Paper 2015/591

How much randomness can be extracted from memoryless Shannon entropy sources?

Maciej Skorski

Abstract

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 $n-O(\sqrt{\log (1/ϵ)n})$, where $ϵ$ is the distance of the extracted bits from uniform. In this paper we show a matching $$ upper bound. Therefore, the loss of $$ 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 $$, no matter what the extractor is. Thus, the ``AEP extracting heuristic'' should not be used without taking the precise error into account.