Improved Extractors for Small-Space Sources

Chattopadhyay, Eshan; Goodman, Jesse

doi:10.1109/focs52979.2021.00066

Cited by 6 publications

(12 citation statements)

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“…Chattopadhyay and Li [CL16b] then constructed an extractor with error n −Ω(1) for space-s source with entropy k ≥ s 1.1 2 log 0.51 (n) based on their sumset source extractors. Recently, based on a new reduction to affine extractors, Chattopadhyay and Goodman [CG21] improved the entropy requirement to k ≥ s • polylog(n) (or k ≥ s log 2+o(1) (n) if we are only interested in constant error and one-bit output). 2 With our new extractors for sum of two sources and the reduction in [CL16b], we can get extractors for space-s source with entropy s log(n) + polylog(n), which is already an improvement over the result in [CG21].…”

Section: Small-space Sourcesmentioning

confidence: 99%

“…Recently, based on a new reduction to affine extractors, Chattopadhyay and Goodman [CG21] improved the entropy requirement to k ≥ s • polylog(n) (or k ≥ s log 2+o(1) (n) if we are only interested in constant error and one-bit output). 2 With our new extractors for sum of two sources and the reduction in [CL16b], we can get extractors for space-s source with entropy s log(n) + polylog(n), which is already an improvement over the result in [CG21]. In this work we further improve the reduction and obtain the following theorems.…”

Section: Small-space Sourcesmentioning

confidence: 99%

“…However, such a reduction does not work for entropy smaller than √ n (cf. [CG21]). Chattopadhyay and Li [CL16b] observed that with a sumset source extractor we can extract from the concatenation of independent sources with unknown and uneven length.…”

Section: Reduction From Small-space Sources To Sumset Sourcesmentioning

confidence: 99%

“…They then showed that with a sumset source extractor, we can "adaptively" pick which time steps to condition on and break the √ n barrier. Chattopadhyay and Goodman [CG21] further refined this reduction and showed how to improve the entropy requirement by reducing to a convex combination of affine sources. The reductions in [CL16b] and [CG21] can be viewed as "binary searching" the correct time steps to condition on, so that the given source X becomes the concatenation of independent blocks (X 1 , .…”

Section: Reduction From Small-space Sources To Sumset Sourcesmentioning

confidence: 99%

“…Here we focus on the small-space extractors which minimize the entropy requirement. For small-space extractors with negligible error, see[CG21] for a survey.…”

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Extractors for Sum of Two Sources

Chattopadhyay¹,

Liao²

2021

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We consider the problem of extracting randomness from sumset sources, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An (n, k, C)-sumset source X is a distribution on {0, 1} n of the form X1 +X2 +. . .+XC, where Xi's are independent sources on n bits with min-entropy at least k. Prior extractors either required the number of sources C to be a large constant or the min-entropy k to be at least 0.51n.As our main result, we construct an explicit extractor for sumset sources in the setting of C = 2 for min-entropy poly(log n) and polynomially small error. We can further improve the min-entropy requirement to (log n) • (log log n) 1+o(1) at the expense of worse error parameter of our extractor. We find applications of our sumset extractor for extracting randomness from other well-studied models of weak sources such as affine sources, small-space sources, and interleaved sources.Interestingly, it is unknown if a random function is an extractor for sumset sources. We use techniques from additive combinatorics to show that it is a disperser, and further prove that an affine extractor works for an interesting subclass of sumset sources which informally corresponds to the "low doubling" case (i.e., the support of X1 + X2 is not much larger than 2 k ). * Supported by NSF CAREER award 2045576 1 Supp(X) denotes the support of X. We use log to denote the base-2 logarithm in the rest of this paper.Definition 1.1. Let X be a family of distribution over {0, 1} n . We say a deterministic function Ext : {0, 1} n → {0, 1} m is a deterministic extractor for X with error ε if for every distribution X ∈ X ,We say Ext is explicit if Ext is computable by a polynomial-time algorithm.The most well-studied deterministic extractors are multi-source extractors, which assume that the extractor is given C independent (n, k)-sources X 1 , X 2 , . . . , X C . This model was first introduced by Chor and Goldreich [CG88]. They constructed explicit two-source extractors with error 2 −Ω(n) for entropy 0.51n, and proved that there exists a two-source extractor for entropy k = O(log(n)) with error 2 −Ω(k) . Significant progress was made by Chattopadhyay and Zuckerman [CZ19], who showed how to construct an extractor for two sources with entropy k = polylog(n), after a long line of successful work on independent source extractors (see the references in [CZ19]). The output length was later improved to Ω(k) by Li [Li16]. Furthermore, Ben-Aroya, Doron and Ta-Shma [BDT19] showed how to improve the entropy requirement to O(log 1+o(1) (n)) for constant error and 1-bit output. The entropy requirement was further improved in subsequent works [Coh17, Li17], and the state-of-the-art result is by Li [Li19], which requireslog log log(n) ). For a more elaborate discussion, see the survey by Chattopadhyay [Cha20]. Apart from independent sources, many other classes of sources have been studied for deterministic extraction. We briefly introduce some examples here. A well-studied class is oblivious bit-fixing sources [CGH + 85, GRS06, KZ...

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Section: Small-space Sourcesmentioning

confidence: 99%

Section: Small-space Sourcesmentioning

confidence: 99%

Section: Reduction From Small-space Sources To Sumset Sourcesmentioning

confidence: 99%

Section: Reduction From Small-space Sources To Sumset Sourcesmentioning

confidence: 99%

“…Here we focus on the small-space extractors which minimize the entropy requirement. For small-space extractors with negligible error, see[CG21] for a survey.…”

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confidence: 99%

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