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arXiv:1211.0651 · Lecture notes in computer science

Non-malleable Condensers for Arbitrary Min-entropy, and Almost Optimal Protocols for Privacy Amplification

Xin Li

Published Jan 1, 2015• 23 citations
arXivHTMLPDF
DOI
10.1007/978-3-662-46494-6_21

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Design Extractors, Non-Malleable Condensers and Privacy Amplification.

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We introduce a new combinatorial object, called a design extractor, that has both the properties of a design and an extractor. We give efficient constructions of such objects and show that they can be used in several applications. Improving the output length of known non-malleable extractors. Non-malleable extractors were introduced in [DW09] to study the problem of privacy amplification with an active adversary. Currently, only two explicit constructions are known [DLWZ11, CRS11]. Both constructions work for n bit sources with min-entropy k > n/2. However, in both constructions the output length is smaller than the seed length, while the probabilistic method shows that to achieve error e, one can use O(log n+log (1/e)) bits to extract up to k/2 output bits. In this paper, we use our design extractor to give an explicit non-malleable extractor for min-entropy k > n/2, that has seed length O(log n+log (1/e)) and output length Ω(k). Non-malleable condensers. We introduce and define the notion of a non-malleable condenser. A non-malleable condenser is a generalization and relaxation of a non-malleable extractor. We show that similar as extractors and condensers, non-malleable condensers can be used to construct non-malleable extractors. We then show that our design extractor already gives a non-malleable condenser for min-entropy k > n/2, with error e and seed length O(log (1/e)). A new optimal protocol for privacy amplification. More surprisingly, we show that non-malleable condensers themselves give optimal privacy amplification protocols with an active adversary. In fact, the non-malleable condensers used in these protocols are much weaker compared to non-malleable extractors, in the sense that the entropy rate of the condenser's output does not need to increase at all. This suggests that one promising next step to achieve better privacy amplification protocols may be to construct non-malleable condensers for smaller min-entropy. As a by-product, we also obtain a new explicit 2-round privacy amplification protocol with optimal entropy loss and optimal communication complexity for min-entropy k>n/2, without using non-malleable extractors.

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

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We make progress in the following three problems: 1. Constructing optimal seeded non-malleable 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, and by Cohen all require seed length and min-entropy with quadratic loss in parameters. As a result, the best known explicit privacy amplification protocols with an active adversary, which achieve two rounds of communication and optimal entropy loss was sub-optimal in the min-entropy of the source. In this paper we give an explicit non-malleable extractor that works for nearly optimal seed length and min-entropy, and yields a two-round privacy amplification protocol with optimal entropy loss for almost all ranges of the security parameter. For the third problem, we improve upon a very recent result by Cohen and Schulman and give an explicit extractor that uses an absolute constant number of sources, each with almost logarithmic min-entropy. The key ingredient in all our constructions is a generalized, and much more efficient version of the independence preserving merger introduced by Cohen, which we call non-malleable independence preserving merger. Our construction of the merger also simplifies that of Cohen and Schulman, and may be of independent interest.

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We introduce a new combinatorial object, called a design extractor, that has both the properties of a design and an extractor. We give efficient constructions of such objects and show that they can be used in several applications. Improving the output length of known non-malleable extractors. Non-malleable extractors were introduced in [DW09] to study the problem of privacy amplification with an active adversary. Currently, only two explicit constructions are known [DLWZ11, CRS11]. Both constructions work for n bit sources with min-entropy k > n/2. However, in both constructions the output length is smaller than the seed length, while the probabilistic method shows that to achieve error e, one can use O(log n+log (1/e)) bits to extract up to k/2 output bits. In this paper, we use our design extractor to give an explicit non-malleable extractor for min-entropy k > n/2, that has seed length O(log n+log (1/e)) and output length Ω(k). Non-malleable condensers. We introduce and define the notion of a non-malleable condenser. A non-malleable condenser is a generalization and relaxation of a non-malleable extractor. We show that similar as extractors and condensers, non-malleable condensers can be used to construct non-malleable extractors. We then show that our design extractor already gives a non-malleable condenser for min-entropy k > n/2, with error e and seed length O(log (1/e)). A new optimal protocol for privacy amplification. More surprisingly, we show that non-malleable condensers themselves give optimal privacy amplification protocols with an active adversary. In fact, the non-malleable condensers used in these protocols are much weaker compared to non-malleable extractors, in the sense that the entropy rate of the condenser's output does not need to increase at all. This suggests that one promising next step to achieve better privacy amplification protocols may be to construct non-malleable condensers for smaller min-entropy. As a by-product, we also obtain a new explicit 2-round privacy amplification protocol with optimal entropy loss and optimal communication complexity for min-entropy k>n/2, without using non-malleable extractors.
We make progress in the following three problems: 1. Constructing optimal seeded non-malleable 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, and by Cohen all require seed length and min-entropy with quadratic loss in parameters. As a result, the best known explicit privacy amplification protocols with an active adversary, which achieve two rounds of communication and optimal entropy loss was sub-optimal in the min-entropy of the source. In this paper we give an explicit non-malleable extractor that works for nearly optimal seed length and min-entropy, and yields a two-round privacy amplification protocol with optimal entropy loss for almost all ranges of the security parameter. For the third problem, we improve upon a very recent result by Cohen and Schulman and give an explicit extractor that uses an absolute constant number of sources, each with almost logarithmic min-entropy. The key ingredient in all our constructions is a generalized, and much more efficient version of the independence preserving merger introduced by Cohen, which we call non-malleable independence preserving merger. Our construction of the merger also simplifies that of Cohen and Schulman, and may be of independent interest.