Section 16.5 due on December 11

I'm not sure if I understand why s needs to be mod n in the ElGamal signature scheme.  Other than that, this section made sense to me.

It's interesting how similar ElGamal and elliptic curve ElGamal are, and how the principles of one can be applied to the other.

Section 16.4, due Monday, December 9

This actually made a surprising amount of sense, though I'm confused how to determine the set of possible x values to put in an elliptic curve based on a finite field.  Do we just make one of the finite fields based on the type of finite fields we made before, where they were the set of polynomials modulo a polynomial with binary coefficients?

It was interesting to see how a finite field is used in this context, and also to see more applications of this concept.

Section 16.3, due Friday, December 6

This section was interesting, but I don't understand how it adds anything new to the factoring realm.  I think I'm not getting the bigger picture of how elliptic curves help us factor other than providing interesting opportunities to compute a GCD.

It is interesting to see the modern methods of factoring, and recognize that factoring is a more complex process than we recognize.

Section 16.2, due December 4

I don't understand how plaintext is encoded by elliptic curves.  I see how it's put in the x-point and how you have to be careful about making sure there's a square root to form the y-coordinate, but other than that, I'm lost.

It is interesting to think about the implications of elliptic curve cryptography.  I didn't expect to see it have ways to help with factoring, but I'm excited to see how it can lead to factoring algorithms.

Section 16.1, due on December 2

This section on elliptic curves is fascinating as well as being understandable.  The only main part that didn't make sense were some of the computations that were done in finding the third point in the addition algorithm.  Other than that, everything else made sense.  I am getting the feeling though that some things in here will use some more difficult concepts from abstract algebra.

This seems like a really interesting subject, and I'm interested to see how these curves will be applied to cryptography.  It seems like they aren't that connected, but I'm sure I'll see how wrong that thought is soon.

Assignment 37

Error correcting codes were rather hard for me to understand.  It makes sense why we need them, but how they work in practice did not make sense to me.  I don't understand how the error detecting and correcting algorithms work.

It was interesting to see how necessary these codes are.  I had never thought about the dangers of a noisy channel, but it makes sense that such error correcting codes would be needed in order to ensure clear communication.

Assignment 36

The enigma machine is quite an enigma to me.  I understand that it's mechanically generated, but I don't quite understand how the rotors produce different permutations of letters.  I understand that the rotors move, but I don't get how that creates different possibilities.

It is interesting to see an example of using technology in encrypt.  It's also amazing to see how much encryption has advanced, technology-wise, just in the past 70 years.

Assignment 35

Shor's Algorithm is very hard to understand, even from the nonmathematical perspective provided by the blog I read.  I especially don't understand the quantum Fourier transform.  This transform, and how it revealed the exponential period, didn't really make sense to me.

However, it was interesting to read about how quantum principles can be used to factor large numbers, especially since the advance isn't just in computing power, it's in how the computing is done.

Assignment 34

I'm not very clear on how these vectors in the complex plane work, or how the qubits are changed by the way in which they are measured.  It make sense that the particles are altered by the way they are measured, but I still don't get the nitty-gritty details behind all of this.

It was interesting to finally learn about the basis behind using a quantum computer, and the thought behind them.  I've always heard about them in science fiction books, but never have quite understood the theory behind them.

Assignment 32

What topics do I think are most important?  I think the basic principles behind RSA and the ElGamal encryption systems, as well as the principles of public key cryptography in general are important.  Also, the potential weaknesses of these systems are rather important to understand.

What kind of questions do I expect?  Perhaps using simplified versions of the cryptosystems we have studied recently, as well as applying the number theory we have learned behind the systems. I can also see primality testing being used.

What do I need to work on understanding?  I need to remember the continued fraction attack, as well as our methods for computing discrete logarithms.  I also need help with primality testing.

Assignment 31

I mainly got confused by the various names associated with the different methods.  The methods made sense, but the names and terminology associated with each of them was a little difficult to decipher.  Perhaps as I work through examples it will make more sense.

What was interesting about this section was that I had read about something similar in a Math Education magazine a week or two ago.  They talked about they system of creating lines so two people would be needed to find the secret.  It gave me a nice quick start into understanding this section.

Assignment 30

I am a little confused by the birthday attack on signatures.  I'm still not sure how that works in practice, or how one would do it.  I do understand that it is probabilistic, but I'm still not completely sure how it works.

Signatures in general are very interesting.  I'm surprised to see how much our public key encryption systems can be used in clever ways to produce signatures.  I was also surprised to see how a small change to a document could protect the signing.  I'll have to remember that trick to protecting one's self against fraudulence in an electronic document.

