Shannon Entropy and Beyond: An Information-Theoretic Framework for Randomness Pre-Screening

Shannon entropy is the most common measure that one could use to check if a data source has random behaviour or not. A value close to the maximum is usually considered as evidence that the source is “random enough”. The present paper shows that this criterion alone is not enough. A deterministic logistic map driven at r=3.9999 reaches 94.97% of the Shannon maximum, yet it is fully predictable once we look at the built-in patterns: its permutation entropy drops to 77.01% of the maximum and its sample entropy falls to 0.67, against 2.33 for a high-quality pseudo-random generator (PRNG). Building on this observation, we combine four entropy measures—Shannon, Rényi, permutation, and sample—into a single diagnostic profile of the analyzed source. In order to validate our approach with practical, real life data, we test it on 2538 official draws of the Romanian Loto 6/49 lottery, recorded between August 1993 and April 2026. The lottery historical data set is very close to a high-quality PRNG (pseudo-random number generator) from the point of view of all four measures. We also observe that the entropy deficit of both the lottery and the PRNG decays as a power law with exponent α≈−0.96; in contrast, the logistic map sits at α≈−0.07. A Random Forest classifier trained only on the entropy profile reaches 78% accuracy on the analyzed four-way classification task (lottery, PRNG, logistic map, biased distribution), but scores 55.7% on the binary lottery-versus-PRNG task, consistent with chance. The method introduced in this study is domain-independent and applies directly to RNG certification, cryptographic key auditing, and any setting where structured pseudo-randomness has to be ruled out.