EC Cryptography Tutorials - Herong's Tutorial Examples - Version 1.00, by Dr. Herong Yang
Same Point Addition on an Elliptic Curve
This section describes how to perform the addition operation of a point P to the same point P on an elliptic curve. In this case, we will draw a straight line that passes P and tangent to the curve to find -R.
What happens if we want add a point P on an elliptic curve to itself?
In this case, we need to modify our geometrical algorithm as the following to find R:
1. Draw a straight line passing P and tangent to the elliptic curve.
2. Mark the second intersection of the straight line and the elliptic curve as -R.
3. Mark the symmetrical point of -R on the other side of the x-axis of the elliptic curve as R.
If we use the plus sign "+" as the addition operator, the addition operation of a point P to itself on an elliptic curve can be expressed as:
P + P = R
The above expression can abbreviated as the following, if we follow algebraic formula convention:
P + P = 2P = R
Here is a diagram that illustrates how to perform the addition operation on an elliptic curve geometrically (source: hackernoon.com):
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