16.3, due on November 30

1. (Difficult): How is elliptic curve factoring like the p-1 factoring algorithm?

And why do we have to invert 599 when computing 8*7!?

And 8!P is not equal to infinity mod 761 because 8! is not a multiple of 777, but why must only multiples of 777 give inifinity mod 761 again; how is this related to cycles mod any number? I need to understand this example (starting on page 357) better to understand the general case I think.

And I have lots of questions about singular curves, so I'll just be brief and say I didn't really understand singular curves.

2. (Reflection): The elliptic curve factorization method seems pretty strong.

Also, so addition and multiplication on using points on an elliptic curve is analogous to multiplication and exponentiation using modular arithmetic. I think that's what the book is saying.

The last statement I think is a cool discovery that the p-1 and division trial methods are both encompassed by the elliptic curve factorization method. (360)

16.2, due on November 28

1. (Difficult): Half the numbers mod p are squares? Really? How do we know that?

Also, how can n*A = infinity? In fact the whole section on attacks for discrete logarithms for elliptic curves is fuzzy, especially because I really never understood the index calculus attack.

And, why do we take m=floor(x/K) instead of solving the equation x=mK+j like we normally would?

2. (Reflection): Just so I can solidify, when encoding using an elliptic curve we can make the probability of failure to encode the message as small as we would like. That's nice.

Lastly, elliptic curves are still very strange to me.

16.1, due November 26

1. (Difficult): What is a component of a cubic polynomial? And what does the top and bottom of the y-axis mean? Also, an example of change of variables would be good.

Lastly, what's the point of the Law of Addition in regards to elliptic curves? And the "point" infinity from which we define negatives and subtraction is still confusing. I'M NOT GETTING THE GEOMETRICAL MEANING BEHIND THE LAW OF ADDITION, INFINITY, SUBTRACTION AND ANYTHING ASSOCIATED WITH ELLIPTIC CURVES.

2. (Reflection): The computational summary at the end is nice. I would still like to understand this subject thoroughly by knowing why the formulas work by understanding the real geometrical meaning behind elliptic curves.

Furthermore, how is this going to help with cryptography? Is it going to be for more secret splitting or something like that where 3 people have parts of the secret and it takes two to get the secret (two points determine a third point and the elliptic curve)?

2.12, due on November 20

1. (Difficult): Actually I think I understood it all. I would like to try some examples myself though. Seeing some in class would also help.

2.(Reflection): Cryptology is cool because of stories like Rejewski and and the other Polish and British breaking the German's codes during WWII. Wow, they were patient and diligent to find all 105456 initial settings of the Enigmas! Cool stuff.

Non-math explanation and 19.3, due on November 19

1. (Difficult): How does a quantum computer transfer qubits without measuring them? Is this what a quantum computer does? What's the trick to designing the quantum computer?

Also, the notation is difficult. I'm not use to it.

2. (Reflection): Do you think we'll ever invent a good quantum computer?

*** I read some of the assignment, skimmed other parts, and skipped other parts (I'll have to go back to). It was just too much reading for me right now. ***

19.1 and 19.2, due on November 18

1. (Difficult): The whole quantum mechanics idea makes sense, but I can't explicitly explain it. It just kind of intuitively makes sense. So reading a photon changes the photon? So why do observed horizontal photons stay horizontal and vertical photons change?

Also, when Eve eavesdrops Bob will be wrong 25% of the time EVEN IF HE USES THE CORRECT BASES, correct? So really he has a 3/8 chance of getting the right bit while without Eve's eavesdropping he has 1/2 chance of getting the right bit. Correct?

2. (Reflection): I never would have thought of using quantum mechanics in cryptography!

14.1 and 14.2, due on November 14

1. (Difficult): Why doesn't Peggy want Victor to know which way she goes through the door? Didn't follow that in the opening door example.

I would like an example for 14.2.

Why is the second method in 14.2 faster than the first method?

About Arthur, I'm not understanding the part about the probability of square roots mod p or q or n.

2. (Reflection): Do ATM's work this way?

So much cryptography just seems like puzzles. They are intriguing. Difficult sometimes, but intriguing.

12.1 and 12.2, due on November 12

1. (Difficult): I'm not following why in section 12.1 the r's (or s) need to be random and even if they must be random, why can't we choose r's random so that each number is equally likely? I think I need an example or discussion because a worded explanation is confusing to me.

For section 12.2 the Shamir threshold scheme, why do we know p(x) = s(x)? And why is the formula for M so complicated? Why isn't is more like the linear system approach? And why do we need t pairs? I'm not quite getting this but sort of.

2. (Reflection): I like the part at the very end of 12.2 about sharing a secret among two companies A and B such that they need each other to find the secret. I think it's clever and beautiful.

Midterms 2 Study Quesitons, due on November 9

1. The midterm exam will be Monday and Tuesday, November 12 and 13, in the testing center, and will cover chapters 6, 7, 8, 9, and sections 3.4-3.10 and 3.12. For Friday November 9, as you study for the exam, write responses to some or all of the following questions.
• Which topics and ideas do you think are the most important out of those we have studied?  Factoring modulo n and finding squares modulo n are important because both of these help find each other and we can then use factoring to try and break RSA. Also, knowing the workings and attacks for RSA and discrete logarithms, like the ElGamal system, will be really important. 
• What kinds of questions do you expect to see on the exam? I expect to see a few shorter questions about the Chinese remainder theorem, Euler's theorem, Jacobi symbol calculations, ... and such math preliminary to understanding our major coding systems towards the beginning of the test. Then I expect questions about making and breaking RSA codes and discrete logarithm codes and variations of such in the second half of the test. Somewhere there will probably be a digital signature question and a primality test question too.
• What do you need to work on understanding better before the exam? I need to understand the continued fraction low exponent attack on RSA better and all the primality tests the most.

8.3 and 9.5, due on November 7

1. (Difficult): Hexadecimal notation is confusing to me. And I would like to see an example of SHA-1, even the diagrams are confusing. Also, I would like to see examples of the operations on strings of 32 bits. (pg225)

Lastly, what does "with appendix" mean again with regards to signature schemes?

2. (Reflection): DSA, another nice signature procedure. I kind of like the signature methods. I think they're clever.

9.1-9.4, due on November 5

1. (Difficult): 9.1/9.2- So now we don't care about giving m away, we just care about the signature? Because m is essentially public once (m,y) is made public.

9.2- Why is finding r more difficult than a discrete log problem.? Why does finding d allow Eve to compute alpha to the k?

9.4- Why does the birthday attack halve the number of bits under attack? Wasn't 2^30 arbitrary? Couldn't he find more or less places to change than 30?

2. (Reflection): What real world examples are there of blind signatures?

Also, I like the hash function, signatures scheme combo. Great solution so our signatures isn't forever long.

Finally, I like that Alice foiled Fred anyway. Haha!

8.4-8.5 and 8.7, due on November 2

1. (Difficult): In 8.4 where does the ln2 come from in the explanation of equation (8.1)? Why did we use that magic number?

Also, an n-bit function has output n or input n? Output n, right?

Finally, 8.5 and 8.7 mostly make sense but examples would help a lot.


2. (Reflection): I like thinking about problems like the birthday and license plate problem! I, for whatever reason, find it fascinating  Now maybe that's why I'm a math major, but I think everyone likes knowing these random things for some reason. That's why people like statistics and trivia. However, these brainteaser problems are more fun because it's not just about collecting information, it's about breaking the puzzle and deriving the answer. That's much more satisfying!

Also, the BSGS system might be easier and faster to compute (and take the same storage space as the birthday attack) but in my opinion the birthday attack is easier to understand.