Question on RSA trapdoor permutation
My own answer would be: 2048, 2048, and still 2048 bits.

Why ? Because:


2048-bit is the current "standard recommendation"; it has been so for quite some time, and is likely to remain so for quite some time (decades). See this site for pointers.
There are plans for removing support for keys shorter than 2048 bits in some widespread software, e.g. Firefox. Using shorter keys invites future interoperability trouble and maintenance headaches.
There are widespread systems (in particular hardware, such as smart cards and HSM) which have trouble with keys larger than 2048 bits. Even when longer keys are supported, the computational cost rises quite sharply with the key size, so you don't want to overdo it (if your algorithm is super-secure but too heavy, then people will look for ways not to use it, and so goes security down the drain).
If your enemy has access to a quantum computer, then he has resources which are way beyond your wildest dreams. Remember that hiring a few thugs to break your kneecaps is cheap; an attacker with a budget of more than 10k$ will just do that, instead of trying to unravel your public key with computers. Then an attacker who can afford a machine that does not seem to exist at all... Contrary to a quantum computer, a tiny camera concealed in your own glasses is doable with existing technology, and will reveal all your secrets. Worrying about quantum computers is therefore misdirected paranoia.


Your examiner would probably dislike this answer, though. He probably expects that you answer "don't use RSA" when the attacker has a quantum computer, so that you show that you have understood that a quantum computer will make short work of RSA keys of any size.