What Is Reduced Elliptic Curve Group

This section describes Reduced Elliptic Curve Groups or Elliptic Curve over Prime Field GF(p), denoted as Ep(a,b), which uses elliptic curve equations reduced by modular arithmetic of prime number p to define the group element set, and uses point addition operation based on the rule of chord reduced by the same modular arithmetic as the group operation.

After applying modular arithmetic reduction of prime number p to the elliptic equation and the point addition operation, I think we have found the complete definition of a new type of elliptic curve Abelian groups, we can call them reduced elliptic curve groups for now.

1. Group Element Set - All integer points P = (x, y) that satisfy the elliptic equation of coefficients a and b, reduced by modular arithmetic of prime number p as expressed below:

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

2. Group Operation - The elliptic curve point addition operation based on the rule of chord reduced by modular arithmetic of prime number p as expressed below:

Additive 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 p)     (11)
   yR = m(xP - xR) - yP (mod p)  (12)   
   
If P != Q, m is determined by:
   m(xP - xQ) = yP - yQ (mod p)  (18)

If P = Q, m is determined by:
   2m(yP) = 3(xP)2 + a (mod p)   (19)

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

I don't have any reference resource that proves the above definition produces Abelian group. But I will provide some verifications of some specific cases of p, a and b.

Reduced elliptic curve groups are called with different names in different reference sources like: Elliptic curves over prime filed GF(p) as in "Elliptic Curve Cryptography: finite fields and discrete logarithms" at andrea.corbellini.name/2015/05/23 /elliptic-curve-cryptography-finite-fields-and-discrete-logarithms/, and in "Elliptic Curve Cryptography" at https://slideplayer.com/slide/6610565/.

A reduced elliptic curve group can be denoted as Ep(a,b) for a specific set of p, a and b. It is also called an elliptic curve over a finite field.

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