  The non-local correlations exhibited when measuring entangled particles can
be used to certify the presence of genuine randomness in Bell experiments.
While non-locality is necessary for randomness certification, it is unclear
when and why non-locality certifies maximal randomness. We provide here a
simple argument to certify the presence of maximal local and global randomness
based on symmetries of a Bell inequality and the existence of a unique quantum
probability distribution that maximally violates it. Using our findings, we
prove the existence of N-party Bell test attaining maximal global randomness,
that is, where a combination of measurements by each party provides N perfect
random bits.
