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

Doubling or Squaring in Abelian Group

This section describes what is Doubling or Squaring in Abelian Groups. When performing the Abelian Group operation on two identical elements, this operation is called Squaring in multiplicative notation or Doubling in additive notation.

In order to understand the Discrete Logarithm Problem (DLP) in Abelian Groups, we need to introduce the Doubling or Squaring operation first:

In order to understand the in Abelian Groups, we need to introduce the Doubling or Squaring operation:

**What Is the Doubling or Squaring Operation?**
Doubling or Squaring is an operation derived from the base operation defined
in an Abelian Group.
The Doubling or Squaring operation of an element returns the same result
as the base operation using the same element as both operands.

In additive notation, we call it as the Doubling operation, because we add the element to itself. And we express it using the factor of 2 notation. For example,

The Doubling operation of an element P is expressed as: 2P = P + P

In multiplicative notation, we call it as the Squaring operation, because we multiplying the element to itself. And we express it using the square notation. For example,

The Squaring operation of an element P is expressed as: P^{2}= P * P

Table of Contents

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

►Discrete Logarithm Problem (DLP)

►Doubling or Squaring in Abelian Group

Scalar Multiplication or Exponentiation

What Is Discrete Logarithm Problem (DLP)

Examples of Discrete Logarithm Problem (DLP)

Scalar Multiplication on Elliptic Curve as Trapdoor Function

Generators and Cyclic Subgroups

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)