Section 16.5 LAST BLOG EVER!

It is interesting that modular multiplication and modular exponentiation can be made to be analog and used with elliptic curves.

It was difficult to follow the explanation of the elliptic curve way of doing a diffie-hellman key exchange.

Section 16.4

It is interesting that elliptic curves must be changed slightly when working mod 2. This is because of singularities.

I didn't follow the explanation of why GF(4) is necessary for elliptic curves.

Section 16.3

It is interesting that a number with only small factors is smooth and if all are less than B it is B-smooth. This makes a lot of sense in the context of elliptic curves.

It was hard to follow the exact algorithm to factor a number using elliptic curves.

Section 16.2

I easily understood the explanation of elliptic curves mod p and how to solve them by following the example.

However the example given representing some plaintext was overly complicated and i couldn't not follow the result.

Section 16.1

The discussion on elliptic curves was very interesting. I was particularly intrigued by the use of elliptic curves to find a point on the curve using two other points.

I didn't really follow the examples on how to solve an elliptic curve mod p.

Section 2.12

Enigma is a very interesting system. It seems to be the first crypto system that actually uses a machine. It seems incredible that it could have been broken. Since not even the user knows the exact connection between the plain and cipher texts.

It was hard to follow the attack that broke Enigma. I didn't understand what the different letters represented and how the machine could be used to solve the problem.

Shor and 19.3

It was very interesting to read about the way that the period was describe without math. This helps to visualize the real meaning of the algorithm that we use.

It was difficult to follow the explanation of the actual algorithm in the book. Mostly because i did not understand all the notation.

Section 19.1-19.2

It was interesting to read about the "hypothetical" quantum computer. It is incredible that we can study to a certain extent the quantum mechanical phenomena that take place on the atomic level.

It was difficult to understand how quantum bits are much more efficient than binary bits and how to implement their use.

Section 14.1-14.2

It was easy to understand the reasons for zero knowledge techniques because of the illustration of a real life problem in the beginning. The algorithm makes sense when considering the preceding explanation.

However the fiege-fiat-Shamir algorithm is harder to understand and follow.

Section 12.1-12.2

I learned about the tw threshold scheme which requires a certain number of participants to reconstruct the message but no less. This is an interesting approach that allows for some leeway.

However, It was difficult to follow the approach described for the Shamir threshold scheme.

Midterm Review

RSA, Factoring and discrete logs have been the most common of the things we've discussed since the last midterm.

I expect to see RSA, factoring, discrete logs, different ways of solving weak systems, and birthday attacks on the test.

I need to be able to remember how to do a crypto-system and recognize it's name. I also need to review hash functions.

Section 8.3 and 9.5

It was very difficult to follow the technical explanation of the SHA.

It was interesting to learn about the process for creating a digital signature. It is helpful because it is faster than other methods we have studied.

Section 9.1-9.4

It is interesting to recognize the importance and uses of digital signatures. Hash functions make the use of a digital signature even easier, but with the same security.

I don't really understand how a few extra spaces can make a document fraudulent and how the birthday attack affects this scheme.

Section 8.4, 8.5, 8.7

It was interesting to learn about birthday attacks. With so many options for birthdays i had no idea that the odds were so high to have a birthday match up.

It was difficult to understand the steps that are taken to avoid multi-collisions.

Section 8.1-2

It was interesting to read about how a hash function can be strongly collision free. It seems that in most cryptosystems it is highly unlikely that two ciphertexts will match when the messages do not. For hash functions it is hard to ensure this.

It is difficult to understand how discrete logs apply to hash functions and how the proposition described in 8.1 works.

Section 7.3-7.5

It is interesting that a way has been devised to commit to the information that is transferred as in the dilemma with football games, so that neither party can change the message. It seems that cryptographers have thought of everything.

It was very confusing to read through the explanation of El Gamal. It seems very similar to RSA, but using discrete logs.

Section 7.2

I am thoroughly confused by procedure illustrated for pohlig-Hellman algorithm.

The part about the index calculus was very interesting. It is interesting that series of relations allows us to solve for the discrete log.

Sections 6.5-7 and 7.1

It was really interesting to read about trapdoors and their usefulness and pitfalls.

I was intrigued by discrete logs but i didn't really understand the discussion of one-way functions and why they are only "probably" one way?

Section 6.4.1 and 6.4.2

The quadratic Seive is a very interesting method. It is interesting that the setup works for any modulus. However, it seems difficult to find the squares.

The universal exponent factorization method is very complicated. I didn't understand where the value of u came from.

Section 6.4

I was thoroughly confused by the p-1 factoring algorithm. What is B! And how does it affect the factors of n?

It is interesting to see the connection between a difference of squares and factoring numbers.

Section 6.3

I think it is interesting that the Fermat primality test tells when a number is composite and when its not true tells that a number is probably prime. This seems like a very vague condition for mathematics.

I didn't understand the Miller-Rabin Primality test. I don't understand why the process associated with it. 

Section 3.10

It is very interesting how the properties of the Jacobi symbols allows us to solve a large value problem in only a few steps. The definition is barely used.

I didn't understand the final explication about how jacobi and legendre symbols applies to RSA. What is quadratic residuosity?

