Prove of Elliptic Curve Addition Operation

This section describes how to prove that the addition operation on an elliptic curve can be successfully performed geometrically.

To prove that each and every addition operation on an elliptic curve can be successfully performed, we need to prove the following:

1. Every straight line that passes through two points P and Q on an elliptic curve must intersect with the same curve at a third point -R. Otherwise we will not be able to find -R. I need to do more research to prove this.

2. Every straight line that passes through two points P and Q on an elliptic curve must intersect with the same curve in at most three points. Otherwise we may find multiple possible points of -R. This can be approved based on the first property of elliptic curves described earlier.

3. Every point -R on an elliptic curve must have an x-axis symmetrical point R on the same curve. Otherwise we will not be able to find R. This can be approved based on the second property of elliptic curves described earlier.

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