 Alright, so something else I want to be able to do is to take a look at very high multiples of a point on an elliptic curve, and what we can do is we can do the same thing we did when looking at high powers of a number, which is use the fast powering algorithm. And that's essentially the same in both cases. So here I have an elliptic curve, mod 17, and I want to find 9 times 1, 6. In other words, I want to find the sum of 1, 6, 9 times. And I can use my fast powering algorithm in essentially the same way. First I'll note that 1, 1, 6 is just going to be the number itself. And now I can find 2, 1, 6. So here I have a point repeated, so I use the derivative to find the slope, substitute that in, slope is congruent to 1, and my elliptic curve point addition formula tells me that that 2, 1, 6 is going to be the point 16, 13. And well, I can continue the fast powering algorithm again if I want to find 4, 1, 6. It's the same as twice this, and so that's going to be twice 16, 13. Again, it's the same point repeated, so I use the derivative to find the slope. I apply the elliptic curve addition formula, and I find where that point is located. And again, I want to find twice that again, so I'm going to find 2 times 0, 14. I'm going to add that point to itself, and again, find the slope, find the coordinates x3 and y3, and there's my point. Now I want to find 9 times 1, 6, which means I want to add this point to this point. And now I have two points on an elliptic curve, and I can apply the straightforward addition formula. Now, since the points are distinct, then I find the slope in the ordinary fashion, that would be 2, and find my x3, and that gives me my y3 value, 9, 1, 6 is going to correspond to the point 5, 3.