To calculate the public keys of Alice and Bob, as well as the shared secret point, we can use the elliptic curve group defined by the equation y^2 ≡ x^3 + ax + b (mod p), where a = 3267, b = 695, and p = 3623. The subgroup generator is P = (0, 858).
First, let's calculate the public key of Alice (A) by multiplying her private key (38) with the generator point P:
A = 38P
Similarly, we calculate the public key of Bob (B) by multiplying his private key (11) with the generator point P:
B = 11P
Next, we need to derive the shared secret elliptic curve point TAB by multiplying Alice's public key (A) with Bob's private key (11):
TAB = 11A
Finally, the shared secret will be the x-coordinate of TAB.
To perform these calculations, we can use mathematical software such as the Sage Cell Server (SCS) as mentioned in the instructions. Here's an example of how you can use SCS to compute the values:
```
# Define the elliptic curve group
G = EllipticCurve(GF(3623), [3267, 695])
# Set the generator point P
P = G(0, 858)
# Calculate the public key of Alice (A)
A = 38 * P
# Calculate the public key of Bob (B)
B = 11 * P
# Derive the shared secret elliptic curve point TAB
TAB = 11 * A
# Compute the x-coordinate of TAB as the shared secret
shared_secret = TAB[0]
shared_secret
```
Running this code will give you the x-coordinate of the shared secret point, which is the value you are looking for in the Elliptic Curve Diffie-Hellman Key Exchange.