Section 16.5, due December 5

The hardest part of the section was the explanation of the El Gamal digital signature algorithm for elliptic curve cryptosystems.

It's interesting how much simpler certain cryptographic algorithms can be when they are implemented using elliptic curves.  This is because the operations that are analogous to modular multiplication and modular exponentiation on elliptic curves are much easier to perform, but are just as strong cryptographically as systems implemented using large prime numbers.  It's interesting to see alternate ways of implementing familiar cryptosystems.

Section 16.4, due December 3

This was a pretty confusing section.  I'm having a hard time understanding elliptic curves over finite fields.  An longer explanation of the example in the section would really help.

Elliptic curves are interesting but they just seem to become less and less and intuitive the more I learn about them.  It is interesting that NIST recommended 15 specific elliptic curves (mod 2) for use in elliptic curve cryptosystems.  I might be wrong, but it seems that ECC is easier to implement than RSA since you choose from fewer curves, and key generation is easier than RSA.  But it's still a strong system.  That's just my initial thought.

Dr. Cristian Tomasetti, Colloquium, Extra Credit

I didn't fully understand everything that Dr. Tomasetti talked about.  I was unfamiliar with some of the math that he talked about, so probably the most difficult part of the talk was just when he was going over some of the formulas he derived.

I really enjoyed Dr. Tomasetti's presentation.  I've been interested in cancer research for a long time, and I would definitely like to go into mathematical biology to have a chance to do the kind of work that Dr. Tomasetti does.  It was very fascinating to learn of the correlation between the age of cancer patients upon diagnosis and the number of pre-cancer phase mutations in those same patients.  I would definitely be interested in learning more about what researchers are doing to find ways to differentiate between which mutations lead to proliferation of passengers and which lead to proliferation of drivers.

Section 16.2, due November 28

The hardest part of the reading was the part about using elliptic curves to represent plaintext. 

Elliptic curves are definitely strange and not one of the most intuitive mathematical concepts that I've encountered.  But I definitely do want to know more about how they can be used in a cryptosystem.  Looking at elliptic curves mod p seems like it is definitely be a lot simpler than just looking at elliptic curves on the real numbers. 

Dr. Dave Richeson, Focus on Math, Extra Credit

The most difficult part of Dr. Richeson's talk to understand was some of the explanations of how certain constructions were done.  A few of them were just a little hard to follow.

I enjoyed Dr. Richeson's talk.  I'm taking a class on the history of math this semester, so a lot of his talk dovetailed nicely with what I've learned about geometry and geometric constructions.  They do seem like very elementary problems (even though they're impossible), and they might not have important applications to the real world, but it is still very interesting to see what mathematics is (and is not) capable of.

Section 16.1, due November 26

The most difficult part of the section was the part about the addition law for elliptic curves.  It wasn't super difficult to understand, but an example would be very helpful.

I think the most important part of the section, at least as it pertains to cryptography, is the fact that elliptic curve systems can do with 313 bit keys what certain conventional systems can only do with 4096 bit keys.  The is vast improvement in efficiency, made possible by a creative cryptosystem built on powerful mathematics.

Section 2.12, due November 20

This wasn't a very difficult section, but a more in depth explanation of how the enigma worked would be helpful, though I think I basically understand it.

But I have always been pretty interesting in the Enigma system.  I read The Code Book a long time ago and had the chance to read about the history of the Enigma and its effect on World War II.  Although cryptography has many non-civilian applications, it seems that it will always be an important part of warfare, for good or ill.

Section 19.3 and Blog Post, due November 19

The most difficult part of the reading was the explanation of the Quantum Fourier transform.

I had never thought about how superposition of states could be used as an effective tool in solving the problems of RSA (namely, factoring a large integer).  However, it is an idea that correctly applied has the potential to work.  Indeed, it has been shown that it can be solved provided a quantum computer.  So the big obstacle here will just be actually building one, which could be a huge obstacle.  Or maybe the government already has one and we just don't know it...

Section 19.1 and 19.2, due November 16

I had a little trouble understanding the explanation in the beginning about the polarity of particles and how that applies to sending messages through quantum cryptography.

