Commutativity of Elliptic Curve Operations

This section describes the associativity of the addition operation on an elliptic curve. P + (Q + S) = (P + Q) + S is true.

With the "infinity point" identified as the "identity element" of our elliptic curve addition and subtraction operations, we can look at the "commutativity" property of those operations. In other words, is the following statement true or not: true or not:

P + Q = Q + P

It turns out that the above statement is absolutely true.

If we look at the first step in our addition operation definition, "Draw a straight line passing P and Q". It doesn't matter if you draw a straight line "from P to Q", or "from Q to P", you will end up with the same straight line, because there is only one straight line passing through P and Q.

If P and Q are the same point, the first step of our addition operation definition says "Draw a straight line passing P and tangent to the elliptic curve". You will end up with the same straight line too, no matter how you draw it.

So our addition operation on an elliptic curve is "commutative".

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