Consider the BBS pseudorandom bit generator with Blum integer n $=$ pq$.$

$($a$)$ Determine the period of the generator as a function of n and s$0.$ Explain why you should

gcd$($p $1,$ q $1)$ be small.

$($b$)$safe primes$$ are special prime numbers of the form p $=2$p

$+1$ where p

$$

is also first.

We call a safe prime p $=2$p a $$SafeSafe prime$.$

$+1$ for which p

$$

it is also safe

$2$

prime and p

$1($mod $4),$ where p

$=($p

$1)/2.$ What is the maximum period of the generator at

case where both p and q are $$SafeSafe$$ primes?Prove your claim.