Strong chain rules for min-entropy under few bits spoiled

M. Skórski, in:, 2019 IEEE International Symposium on Information Theory, IEEE, 2019.


Conference Paper | Published | English

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Abstract
It is well established that the notion of min-entropy fails to satisfy the \emph{chain rule} of the form H(X,Y)=H(X|Y)+H(Y), known for Shannon Entropy. Such a property would help to analyze how min-entropy is split among smaller blocks. Problems of this kind arise for example when constructing extractors and dispersers. We show that any sequence of variables exhibits a very strong strong block-source structure (conditional distributions of blocks are nearly flat) when we \emph{spoil few correlated bits}. This implies, conditioned on the spoiled bits, that \emph{splitting-recombination properties} hold. In particular, we have many nice properties that min-entropy doesn't obey in general, for example strong chain rules, "information can't hurt" inequalities, equivalences of average and worst-case conditional entropy definitions and others. Quantitatively, for any sequence X1,…,Xt of random variables over an alphabet X we prove that, when conditioned on m=t⋅O(loglog|X|+loglog(1/ϵ)+logt) bits of auxiliary information, all conditional distributions of the form Xi|X<i are ϵ-close to be nearly flat (only a constant factor away). The argument is combinatorial (based on simplex coverings). This result may be used as a generic tool for \emph{exhibiting block-source structures}. We demonstrate this by reproving the fundamental converter due to Nisan and Zuckermann (\emph{J. Computer and System Sciences, 1996}), which shows that sampling blocks from a min-entropy source roughly preserves the entropy rate. Our bound implies, only by straightforward chain rules, an additive loss of o(1) (for sufficiently many samples), which qualitatively meets the first tighter analysis of this problem due to Vadhan (\emph{CRYPTO'03}), obtained by large deviation techniques.
Publishing Year
Date Published
2019-07-01
Proceedings Title
2019 IEEE International Symposium on Information Theory
Article Number
8849240
Conference
ISIT: International Symposium on Information Theory
Conference Location
Paris, France
Conference Date
2019-07-07 – 2019-07-12
IST-REx-ID

Cite this

Skórski M. Strong chain rules for min-entropy under few bits spoiled. In: 2019 IEEE International Symposium on Information Theory. IEEE; 2019. doi:10.1109/isit.2019.8849240
Skórski, M. (2019). Strong chain rules for min-entropy under few bits spoiled. In 2019 IEEE International Symposium on Information Theory. Paris, France: IEEE. https://doi.org/10.1109/isit.2019.8849240
Skórski, Maciej. “Strong Chain Rules for Min-Entropy under Few Bits Spoiled.” In 2019 IEEE International Symposium on Information Theory. IEEE, 2019. https://doi.org/10.1109/isit.2019.8849240.
M. Skórski, “Strong chain rules for min-entropy under few bits spoiled,” in 2019 IEEE International Symposium on Information Theory, Paris, France, 2019.
Skórski M. 2019. Strong chain rules for min-entropy under few bits spoiled. 2019 IEEE International Symposium on Information Theory. ISIT: International Symposium on Information Theory
Skórski, Maciej. “Strong Chain Rules for Min-Entropy under Few Bits Spoiled.” 2019 IEEE International Symposium on Information Theory, 8849240, IEEE, 2019, doi:10.1109/isit.2019.8849240.
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