Elliptic Curves in Integer Space

This section describes the fact that elliptic equations in 2-dimensional integer space can not be used to construct Abelian groups.

Earlier in the book, we learned how to use elliptic equations in 2-dimensional real number space to construct Abelian groups.

Can we use elliptic equations in 2-dimensional integer space to construct an Abelian group?

Let's try to define an element set with elliptic equations in 2-dimensional integer space and the same rule of chord operation.

Element Set: 
   All P = (x,y), such that:
      y2 = x3 + ax + b
   where:
      a and b are integers
      4a3 + 27b2 != 0
      x and y are integers
      
Operation: 
   Rule of chord
   
Identity Element: 
   The infinite point of 0 = (∞, ∞)

Is the above definition provides any Abelian Groups? The answer is no.

Let's take (a,b) = (1,4) as an example:

All P = (x,y), such that:
   y2 = x3 + x + 4
where:
   x and y are integers

We can see that P = (0,2) is a valid element in the set, because it satisfy the curve equation:

   22 = 03 + 0 + 4

If P = (0,2) in the element set, 2P = P + P must be in the element set too because of the "Closure" condition of Abelian groups. So let's verify this by calculating 2P using the algebraic equations provided earlier in the book, assuming 2P = R = (xR, yR):

   xR = m2 - 2xP                  (4)
   yR = m(xP - xR) - yP           (5)

       3(xP)2 + a
   m = ---------                 (6)
         2(yP) 

Here is the result of the calculation:

Calculation using equations (4), (5), and (6):
   m = (3*0*0 + 1)/(2*2) = 1/4
   xR = (1/4)*(1/4) - 2*0 = 1/16
   yR = (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64

   Result: 
   2P = R = (xR, yR) = (1/16, -129/64)
   
Verification to see if (x, y) = (1/16, -129/64) is on the curve:
   y2 = x3 + x + 4
   (-129/64)*(-129/64) = (1/16)*(1/16)*(1/16) + 1/16 + 4
   16641/4096 = 1/4096 + 256/4096 + 16384/4096
   16641/4096 = 16641/4096
   
   Result:
   (x, y) = (1/16, -129/64) is on the curve

But (x, y) = (1/16, -129/64) is not in the element set, because its coordinates are not integers!

So elliptic equations in 2-dimensional integer space can not be used to construct Abelian groups.

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