Section 7.2

The general idea of computing discreet logs seems rather doable.  Nothing too crazy at least in the work.  The part that looked like it will take a little bit more practice is using the Pohlig-Hellman algorithm.
I am also not entirely sure what a precomputation is.   I guess its just like extra computations that are done before, but wouldnt that just make it a computation....
And then just to check my understanding, we have the log 4 algorithm to fill in a gap in the pohlig algorithm is that correct?

Section 6.5-6.7 and 7.1

It was interesting to read about real examples of the RSA system, and I understand that it is widely used still but reading about it is interesting.  It was cool to see that in some situations it is able to be broken, and that you could win all of $100 for doing so...  Also it was interesting to read about the different "trapdoors" that are built into public key systems, and they are necessary but also scary.  That's what all the allegations against the NSA are about right?

Discreet logs look like they are going to be almost as much fun as factoring large primes... Cryptography moves from one tricky thing to the next.  I guess thats the whole point of it thought isn't it.

Sections 6.4.1 and 6.4.2

In this section it talked about finding linear dependencies, and does that just mean where they are congruent to 0 mod whatever the current mod is? Or what exactly does that mean?  I guess the whole concept of a factor base is still kinda of fuzzy for me.

And these theoretical methods we covered in class, so i feel pretty comfortable with them at this point.  Your lectures really make the material that we read make a lot more sense.  Im not sure what exactly it is but thank you for putting in the time to make everything come together.

Section 6.4

Now say what? If a quantum computer were built then we could factor.  Well, is that all we need? Easy Peasy.... Might be better to just work on it from that end....

The factoring algorithm leaves something lacking... I can understand the idea of it all but when it comes to using it, i feel like it is another clunky tool.  And if I understand it right, it is based on how we choose B?  But there arent really specific rules for that? Just make it n? And like Vincent from DES said, there is always a tradeoff between security and speed.  So now if we want to have a higher chance of success we have to wait longer.

Section 6.3

Here is a nice like theorem that I wish I would have noticed for the homework that was due on Monday.  That basically solves two of the problems right there.  It is interesting that we can figure out that numbers can be factored without ever actually factoring them.

It is also interesting in the fermat primality test that "then n is probably prime" so even with our test we arent exactly sure about what is going on.  Also how likely is very likely that its true?  Are we talking like usually, 55%? Or like very probably like 95%?

Is the Miller Rabin test really saying that you just randomly choose one of the numbers and then see if it works and if not then try another?   That seems like an inefficient way first of all, and second does the randomization help in any way? Also what does it mean when they reach mod 1 but not all at the same time?

Section 3.10

I was wondering since this method is more general does that mean that we might as well jump right into this method when trying to decide if a number is a square or is it worth it to try the method we used last time and then if we have to come here?  

I am also a little confused on parts 4 and 5 of the jacobi symbols part, it seems like it might be building on what we did last time, but I got a little lost reading about it.

In the end it seems like everything we do, is basically saying that its still really hard to factor n.  You have some things that are like factoring n, but we still cant do those things either.  Kind of a frustrating approach to solving a problem, probably worse for those more invested in it than I...

Section 3.9

This section looks like the math part of another attack that we will try against the RSA algorithm.  It again looks like you can choose your n in such a way that this wont really work, but it could work.  The idea seems to make sense in my head, the reasoning behind it at least.  Putting it into practice on paper will probably be less fun than just reading about it though.
Also is there a difference between moduli and modulus? I thought that I had heard both but now I'm not so sure.