We consider a DHKE protocol over a Galois fields GF$(2^$m$).$ Up to now, we have been using groups but it is possible $($and simple$)$ to use GF for DHKE. Here, the generator is a polynomial $($denoted $a5g(x)$ and we have an irreducible polymomial instead of our public prime $p($denoted as $p(x)).$

$($a$)$ In our example here, all arithmetic is done in GF $(2^{n}5)$ with $p(x)=x^{n}5+x^{n}2+1$ as an irreducible field polynomial.

$($b$)$ The primitive element for the Diflie$-$Hellman scheme is $g|x|=x^2.$ The private keys are $a=3$ and $b-12.$ What is the session shared ker kae?

Hint: Derive the publickey of Alice $[$A$]$ by using the generator and her private key $(A-$g $(x{)}^2$ a modp$(x)$ in GF$(2*5)).$ Do not forget to reduce using $p(x).$ Bob can now find the session key kas through another exponentiation. Do not forget to reduce.