**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.00, 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):

Table of Contents

►Geometric Introduction to Elliptic Curves

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)

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)