Assignment 29

In reading this section, I didn't quite understand how the birthday attack was to be used on weak hash functions, or how the multicollisions described would occur with such frequency.  Also, the reasoning behind why concatenating weak hash functions doesn't increase the strength was rather mysterious to me.

Still, it was interesting to see how hash functions could be applied crytographically.  I think the it's interesting to try to find things that are pseudorandom or appear random, and thus can be used in cryptography.

Assignment 28

I am still surprised by the nature of one way functions.  I'm a little confused though as to how the exponential hash function example provides a way for us to solve the discrete log problem, and how the parts of it are so correlative to it.  I'm not sure if they're providing a way to solve that is computationally unfeasible or just unlikely to occur.

Hash functions are quite interesting because they are not technically bijections, but they behave like them, and it is in computational power that they become difficult to break.  It's interesting how we can use computational prowess to get around logical difficulties.

Assignment 27

This sections was interesting in its conceptual implications.  However, bit commitment was a bit confusing to me.  I'm don't completely understand how the discrete logarithm is applied to ensure bit commitment.

As a whole, this section was rather interesting in its practical applications.  Bit commitment and key generation are important things to do in cryptography; it's interesting to see these practical problems solved in a clever, mathematical way.

Assignment 26

Well.  This assignment had some crazy algorithms.  I did not understand the Pohlig-Hellman Algorithm or the Index Calculus.  In Pohlig-Hellman, I don't understand where they get the strange expansion of x.  I don't understand that congruence.  Also, in the index calculus, I got lost right at the beginning of reading it and didn't really get farther.  I see they're using logarithms in their factor base, but I don't get it from there.

Overall, this was interesting to see how there are different ways to get around the big discrete log problem.  Still, I'm interested to see better how these work, and how this public key system compares to RSA.

Assignment 25

I found the one-way functions rather interesting in the reading today.  Also, the notion of one-way functions is interesting.  It was a little difficult to understand the theory behind one-way functions and public key theory in general.  I'm wondering if there will ever be a time that our "one-way" functions are no longer one way, either due to our computing power or mathematical skills.

It is interesting to recognize that one-way functions exist, where the original function is very accessible but the inverse is not.  I wonder if there is an efficient one-way function out there; I know RSA is sometimes unfeasible to use because of time constraints.  Perhaps there are more feasible public-key systems that are more commonly used.

Assignment 24

The quadratic sieve presented in this section was rather interesting.  I'm still unclear on how they choose the various values of i and j to produce the matrix in which we look for linear combinations.  Also, I'm curious if there's an easy way to find linear combinations among the rows in the matrix we produce.

This section provided some interesting thinking material when it comes to various methods of factoring.  I'm surprised at how creative one can get in factoring just using the basic principle we've been using.  Are there other "basic principles" we could use to build factoring methods?

Assignment 23

I was perplexed by the computations involved in the p-1 factoring algorithm.  I wasn't sure if the B! means that we eventually hit B factorial or if they are describing a process that is similar to factorial.  I'm also not sure how we use b, once we compute it.

I was interested by all the different methods available for factoring.  While the basic factoring method we have always used is very straightforward, it's not very functional for large numbers.  It's not too bad to lift a 50 lb bag of potatoes by hand, but lifting an engine block by hand, while being straightforward, is not always the best idea.  I feel like the factoring methods we're learning are going to help prevent us from lifting mathematical engine blocks by hand.

Assignment 22

For this reading assignment, one of the more difficult things was understanding how the primality tests work, and how they produce pseudoprimes.  The sequences that reach 1 and -1 at different times were confusing to me, especially as to how they affected the output of our tests.

What I thought was interesting is that the tests we used did not prove primality, they could only establish a number being composite or a high probability of a number being prime.  It makes sense that these faster algorithms would not be able to provide proofs, but would be able to point us in the right direction.

Assignment 21

The notation developed in this chapter was a little hard to piece together, though it seems rather powerful.  I didn't quite get how to do the calculations for the Legendre symbol, but I'm sure it'll come with practice.  The properties of the symbol don't quite make sense to me, to be honest.

This was an interesting example of how powerful notation can be.  These symbols don't just appear; they were invented by Legendre and Jacobi, yet they have far reaching theoretical effects.  It's interesting to me to see how such little things can do so much in the world of mathematics.

