Finite Elliptic Curve Group, Eq(a,b), q = p^n

This section describes finite elliptic curve groups constructed with modular arithmetic reduction of prime power numbers, p^n.

Finite elliptic curve groups can also be constructed using modular arithmetic reduction of prime power numbers.

Let's assume we have an elliptic curve, E(a,b):

```   y2 = x3 + ax + b
where:
a and b are integers
4a3 + 27b2 != 0
```

Let's assume we have a prime number p, and a prime power number, q = pk. An Abelian group can be defined as:

1. Group Element Set -

```All P = (x,y), such that:
y2 = x3 + ax + b (mod q)
where:
q = pk
k is positive integer
p is a prime number
a and b are integers
4a3 + 27b2 != 0
x and y are integers in {0, 1, 2, ..., q-1}
```

2. Group Operation -

```For any two given points on the curve:
P = (xP, yP)
Q = (xQ, yQ)

R = P + Q is a third point on the curve:
R = (xR, yR)

Where:
xR = m2 - xP - xQ (mod q)     (20)
yR = m(xP - xR) - yP (mod q)  (21)

If P != Q, m is determined by:
m(xP - xQ) = yP - yQ (mod q)  (22)

If P = Q, m is determined by:
2m(yP) = 3(xP)2 + a (mod q)   (23)
```

3. Identity Element - The infinite point of 0 = (∞, ∞).

Examples of finite elliptic curve groups using prime power numbers will be provided later.