Shannon Entropy Versus Renyi Entropy from a Cryptographic Viewpoint
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How accurately does Shannon entropy estimate uniformity? Concretely, if the Shannon entropy of an n-bit source X is \(n-\epsilon \), where \(\epsilon \) is a small number, can we conclude that X is close to uniform? This question is motivated by uniformity tests based on entropy estimators, like Maurer’s Universal Test.
How much randomness can we extract having high Shannon entropy? That is, if the Shannon entropy of an n-bit source X is \(n-O(1)\), how many almost uniform bits can we retrieve, at least? This question is motivated by the folklore upper bound \(O(\log (n))\).
Can we use high Shannon entropy for key derivation? More precisely, if we have an n-bit source X of Shannon entropy \(n-O(1)\), can we use it as a secure key for some applications, such as square-secure applications? This is motivated by recent improvements in key derivation obtained by Barak et al. (CRYPTO’11) and Dodis et al. (TCC’14), which consider keys with some entropy deficiency.
Our approach involves convex optimization techniques, which yield the shape of the “worst” distribution, and the use of the Lambert W function, by which we resolve equations coming from Shannon Entropy constraints. We believe that it may be useful and of independent interests elsewhere, particularly for studying Shannon Entropy with constraints.
KeywordsShannon entropy Renyi entropy Smooth renyi entropy Min-entropy Lambda w function
The author thanks anonymous reviewers for their valuable comments.
- [AIS11]A proposal for: Functionality classes for random number generators1, Technical report AIS 30, Bonn, Germany, September 2011. http://tinyurl.com/bkwt2wf
- [AOST14]Acharya, J., Orlitsky, A., Suresh, A.T., Tyagi, H.: The complexity of estimating renyi entropy, CoRR abs/1408.1000 (2014)Google Scholar
- [BK12]Barker, E.B., Kelsey, J.M.: Sp 800–90a recommendation for random number generation using deterministic random bit generators, Technical report, Gaithersburg, MD, United States (2012)Google Scholar
- [DPR+13]Dodis, Y., Pointcheval, D., Ruhault, S., Vergniaud, D., Wichs, D.: Security analysis of pseudo-random number generators with input: /dev/random is not robust. In: Proceedings of the 2013 ACM SIGSAC Conference on Computer and Communications Security, CCS 2013, pp. 647–658. ACM, New York (2013)Google Scholar
- [HILL88]Hstad, J., Impagliazzo, R., Levin, L.A., Luby, M.: Pseudo-random generation from one-way functions. In: Proceedings of the 20TH STOC, pp. 12–24 (1988)Google Scholar
- [Hol11]Holenstein, T.: On the randomness of repeated experimentGoogle Scholar
- [LPR11]Lauradoux, C., Ponge, J., Röck, A.: Online Entropy Estimation for Non-Binary Sources and Applications on iPhone. Rapport de recherche, Inria (2011) Google Scholar
- [RW04]Renner, R., Wolf, S.: Smooth renyi entropy and applications. In: Proceedings of the International Symposium on Information Theory, ISIT 2004, p. 232. IEEE (2004)Google Scholar
- [Shi15]Shikata, J.: Design and analysis of information-theoretically secure authentication codes with non-uniformly random keys. IACR Cryptology ePrint Arch. 2015, 250 (2015)Google Scholar
- [VSH11]Voris, J., Saxena, N., Halevi, T.: Accelerometers and randomness: perfect together. In: Proceedings of the Fourth ACM Conference on Wireless Network Security, WiSec 2011, pp. 115–126. ACM, New York (2011)Google Scholar