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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