Randomness Extraction from Somewhat Dependent Sources
We initiate a comprehensive study of the question of randomness extractions from two somewhat dependent sources of defective randomness. Specifically, we present three natural models, which are based on different natural perspectives on the notion of bounded dependency between a pair of distributions. Going from the more restricted model to the less restricted one, our models and main results are as follows.
1) Bounded dependence as bounded coordination: Here we consider pairs of distributions that arise from independent random processes that are applied to the outcome of a single global random source, which may be viewed as a mechanism of coordination (which is adversarial from our perspective).
We show that if the min-entropy of each of the two outcomes is larger than the length of the global source, then extraction is possible (and is, in fact, feasible). We stress that the extractor has no access to the global random source nor to the internal randomness that the two processes use, but rather gets only the two dependent outcomes.
This model is equivalent to a setting in which the two outcomes are generated by two independent sources, but then each outcome is modified based on limited leakage (equiv., communication) between the two sources.
(Here this leakage is measured in terms of the number of bits that were communicated, but in the next model we consider the actual influence of this leakage.)
2) Bounded dependence as bounded cross influence: Here we consider pairs of outcomes that are produced by a pair of sources such that each source has bounded (worst-case) influence on the outcome of the other source. We stress that the extractor has no access to the randomness that the two processes use, but rather gets only the two dependent outcomes.
We show that, while (proper) randomness extraction is impossible in this case, randomness condensing is possible and feasible; specifically, the randomness deficiency of condensing is linear in our measure of cross influence, and this upper bound is tight. We also discuss various applications of such condensers, including for cryptography, standard randomized algorithms, and sublinear-time algorithms, while pointing out their benefit over using a seeded (single-source) extractor.
3) Bounded dependence as bounded mutual information: Due to the average-case nature of mutual information, here there is a trade-off between the error (or deviation) probability of the extracted output and its randomness deficiency. Loosely speaking, for joint distributions of mutual information t, we can condense with randomness deficiency O(t/ε) and error ε, and this trade-off is optimal. All positive results are obtained by using a standard two-source extractor (or condenser) as a black-box.
Randomness Extraction
min-entropy
mutual information
two-source extractors
two-source condenser
Theory of computation
12:1-12:14
Regular Paper
This work was supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) via Contract No. 2019-1902070006. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either express or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.
Full versions posted on ECCC as TR19-183.
https://eccc.weizmann.ac.il/report/2019/183/
Marshall
Ball
Marshall Ball
Computer Science Department, Columbia University, New York, NY, USA
https://orcid.org/0000-0002-4236-3710
Partially supported by an IBM Research PhD Fellowship.
Oded
Goldreich
Oded Goldreich
Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel
https://orcid.org/0000-0002-4329-135X
Partially supported by an ISF grant number (Nr. 1146/18) and received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819702); research was conducted while he enjoyed the hospitality of the computer science department at Columbia University.
Tal
Malkin
Tal Malkin
Computer Science Department, Columbia University, New York, NY, USA
10.4230/LIPIcs.ITCS.2022.12
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Marshall Ball, Oded Goldreich, and Tal Malkin
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