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.