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.

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)

 Finite Fields

 Generators and Cyclic Subgroups

Reduced Elliptic Curve Groups

 Converting Elliptic Curve Groups

 Elliptic Curves in Integer Space

 Python Program for Integer Elliptic Curves

 Elliptic Curves Reduced by Modular Arithmetic

 Python Program for Reduced Elliptic Curves

 Point Pattern of Reduced Elliptic Curves

 Integer Points of First Region as Element Set

 Reduced Point Additive Operation

 Modular Arithmetic Reduction on Rational Numbers

 Reduced Point Additive Operation Improved

 What Is Reduced Elliptic Curve Group

 Reduced Elliptic Curve Group - E23(1,4)

 Reduced Elliptic Curve Group - E97(-1,1)

 Reduced Elliptic Curve Group - E127(-1,3)

 Reduced Elliptic Curve Group - E1931(443,1045)

 What Is Hasse's Theorem

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

 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