– How does Elliptic Curve Cryptography work?

Question

– How does Elliptic Curve Cryptography work?

In: 4

Answer 1

An elliptic curve, with respect to cryptography, is a special mathematical curve that looks something like this:

[https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig02.jpg](https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig02.jpg)

What you do is you pick two points on the curve. In the above diagram the points we pick out are P and Q. You then draw a line through P and Q and it will intersect the curve at a third point. This is -R in the diagram. We then draw a vertical line through -R until it intersects the curve on the other side, and this is R.

However, in reality we don’t use the curve like that. We’re only interested in points on the curve where the x and y coordinates are integers. Also, we set boundaries, where if the curve goes past a certain point in the x- and y- directions, we wrap it around from the beginning. That turns the above curve into something that looks like this:

[https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig07.jpg](https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig07.jpg)

However, it still retains the property above, about choosing points which produce a line that intersects at a third point. It’s just that this line wraps around as well:

[https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig09.jpg](https://www.allaboutcircuits.com/uploads/articles/Curve_Cryptography_fig09.jpg)

(Ignore the donut).

What’s great about this is we can produce a final result (R) from which it is really hard to figure out what the starting points were (P and Q). This makes it useful for assymmetric cryptography where you have to share this result without revealing how it was arrived at.