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 proposed to 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 best precise results of this form give only \(n-O(\sqrt{\log (1/\epsilon ) n})\) bits, where \(\epsilon \) is the distance of the extracted bits from uniform. In this paper we show a matching \( n-\varOmega (\sqrt{\log (1/\epsilon ) n})\) upper bound. Therefore, the loss of \(\varTheta (\sqrt{\log (1/\epsilon ) n})\) 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.
M. Skorski–This work was partly supported by the WELCOME/2010-4/2 grant founded within the framework of the EU Innovative Economy Operational Programme.
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Notes
- 1.
Consider simply a n-bit distribution X which puts the weight 0.5 on the string \(0^{n}\) and is uniform elsewhere.
- 2.
Because \(\epsilon \approx 2^{-kn}\) provides exponential security which is already overkill in most cases.
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Acknowledgments
The author is grateful to the organizers of the 8th Annual North American School of Information Theory in San Diego, and the organizers of the conference LATINCRYPT 2015 in Guadalajara, for the opportunity to present preliminary versions of this work.
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Skorski, M. (2016). How Much Randomness Can Be Extracted from Memoryless Shannon Entropy Sources?. In: Kim, Hw., Choi, D. (eds) Information Security Applications. WISA 2015. Lecture Notes in Computer Science(), vol 9503. Springer, Cham. https://doi.org/10.1007/978-3-319-31875-2_7
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