The most interesting part of this section was definitely the mention of the possibility of factoring integers in polynomial time using an appropriate quantum computer.  If this exists now, that could mean that RSA is totally open to anyone who has it.  However, I have no idea what the actually likelihood is of the existence of a quantum computer is, or what a quantum computer is even, but I definitely want to know more.

Sections 12.1 and 12.2, due November 12

The most difficult part of the reading was the part about the method to find the secret message using the of the message pairs and the Lagrange interpolating polynomial.  An in class example will help a lot.

Secret splitting is a really interesting principle, and one that I had never really thought about.  It can be implemented pretty simply and it would be infeasible to find the secret message without the correct amount of people.  Also, I learned about Lagrange polynomials in my numerical methods not too long ago.  I definitely didn't think there would be an interesting cryptographic application of Lagrange interpolation.

Section 8.3 and 9.5, due November 6

I think the most difficult part of the reading was the Secure Hash algorithm.  A better explanation of it would be great.

I like the digital signature algorithm.  It's simple to understand and a great application of RSA.  It adds a deeper level of security to RSA and makes it that much more powerful.

Section 9.1-9.4, due November 5

The most difficult part of the reading was the algorithm behind the ElGamal signature scheme and exactly how it uses discrete logarithms to generate signatures.

I really enjoyed the article about Zach Harris.  It definitely helped me to understand the importance of digital signatures for emails.  It also helped me to understand that internet security isn't perfect, and that people are always looking for ways around it, and sometimes a way around can be pretty simple and can be easily overlooked.

Dr. Chin Ling Guo, Math Biology Seminar, Extra Credit

On Thursday Nov. 1st I attended a Math Biology Seminar presented by Chin Ling Guo.  It was about a project he's been doing in simulating the self-organization of epithelial tubules.

The most difficult aspect of the presentation was simply that the presenter had a more biological than mathematical background, so while the presentation was not terribly difficult to understand, some of the background explanations of the biology were a little difficult to understand.

The work that the presenter has been doing focuses on using mechanical cell-cell interactions in order to cause epithelial cells to self-organize into long tubules.  Up until recently, many scientists have thought that causing this self-organization would require chemical cues, but by allowing cells to associate in the right type of extracellular matrix, mechanical associations allow the cells to self-organize properly.

8.1-8.2, due October 31

I'm not sure I really understand what a collision is, or what a weakly collision-free hash function is.  Also, I'm not sure I quite understand what the purpose of a hash function is.

Hash functions are a neat application of functions that are easy to evaluate, but difficult to invert, since we don't know what the preimage of the function is.  Cryptography is built on some very simple principles, but all the same, they provide the necessary complexity for message security, or in this case to produce a digital signature for instance.  With hash functions, the most simple properties of functions provide the basis for an important cryptographic implementation.

7.3-7.5, due October 29

The most difficult part of this reading assignment was the section about security of ElGamal ciphertexts. Mainly I just had difficulty understand the proposition given in section 7.5.1.

I enjoyed reading about other ways in which the principles of public key cryptography can be implemented; specifically through Diffie-Hellman and ElGamal.  The security of these systems, as in the case of RSA, is built upon mathematical operations which are computationally infeasible.  This has made me wonder about other mathematical operations which are computationally infeasible, and thus could somehow serve as the basis of a public key cryptography algorithm.

7.2, due October 26

The most difficult part of section 7.2 to understand is the part about finding discrete logarithms using the index calculus.

Discrete logs are a completely new  topic for me, though I feel like I've heard the term a lot before.  Anyway it's totally different from what I thought it was.  The analogs of certain operations, such as exponentiation, taking square roots, and taking logarithms, in modular arithmetic are very interesting to me.  I'm encountering most of them for the first time in this class, but it's definitely helping me to realize that modular arithmetic is a broad field of study with lots of important applications.

6.5-6.7, 7.1 due October 24

The most difficult part of the reading was the application about treaty verification.  I just wasn't really sure what was meant by using RSA in reverse.

