16.5 Elliptic Curve Cryptosystems

• I thought it was very cool to see how we can use elliptic curves and apply the things we've already learned to get cryptosystems.
• All in all, the individual cryptosystems made sense to me but I don't get how they could calculate a point multiplied by an integer so fast on the elliptic curve. I guess it wouldn't take too long if you put the integer in terms of powers of 2 (this way you could do the thing similar to successive squaring but in adding points).

16.4 Elliptic Curves in Characteristic 2

• I found it interesting that they use finite fields over GF(2^n).
• However, it was semi hard to visualize what this exactly means. I got confused on the fact that the derivatives are vertical lines because the second derivatives are all zero. I'm not sure why that was pertinent either.

16.3 Factoring with Elliptic Curves

• I found it interesting to find that elliptic curve factorization method is more dependable than the p-1 factorization method since it only requires that there be enough smooth numbers around p opposed to p having to be smooth (like in the p-1 method).
• I was a little confused on the relationship between addition in the elliptic curve and the multiplication of corresponding numbers. There was an example in the book but I didn't fully understand it.
  

16.2 Elliptic Curves Mod p

• It was interesting to see how we can use elliptic curves to encode a message.
• I was confused on how the decryption of the cipher text would work. Unless K was known by the decoder, how would the point (x,y) be useful? Does this mean that this is a private key cryptosystem?