How Much Randomness Can Be Extracted from Memoryless Shannon Entropy Sources?

We revisit the classical problem: given a memoryless source having a certain amount of Shannon Entropy, how many random bits can be extracted? This question appears in works studying random number generators built from physical entropy sources. Some authors use a heuristic estimate obtained from the Asymptotic Equipartition Property, which yields roughly n extractable bits, where n is the total Shannon entropy amount. However the best known precise form gives only n − O( √ log(1/ )n), where is the distance of the extracted bits from uniform. In this paper we show a matching n−Ω( √ log(1/ )n) upper bound. Therefore, the loss of Θ( √ log(1/ )n) bits is necessary. As we show, this theoretical bound is of practical relevance. Namely, applying the imprecise AEP heuristic to a mobile phone accelerometer one might overestimate extractable entropy even by 100%, no matter what the extractor is. Thus, the “AEP extracting heuristic” should not be used without taking the precise error into account.