Point doubling

The other group operation on elliptic curves is called point doubling. This is a process where P is added to itself. In this method, a tangent line is drawn through the curve, as shown in the following graph. The second point is obtained, which is at the intersection of the tangent line drawn and the curve.

This point is then mirrored to yield the result, which is shown as 2P = P + P:

In the case of point doubling, the equation becomes:

Here, S is the slope of tangent (tangent line) going through P. It is the line shown on top in the preceding diagram. In the preceding example, the curve is plotted over real numbers as a simple example, and no solution to the equation is shown.

The following example shows the solutions and point doubling of elliptic curves over finite field F₂₃. The graph on the left side shows the points that satisfy the equation:

As shown on the right side in the preceding graph, the calculation that finds the R after P is added into itself (point doubling). There is no Q as shown here, and the same point P is used for doubling. Note that in the calculation, l is used to depict the tangent line going through P.

In the next section, an introduction to the discrete logarithm problem will be presented.