EC Cryptography Tutorials - Herong's Tutorial Examples - v1.02, 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)