  The generation of series of random numbers is an important and difficult
problem. Even the very definition of random is difficult. Appropriate
measurements on entangled states have been proposed as the definitive solution
to produce series of certified randomness. However, several reports indicate
that quantum based devices show a disappointing rate of series rejected by
standard tests of randomness. This problem is usually solved by using
algorithms named extractors but, if the extractor were known by an eavesdropper
(a situation that cannot be ruled out) the key security in QKD setups may be
menaced. We use a toy fiber optic based setup, similar to a QKD one to be used
in the field, to generate binary series, and evaluate their level of randomness
according to Ville principle. Series are tested with a battery of standard
statistical indicators, Hurst exponent, Kolmogorov complexity, minimum entropy,
Takens dimension of embedding, and Augmented Dickey Fuller and Kwiatkowski
Phillips Schmidt Shin to check stationarity. A theoretically predicted
relationship between complexity and minimum entropy is observed. The good
performance of a simple method to get useful series from rejected series,
reported by Solis et al, is confirmed and supported with additional arguments.
Regarding QKD, the level of randomness of series obtained by applying Toeplitz
extractor to rejected series is found to be indistinguishable from the level of
non-rejected raw ones.
