**EC Cryptography Tutorials - Herong's Tutorial Examples** - Version 1.00, by Dr. Herong Yang

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: y^{2}= x^{3}+ ax + b where: a and b are integers 4a^{3}+ 27b^{2}!= 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: y^{2}= x^{3}+ 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:

2^{2}= 0^{3}+ 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 = (x_{R}, y_{R}):

x_{R}= m^{2}- 2x_{P}(4) y_{R}= m(x_{P}- x_{R}) - y_{P}(5) 3(x_{P})^{2}+ a m = --------- (6) 2(y_{P})

Here is the result of the calculation:

Calculation using equations (4), (5), and (6): m = (3*0*0 + 1)/(2*2) = 1/4 x_{R}= (1/4)*(1/4) - 2*0 = 1/16 y_{R}= (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64 Result: 2P = R = (x_{R}, y_{R}) = (1/16, -129/64) Verification to see if (x, y) = (1/16, -129/64) is on the curve: y^{2}= x^{3}+ 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

Geometric Introduction to Elliptic Curves

Algebraic Introduction to Elliptic Curves

Abelian Group and Elliptic Curves

Discrete Logarithm Problem (DLP)

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)

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

tinyec - Python Library for ECC

ECDH (Elliptic Curve Diffie-Hellman) Key Exchange

ECDSA (Elliptic Curve Digital Signature Algorithm)