[muldvarp]

> Back in university I learned the in and outs of RSA, and to be honest, it seemed simple, understandable, and frankly, quite solid.

If you have a degree in computer science and not in cryptography it is quite frankly unlikely that you learned more than the basics of RSA.

> It absolutely depends that p and q are chosen at random, but besides that, should be basically uncrackable.

No. There are many things that can go wrong even if you choose p and q adequately.