16.5 due December 11

1. While I understand how to send a message through an elliptic curve, I am not comprehending how our message can be a point x on our elliptic curve. Wouldn't the message then just be some (x,y)? How is that a meaningful message at all? I can understand key exchanges and signatures well enough, but the whole "Send a message as a point x" does not seem useful, even though it is said to be a lot easier to use this method with smaller primes.
2. After our last hw, and what is said in this chapter, it is really cool to see how we can use smaller primes for an elliptic curve equation! I think it is a pretty good step forward, because I'm sure a lot of the primes we are using are so crazily big when they don't need to be. It's been really fun this semester to see how cryptography works, and to be able to experience it first hand!

16.4 due December 9

1. This section was pretty straightforward. I think the only thing I would really like for clarification in class is the additions for GF(4), like why (0,w^2) + (w,w^2) = (w,1). It almost makes sense, but just a little more in depth as to why this, and the other examples work that that would be great!
2. When they mentioned in this chapter that we can work things out pretty well in mod2, I am wondering how this will work? I think they could mean in binary, like we used before, which would make sense. I hope that I am right!

16.3 due December 6

1. Why does it work that we can factor n using elliptic curves? I am trying to follow the logic that when we find 2P we find the inverse and if it is a number not 1, then it is our factor? How does that work? I feel like they didn't really explain it in the book very well.
2. I really hope that we didn't save a super easy method for last when finding factors! I don't think this will work for all cases, but I think it would be really unfair if it turns out that it is so much easier to factor this way! It definitely seemed easier in the example they gave us!

16.2 due December 4

1. Why are the number of points mod p going to be p? I thought in the previous section we decided that there would be 3 points, sometimes including infinity? And is Hasse's Theorem just a proof that there are going to be at least p points, or is that showing that there can be less than p points? Also, is it possible for there to be less than p points? What about significantly less than p?
2. I think this chapter is going to be a bit rough for me. What I am thinking right now is still how are we going to put a message into an elliptic curve equation? I feel that it seems a bit impractical, even with a fast working computer. Although I am not very knowledgeable with how this works still, I think that it would take more time to encode, and possibly be easier to cipher then an RSA or something along those lines.

16.1 due December 2

1. I don't think they really got into this, but I was a little confused at what our message is going to be in our elliptic curve equation. Is it going to be whatever x is? Will it be the a, b, and c? I feel that they didn't really go in depth on why this is going to be important for us as cryptographers.
2. Honestly, this section was not very interesting. I am a bit nervous on how we will be using these elliptic curve equations in the future. It seems to me like they will either be super easy, or super hard. I really cannot see an inbetween on this! I really hope somewhere in this chapter there is some sort of history lesson! I think those are the most interesting parts for me, and they motivate me to actually use what we learned!