Please answer the following questions.

$($a$)$ What are the essential ingredients of a symmetric cipher?

$($b$)$ List ways in which secret keys can be distributed to two

communicating parties.

$($c$)$ How many keys are required for twenty four people to communicate

via a symmetric cipher?

$($d$)$ List ways in which secret keys can be distributed on network to two

communicating parties.

$($e$)$ Determine if the following assertions are true.

a$.-2-=31(mod11)$

b$.77-=5(mod12)$

c$.1111-=11(mod111)$