16.5 due on December 8

1. I think I understood most things which scares me, because I normally don't. I am still not comfortable with adding points and multiplying points with an integer. The the notation for the key exchange is not sitting well with me. I don't understand what the N_A or N_B means. They didn't really explain that.
2. It blows my mind how similar elliptic curves are to discrete logs. From the reading it sounds like a superior system. We probably aren't going to go over weakness, but there has to be some weaknesses of elliptic curves. I am curious because they seem quite useful in many ways.

16.4 due on December 6

1. It must have been a long weekend because I thought I was understanding elliptic curves, but after reading this section, I no longer feel like that. I think the confusion comes from the equation E: y^2+a_1*x*y+a_3*y=x^3+a_2*x^2+a_4*x+a_6. I understand that taking the derivative of the normal form in mod 2 would cause problems, so there needs to be this other form. I just don't under stand how to solve for the points and such. This carried over to the finite fields.
2. I am curious to see how elliptic curves play into the cryptosystems. I guess that is the next section, so I won't have to wait much longer for my curiosity to be soon satisfied. I am always curious how things come about, the historical background of things.

16.3 due December 3

1. I had some difficulty understanding the whole of section 16.3.1; I just couldn't really figure out what the author trying to convey. I also had some trouble understanding the example in 16.3 where factorials where introduced. I also didn't see how that related to the p-1 method and smooth factors and elliptic curves.
2. I did understand and found it cool the part where the reading noted that using elliptic curves to factor a composite number succeeds much more often than the p-1 method. I also just think is really cool that elliptic curves can be used to factor composite numbers. I wonder if it was a purposeful exploration of elliptic curves, or if one just happened to stumble on the fact that elliptic curves are effective in factoring numbers.

16.2 due on December 1

1. I didn't quite follow how the explanation for an approximate value for number of points mod p. I also didn't understand the discrete logarithms on elliptic curves. I don't see how B=kA for some integer k relates to a discrete logarithm problem.
2. I actually understood the elliptic curve cryptosystem mentioned worked for the most part. It seems kind of like it isn't all that efficient as there is a possibility of performing many operations to find a square root of x^3+bx+c for x=mK+j. I am interested in an example when there is a mod of a composite such as the example in 16.1 and how elliptic curves can lead to factoring the composite.

2.12 due on November 23

1. I had some difficulty understand the attack. The first part was the daily settings and how that was transmitted. It sounded like it was in a book, so why must that be transmitted? Then I didn't understand at all how the permutation cycles were used to show what letters were mapped to other letters.
2. I found it interesting that the British had been breaking the Enigma throughout World War II. I guess really all I have about the Enigma is from U-571. The movie made it seem like it it was much later in the war that the Enigma was broken, but the book said the British new how to two months before Germany invaded Poland. That was the onset of World War II. 

19.3 and Shor's Explanation due on November 22

1. Shor's algorithm still doesn't make sense to me. The blog helped shed a little bit of light on it, especially the analogy with the clocks on the wall. But, I still don't understand how that helps factor a number n. It appears to me to be a probabilistic algorithm where there are still possibilities of not find a factor of n, so what good is it compared to the classical computer.
2. The theoretical capabilities of quantum computer sounds exciting and cool. I can see how it has become and becoming a popular area of research. I have heard some things about quantum computing it the past but have not really understood. As a soon to be clueless to a career path math graduate, it may be a possible area to pursue.

19.1-19.2 due on November 19

1. I don't quite understand how basis of vector spaces work into quantum computing. Hence, the quantum key distribution did not make much sense. Basis of vector spaces wasn't my strong point of linear algebra as well. I also have difficulty understanding how the quantum key distribution is any different from the digital computing of now; it just seem like they were using different symbols for 0 and 1.
2. This whole topic is interesting; the possibilities that are out there if quantum computing does  work out. I found it interesting, from the quantum key distribution that if Eve listens in, it changes the data sent or the state of the photon. This seems that it quantum computing may offer new security as well as the power to destroy security. It would be a whole new ball game.

