5. Generating primes with specific properties Often, one wants to generate primes with some specific properties. For example, you are asked to generate two large primes p and q such that:

(a) the primes p and q differ in length by only a few binary digits;

(b) the numbers p - 1 and q - 1 contain large prime factors; and

(c) the greatest common divisor (see Definition 2.22) of p - 1 and q - 1 is small.
Add another layer to your implementation of generating p and q that realizes these three conditions (the first and third conditions are easily checked). For the second condition, generate a large prime p' and then test the numbers p' + 1, 2. p' + 1, 3. p' + 1,... in sequence for primality. Choose p as the first one that Miller-Rabin classifies as a prime number. Do the same for q.