**EC Cryptography Tutorials - Herong's Tutorial Examples** - v1.03, by Herong Yang

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

►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)