Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors

Marius Zimand

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arXiv · 1103.5669

Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors

Marius Zimand

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Abstract

We prove a strong Symmetry of Information relation for random strings (in the sense of Kolmogorov complexity) and establish tight bounds on the amount on nonuniformity that is necessary for extracting a string with randomness rate 1 from a single source of randomness. More precisely, as instantiations of more general results, we show: (1) For all n-bit random strings x and y, x is random conditioned by y if and only if y is random conditioned by x, and (2) while O(1) amount of advice regarding the source is not enough for extracting a string with randomness rate 1 from a source string with constant random rate, ω(1) amount of advice is. The proofs use Kolmogorov extractors as the main technical device.

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BibTeX RIS

Marius Zimand. 2011-03-29. Symmetry of information and bounds on nonuniform randomness extraction via Kolmogorov extractors. https://arxiv.org/abs/1103.5669

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