14.1-14.2 due on November 17

1. I thought I understood Identification scheme until it got to the point of where it asks 'what if Peggy doesn't know a square root of y?' I didn't follow the guessing process. Then in the actual Feige-Fiat-Shamir Identification Scheme, I came to realize that I really didn't understand it at all. I think it is compute y that gets me.
2. The story about the fake atm was pretty clever. I think it was just recently that there were people installing code on atms that would record and send peoples' information to the thieves. The Zero-Knowledge Proofs are pretty interesting; it seems like it is based of probability. Yet, there are some differences I feel. 

12.1 - 12.2 due on November 15

1. I didn't understand the Shamir Threshold Scheme at all. I was hoping the example would help me understand, but it didn't. I got lost with the Lagrange interpolating polynomial. With the linear system approach I didn't understand the construction of the matrix. I followed the Blakely method better, but I didn't follow how the plane equations were constructed as well.
2. The whole idea of Threshold schemes are pretty cool. I never really thought about situations that would need such a system, but now after reading, it applies to a lot of different life situations. They seem like a very nice solutions to the problems. 

8.3, 9.5 due on November 10

1. When the author said in 8.3 that "the reader is warned that discussion that follows is fairly technical", I knew I was in trouble. I understood until that padding portion of the SHA-1. I didn't get much after that. With the DSA, I didn't quite understand the verification process and the explanation why it works.
2. It is interesting that both sections we read, that speed was mentioned. The DSA removes one step of modular exponentiation from the ElGamal scheme so it is faster. How much faster are we talking about: milliseconds, seconds, minutes? I am sure it depends on the size of message, so what is the Big-O of it is probably the better question to ask.

8,4-8.5, 8.7 due on November 5

1. When I finished the reading, my summary thoughts were "I guess I don't understand how hash functions work." In 8.7 it didn't sound like like the hash function was much different that the modes of operation found in chapter 4. I also didn't understand the multi-collisions section.
2. The birthday attack is really interesting. It is cool that it is using a probability concept in cryptography. It sounds like there a 70% chance that there are two people in the class with the same birthday.

8.1-8.2 due on November 3

1. I have some difficulty understanding the difference between strongly collision-free and weakly collision-free. They seem to be saying the same thing, but I know there is a significant difference. I also had some difficulty following the example of the discrete log hash.
2. I have heard of hash functions and of MD5 separately. I knew that MD5 had something to do with data integrity. It is cool to piece it all together and get a better understanding. It is interesting that some of the popular hash functions have turned out to fail the strong collision-free requirement. It obviously must be hard to determine. I noticed the text said the the discrete log hash function is "probably" strong collision-free.

7.3-7.5 due November 1

1. I had difficulty understanding how Bob finds the value of b by looking at x mod 4. I followed the example except for that one part. I also didn't understand the Computational Diffie-Hellman Problem and the Decision Diffie-Hellman Problem. What I didn't understand was the significance or how they fit into the rest of it.
2. I was pretty interested in the name of ElGamal. All the algorithms we have covered were named after people. This looked like a code name in Spanish or something. A quick google search revealed this algorithm wasn't any different; it was developed by Dr. Taher ElGamal of Ciaro Egypt, a computer scientist.

7.2 due on October 29

1. What was difficult to understand ... It started at "7.2 Computing Discrete Logs". I felt like I kind of knew what was going in the introductory text, but when I got the the Pohlig-Hellman algorithm, I just got lost. I think the other stuff would follow.
2. Though I didn't understand 7.2.4 the author said that "there is a philosophical reason that we should not expect such an algorithm." It makes wish I understand even more.

6.5-6.7, 7.1 due on October 27

1. I had some difficulty understanding how the treaty verification worked. It raises the question to me is RSA still secure if you compute d and let (n,d) be public but keep e private? It seems me that country B can just gather their own x data and then use it to find d. After that country B can do what they want.
There was also some difficulty understanding the method of authentication and non-repudiation. I didn't follow their explanation.

2. The RSA challenge if funny. I think it would take me a little more than $100 to put in a serious effort to try to decrypt the message. The people that worked on it must have been looking for a challenge. Trapdoors are interesting. They seem like a very easy security breach on a cryptosystem if word ever got, but after reading and thinking about it that is pretty much the case with any cryptosystem.

