The Cloudflare ECC introduction is well written and very accessible - however I notice that it omits a very crucial bit of explanation.
In the billards example it's stated that for a given player starting at point A:
"It is easy for him to hit the ball over and over following the rules described above."
arriving at some point B after some number of hits N. However, regarding some other player who knows point A and B
"they cannot determine the number of times the ball was struck to get there without running through the whole game again"
The question of course, is "Why not? Why can't the second player just start at point A and keep hitting the ball until it gets to point B, and count the hits?
The answer is that the first player does not run through the the sequence one hit at a time! Knowing N, you can take a mathematical shortcut from A to B. The simplest of these is known as "double and add" for elliptic curves. This is similar to the "square and multiply" method for fast exponentiation. The shortcut is the critical advantage, because it's the enormous computational complexity of running through the entire game without the shortcut that is the basis of ECC's security.
In the billards example it's stated that for a given player starting at point A:
"It is easy for him to hit the ball over and over following the rules described above."
arriving at some point B after some number of hits N. However, regarding some other player who knows point A and B
"they cannot determine the number of times the ball was struck to get there without running through the whole game again"
The question of course, is "Why not? Why can't the second player just start at point A and keep hitting the ball until it gets to point B, and count the hits?
The answer is that the first player does not run through the the sequence one hit at a time! Knowing N, you can take a mathematical shortcut from A to B. The simplest of these is known as "double and add" for elliptic curves. This is similar to the "square and multiply" method for fast exponentiation. The shortcut is the critical advantage, because it's the enormous computational complexity of running through the entire game without the shortcut that is the basis of ECC's security.