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

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∟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 = p^k. 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 = p^k
      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 = (x_P, y_P)
   Q = (x_Q, y_Q)

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

Where:
   x_R = m² - x_P - x_Q (mod q)     (20)
   y_R = m(x_P - x_R) - y_P (mod q)  (21)

If P != Q, m is determined by:
   m(x_P - x_Q) = y_P - y_Q (mod q)  (22)

If P = Q, m is determined by:
   2m(y_P) = 3(x_P)² + a (mod q)   (23)