  In this paper we propose a quantum random number generator (QRNG) which
utilizes an entangled photon pair in a Bell singlet state, and is certified
explicitly by value indefiniteness. While "true randomness" is a mathematical
impossibility, the certification by value indefiniteness ensures the quantum
random bits are incomputable in the strongest sense. This is the first QRNG
setup in which a physical principle (Kochen-Specker value indefiniteness)
guarantees that no single quantum bit produced can be classically computed
(reproduced and validated), the mathematical form of bitwise physical
unpredictability. The effects of various experimental imperfections are
discussed in detail, particularly those related to detector efficiencies,
context alignment and temporal correlations between bits. The analysis is to a
large extent relevant for the construction of any QRNG based on beam-splitters.
By measuring the two entangled photons in maximally misaligned contexts and
utilizing the fact that two rather than one bitstring are obtained, more
efficient and robust unbiasing techniques can be applied. A robust and
efficient procedure based on XORing the bitstrings together---essentially using
one as a one-time-pad for the other---is proposed to extract random bits in the
presence of experimental imperfections, as well as a more efficient
modification of the von Neumann procedure for the same task. Some open problems
are also discussed.
