 So one of the new areas of cryptography involves the finding of rational points on elliptic curves. And this emerges as follows. Suppose I have some curve defined implicitly, f of x, y equals zero, and it's some curve, and I have these points x, y in R2, and what I would call the rational points are going to be the points on the curve whose coordinates are both rational numbers. And there's a couple of ways we can approach this. Sometimes, once we have one rational point on the curve, we can use it to find others. And for example, if P is a rational point on a circle, then any non-tangent line with a rational slope that passes through P will intersect the circle at another point Q, which will also turn out to be a rational point. This is actually something that's fairly easy to prove, and I suggest you try and work your way through a proof because it's a fairly nice proof that involves a little bit of algebra and a little bit of the theory of equations. For elliptic curves, for historical reasons, curves of the form y squared equals xQ plus ax plus b are known as elliptic curves. They actually have almost nothing to do with ellipses, but suppose I have a rational point on an elliptic curve. And for right now, we'll assume that we're not dealing with the x-intercept because that'll cause a little bit of a complication. So I have some rational point on the elliptic curve, and so what I'm going to do is I'm going to draw the tangent line through that point. And this tangent line has to intersect the curve at some other point, and it turns out that this other point of intersection will also be a rational point. Again, there's a fairly short proof of this that, again, involves a little bit of calculus, a little bit of algebra, and a little bit of the theory of equations, and it's a nice little proof that you should be able to work your way through. Now because of the symmetry of the elliptic curve, y squared equals xQ plus ax plus b, this y squared means that the curve is going to be symmetric about the x-axis. So if I have my point Q as a rational point, its reflection, Q prime, is also going to be a rational point. So now given this first rational point, I now have another, and I have another. Now I have two distinct rational points on the curve, and what I can do is I can draw the line through that point, and this will intersect at another point, r, which turns out to be rational, and again its reflection is also going to be a rational point, r prime. So now I have a bunch of distinct rational points, and I can do the same thing once again. And the proof that the line PQ prime will intersect at another rational point, fairly easy proof relying on the properties of algebraic equations. And what this means is that given one rational point on the elliptic curve, I can find many. So now I have this point P, I have this point r prime, the line between these two will intersect at another point, and again the reflection and the point itself will also be rational. Now what you should notice here is that what we're doing is we're taking, in some sense, we're taking two points, P and Q prime, and we're adding, for want of a better term, these two points, and we're constructing a third point, r prime, or r, depending on how you want to view that. And so what we have is we have the beginnings of an arithmetic of the rational points on an elliptic curve. And that arithmetic will emerge as follows. So let's say I take two points on an elliptic curve, I'm going to define the sum of those two points in the following way. So to begin with, if P is not equal to Q, for example, here's another elliptic curve, and I have P and Q distinct points, then what I'm going to do is I'm going to draw the line from P to Q, and it's guaranteed to intersect the curve at some point r, which will be a rational point. Again, simple proof verifies this. And I'm going to take the reflection of r across the x-axis to r prime. So I'm going to find that point of intersection. I'm going to reflect it. And this reflection is going to be what I'll define as the sum of the two points P plus Q. Now if P is the same as Q and P is not on the x-axis, so here I have P and Q being both at the same point, then what I'm going to do is I'm going to define the tangent line, and I'm going to extend it until it meets the curve at r. And again, what I'm going to do at that point is I'll reflect it across the x-axis to r prime, and I'll define P plus P to be r prime. Finally, the peculiar case we have if P is on the x-axis, if P is actually one of these points on the x-axis, the tangent line is going to be vertical. I'll draw the vertical tangent. And I'm going to define a new point, E, which is the point, so to speak, at infinity and its way up there someplace. And that's going to be what I'll define as the sum of P with itself in the case where P is the point on the x-axis. Now what's worth noting here is that I can do these things purely geometrically. These have meaning whether or not P, Q, and R are rational points. This is actually an arithmetic that holds for any point on an elliptic curve, whether they're rational or not. Now our interest is rational points on the elliptic curve, and for that we have to use a little bit of algebra. And so we'll take a look at that in the next video.