Elliptic curves (ECC) are a plane algebraic curve with the form:
where and are elements such that does not have double root.
ECC as a group
The curve is symmetrical about the x-axis, so given point , we can take .
if and are two points on the curve, operation is defined as the intersection point of the line , it generates a third point , then we take .
Point at infinity, ,
is the inverse of the intersection of the line with the elliptic curve. Let be .\\
Points of intersection:
We substitute the first into the second equation to get:
The three solutions to that cubic equation give the x-coordinate ,, of the three points of intersection of the line with the curve.\\
From Vieta’s first formula, we see the sum of those x-coordinates in so that . When we reflect over the x-axis, the x-coordinate does not change, so . Thus,
Using the equation of the line,
When we reflect over the x-axis, the sign of the y-coordinate changes
We need to calculate the slope () at a point
We can use use the equation we have deduced before: