Point addition

Point addition is shown in the following diagram. This is a geometric representation of point addition on elliptic curves. In this method, a diagonal line is drawn through the curve that intersects the curve at two points P and Q, as shown in the diagram, which yields a third point between the curve and the line. This point is mirrored as P+Q, which represents the result of the addition as R.

This is shown as P+Q in the following diagram:

The group operation denoted by the + sign for addition yields the following equation:

P + Q = R

In this case, two points are added to compute the coordinates of the third point on the curve:

P + Q = R

More precisely, this means that coordinates are added, as shown in the following equation:

(x₁, y₁) + (x₂, y₂) = (x₃, y₃)

The equation of point addition is as follows:

X₃ = s2- x₁ - x₂ mod p

y₃ = s (x₁ - x₃) - y₁ mod p

Here, we see the result of the preceding equation:

S in the preceding equation depicts the line going through P and Q.

An example of point addition is shown in the following diagram. It was produced using Certicom's online calculator. This example shows the addition and solutions for the equation over finite field F₂₃. This is in contrast to the example shown earlier, which is over real numbers and only shows the curve but provides no solutions to the equation:

In the preceding example, the graph on the left side shows the points that satisfy this equation:

There are 27 solutions to the equation shown earlier over finite field F₂₃. P and Q are chosen to be added to produce point R. Calculations are shown on the right side, which calculates the third point R. Note that here, l is used to depict the line going through P and Q.

As an example, to show how the equation is satisfied by the points shown in the graph, a point (x, y) is picked up where x = 3 and y = 6.

Using these values shows that the equation is indeed satisfied:

The next subsection introduces the concept of point doubling, which is another operation that can be performed on elliptic curves.