Please answer the following questions.

$($a$)$ Define quadratic congruence and its application in cryptography, if

any.

$($b$)$ What is primality testing, such as Fermat Test, Miller$-$Rabin Test, and

AKS $($Agrawal$,$ Kayal, and Saxena$)$ Test? $($Hint: Do a computer search$)$

$($c$)$ What is the role of a compression function in a hash function?

$($d$)$ What are the three broad categories of applications of public$-$key

cryptosystems?