If MtGox either
a) doesn't have the private keys for the coins in their wallet
or
b) they have the private keys but the coins have been moved (given to attackers, stolen years ago in prior hacks, embezzled)
there is nothing short of a hard fork to mint new coins for MtGox that anyone can do.
I have a question:
I do understand that private keys are random numbers, but at same point they have to be created on deterministic factors like time etc. So when the private key would be let's say a hash of the "magic_number"+timestamp, then there is a chance to rebuild the private key, if you have the magic number and the timestamp. For the sake of simplicity let's assume the bug is that a letter x got added to the private key. So no private key would work anymore.
-> So in this case there would be a feasible way to compute the private keys.
But I have no idea how MtGox created the private key. Maybe they took something that can't be found like Marc used 500 random characters (just pushed his hands on the keyboard).
So based on what private keys get created in the qt wallet?
The QT client uses the OS level random number generator to generate private keys. If values of PRNG could be recomputed, then anyone could recompute any other persons private keys. Bitcoin would fail and so would essentially all other crypto. PRNG while not truly random are designed to make such recomputation infeasible. They don't just use a timestamp, they pull data from an entropy pool which is filled with sources like # of disk I/O failures in last x seconds, random noise from sound card DAC, temperature of processor, the timing (in milliseconds) between keystrokes on the keyboard, the mouse movement data, the latency recorded on IDE calls.
To recompute a PRNG value would require not just the timestamp of the value but recreating the system in the exact same configuration as it was at the time the random value was requested. This is nearly impossible unless there is some flaw in the PRNG and even then you would need some extensive cryptanalysis and a lot of computing power (i.e may take quadrillions of attempts to recompute the target value).
Of course it is also possible to generate private keys using a true hardware random number generator (quantum random number generator is one example). For those there is no method, not even theoretical to recompute the generated number.