**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.00, by Dr. Herong Yang

Addition Operation on an Elliptic Curve

This section describes the addition operation on an elliptic curve geometrically. The addition of points P and Q on an elliptic curve is a point R on the curve, which is the symmetrical point of -R, which is the third intersection of the curve and the straight line passing through P and Q.

The addition operation on an elliptic curve is defined geometrically as below:

For any given two points P and Q on an elliptic curve, the addition operation of P and Q will result a third point R on the curve by running the following geometrical algorithm:

1. Draw a straight line passing P and Q

2. Mark the third 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.

The above geometrical algorithm is also called "rule of the chord".

If we use the plus sign "+" as the addition operator, the addition operation of points P and Q on an elliptic curve can be expressed as:

P + Q = R

Here is a diagram that illustrates how to perform the point addition operation on an elliptic curve geometrically (source: stackoverflow.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)