Identity Element on an Elliptic Curve

This section describes the 'identity element', which is the 'infinity point' in our addition and subtraction operations on an elliptic curve.

Let's take a quick break from our elliptic curve discussion to introduce the concept of "identity element" of addition and subtraction arithmetic operations on a set of elements.

The "identity element" must satisfy the following rules:

1. Any element, E, adding to the "identity element" equals to the element itself. In other words:

E + I = E, "I" is the "identity element"

2. Any element, E, subtracting from the element itself equals to the "identity element".

 
E - E = I, "I" is the "identity element" 

For example, the number zero, 0, is the "identity element" of addition and subtraction arithmetic operations over the integer set.

Now let's go back our elliptic curve discussion. You can see that "infinity point" perfectly matches the above definition of the "identity element" for addition and subtraction operations introduced on an elliptic curve.

1. Any point, P, on an elliptic curve adding to the "infinity point" equals to the point itself. In other words:

P + O = P, "O" is the "infinity point"

2. Any point, P, on an elliptic curve subtracting from the point itself equals to the "infinity point".

 
P - P = O, "O" is the "infinity point"

Conclusion: the "infinity point" is the "identity element" of our addition and subtraction operations on an elliptic curve.

By the way, the term "identity element" is more accurate for the number 1 in multiplication and division operations over the integer set.

Maybe, the term "zero element" is more accurate for the number 0 in addition and subtraction operations over the integer set.

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