(i) Explain briefly how and why a signature scheme is used. Describe the el Gamal scheme,

(ii) Define a cyclic code. Define the generator of a cyclic code and show that it exists. Prove a necessary and sufficient condition for a polynomial to be the generator of a cyclic code of length $n$.

What is the $\mathrm{BCH}$ code? Show that the $\mathrm{BCH}$ code associated with $\left\{\beta, \beta^{2}\right\}$, where $\beta$ is a root of $X^{3}+X+1$ in an appropriate field, is Hamming's original code.