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.