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:
   P2 = P * P 

Last update: 2019.

Table of Contents

 About This Book

 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)

 What Is Trapdoor Function

 DLP And Trapdoor Function

 Scalar Multiplication on Elliptic Curve as Trapdoor Function

 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