Assignment 20

This assignment was a little difficult in understanding why a would be equivalent to b (mod p) but a is also equivalent to -b (mod q) when a and b are square roots of some number modulo n.  I understand how to use these facts, but I'm still a little shaky on why they work.

In looking at this section, I'm fascinated by the fact that these problems can have 4 square roots instead of the normal 2 we would expect.  Modular arithmetic just always seems to be full of surprises.

Assignment 19

In reading this assignment, I was surprised at the unorthodox methods possible in attacking RSA.  The one that was difficult to understand was how the variance helped us use timing to attack RSA.  It makes sense to use the mean to attack RSA, so you know how mush time one would expect each number to take in the decryption process, but I didn't quite follow the use of the variance.

Still, the timing attack to me was fascinating.  It just goes to show that often there are unforeseen holes in the things we use in life.  Something that is very strong can have a weakness in a part you didn't expect.  We can't put too much trust in security systems; as their improved, people will find more and more ingenious ways of attacking them.  Cryptography is no simple game.

Assignment 18

So this was an interesting section to read.  It was a little difficult to follow the algorithm described in the chapter for building the continued fraction, but I'm sure that will be resolved with practice.  The random theorem given with the absolute value and the s^2 is a little interesting, I'm not sure how it's useful.  But I'm sure I'll find out.

It's been bothering me trying to remember when I've done continued fractions before.  I think we did it in history of math, but I can't remember why.  Either way, I'm interested to see how this applied to cryptography.  Much of the number theory we've done before was a little more obviously applicable.  I'm interested to see how this one fits in.

Assignment 17

RSA encryption is an amazing thing.  However, I am a little confused by how to find d, the inverse of e mod (p - 1)(q -1).  I think that will be resolved just by seeing it done/actually doing it myself.

I was just amazed by how simple RSA is compared to DES and AES.  It is such a simple algorithm.  It's a beast to compute, but it's so simple, yet so secure.  I'm just impressed by how slick RSA is.

Assignment 16

The most difficult thing for me was understanding the three-pass protocol.  It looks like their taking primes to powers now that I'm looking at it.  Okay, now that I'm blogging about it and thinking about it more, it's starting to make sense.  Alrighty then.  It's Euler's phi function that's tripping me up the most now.  Do we need the prime factorization of the number of do it?

What was interesting to me was the concept of primitive roots.  It seems related to the multiplicative generators we found in our modular arithmetic with polynomials.  It seems to me that when we were finding generators, we were really finding primitive roots of the polynomials.  It's nice to see the numerical equivalent of these roots.  I'm excited to learn more about these primitive roots.  And while the three-pass protocol was a little hard to decipher, it's really quite clever now that I think about it.  I'm impressed with those who came up with it, especially since it allows Alice and Bob to exchange a key over an open channel.

Assignment 15

It was difficult to understand exactly how the proof to the Chinese Remainder theorem works, those I supposed that won't be too essential for our work.  Also, I still don't quite get how the x^2 problem in the book yielded four solutions.  That was a little confusing, as well as the general rule for such problems.  I wasn't sure why that worked, or how it was a consequence of the general form of the Chinese Remainder Theorem.

What was interesting was the fact that I got to finally learn about the Chinese Remainder Theorem.  It's something I've heard about for years but never actually learned about.  It's nice to finally have the mystery of this theorem solved in my life.  I've wondered for quite awhile what this theorem is, and now I finally know.

Assignment 14

Out of the topics we've studied, I would say division in modular arithmetic, as well as the specific coding systems of DES of AES are the most important.  While both DES and AES are a little complicated to memorize, I think it's good to understand their basic structure in order to understand how each cryptosystem generates confusion and diffusion yet still works in an orderly enough manner that a computer can accomplish the encryption and decryption process.

I expect, on the exam, to see questions similar to those on the homework, such as finding GCDs using the the extended Euclidean algorithm, solving modular algebraic expressions, and perhaps breaking a simple substitution cipher.  In relation to the most recent homework, I can see generating simple s-boxes as possible exam questions, as well as finding and using a generator for a finite field.

I personally need to work on assimilating the knowledge into a cohesive whole.  I will be missing the exam review due to marching band, but I think if I can make sure I am comfortable with finite fields and the modular stuff, I should be okay.  I'm definitely going to be going through the study guide on the band bus.  It's going to be a fun road trip.  I need to personally figure out the modes of operation, and make sure I understand how each of those work, and be able to apply them in a simple case.  I think that will be the big part for me.  I hope this exam goes well.

