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:
   Pn = 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 = Pn "Exponentiation" in multiplicative notation
P * P = P2           "Squaring" in multiplicative notation
P = P1               "Self" element
1 = P0               "Identity" element
P-1                  "Inverse" element
P-1 * P-1 = P-2
P1 * P1 = P2

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