Q$7.$ Answer the following about Zero$-$knowledge proofs.

$($a$)$ What is the main reason behind the high cost associated with ZKPs in general?

$($b$)$ Consider the zero$-$knowledge proof of graph$-$isomorphism. If each graph has $64$ nodes, compute the optimal number of rounds.

$($c$)$ Explain why ZKPs are needed in electronic election systems.

$($d$)$ Besides electronic election, name three other applications that use ZKPs and brielfy explain why ZKP is needed in each case.

$($e$)$ What are the main requirements of a zero$-$knowledge proof?