I loved the section about the RSA challenge.  It serves as an awesome demonstration of just how powerful RSA can be, but also how incredibly ingenious mathematicians, engineers, and computer scientists can be when doing their best to solve a difficult problem.  The statement of the challenge itself is so simple, as is the answer.  Howeve, tt took 600 people with 1600 computers working in spare time to generate the necessary information to decrypt the message.  Amazing.

6.4.1 and 6.4.2, due October 21

The most difficult part of the reading was the part about the matrix formed using the squares of prime factors of certain numbers used for the quadratic sieve, and what it means to look for linear dependencies (mod 2).

I loved the seeing the table that listed factorization records up to 200 digits, although I'm guessing that a lot of these cases where such large numbers were factored (129 and greater) had a good bit to do with being extremely lucky, and having lots of parallel processing power working on factoring a single number for a long time.  Still it's pretty amazing that someone has been able to factor a 200 digit number.

6.4.0, due October 19

I think one of the things I was having difficulty understanding is why we're learning so much about factoring, when in practice, the kinds of integers that will actually be used in implementing RSA cannot be factored in time shorter than the life of the universe.  Although when I think about it, they are important principles of number theory that I'm sure can help with understanding other concepts of cryptography. 

I never would have imagined when I first learned about something as simple as factoring numbers that it could become impossible, at least in a reasonable amount of time for numbers with many digits.  It seems like something that should be simple, like dividing or multiplying.  The fact that it is so difficult is also good for cryptography since it makes RSA a useful algorithm.  I guess one day when a way to factor massive integers has been discovered, we'll just have to start using a new algorithm.  Although I have no idea how close that day actually is.  That will be amazing though.

6.3, due October 17

The most difficult part of this reading was the part about the Miller-Rabin primality test.  I just didn't really understand how it's performed or the part about probability of getting a prime around a certain number.

It is really interesting to know that there are methods for testing whether a large integer is composite.  Though its not the same as factoring it, that can be still a very useful piece of knowledge.  Such as when the problem of the week asks you to determine whether a large integer is prime or composite.  But it's also plain to see that this is useful in studying RSA, and its good to see a useful application of Legendre and Jacobi symbols.

3.10, due 10/15/12

The reading wasn't too difficult, but it would be helpful to see examples in class of how to find Jacobi symbols.

The most interesting part of this section is just to see how principles of number theory can be used to determine characteristics of large primes.  I'm sure that without properties of Legendre and Jacobi symbols, determining where a number a is a square mod p (p being a large prime) would be difficult indeed.

3.9, due October 12th

The most difficult part of the reading in this section was just trying to understand how finding the four solutions to our congruence (mod n) where n = pq is equivalent to factoring pq. 

The fact that the section is summed up by explaining that solving these congruences is equivalent to factoring some n = pq where p and q are primes makes it obvious that we can examine this method in the context of RSA.  However, I'm guessing that the mechanics of RSA will make it so that this method will not be very useful.

6.2, due October 8

The most difficult part of the reading was the part about timing attacks.  After reading that part I wasn't even really sure what a timing attack is.

This section changed some of my perceptions about RSA.  Ever since I had learned about it, I had kind of just figured that it was basically unbreakable as long the primes p and q chosen to generate n are really large.  But after reading this section, I can see that RSA can be broken if it is implemented sloppily, and there are actually effective ways to factor n into its original primes if n satisfies certain properties.  Very interesting.

3.12, due October 8

This was a simple enough section.  It wasn't exactly clear to me how the theorem presented in the section actually applies to continued fractions.  An explanation of that would be helpful.

It was helpful that the authors indicated that numbers used to construct a continued fraction can be obtained by doing the Euclidean algorithm and taking the partial quotients from each step.  This seems to be a very interesting and intuitive way to construct continued fractions.

6.1, due October 5

This chapter wasn't too hard to understand, and as usual I just want to see examples.  An actual example using the RSA algorithm will be a lot more helpful than what is in the book.  I tend to get lost when trying to read their explanations.

