Q$4.$ Answer the following questions.

$($a$)$ Design a LFSR using the primitive polynomial $p(x)=x^3+x+1$ and generate a pseudorandom string of length $14.$ What is the period of the LFSR$.$

$($b$)$ Perform the the multiplication operations for $(111)inGF(2^3)$ and $(101)$ inGF$(2^3{)}^{\text{'}}$ using the primitive polynomial $x^3+x^2+1.$

Q$5.$ Answer the following questions.

$4+2=6$

$($a$)$ Generate all the elements of the field $GF(2^3)$ using the irreducible polynomial $f(x)=x^3+x^2+1.$

$($b$)$ Find the multiplicative inverse of the element $g^3$inGF$(2^3).$