Talk on Math Minimal Surfaces October 21

1. The lecture seemed quite understanding and fun. My only difficulty is that what do you what do with minimal surfaces and how did they study of minimal surfaces start? I guess the end of the lecture hit home. I am pretty close to be graduating with a math degree, and I have not a clue what to afterward. That is probably the most difficult thing for me chew on.

2. Soap films seem like a lot of fun, but not quite as fun as running through encryption levels of AES. The idea of research sounds fun. I enjoy listening to new things discovered. I am signed up for newsletter that latest breakthroughs in the computing world. I don't know if I could handle doing research especially in mathematics. It seems like there would be a lot of drudgery. I have a hard time getting through homework assignments. Research is probably different from homework though.

6.4 due on October 22

1. I didn't follow the p-1 Factor Algorithm. I keep getting lost with all the subscripts and superscripts. The explanation following didn't make much sense as well; though for some reason the explanation for choosing B did make sense. Maybe because it was left up to the situation.
2.I have heard of elliptic curves used in cryptography, but I don't know much about them. I have also heard a bit about quantum computing as well. I am interested in both of those topics. Hopefully we will cover them ...

6.3 due October 20

1. I did pretty well following the Miller-Rabin Primality Test. I had difficulty following and understanding the material after the pseudo-primes to the Solovay-Strassen Primality Test, the explanation to why the Miller-Rabin test works.
2. It think the important thing to take from the reading is that these are "composite testing" methods. The methods don't really prove something is prime but is a composite. It is interesting how the problem lies in factoring numbers, and it seems like there is a lot of room for finding better algorithms.

3.10 due on October 18

1. I don't quite follow this section. I read over it and got to the end, and I was lost and tried reading it again. The Legendre symbol sort of made sense but then it seemed like the Jacobi symbol was the same symbol but meant something different. I don't kind of lost in those definitions and examples.
2. During the math lecture on primes the Legendre was mentioned. I thought I would gain a bit more understanding, but didn't. I am amazed how people figure this stuff out. Gauss is mentioned in both physics and math; Euler, Fermat. Brains.

Focus On Math - "A Brief History of Primes" by John Friedlander

1. Dr. Friedlander mentioned in  his presentation that checking if a number is prime is fast. I must have dozed off or missed something because it seems like to me that one would have to check a number with all previous primes  and compare the gcd. That doesn't seem very quick.
2. I wish Dr. Friedlander would have gotten into more of what he does. Much of the material covered has been covered in previous classes. Maybe have a brief portion on the future of primes. I have heard of the Riemann Conjecture for some time, but what it was about. It is interesting that the conjectures he presented haven't been proved. They seem so simple, but those are usually the hardest problems.

3.9 due on October 15

1. I didn't follow the follow the portion where the modulus was a composite. I understood how the square roots were solved for the individuals mods, but then the combined part to get the four solutions lost me. The concept or the process of finding the square roots mod n is giving my brains some fits.
2. I thought it was interesting that given n=pq where p,q are primes, when a person finds the solutions to the squares, he/she finds the factors of n. Seeing how we are working with the RSA, this might be another way to attack the algorithm.

6.1 due on October 13

1. I had difficulty understanding the theorems in 6.2.0. I am not sure how they work out. I am sure we'll go over them in class. I was also really confused by the "more sophisticated method of preprocessing the plaintext". It seemed like they jumped from a plaintext attack to this method, and I didn't see quite the need for the method.

2. I seems like RSA is not so secure afterwards though I am sure it is still quite secure. I am in a stats class right now so the last method was intrigueing. It was using some sample variables, means, and deviations to calculate the processing times of the computer. It's genius. Is it an effective attack? I didn't quite figure out how they actually timed the computer, but the concept is cool.

6.1 due on October

1. I don't understand where the polynomial X^2-(n-phi(n)+1)X+n came from. It seems like a real useful polynomial for RSA.  The rest of the section was straight forward after the previous class lectures.
2. The RSA is neat in how simple the concepts are quite simple when compared to DES and AES. It seems like it would be easy to break all you have to do is factor a number, but it isn't the case. It is cool they use just a couple of math theorems for the method.

