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.