This is an awesome cryptographic algorithm.  It's really cool to see to see principles of number theory and abstract algebra put to use in such an amazing, practical way.  I think it's interesting that it seems to but a shorter, less-involved algorithm than AES or DES, and yet is so much powerful because of the principles it uses.

3.6-3.7, due October 3

It would be very helpful to see examples involving Euler's formula, as well as a better explanation of the Three-Pass protocol.

Although I didn't understand it fully, the Three-Pass protocol is a neat solution to the problem of securely delivering a key to the recipient of an encoded message.  Since work in cryptography assumes Kerckhoff's principle, the security of any method is derived from the key, which means methods of public-key cryptography are very secure.

3.4-3.5 due October 1

The most difficult part of the reading was trying to figure out how to actually do examples using the Chinese Remainder Theorem.  Seeing one or two in class would be super helpful.  The theorem and the proof are not that bad.

I really liked seeing how modular exponentiation can be used to find congruences for a massive power of 2 mod some number, without ever using a very large number.  It's just a really neat mathematical "trick", and I'm guessing one that will come in handy when we discuss the RSA algorithm.

Questions about the exam, due September 28

• Which topics and ideas do you think are the most important out of those we have studied?
• I think the most important thing to know for the exam will just be to have a general idea of how each of the encryption algorithms we have discussed work.  This might seem obvious, since it's the core focus of the class, but this will the first thing I study for the exam.  I want to be explain simply to myself how each of the encryption and decryption algorithms work.  Also, I will make sure to study the ideas of number theory we have covered thus far, especially properties of modulo arithmetic and the Euclidean algorithm.
• What kinds of questions do you expect to see on the exam?
• I expect there will be computation questions dealing with modulo arithmetic, such as finding the inverse of a number (mod n) or solving equations involving modulo arithmetic.  I also expect there will questions where I will be required to correctly implement certain encryption algorithms, as well as questions where I will have to decrypt simple messages by hand.  Basically I expect the exam to lean heavily toward computation and algorithm implementation, as well possibly some basic proofs of theorems from number theory.
•  What do you need to work on understanding better before the exam?
•  I think the concepts I understand the weakest are finite fields, LFSRs (how to generate and decrypt them), as well as the DES and AES algorithms.  Otherwise I feel  fairly confident in the material that we have covered thus far.

5.1-5.4, due September 26

Rijndael has a pretty involved algorithm, but I felt like I understood the encryption algorithm well enough.  I think the most difficult part to understand was certain aspects of the decryption algorithm, and how to invert some of the processes from the encyption algorithm.

I was glad to see that the authors mentioned others algorithms that were submitted as replacement standards for data encryption.  I would be interested to look them up and see how the algorithms work.  I also found it interesting that Rijndael is vulnerable to certain attacks up to six rounds, but that any amount greater than seven has no known attacks.  Adding rounds to the encryption seems simple, but apparently adds a lot security to the system.

Questions, due September 24

I have spent on average about 4 hours on the homework assignments.  The reading and lecture have been preparing me to work on the assignments, but I just haven't putting in enough effort to work on them.  Also going to Chris's office hours would really help me.

I think the examples are the most helpful thing we do in class.  I understand the algorithms that are part of our reading assignments well enough, though a little extra explanation in class helps, and then actually going through examples is VERY helpful.

I think the things I need to do most are just not procrastinating the homework assignments, and going to office hours to get help on the homework. 

3.11, due September 21

I felt the most lost in this reading when the authors were explaining the application of finite fields to LFSR sequences.   

This section was a good review for some material from abstract algebra.  Just like the authors, I found the analogous relationship between integers mod a prime and polynomials mod an irreducible polynomial to be pretty remarkable.  If I learned that in 371, I did not remember it.

4.1-4.2,4.4, due September 17

The most difficult part of the reading, which also happened to be a big part of the reading, was the overview of the DES algorithm.  I just don't think I followed it very well.  A slower walkthrough of the algorithm would definitely be helpful.

The most interesting part of the reading was the part about how DES is not a group.  I enjoy abstract algebra, so it was interesting to see how abstract algebra can be applied to a consideration of the DES algorithm.

