16.5, due on December 5

1. (Difficult): I don't remember why x congruent to y2*y1^-a decrypts for the ElGamal Cryptosystem and I forgot how to encrypt a message to a point using Elliptic Curves. The former I can review myself I think but a review of the latter in class would be helpful.

Also, I didn't understand the end of the ElGamal Digital Signatures when it says n=p-1 and when it talks about arbitrary assigned integers to points.

2. (Reflection): The  Elliptic Curve Diffie-Hellman Key Exchange I got! And elliptic addition/multiplication being analogous to regular multiplication/exponentiation makes some sense. That's good.

This is our last blog. Wow time flies!

16.4, due on December 3

1. (Difficult): What does an elliptic curve mod 2 look like? How does it relate to the graph of our other elliptic curves we've dealt with? (How did we know the intersection P=(0,1) on page 361?)

I didn't quite follow the new law of addition. How does it work?

Also, I think I understand the last example of addition in GF(4), but a picture would be really helpful so I know if I'm getting it our not.

2. (Reflection): It is crazy how math builds on each other. Elliptic curves mod 2 seem like such a crazy idea, but it's just a bunch of simple math concepts combined.

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. ***