Section 16.3, due December 4

It seems like finding these points and finding factors would take longer than the book says. I wasn't quite sure when the book said that they looked at 10000!P. Wouldn't that take a long time to do? This seems like one of those things where you just have to try it yourself to see that it works. Did the book choose convenient cases, or is it pretty fast to factor smaller numbers?

I got a little lost when the book started talking about singular curves. I understand that the equation has multiple roots, but should we try to use a singular curve to factor n or should we avoid them?

Section 16.2, due December 2

Is there any graphical interpretation that we can use to think of elliptic curves mod n? When I graphed the points in the example, I didn't seem to get anything.

It seems like it takes some work just changing our message into plaintext. Most other systems that we have looked at don't seem to have that problem. Does that add a lot of time or is it rather inconsequential?

Section 16.1, due November 30

I thought that was a very good introduction to elliptic curves. I did not realize that the form for an elliptic curve was y^2=x^3+... They only showed a couple of graphs of what elliptic curves can look like. I know that they said that it only makes sense graphically in the rationals or reals, but could you show more example graphs of elliptic curves in class so we could have an idea of the different possibilities they can look like.

The book said that elliptic curves can be thought of rational points, reals, or even mod p. Can you have elliptic curves in the complex plane? The book did not mention that. Would the form still be the same as a standard elliptic curve and would it look the same, too?

Section 19.3 and blog, due November 23

I must admit I did not understand a lot of this. I even found the explanation on the blog kind of confusing. I did understand, though, that we need to find the period and then can use universal exponent factorization method to factor n.

I think that this is very interesting. For a while, before I learned about PKC I wanted to be quantum physicist. I really do want to understand this better and I am glad we will be going over it in class tomorrow.

19.1-2, due November 20

Quantum cryptography is pretty cool. It is definitely the future of cryptography. I hope that we get to learn how factoring works with quantum computers. It will be fun to see the light filters tomorrow in class.

This section didn't seem very difficult. I am sure that the specifics of it are a lot more complicated. From what I have read about quantum mechanics, there are a lot of things happening that just goes against all common sense.

Section 14.1-2,due Nov. 18

The idea of a zero-knowledge protocol is very interesting. You need to be careful that you don't inadvertently give away some information in your "proof." I definitely do not like the loose application of "proof" in this section.

While the concept is easy to understand, the application seemed a bit more complicated. I had some trouble understanding what 14.2 was doing. I understand that we want to do this process in the fewest amount of applications as possible.

Section 12.1-2, due November 16

I thought that this whole idea was really interesting. Before I read the section I tried to think of how one would go about solving the problem of letting any t people know that secret, but I could not think of one. But the method is so simple that it almost seems obvious afterward.

Nothing in these sections seemed that difficult and the implementation seemed pretty straightforward.

Test Review Questions

Which topics and ideas do you think are the most important out of those we have studied?
Probably discrete logarithms and how to factor big numbers, and hash functions to a lesser extent.

What kinds of questions do you expect to see on the exam?
I figure there will be questions on there asking about jacobian and legendre symbols. Maybe there will be stuff on how to use El Gamal.

What do you need to work on understanding better before the exam?
I understand how the El Gamal system works, but I am not for sure who sends what to whom and how to encrypt m and decrypt c.

Are there topics you are especially interested in studying during the rest of the semester? What are they?
I am really interested in elliptic curves and how they are used in the field of cryptography.

Section 8.3 and 9.5, due November 11

I thought that it was kind of hard to understand how exactly the SHA-1 algorithm works. It reminds me a lot of DES and AES where we are just doing a bunch of operations to make things hard to compute backwards.

The Digital Signature Algorithm was a lot easier and simpler to understand. I was wondering if there are any algorithms for digital signatures that don't depend on modular arithmetic. They also talked about the NIST adopting it as a standard. Did they have one before 1991? This book is three years old, so are there other algorithms being considered to replace it currently?

Section 7.3-7.5, due Novemeber 2

I think that it is kind of cool that we can use discrete logs for other things besides sending messages. I think that bit commitment is close to the same thing if Alice sent Bob her secret p and q, the factorization of n. He could then decrypt he message she had sent previously.

ElGamal makes sense mathematically but it might take a bit to remember how to use it. The key used in it is a lot different than RSA. Then again, the algorithm has a lot of similarities and also a lot of differences.

Sections 6.5-7, 7.1, due October 28

I think that it is cool that PKC can be extended just a little more to provide digital signatures and verifications. I guess I never thought of using d to encrypt and e to decrypt, but of course it works exactly the same.

I am also very interested in learning about discrete logs. I don't exactly understand how they are used yet, or why it is hard to do them in reverse. I suppose, though, that we will be spending plenty of time on them in the future.

Section 6.3, due October 21

Oops. I read the wrong section last time. But I really liked this reading on primality testing. We talked about the jacobi symbols, but not really what they meant. It was neat to see how they fit into primality testing.

I am not really sure what "probably prime" means. I understood that if these tests told us a number was composite, then we knew that for sure. When using primes for an RSA encryption, I doubt that they just use a number that is pseudoprime, or even strong pseudoprime. What other tests do they do? Are they very complicated? I remember reading this last summer about some primality tests and they went way over my head.

Section 6.4.1, due October 19

I think that it is really interesting the methods that are devised to break any cryptological system. It takes a lot of ingenious thinking. It is the same in this case. My hat is off to whoever invented the Quadratic sieve.