3.6-3.7 due on October 5

1. I think Fermat's Little Theorem and Euler's Theorem are pretty cool. The really simplify congruences. It is amazing to see that discovered quite a while ago, these theorems play in important role in modern day cryptography.

2. I followed the three-pass protocol the nonmathematical way but the mathematical way was a little more hard to follow. Then primitive roots sections was murky. I understood the example and understand what a primitive root is ... sort of, but the propositions were not so clear.

Review due on October 1

• Which topics and ideas do you think are the most important out of those we have studied?
• I think probably types of attacks, gcd, euclidean algorithm, congruences, finite fields, modes of operation.
• What kinds of questions do you expect to see on the exam?
• Maybe some gcd, euclidean algorithm, and congruences questions. I have no idea what to expect.
• What do you need to work on understanding better before the exam?
• I haven't been able to even review yet because I have been doing the homework assignment, so I don't know what I need work on. 

5.1-5.4 due on September 29

1. I had some difficulty with the key schedule. I didn't understand the whole process of how the key was selected from the original key. I also had some trouble following the decryption; especially the part about why the MC step was not used on the last round of the encryption process.

2. I found it interesting that the other four algorithm methods where secure and could have future possible use. What was the criteria for selection? Why was the Rijndael picked? I also found interesting how the S-box construction was carefully chosen to maximized bit diffusion and open so to avoid the mystery that was common with DES.

Course Feedback due on September 26

• How long have you spent on the homework assignments? Did lecture and the reading prepare you for them?
• I have probably spent about 4-6 hours, I think. I think the lectures and the reading were good preparations to the homework. I think the homework also helped clarify things e.g. the LFSR sequence. 
• What has contributed most to your learning in this class thus far?
• I think lectures have been the most beneficial. I feel the book sometimes is a bit confusing and doesn't use good examples. During lectures, my classmates asks questions in which the answers help me understand as well. 
• What do you think would help you learn more effectively or make the class better for you?
• I have thought it would be nice to do a small scale real encryption, decryption and break, though it would seem not plausible to do a decent size plaintext. Seeing the actual process helps me understand. In the book the examples are still quite abstract, and I don't quite follow the process as well. 

3.11 due on September 24

1. I did pretty well understanding until 3.11.3 the LFSR Sequences, no comprende! I am still trying wot work out normal LFSR Sequences. I think I am getting hung up on the notation with the original and now they throw in all this other matrix notation with P(X) stuff, and I can't keep it straight.

2. I thought the addition and multiplication operations of GF(2^8) really interesting. It reminded me of linear algebra taking the coefficients and then XOR them. The idea of modding polynomials is a bit tripping but cool as well.

Sorry this is late. I started this post Wed night and ran out of time, so I saved the draft meaning to come back and finish it later. Since I had went through the action of blogger, my mind kind of checked it off, and I forgot that it needed to be finished.
Also now after doing the homework, I have a better understand how the LFSR works, so 1 is really not so relevant now.

4.5-4.8 due on September 22

1. I have difficulty seeing how errors are generated in the encryption process, for this reason, I have some trouble seeing the difference in the ability of the OFB to be better at error correction than the other modes. To me it seems as though if you program a computer to do something it does it exactly as it was told, so if there were errors, than the encryption prgram needs to be modified. I also had b it of trouble following the password security sections. I think my brain was shot after trying to follow the modes of operation. I think an example would be nice.
2. I found the section about breaking DES interesting, especially that even though it was known that DES was weak and vulnerable, it was still about for over 10 years. It is amazing to see the the difference in technology from when Rocke Verser broke the DES in '97 to the 39 days it took in '98.

4.1,4.2, and 4.4 due on September 20

1. I was lost most of the reading. I think what throws me off the worse it the S-boxes. I just can't quite figure out how they are used. They send in 4 bits, but how do they chose they out put. The example wasn't clear to me how that was done.

2. Cryptography uses a lot of different areas of mathematics. When I was reading section 4.4 DES Is Not a Group. I had to pause and recall what a group was from Math 371. I am going to have to pull out my abstract algebra book and review the properties of a group.

