Consider the elliptic curve group based on the equation

$23++$mod$$

where $=1837$

$,=560$

$,$ and $=2659$

$.$

We will use these values as the parameters for a session of Elliptic Curve Diffie$-$Hellman Key Exchange. We will use $=(0,1052)$

as a subgroup generator.

You may want to use mathematical software to help with the computations, such as the Sage Cell Server $($SCS$).$

On the SCS you can construct this group as:

G$=$EllipticCurve$($GF$(2659),[1837,560])$

Here is a working example.

$($Note that the output on SCS is in the form of homogeneous coordinates. If you do not care about the details simply ignore the $3$rd coordinate of output.$)$

Alice selects the private key $10$

and Bob selects the private key $31$

$.$

What is $$

$,$ the public key of Alice?

What is $$

$,$ the public key of Bob?

After exchanging public keys, Alice and Bob both derive the same secret elliptic curve point $$

$.$ The shared secret will be the $$

$-$coordinate of $$

$.$ What is it$?$vv