Section 3.9

It was hard to understand the proof for square roots mod n. I don't follow the logic.

It is interesting that finding the square roots of a number that is relatively prime to n is as easy as factoring n.

Mathematics of Voting procedures

Dr. Jenkins,
I wrote this last week, but i just realized it didn't post. Sorry.

Dr. Saari, explained a way that each of three "candidates" can win based on the type of voting method, plurality, pairwise, or a runoff election. The question is what does that do to democracy? These types of phenomena is surprisingly likely. Plurality method is one point for first place and zero for all others. Another method is to weight each place with a certain number of points. One conclusion is that anything is possible no matter the voting method. Dr. Saari could fix any election using mathematics.
One way to measure good voting procedures is to see if actual ties are made to be ties using the voting procedures. In the end, it is clear that voting is arbitrary and very vulnerable and that Voting is everywhere.

Section 6.2

The discussion of timing attacks was very interesting. Who knew that that kind of thing was possible.

It was hard to follow the short plaintext attack. I didn't understand about OAEP.

Section 3.12

The theorem in this section is very interesting. It is a very interesting way to express an irrational number.

However, I'm still wondering how we are going to use repeated fractions in cryptography.

Section 6.1

It was super interesting to read about RSA and its capability to be decrypted by the receiver, but not cracked if intercepted by the enemy.

I understood how the algorithm worked, but i just couldn't understand why it works.

Sections 3.5-3.6

I was very interested to learn about primitive roots. It is very cool that numbers can work that way. Is it just a coincidence? How does that work?

I didn't understand the proof of Euler's function.

Test Review

I think the most important things we have learned is the names and algorithms for encrypting using different ciphers. The most common of the things we've learned are DES and AES, so we should probably know those.

I expect proofs and questions about what type of attack will work on which cipher.

I need to study more about the modes of operation and the things that we studied this week.

Sections 5.1-5.4

It was interesting to read that AES was chosen from 15 crypto-systems. I wonder what was different and inherently better about AES than the other options.

It understood most of the reasons that AES was designed in this way, but i had one questions. Why is it that if you change one input byte all four outputs are changed but if you change two, then at least three are changed?

Questions

Each homework assignment took around 5 -7 hours. Each assignment was a learning opportunity. I felt that i knew where to look when i struggled with the assignments because of the reading and the lectures  introduced the concepts but the assignments required me to delve in deeper.

I think lectures make all the difference. Each unclear concept from the reading is clearly explained in class.

I need to start the assignments earlier in the week in order to not be rushed to finish them.

Section 3.11

I found this section very similar to abstract algebra. It was interesting to see how finite fields apply to cryptography.

I was confused by the GF field and its properties.

Section 4.5-4.8

I was interested to learn a little history about the breaking of DES.  It is clear how hard it is to change the standard in encryption because of the many problems it solves. It is interesting to read about salt and how it can make DES more secure. I understand a little more why my passwords must be secure.

It was difficult to understand the different modes of operation, such as Cipher Feedback and Output feedback. I don't understand how the functions work.

Sections 4.1,4.2,4.4

I found the discussion on DES very interesting. I like how the many permutations kept it secure for so long.

I was thoroughly confused by the example given to illustrate the DES algorithm.

Section 2.9-2.11

I was thoroughly intrigued by the one way function that creates randomness. I am interested to understand more about random numbers and how to find them.

On the other hand, i was thoroughly confused by LFSR sequences and how the recurrence formula's describe them.

Sections 3.8, 2.5-2.8

The Sherlock Holme's story was very interesting and it was interesting to read how each of the ciphers discussed can be broken.

I do not understand the concept of binary, so it was difficult to follow the section about binary.

Section 2.3 Due September 10

I thought it was very interesting how using process of elimination and the frequency of letters, the encryption key could be found fairly easily. But this method seems to rely heavily on the frequency being known and perfectly followed.

I was not able to understand the second method for finding the key. How did the knowledge that the larges dot product .0713 lead the author to the first shift being 2??

3.2-3.3 due 8/31

There was nothing to difficult in this section because I feel that I've learned it all in 371. However I feel that it will get complicated dealing with fractions mod n.

I thought it was very interesting that the book discussed division in mod n with out even bring up rings or fields. It shows that the concept is not the important part but the application is.

Section 1.1, 1.2 and 3.1 due on 8/29

Reading about the types of keys, ciphers, attacks, and applications was very confusing, it was hard to keep track of each separate definition and description.

I thought it was very interesting that factoring is the key to a difficult to factor code. It is something we can easily do with low numbers but it becomes exponentially harder as the numbers get larger.

Introduction Due 8/29

I am a Senior majoring in Math Education.
I have taken, 290, 313, 314, 334, and 371.
I think the topic is super interesting and can be a very fun application for mathematics.
I have used Maple and Mathematica, but I don't remember much. I will refresh myself before using them for homework.
Dr. Jenkins, you have been my very best math professor at BYU. Your explanations are very clear and concise. I was thoroughly challenged in Math 371, but i never felt like i was drowning.
I served a mission in Curitiba, Brazil.
I am able to come your office hours.