I came across these questions while studying for a crypto course, does anyone have any ideas on how to answer these?
(a) Random prime numbers of size 1536 bits are chosen to generate an RSA modulus of size 3072 bits. Knowing that the number field sieve factorization algorithm would succeed in factorizing an RSA modulus of this size in $2^{128}$ elementary operations on average, compare the two following approaches for cracking this RSA key:
i. factorizing the modulus using the aforementioned number field sieve factorization algorithm; or
ii. factorizing the modulus by enumerating candidate prime factors and looking for one that divides the modulus.
(b) Same question as above, but this time the primes p and q that make up the modulus n = pq are no longer randomly chosen, but derived deterministically from a user-supplied password. (E.g., we can use a pseudo-random generator to map the password to a pseudo-random number r of the right length, then use primality testing to find the first prime that is greater than r.) Feel free to make any reasonable assumption you need regarding the password.