**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

Scalar Multiplication or Exponentiation

This section describes what is Scalar Multiplication or Exponentiation in Abelian Groups. They are used represent the process of performing Abelian Group operations consecutively n times with the same element.

If we generalize the concept of Doubling or Squaring or operation we will have the Scalar Multiplication or Exponentiation operation.

**What Is the Scalar Multiplication or Exponentiation Operation?**
Scalar Multiplication or Exponentiation is an operation derived from
the base operation defined in an Abelian Group.
The Scalar Multiplication or Exponentiation operation
an element returns the same result as a sequence of base operations using
the same element multiple times.

In additive notation, we call it as the Scalar Multiplication operation, because we add the element to itself multiple times. And we express it using the factor of n notation. For example,

The Scalar Multiplication operation of an element P is expressed as: nP = P + P + ... + P (adding P n times)

Actually, the "Scalar Multiplication" concept can be used to cover a number of special cases in additive notation:

P + P + ... + P = nP "Scalar Multiplication" in additive notation P + P = 2P "Doubling" in additive notation P = 1P "Self" element 0 = 0P "Identity" element -P "Inverse" element (-P) + (-P) = -2P

In multiplicative notation, we call it as the Exponentiation operation, because we multiplying the element to itself multiple times. And we express it using the power of n notation. For example,

The Squaring operation of an element P is expressed as: P^{n}= P * P * ... * P (multiplying P n times)

Actually, the "Exponentiation" concept can be used to cover a number of special cases in multiplication notation:

P * P * ... * P = P^{n}"Exponentiation" in multiplicative notation P * P = P^{2}"Squaring" in multiplicative notation P = P^{1}"Self" element 1 = P^{0}"Identity" element P^{-1}"Inverse" element P^{-1}* P^{-1}= P^{-2}P^{1}* P^{1}= P^{2}

Last update: 2019.

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)