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):

Same Point Addition on an Elliptic Curve
Same Point Addition on an Elliptic Curve

Table of Contents

 About This Book

Geometric Introduction to Elliptic Curves

 What Is an Elliptic Curve?

 Elliptic Curve Geometric Properties

 Addition Operation on an Elliptic Curve

 Prove of Elliptic Curve Addition Operation

Same Point Addition on an Elliptic Curve

 Infinity Point on an Elliptic Curve

 Negation Operation on an Elliptic Curve

 Subtraction Operation on an Elliptic Curve

 Identity Element on an Elliptic Curve

 Commutativity of Elliptic Curve Operations

 Associativity of Elliptic Curve Operations

 Elliptic Curve Operation Summary

 Algebraic Introduction to Elliptic Curves

 Abelian Group and Elliptic Curves

 Discrete Logarithm Problem (DLP)

 Finite Fields

 Generators and Cyclic Subgroups

 Reduced Elliptic Curve Groups

 Elliptic Curve Subgroups

 tinyec - Python Library for ECC

 EC (Elliptic Curve) Key Pair

 ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

 ECDSA (Elliptic Curve Digital Signature Algorithm)

 ECES (Elliptic Curve Encryption Scheme)

 Terminology

 References

 Full Version in PDF/EPUB