Assignment 12

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

Personally, I tend to spend 1-2 hours on my homework assignments.  Lecture has been my biggest preparation for homework assignments, though in the last couple of assignments, the book has been more of a help.  I was able to look up the things I didn't understand in lecture and apply them in the homework.

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

Honestly, I would say the homework has contributed the most to my learning.  I usually understand the lecture, the reading is decently comprehensible, but when I get to apply it in the homework, things make much more sense.

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

The one thing I can think of is more example problems in class.  I know when we work on problems in class, or split into groups like we did for the substitution cipher we did on the first day of class, I understand the material better because I'm able to apply it.  However, I do understand that time may be an issue in putting that into practice.

Assignment 11

I have never worked with fields before, so this whole discussion about Galois fields was pretty crazy to me. The definition of a field makes sense to me, though.  I'm struggling with seeing how the Euclidean Algorithm will work on these binary polynomial fields, but I suspect as I work with them more, I'll figure them out.

It was good to finally read a good discussion about a field.  Fields are things I've heard about in other math classes, but I've never had one rigorously defined for me.  I'm excited to learn more about fields, and how they apply to cryptography.  I'm worried my lack of experience with group theory might make things difficult, but I'm sure things will work out in the end.  This is getting interesting; the mathematics are definitely ramping up from what they've been before.

Assignment 10

It was difficult for me to understand OFB.  I understand that it is a way of implementing DES, but I don't get it.  I know it's good for error correction.  I didn't understand most of this DES section, but it was all very interesting.  The five methods of implementing DES were confusing to me, though it made sense that we would sometimes need to correct for messages that were not perfect multiples of the block length.

I did not realize how much computers have improved when it comes to an ability to do calculations.  It really shouldn't surprise me, just with all the graphics available, but it's amazing how much a well-programmed computer can do.

Assignment 9

Perhaps the most difficult part of this reading assignment was the proof at the end, where we proved that DES is not a group.  I didn't quite understand that proof, but I assume that is mostly because I have not taken abstract algebra yet.  In general, I understood the section, but barely.  I feel like I have a tenuous hold on what the text was trying to say.

What was quite interesting to me was actually the proof I didn't understand.  The last part of the proof, where they show that m has to be greater than 10^(277), thus making DES not closed under composition was very interesting to me.  The proof used mainly a numerical method to show a theoretical result, or at least that's how it seemed to me.  Perhaps I misunderstood the proof, which is entirely possible, but either way, the method for doing this proof was quite interesting to me.  There are definitely more ways to prove things than I had before imagined.

Assignment 8

I will be completely honest.  I didn't really understand this reading assignment at all.  Specifically, I had trouble understanding a linear feedback shift register.  I understand that it's supposed to be a fast form of encryption, but it did not make sense to me.

On the flip side, it was interesting to learn about the one-time pad.  I have heard a lot about this cipher, and have always wondered what makes it so unbreakable.  The concept is fascinating, and it makes sense why it is so hard to use and thus often not used.  I think a lot of the mathematics of the coming sections are about to get more interesting than the mathematics we have dealt with so far.

Assignment 7

This reading was a little more difficult than past readings.  Maybe it's because it's been awhile since I've taken linear algebra, but block ciphers just did not make sense to me.  Maybe it's because I didn't quite get how matrix inverses mod n work, but I think I'm getting there.  I can tell it has to do with finding the adjugate matrix, which I may have to review in order to figure this out.  But this looks interesting.

I liked learning about the Sherlock Holmes cipher, and the errors the author made in the dancing-man ciphertext.  I think that's a good warning to us, to be sure not to take our codes too lightly.  It can be easy to get too confident and mess things up.  Also, it reminded me of the code we had to break for homework, where in breaking one of the letters we found that the frequency of e in that string was actually lower than usual.  Things are not always what they seem in cryptography; that's what makes it such an interesting field of study.

Assignment 6

This sections was very interesting to read, though the methods of decrypting a Vigenere cipher are still a little fuzzy to me.  It makes sense how the cipher works, but the method of finding the key using dot products still doesn't exactly make sense to me.  I kind of get it, but it's a little confusing why the maximum dot product appears when the proper shift is found.

However, this code seems very usable to me.  It sounds quite familiar, actually.  I think it might have been used the movie "The Prestige" to encode a journal.  I'm not sure, though.  It makes sense now how a cipher can be developed that uses multiple keys.  In fact, encryption keys in general make a lot more sense because of this cipher.  I like it.  I kind of want to start sending messages with this cipher.

