TITLE: Who generates the prime numbers for encryption?
QUESTION [5 upvotes]: I was talking to a friend of mine yesterday about encryption. I was explaining RSA and how prime numbers are used - the product $N = pq$ is known to the public and used to encrypt, but to decrypt you need to know the primes $p$ and $q$ which you keep to yourself. The factorization of $N$ is the hard part, and that's why RSA is safe.
Then I was asked: Who actually calculates these primes, and how? They're huge, so can you do it on just a normal computer (in reasonable time)? And if not, and encryption software gets the primes from somewhere else, this third party would have a list of primes (however large) to try to factor $N$ with. Using it would be considerably easier than just brute forcing, trying to divide with every prime number up to $\sqrt{N}$. If they (or someone else) has the list, encryption isn't really safe.
So, how is it actually done?

REPLY [2 votes]: Cryptography libraries like PGP, GPG, Openssl generate the primes used for RSA encryption. They generally won't store the primes directly but compute N and Phi then forget them.
Openssl can generate a 2048 bit RSA key quite fast in just a few seconds on my laptop. On a Linux machine or Mac OS X you can run the command
openssl genrsa and inspect the output.