2.9-2.11 due on September 19

1. Ok the LFSR Sequence ... What is going on? I didn't follow ... well pretty much the whole section, so hopefully we will cover it in class.

2.  I thought the one-time pad interesting, but what struck me most interesting was the section about generating truly random bits. Having some experience using random generator, I have wondered how they worked. I now know how and that they were really not random but pseudo-random.

3.8 and 2.5-2.8 due on September 14

1. I had some difficulty understand the process of encrypting and decrypting the playfair cipher. It seemed like it was  suppose to be put in a matrix but the author didn't chose not to for an explanation. The fraction inverse of (mod n) is still a foreign idea to wrap my brain around.

2. I thought the block ciphers and the use of matrices in cryptography interesting. It seems like linear algebra is used every where in all sorts of places in mathematics, yet there is only one undergraduate course offered. Maybe there should be more linear algebra courses.

2.3 due on September 13

1. I think the most difficult part of understanding was the description of finding the key when the author began on vector Ao, Ai, Aj, etc. At the end, I caught the main jest of it and the probabilities, but it is still hazy.
2. I think the Vigenere Cipher is way cool and how it takes care of the frequency analysis of the letters. Even more cool is the methods for breaking it, even though they are not completely understood. Also the book with no e's is pretty interesting.

2.1-2., 2.4 due on September 10

1. I think the hard thing for me to follow was how they described breaking the affine cipher with ciphertext only. It appeared that it was just a brute force, but then where did the 20 character ciphertext come in from?
2. I found it interesting how modulo was used to describe shift ciphers and affine ciphers. I never have thought of a shift cipher as mod 26, but it sure works good to describe it. I liked how the book described the ways to break it the ciphers using ciphertext, known plaintext, chosen plaintext, and chosen cipher text. It seemed like they were both pretty easy to break.

Guest Lecturer, due on September 10

1. The thing most difficult for me to understand was the last bit she said about the general's wife and the problem with the their encryption system. I think it was just a problem because she was out of time and had to go over it quicker than expected.

2. I found it interesting how the leaders protected the identity of people that were friends of the church, as well has communicating in code for political correspondence. It makes sense with the persecution going on to keep such things hidden. It would be real easy for friends to become enemies after receiving persecution.

3.2-3.3, due on September 3

1. I found the last page and a half difficult to wrap my brain around. The first concept is the solutions to ax congruent to b mod n when the gcd(a,n) is not equal to 1. I think it is the relationship between the x is congruent to c mod m as a solution to ax congruent to b mod n. I just don't see how they relate.
The second concept is working with fractions. I can see the benefit when working with large numbers, but seems like a lot more work in the because you would have to know the multiplicative inverse or all the denominators one could divide by for the modulo.

2. Though the operations with congruences are somewhat difficult to get a wrap around, I find it the most interesting.  Congruences are now more than just an interesting fact using remainders. It will be interesting to see how these are applied.

1.1-1.2 and 3.1, due September 1

1. I had some difficulty when the readings went into detail about measuring the size of numbers. I didn't follow after the measure in the actual number of digits in a number's decimal representation. To me that seems like a good way to measure a number's size.

2. I was excited to see some proofs and theorems from 290 and 371. I wonder often in my math classes what is this, being some theorem, lemma, etc., good for? I often thought it was just for the sake of knowledge. As I read section 3.1, I was excited to see an application of Euclidean algorithm. I am interested in seeing an application of these principles.

Introduction, due on September 1

I am a senior in mathematics.  I have taken Math 290, 313, 314, 334, 341, 371, and 450. In 290 we covered a little about cryptography and have been intrigued ever since. So, I want to learn more about it.
I have some minor experience with Maple from 314. I have some programming experience from CS142. I wouldn't mind using a CAS for homework as long as there is plenty of time to learn how to use it or a good tutorial.
I really enjoyed taking 341 from Dr. Fearnley. He went at our pace. It gave us time to really discover and understand.
Something interesting about me, I can comfortably shoot a target at 1000 yards.