How come keys can stay secure through factorization of primes

Question

Sooo I under n=pq where p and q are both large primes so it making factorizing them inefficient to the point of impossible but there must a a list of prime sets (heard people call them trap doors I think?) That are well know? Like if p and q are both know pairs wouldn't you be able look at n and go well I know this already so I don't need to even work it out I already can tell you it's pq. Or are there an infinite number of primes available and Is it only limited by the how large the key is? Isn't there a finite ammount of primes available in 2048 bits that it wouldn't be that hard? Clearly it's time consuming enough that it hasn't been cracked but I don't get why not since you know how many primes there are and also what they all are.

Answer 1

There are around 10^305 eligible primes for 2048 bit key (1024-bit long primes, as given by the prime counting function approximation), so the total combination count(p*q) is around 10^610. Each key is 2048 bits long, so the whole key space is 10^613 bits. Or 10^562 petabytes of data. This is an impossibly large number of keys, number of atoms in the universe is 10^80.

So no, enumerating this keyspace is out of the question. RSA is weak to quantum implementation of Shor’s algorithm, but we haven’t built a quantum computer big enough for cracking even 10-bit RSA keys.