There has to be a secure encryption so that Alice cannot claim that she did not buy a specific item online.

Introduction, due on September 1

1. This is my third year at BYU, and I am a mathematics major.

2. Post-Calculus Courses: Fundamentals of Math, Linear Algebra, Multi-variable Calculus, Differential Equations, Theory of Analysis I & II, Abstract Algebra

3. I am taking this class because I want to learn something very different than analysis. I enjoy analysis, but I wanted to learn a different kind of mathematics. I have also heard many good reviews about how fun this math class is.

4. I have used a small amount of Maple. I have mainly used Matlab in my math classes and research, but I feel comfortable learning Maple and using it in my homework assignments. I have not taken any programming classes, but I feel comfortable because of my experience using Matlab.

5. Dr. Cardon was one of the most effective professors that I have had. I have had other professors that are not as effective.

6. Dr. Cardon was thorough and very clear. He was allowed many questions if needed and had available office hours. Even though his tests were difficult, Dr. Cardon was very fair. I enjoyed the class because it was challenging, but very rewarding because I felt like I was mastering the material. Other professors are not as effective because they are not as accessible to students, and they do not slow down to clearly describe concepts.

7. I know how to play cribbage.

8. I have a class at 2:00-2:50, but 11:00am-1:00pm or after 3:00 pm would be a good time for me.