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):

Addition Operation on an Elliptic Curve
Addition Operation 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