3.8 and 2.5-2.8, due September 12

The most difficult part of the material covered in these sections was probably just the concept of inverting a matrix mod n.  I've never really considered modulo arithmetic for anything other than integers, so the reading definitely expanded my understanding of the power of modulo arithmetic.

Something in the reading that I found to be quite interesting was the introduction of block ciphers, and the way in which they add complexity to the encryption of a plaintext message.  This is evident from the fact that changing one letter in the plaintext will result in a change of n letters in the ciphertext (depending on the size of the blocks of text, n).  Since the use of blocks of size greater than 3 is resistant to frequency analysis, this method can be much more powerful then substitution or Vigenere ciphers.  It's a pretty simple method in the context of today's methods, but it laid the foundation for powerful ciphers such as DES, AES, and RSA cryptography.

2.3, due September 10

I like the Vigenere cipher more than the other "primitive" methods of encryption that we've talked about thus far.  It seems to be a little more robust, since it requires both frequency analysis, as well as some vector algebra to be broken.  The fact that one has to determine the key length as well as the shifts of each character in the key seems to add a little bit more to the strength of the method, though I'm sure it could be broken easily by a computer nowadays.  The second method of finding the key is a little hard to follow at first, but the summary at the end of the section made it much easier to understand.

2.1-2.2, and 2.4, due September 7

Shift ciphers and substitution ciphers were simple enough to understand, but I had never encountered an affine cipher before.  The principle behind it seems simple enough, but I think I'll just need to work out a few examples to better understand the way to encrypt and decrypt it, as well as how to attack it.  These are really simple methods of encryption that are really easy to break now, but they're interesting to me in a historical sense, since these seem to be some of the oldest methods of encryption.  This is evidenced, for instance, by the fact that Julius Caesar was purported to have used the shift cipher.

Guest Lecture, due on September 7

The lecture wasn't all that difficult to understand, considering most of the techniques of cryptography used by people in the early days of the church were fairly simple.  Though I was really glad to finally see what the actual code names used in those certain sections of the D&C were.  I have wondered about that for years.  I also had never heard of the Deseret alphabet, which I think is a very interesting fact about church history.  I think overall the lecture just helped me to understand that in any group or culture where secrecy of information is needed, cryptography will be useful.

3.2 and 3.3, due on August 31

-I haven't gone super in-depth with division involving congruences so there were some portions of the reading that were somewhat difficult to understand, but I'm sure with more examples that the things I missed will make more sense.
-It was interesting to see how fractions can be used to greatly simplify expressions involving modular arithmetic, especially when working with a large modulus, such as the example where 1/2 and 6173 are both congruent to 1 mod 12345.

1.1-1.2 and 3.1, due on August 29

-The most difficult part of the reading material in these sections was probably the short explanation of public key cryptography, though I know that any of the methods discussed in these section will be discussed at length in coming sections, and this is just a short overview.
-Probably the most interesting part of reading these sections was just seeing how many different methods of cryptography were mentioned that I have never heard of before.  I am excited to learn more about how they work, and how they apply to real world situations.

Introduction due on August 29

-My name is Jeff Adams.
-I am a senior majoring in math and minoring in microbiology.
-Thus far I have taken Math 290, Math 314, Math 334, Math 371, Math 341, Math 342, Math 343, and Math 221.  I am currently enrolled in this class, as well as Math 410 and Math 300.
-I am taking this class since I am done with the core requirements for the major and I am starting to take the classes of my choosing.  Since I'm taking classes more on the applied side, this seemed like an appropriate class to take.  Also I have always wanted to learn more of the mathematics behind cryptography.
- I have experience with Java and some Python experience.  I would be comfortable learning a new language if required.
-My more effective teachers have always been ones who have been passionate about the subject they teach and have a strong desire for their students to gain that passion.  They also challenge students on homework and tests but not to the point of insanity.  My least effective teachers have been those who have made the class unnecessarily challenging and have made students feel less confident as a result.
-Something unique about myself is that growing up I lived seven years in Latin America (Costa Rica, Peru, and Argentina).
-I will be able to come to your office hours.