New independent source extractors with exponential improvement

Li, Xin

doi:10.1145/2488608.2488708

Cited by 25 publications

(16 citation statements)

References 30 publications

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“…If the min-entropy k is very close to log 2 n, then we also have improved results over [Li13b]. In particular, we have the following theorem.…”

Section: Our Resultsmentioning

confidence: 67%

“…In a recent breakthrough [Li13b,Li13a], the author further exploited the properties of somewhere random sources and established a connection between extraction from such sources and the problem of leader election in distributed computing. Based on this connection, the author managed to construct the first explicit extractor that uses only a constant number of sources even if the entropy is as small as polylog(n) [Li13a].…”

Section: Introductionmentioning

confidence: 99%

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Three-Source Extractors for Polylogarithmic Min-Entropy

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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We continue the study of constructing explicit extractors for independent general weak random sources. The ultimate goal is to give a construction that matches what is given by the probabilistic method -an extractor for two independent n-bit weak random sources with minentropy as small as log n + O(1). Previously, the best known result in the two-source case is an extractor by Bourgain [Bou05], which works for min-entropy 0.49n; and the best known result in the general case is an earlier work of the author [Li13a], which gives an extractor for a constant number of independent sources with min-entropy polylog(n). However, the constant in the construction of [Li13a] depends on the hidden constant in the best known seeded extractor, and can be large; moreover the error in that construction is only 1/poly(n).In this paper, we make two important improvements over the result in [Li13a]. First, we construct an explicit extractor for three independent sources on n bits with min-entropy k ≥ polylog(n). In fact, our extractor works for one independent source with poly-logarithmic minentropy and another independent block source with two blocks each having poly-logarithmic min-entropy. Thus, our result is nearly optimal, and the next step would be to break the 0.49n barrier in two-source extractors. Second, we improve the error of the extractor from 1/poly(n) to 2 −k Ω(1) , which is almost optimal and crucial for cryptographic applications. Some of the techniques developed here may be of independent interests.

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“…If the min-entropy k is very close to log 2 n, then we also have improved results over [Li13b]. In particular, we have the following theorem.…”

Section: Our Resultsmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Three-Source Extractors for Polylogarithmic Min-Entropy

2015

2015 IEEE 56th Annual Symposium on Foundations of Computer Science

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“…The flurry of work on non-malleable codes has yielded many surprising connections to other areas of theoretical computer science, including additive combinatorics [ADKO15], two-source extractors [Li12,Li13,CZ16], and non-malleable encryption/commitment [CMTV15, CDTV16, GPR16]. As we discuss in Section 1.2, our work establishes yet another connection-to techniques in unconditional derandomization.…”

Section: This Work: Efficient Non-malleable Codes For Small-depth Cirmentioning

confidence: 69%

Non-Malleable Codes for Small-Depth Circuits

Ball

Dachman-Soled

Guo

et al. 2018

2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS)

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We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. AC 0 tampering functions), our codes have codeword length n = k 1+o(1) for a k-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length 2 O( √ k) . Our construction remains efficient for circuit depths as large as Θ(log(n)/ log log(n)) (indeed, our codeword length remains n ≤ k 1+ε ), and extending our result beyond this would require separating P from NC 1 .We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction. * marshall@cs.columbia.edu, Columbia University.

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“…The above two kinds of extractors are closely related, and in many cases techniques used for one can also be used to improve the constructions of the other. These connections have been demonstrated in a number of works (e.g., [Li12b,Li13b,Li13a,Li15d,CZ16]).…”

Section: Introductionmentioning

confidence: 84%

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

Chattopadhyay

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

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We make progress in the following three problems: 1. Constructing optimal seeded nonmalleable extractors; 2. Constructing optimal privacy amplification protocols with an active adversary, for any possible security parameter; 3. Constructing extractors for independent weak random sources, when the min-entropy is extremely small (i.e., near logarithmic).For the first two problems, the best known non-malleable extractors by Chattopadhyay, Goyal and Li [CGL16], and by Cohen [Coh16a,Coh16b] all require seed length and min-entropy at least log 2 (1/ε), where ε is the error of the extractor. As a result, the best known explicit privacy amplification protocols with an active adversary, which achieve 2 rounds of communication and optimal entropy loss in [Li15c, CGL16], can only handle security parameter up to s = Ω( √ k), where k is the min-entropy of the shared secret weak random source. For larger s the best known protocol with optimal entropy loss in [Li15c] requires O(s/ √ k) rounds of communication.In this paper we give an explicit non-malleable extractor that only requires seed length and min-entropy log 1+o(1) (n/ε), which also yields a 2-round privacy amplification protocol with optimal entropy loss for security parameter up to s = k 1−α for any constant α > 0. For the third problem, previously the best known extractor which supports the smallest min-entropy due to Li [Li13a], requires min-entropy log 2+δ n and uses O(1/δ) sources, for any constant δ > 0. A very recent result by Cohen and Schulman [CS16] improves this, and constructed explicit extractors that use O(1/δ) sources for min-entropy log 1+δ n, any constant δ > 0. In this paper we further improve their result, and give an explicit extractor that uses O(1) (an absolute constant) sources for min-entropy log 1+o(1) n. The key ingredient in all our constructions is a generalized, and much more efficient version of the independence preserving merger introduced in [CS16], which we call non-malleable independence preserving merger. Our construction of the merger also simplifies that of [CS16], and may be of independent interest.

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New independent source extractors with exponential improvement

Cited by 25 publications

References 30 publications

Three-Source Extractors for Polylogarithmic Min-Entropy

Three-Source Extractors for Polylogarithmic Min-Entropy

Non-Malleable Codes for Small-Depth Circuits

Explicit Non-malleable Extractors, Multi-source Extractors, and Almost Optimal Privacy Amplification Protocols

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