Sections 6.5-6.7, 7.1 due on October 30

1. I had a couple questions from this reading. I have learned about public keys before, but what I've learned and what I read don't seem to completely mesh. Hopefully you will go over this in class, and I'll be able to see the connection.

In 7.1, I had a hard time grasping discrete logarithms. We've dealt with exponents before, so I don't understand how this is a new topic. I also don't really know where alpha comes from, and I'm guessing we'll be learning this in the future, but what's the application of discrete logarithms?

2. I was really interested to read about the application of RSA to treaty verification. I can see how that would be very valuable, so that the treaty stays secure and the same as what it was when it was agreed upon. The more I learn, the more it seems that cryptography is applicable in many areas of life.

I also enjoyed learning about the RSA challenge. I don't know if I could have come up with it myself, but I like learning how people break puzzles or use ingenuity to solve a problem.

Sections 6.4.1-6.4.2, due October 28

1.  Why is it important to understand theoretical methods when they can't actually be used in practice? I realize it's important to pay attention to the assumptions we make when we use these tests, such as the Miller-Rabin, but I have a hard time seeing the application of an unusable test. Maybe it's because I come from such an application-based major such as information systems.

2. The quadratic sieve seems very systematic, which I like. It's interesting that we can make a matrix out of the primes and look at factoring that way. This still seems like it would take awhile, which means that there still is not a good system for factorization.

Section 6.4, due October 25

1. The p-1 factoring method seems like it would work well, but I don't understand how or if it's any quicker than Fermat factorization. I don't think I understand it fully.

2. While inefficient for large numbers, the Fermat factorization seems more easy to use. It's a lot of repeated calculations, but it's interesting to me that it would work well, especially on a computer, up until a certain size of number.

Section 6.3, due October 23

1. I was doing okay with the use of the different tests until I got to the Soloway-Strassen Primality Test. I understand that its use is similar to the other tests, but it seems more difficult to use. Also, why do we use primality tests when they can only conclusively tell us if a number is composite and not if it's prime?

2. It's really interesting to me that we can find out if a number is composite much faster than we can factor it. These primality tests are surprisingly helpful, assuming that knowing a number is composite is helpful.

Section 3.9, due on October 18

1. For a composite modulus, we divide a more complex number into two of its factors.  Does it matter which factors we use, or do we just keep trying until there's a root for y mod p for some factor p?

2. I was really excited to learn about this topic because I have been wondering about it, amd I hadn't looked ahead in the book. I especially found the last theorem interesting that compared having a square root mod n to factoring. It doesn't solve our factoring problem, but it will be useful to use.

*Sorry this is late. I am on an interview fly-out trip.

Section 6.2, due on October 16

1. I am having a hard time understanding low exponent attacks. I understand how they are useful, but I am not understanding the theorem that explains how to use them. More specifically, how do you choose d? There are guidelines, but it seems like there would be a lot of choices.

2. I have never heard about a timing attack, and it seems like that greatly jeopardizes the security of RSA. The use of derivatives to find the times also is interesting and is another area of math that I get to be refreshed on. I would be really interested to know what preventative features, specifically, can be added to a physical implementation to prevent the timing attack.

Section 3.12, due October 14

1. I don't really understand the theorem in this section. I get the idea of continued fractions, but I don't really understand how to use the theorem, and more specifically where r and s come from.

2. I think continued fractions are really interesting. I'm most interested in the applications of this idea. I can see how it would be useful to be able to approximate a real number with a rational number. This offers a fairly easy way to do this.

Section 6.1, due October 11

1. I do not understand how the two claims given at the end of the section fit together. I understand them individually, and that their purpose is to show that finding the decryption exponent d is essentially as hard as factoring n. Are they supposed to work together or build on each other? Or are they just two separate claims that when put together prove a point?

2. I really think the idea and use of a present key cryptosystem is interesting. I can definitely see its merits in that you don't have to send a key to the other person, and it's not incredibly difficult to use. I am enjoying learning about the algorithms that go along with the cryptosystems I've only previously heard about from a technology standpoint.

Sections 3.6- 3.7, due October 9

1. I've always had a hard time understanding Euler's Theorem, even though I've learned about it before. I'm especially having hard time keeping all of the symbols and their meanings straight and remembering how the different theorems used in the proof applies to Euler's Theorem.

2. I really liked learning about the three pass protocol. Because of the example given, I understood what was going on when they put it in mathematical terms. Although it is vulnerable to attacks and requires multiple communications, it's useful because both Bob and Alice have to lock/unlock the message.

Sections 3.4-3.5, due on October 7

1. I followed the examples for the Chinese Remainder Theorem, but I got lost at the Lemma and its proof in Section 3.4. I understand its usefulness in understanding the Chinese Remainder Theorem, but I don't get the proof, or why they start the proof where they did.

2. I really thought Section 3.5 was interesting. It combines what I know about binary with what I know about congruences to find answers to congruences, even when working with large numbers. This seems like it will be very useful going forward. The only part I didn't understand was the sentence "If we want to compute a^b (mod n), we can do it with at most 2 log2(b) multiplications.

Exam Prep Questions, due October 4

1. I think the most important topics we have covered are the areas of math we have covered, such as modulus and fields. These repeatedly come up in different cryptosystems, so I think they are the most important in getting an overall understanding of cryptography.
2. I expect to see two questions for each cryptosystem we've covered: one for encryption and one for decryption. For those that we haven't memorized or completely learned,  I expect to see questions about the processes of encryption, if not the specific algorithms,
3. I need to work on understanding decrypting. I can encrypt in any of the systems we've talked about, but I have often had trouble on the homework with decryption.