Consider the elliptic curve group based on the equation

y$2$x$3+$ax$+$bmodp

where a$=85$

$,$ b$=2712$

$,$ and p$=3463$

$.$

We will use these values as the parameters for a session of Elliptic Curve Diffie$-$Hellman Key Exchange. We will use P$=(0,725)$

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$(3463),[85,2712])$

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 $47$

and Bob selects the private key $31$

$.$

a$)$ What is A$,$ the public key of Alice?

b$)$ What is B$,$ the public key of Bob?

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

$.$ The shared secret will be the x

$-$coordinate of TAB

$.$ What is it$?$