Reduced Elliptic Curve Group - E23(1,4)

This section provides an example of a reduced Elliptic Curve group E23(1,4). A detailed calculation of reduced point doubling operation on (0,2) is also provided.

Let's take a look at our first reduced elliptic curve group, E23(1,4), as discussed in "Implementation Of Elliptic Curve Diffie-Hellman and EC Encryption Schemes by Kefa Rabah at docsdrive.com/pdfs/ansinet/itj/2005/132-139.pdf:

```The reduced elliptic curve:
y2 = x3 + x + 4 (mod 23)

The group elements:
( 0, 2)  ( 1,11)  ( 4, 7)  ( 7, 3)  ( 8, 8)  ( 9,11)  (10, 5)
( 0,21)  ( 1,12)  ( 4,16)  ( 7,20)  ( 8,15)  ( 9,12)  (10,18)
(11, 9)  (13,11)  (14, 5)  (15, 6)  (17, 9)  (18, 9)  (22, 5)
(11,14)  (13,12)  (14,18)  (15,17)  (17,14)  (18,14)  (22,18)
( ∞, ∞)
```

As the first verification case, we can perform the point doubling operation of P = (0,2) again using the reduced additive operation:

```Given:
P = (xP, yP) = (0, 2)

Find:
2P = P + P = R = (xR, yR)

Where:
xR = m2 - 2xP (mod p)         (11)
yR = m(xP - xR) - yP (mod p)  (12)
2m(yP) = 3(xP)2 + a (mod p)   (19)

Calculation:
2 * m * 2 = 3 * 0 * 0 + 1 (mod 23)
4 * m = 1 (mod 23)
m = 1/4 (mod 23)
m = 6

xR = 6*6 - 2*0 = 36 (mod 23)
xR = 13

yR = 6*(0 - 13) - 2 = -78 - 2 = -80 (mod 23)
yR = 12

Result:
2P = R = (xR, yR) = (13, 12)
```

Yes, the result of (0,2) + (0,2) = (13, 12) is in the group element set! We Abelian group "Closure" condition is satisfied.

We can also redo the same calculation by using the original equation for the parameter m. We should get the same resulting point:

```Given:
P = (xP, yP) = (0, 2)

Find:
2P = P + P = R = (xR, yR)

Where:
xR = m2 - 2xP (mod p)         (11)
yR = m(xP - xR) - yP (mod p)  (12)

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

Calculation:
m = (3*0*0 + 1)/(2*2) = 1/4

xR = (1/4)*(1/4) - 2*0 = 1/16 (mod 23)
xR = 13

xR = (1/4)*(0 - 1/16) - 2 = -1/64 - 2 = -129/64 (mod 23)
xR = 9 * 1/18 (mod 23) = 9 * 9 (mod 23)
xR = 12

Result:
2P = R = (xR, yR) = (13, 12)
```

Cool. This demonstrates that keep m as a rational number without modular reduction is also provides the same resulting point.

Here is a diagram of all non-infinite points of the E23(1,4) group: