Elliptic curves (ECC) are a plane algebraic curve with the form:
General definition
where and are elements such that does not have double root.
ECC as a group
The curve is symmetrical about the xaxis, 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 .
Homogeneous coordinates
Point at infinity, ,
Operations

Inverse

Addition
Case 1:
is the inverse of the intersection of the line with the elliptic curve. Let be .\\
‘s slope:
Points of intersection:
We substitute the first into the second equation to get:
The three solutions to that cubic equation give the xcoordinate ,, of the three points of intersection of the line with the curve.\\
From Vieta’s first formula, we see the sum of those xcoordinates in so that . When we reflect over the xaxis, the xcoordinate does not change, so . Thus,
Using the equation of the line,
When we reflect over the xaxis, the sign of the ycoordinate changes
Case 2:
Case 3:

Duplicate
We need to calculate the slope () at a point
We can use use the equation we have deduced before:
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