Generators, what for?

Coming up with more questions. Since I was going to observe the behaviour of an elliptic curve for varied generators, I started to re-read about generators and cyclic group.

And here are the questions:

1. How do generators affects the curve? And its security?
2. We need a generator because it generates all elements in the group. Means that we can use all the distinct elements. Is that correct?
3. Why do we use cyclic groups?
4. What is the difference between cyclic groups and cyclic subgroups? When to use?

More questions about "order"

[Hey, my last posting is almost a week ago. What have I been doing? How would I finish the phd in 3 years if the progress is like this???]

Order of an element, order of a group, order of a point. Have read about them but there are several questions in my head:

1. What the order of an element has to do with the security level of a curve, if there’s any relation between them?
2. What’s the use of a generator? It’s been said that a generator has the maximum possible order of elements. So what?

[scratching head, thinking hard, but it’s already late now. the phd student is getting sleepy… zzzzzzzzzzzzzzzz]