Assignment 5

Probably the things I least understood about the readings in section 2.1-2.2 was how the attack method of chosen plaintext and chosen ciphertext work.  The other methods of attack made sense, especially those in the affine section, since they were related to how one would find the equation for the line, given two points.  The only difficulty in the known plaintext method of attack for me was the problem that arose when there were two solutions, I didn't quite understand that method.

Section 2.4 was especially interesting to me, since it showed me the power of a properly used substitution cipher.  I would have never thought to use digrams to help crack a substitution cipher, but it makes so much sense to do so.  There is much more to cryptography than I realize.

Assignment 4: Guest Lecturer

This guest lecturer was actually not my first introduction to cryptography in LDS history.  A few years ago, I read a fictional novel called The Moroni Code, which discussed codes used by prominent church leaders.  The main code discussed in that book was the masonic code we discussed in class.  Still, it was enlightening to see the variety of codes that have been used throughout history.

It makes sense that such codes would need to be used, especially considering the persecution experienced by church leaders.  What was interesting to me about this lecture was how cryptography was used in the Doctrine and Covenants.  I had always seen the section headings that discussed unusual names, but never knew what said unusual names were, and whose names they hid.

In the current day, I wonder what codes the churches use.  I'm sure there are sensitive documents and messages that the church does not want to be intercepted and read.  Thus, the Church must still be using some sort of encryption system, if not more than one.

Assignment 3

The most difficult thing about this section for me is understanding the extended Euclidean algorithm.  The method for using the extended Euclidean algorithm was a little confusing to me.  I supposed if I had to practice using the algorithm it would make more sense.  But, the thing that bothers me more is I still don't fully understand why the extended Euclidean algorithm works.  I like knowing the proofs of things, oddly enough.  I like seeing why something works.

It was interesting to read more about number theory.  I had forgotten how much I love learning about greatest common factors and number theory in general.  Maybe I need to take a number theory class.  I remember working with number theory in a math education class, and seeing how different methods students used to find GCDs turned out to be mathematically correct, and the reasons behind the methods.  I don't think I will use the number theory used in this class much in my future professional life, but I know I will need to understand the number theory in this course in order to properly teach it to my future students.

Assignment 2

Probably the most difficult about this material was understanding how the proof of the theorem about ax + by = d, where d = gcd(a,b), works.  I know I have seen the proof before in 290, but the proof in this book for some reason isn't clicking for me, especially since the proof makes the theorem more general.  It makes sense to me that a and b must differ by a multiple of the gcd, but the proof seems to take a different route than I have before seen in proving this fact.

The part about primes was probably the most interesting to me because that was the part that I could understand the most.  The parts in chapter one about the basis for cryptography were interesting, but I wasn't quite sure how the mathematics would come into play, especially in talking about how the magnitude of the number n correlates to log(n).  The part about prime numbers and greatest common divisors reminded me about discussions we had in my first math education class about finding greatest common divisors and least common multiples.  For some reason, I found prime factorizations fascinating.  I was surprised by all of the methods for finding both the LCM and the GCD, especially in seeing how each method was related.  I'm curious to see if finding LCMs will show up in this class.

Assignment 1: Introduction

I am a junior at BYU studying math education.  Post-calculus, I have Math 290, Math 313, Math 314, Math 341, and Math 362 (Geometry).  I am taking cryptography to fill the Math Ed requirement for a higher math class and it sounded more interesting than the other available classes.  I also have some friends taking this class, which makes it nice.  I have no experience with Maple, Mathematica, or SAGE outside of using Wolfram Alpha.  However, I tend to be a fast learner.  In working on the first homework assignment, SAGE started to make sense to me.

A math professor that had a great effect on me was Dr. Pace Nielson, my Math 313 professor.  I think his effectiveness is best illustrateded by something he did after I was done with his class.  I ran into him while I was walking home from campus, and he asked me about my life and we just talked.  He still remembered my name and was interested in my life.  During his class, I could tell that Dr. Nielson wanted us to succeed.  He tried to make sure we as students could succeed while still challenging us.  His tests were not easy, but he always tried to make sure we were prepared for them.

Something interesting about me is that I am a proud member of the BYU Marching Band.  This is my third season with Cougar Band and I love it.  The Cougar Marching Band owns my heart every fall.  I love the camaraderie and the excitement of the band.  I especially love how the band director has the vision of the Cougar Band bringing people to Christ.  We don't play hymns all the time, but we seek to play with precision and accuracy so people can see in us disciples of Christ seeking to excel.

I can come to the office hours listed on the syllabus, though if need be Monday, Wednesday, and Friday at 10 am are also great times for me.