1.1-1.2 and 3.1, due on September 2

1)  In section 1.1.1, I didn't completely understand the description of airplanes determining if another airplane is a friend or a foe.  It says, "send a random message to the plane, which encrypts the message automatically and sends it back.  Only a friendly airplane is assumed to have the correct key.  Compare the message from the plane with the correctly encrypted message.  if they match the plane is friendly.  If not, it's the enemy."
This could be very elementary, so bear with me, but I guess I just don't understand the system.  Anytime you send a message to a plane, it will be sent to their plane and they will encrypt it using their encryption machine and send it back?  Or do they send their response in an encrypted message? I guess I just don't understand what exactly the cipher text would be in this case... the plain text that you just sent them now encrypted?  Or is their response that is encrypted?

I am also not 100% sure on the difference between a key and an algorithm.  What does it mean to keep the algorithm the same but to just change the key?  Does algorithm just mean the type of key?  Like the equation for it?

In sections 3.1.2

I am having a hard time wrapping my head around the Prime Number Theorem.  Why are we trying to prove that pi(x) is the number of primes less than x?  How does this prove that there are infinitely many prime numbers?

I have a surface understanding of the Lemma on pg. 65, and I definitely could not replicate the proof.

Lastly, I just can't understand the Theorem and Corollary on pg. 68, but I will continue trying to read and understand it before class tomorrow.  I don't understand it to the point that I can't even really identify the reason I don't understand it.



2) I loved chapter 1 because it really illustrated the application of this material in our world today.  I am very surprised because I didn't realize how cryptography is so widely used and severely needed in our world of technology, despite the fact that I use it every day.  I like that in 1.2 it broke the applications into four different, clear sections.  I have used and appreciate each part.  I am also interested to see the connection between cryptography and games.

In chapter 3, I felt like I was back in my Math 290 class.  The proofs are simple and beautiful for the most part and I find myself wanting to work out the proof myself before I read the solution.  I am very intrigued by the Euclidean algorithm and I am curious as to how it was discovered.  I am amazed at how quick it is and the fact that you don't need to factor.