Associativity 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.

If we want to support multiple addition and subtraction operations sequentially on an elliptic curve, we must verify the "associativity" property of those operations. In other words, is the following statement true or not:

P + (Q + S) = (P + Q) + S

It turns out that the above statement is true. So our addition operation on an elliptic curve is "associative".

You can verify this by following our geometrical steps that defines the addition operation with several example points.

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