**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 $n-O(\sqrt{\log(1/\epsilon) n})$, where $\epsilon$ is the distance of the extracted bits from uniform. In this paper we show a matching $ n-\Omega(\sqrt{\log(1/\epsilon) n})$ upper bound. Therefore, the loss of $\Theta(\sqrt{\log(1/\epsilon) n})$ 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.

**Category / Keywords: **foundations /

**Date: **received 15 Jun 2015

**Contact author: **maciej skorski at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20150621:162725 (All versions of this report)

**Short URL: **ia.cr/2015/591

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