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 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
.
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 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
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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