Infinity Point on an Elliptic Curve

This section describes how the infinity point is used to represent the intersection of vertical lines and elliptic curves.

In order to completely define the addition operation on an elliptic curve, we need to introduce a special point on the curve, the infinity point.

Consider the following addition operation of P and Q on an elliptic curve, where Q is the symmetrical point of P on the same curve. Or we can express Q as:

Q = -P

In this case, we can follow our geometrical algorithm to find the R = P + Q:

1. Draw a straight line passing P and -P. This will be a vertical line.

2. Mark the third intersection of the vertical line and the elliptic curve as -R, which will be the infinity point of the 2-dimensional space. This is where the vertical line and the elliptic curve will eventually intersect. In other words, -R is the infinity point:

-R = (∞, ∞)

3. Mark the symmetrical point of -R on the other side of the x-axis of the elliptic curve as R which will be the same infinity point. This is because the infinity point has no difference between positive or negative signs. In other words, R is the infinity point:

R = (∞, -∞) = (∞, ∞)

Here is a diagram that illustrates how a vertical line intersects with an elliptic curve at the infinity point.

Infinity Point on an Elliptic Curve
Infinity Point on an Elliptic Curve

Last update: 2019.

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