Pseudo-randomness of observables

There is a somewhat recent paper by Colbeck and Renner that, given the assumptions — 1) QM accurately predicts the correct statistical results in experiment (which so far has always been found to be true) and 2) the measurement can freely be chosen — that no extension to quantum mechanics can have improved predictive power.

I've taken basic QM and I just started taking QFT so understanding the paper is somewhat beyond my skills; so perhaps someone can help me with this situation:

Suppose you have a pseudo-random number generator (e.g. it could be any function really as long as its outputs appear random and the appropriate correlations [entanglement] between different positions exist) that is a function of position and time: $F(\vec{x}, t)$. If you knew the algorithm for this generator, you could plug in the spatial coordinates and time of your experimental measurement*, and get information that allows you to predict the exact* eigenvalue of the linear operator that corresponds to your measurement.

This hypothetical PRNG would be for the whole universe, so I suppose it would count as a non-local hidden variable.

My question is: why does Colbeck and Renner's paper rule out this possibility?

(*Okay, you can't plug in the exact coordinate and time of your measurement because of HUP. Perhaps you just get improved predictive power over QM by giving close enough coordinates and time).

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You might want to look up what is meant by a "cryptographically secure" PRNG: your knowing the algorithm is assumed (all the security resides in the key) and you have a long history of previous outputs and if the generator is secure you can not do better than guessing. 'Course algorithms can not be proven to be secure, but we have a number that have stood up to some pretty sophisticated attacks for years. –  dmckee Aug 25 '13 at 23:59
Adding to @dmckee 's comment: PRNGs are designed for many different purposes: Mersenne Twisters give "random" behavior from the standpoint of making many Monte-Carlo simulations work, but actually encode very pithy, simply rules that can be found out easily in attack. Blum Blum Shub on the other hand also encodes very simple rules, but so far have withstood years of attack. Looking at these two contrasts might help to clarify your ideas - I know that is no direct help to your question. –  WetSavannaAnimal aka Rod Vance Aug 26 '13 at 0:59