Are we just going to be talking about the quadratic sieve? Or will we be talking about number field sieves as well. I get the concept of the quadratic sieve, but there is no way that I would be able to employ it to break a system.

Section 6.1, due October 9

I think that RSA is very elegant. I think it interesting it is from AES. While they are both secure encryption methods, they have almost nothing in common. While AES relies on confusion, RSA relies on only one thing: the difficulty of factoring large numbers.

I am pretty sure that they will talk about this in later chapters, but is it hard to find new primes to use. Obviously, a computer couldn't run through them all because there are so many. I just wonder if there are so many that you could not document them all.

Sections 3.6-7, due September 7

I think that generators are very interesting. I find it very surprising that there in no method to determine the generators of a field without actually testing them. I know it has been proven that one always exists and that there are Phi(p-1) of them but it seems that it shouldn't be that hard to tell which ones they are. But one thing about math is that the seemingly simple can be very complex and visa versa.

Was the three-pass protocol actually put into practice before RSA? Or was it just a stepping stone in PKC that was immediately followed by RSA?

Questions, due September 27

The homework assignments have taken me a 2-3 hours to complete. The lectures and the reading have definitely been beneficial to do the homework.

I have liked how we have looked at simple examples in class, like "baby DES" as it helped to understand it larger version. I think that it is easier to learn in class because I have read beforehand.

I think that it would help if we had homework more than just once a week, so we could better understand some of the alogrithms and ciphers better.

Sections 4.5-4.8, due September 23

I think that double encryption in DES is interesting in that it does not offer double security. Instead of searching for a pair of keys, you rather search for all of the first keys and save them all to memory. Then search for the second key and see if any of them match up, which is called a meet in the middle attack.

I didn't really understand about different kinds of DES. Could you go over the five different modes in class tomorrow. I think that it was just a lot of information to digest all at once.

Sections 4.1,4.2 & 4.4m due September 21

I thought that DES was very interesting. I have never seen a cryptographic algorithm like this before and with it multiple rounds and different steps. I understand how it works, kind of, but I did not completely understand the specific details of the encryption/decryption process.

In the introduction, it said that some people were afraid that IBM or the NSA had put in a trapdoor in the algorithm. This is interesting because this is what Dan Brown's novel Digital Fortress is about. Do you know if there have been recorded instances of people or organizations getting caught with a trapdoor?

Sectiion 4.8-4.11,due September 18

I never realized how difficult it was to create a random sequence. I though a random number generator on the computer would work out just fine. I guess I was mistaken.

If there was a secure way, or less expensive way, to transmit the key one-time pads would not that be ad. For our project, we used a scripture as the key. Agreeing upon a well-known passage would be very cheap, but not very secure, though.

Sections 2.5-2.8,3.8

The cipher used in the Sherlock Holmes book was pretty much a substitution cipher, although it would be impressive that he could solve it with such a short ciphertext. I think it is interesting that we can use matrices to encrypt things as well. I didn't like the ADFGX code very much because it required a keyword and a key matrix in order to make it work.

It does seem kind of foreign doing matrix algebra and modular arithmetic at the same time. I think that it would be good to go over in class tomorrow. It is hard to keep everything straight.

Section 2.3, due September 14

I was kind of hard to understand the second method of finding the key. The first one made sense. But I didn't really get how they used the vectors of each shift to find the key.

I think that it is pretty ingenious some of the ways that cryptanalysts discover to break a code. I can see why it took them a very long time to figure out a mehod on how to find the key length and the actual key.

Sec 2.1-2,4, due September 11

Although the alphine cipher is a lot easier to break using brute force than a substitution cipher, the key used is a lot simpler. I find it interesting that a lot of times there are trade-offs between things. This time it is security vs. simplicity of the encryption key. Other times it is time vs. security.

It is kind of difficult to think about fractions like 1/9 as the inverse of 1/9 rather than the actual fractions. We didn't use fractions at all when dealing with modular arithmetic in math 371.

Wednesday, September 9

One thing that I thought was very interesting was the fact that they used simple substitution ciphers to encrypt their messages. We think of those types of ciphers as easy to solve, but that was all they needed to do to securely hide their message. The jumps in cryptology have been very larger in the last 150 years than all of the years before it.

I think that it would be pretty hard to learn the Deseret alphabet. I agree that once the letters were learned it would be easier reading. Reading a phonetic alphabet would be a lot easier than reading and pronouncing English.

3.2-3, due Friday Sept. 4

I think that it is interesting looking at nonlinear congruences. When x-squared is congruent to 1, it is basically solving for which numbers are their own inverses. The book said that for all odd primes plus or minus 1 mod p are the only solutions. I would like to know if there are similar results for x-cubed and so on.

I think that solving modular equations was the part that was most difficult in these sections. I never remember that they are a lot easier to solve once you have done the extended Euclidean Algorithm.

1.1-1.2 and 3.1, due on September 2

While most of this was review, I did learn that PKC is significantly slower than symmetric algorithms. While PKC is very hard to break, it is only used to send digital signatures or keys to the symmetric algorithm.

I think that it is kind of difficult to understand how some algorithms are slower than others without actually knowing very much about said algorithms. Of course, this will soon go away when we learn more about symmetric algorithms and PKC. I know a little about RSA, but I am excited to learn more about the other kind of PKC algorithms.

Introduction, due Sep. 2

As I have said on numerous occasions, I am very excited and enthusiastic for this class. I think that all of the information you asked for you already know about me. I am glad that we are doing blogs for this class too, as it helped me quite a bit to keep on top of things last winter for 371.