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arXiv:1505.00107

Non-Malleable Extractors and Codes, with their Many Tampered Extensions

Eshan Chattopadhyay, Vipul Goyal, Xin Li

Published May 1, 2015• 3 citations
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Abstract

Randomness extractors and error correcting codes are fundamental objects in computer science. Recently, there have been several natural generalizations of these objects, in the context and study of tamper resilient cryptography. These are seeded non-malleable extractors, introduced in [DW09]; seedless non-malleable extractors, introduced in [CG14b]; and non-malleable codes, introduced in [DPW10]. However, explicit constructions of non-malleable extractors appear to be hard, and the known constructions are far behind their non-tampered counterparts. In this paper we make progress towards solving the above problems. Our contributions are as follows. (1) We construct an explicit seeded non-malleable extractor for min-entropy $k \geq \log^2 n$. This dramatically improves all previous results and gives a simpler 2-round privacy amplification protocol with optimal entropy loss, matching the best known result in [Li15b]. (2) We construct the first explicit non-malleable two-source extractor for min-entropy $k \geq n-n^{Ω(1)}$, with output size $n^{Ω(1)}$ and error $2^{-n^{Ω(1)}}$. (3) We initiate the study of two natural generalizations of seedless non-malleable extractors and non-malleable codes, where the sources or the codeword may be tampered many times. We construct the first explicit non-malleable two-source extractor with tampering degree $t$ up to $n^{Ω(1)}$, which works for min-entropy $k \geq n-n^{Ω(1)}$, with output size $n^{Ω(1)}$ and error $2^{-n^{Ω(1)}}$. We show that we can efficiently sample uniformly from any pre-image. By the connection in [CG14b], we also obtain the first explicit non-malleable codes with tampering degree $t$ up to $n^{Ω(1)}$, relative rate $n^{Ω(1)}/n$, and error $2^{-n^{Ω(1)}}$.

DOI
10.48550/arxiv.1505.00107

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The known constructions of negligible error (non-malleable) two-source extractors can be broadly classified in three categories: (1) Constructions where one source has min-entropy rate about 1/21/21/2, the other source can have small min-entropy rate, but the extractor doesn't guarantee non-malleability. (2) Constructions where one source is uniform, and the other can have small min-entropy rate, and the extractor guarantees non-malleability when the uniform source is tampered. (3) Constructions where both sources have entropy rate very close to 111 and the extractor guarantees non-malleability against the tampering of both sources. We introduce a new notion of collision resistant extractors and in using it we obtain a strong two source non-malleable extractor where we require the first source to have 0.80.80.8 entropy rate and the other source can have min-entropy polylogarithmic in the length of the source. We show how the above extractor can be applied to obtain a non-malleable extractor with output rate 12\frac 1 221​, which is optimal. We also show how, by using our extractor and extending the known protocol, one can obtain a privacy amplification secure against memory tampering where the size of the secret output is almost optimal.
We construct several explicit quantum secure non-malleable-extractors. All the quantum secure non-malleable-extractors we construct are based on the constructions by Chattopadhyay, Goyal and Li [2015] and Cohen [2015]. 1) We construct the first explicit quantum secure non-malleable-extractor for (source) min-entropy k≥poly(log⁡(nε))k \geq \textsf{poly}\left(\log \left( \frac{n}ε \right)\right)k≥poly(log(εn​)) (nnn is the length of the source and εεε is the error parameter). Previously Aggarwal, Chung, Lin, and Vidick [2019] have shown that the inner-product based non-malleable-extractor proposed by Li [2012] is quantum secure, however it required linear (in nnn) min-entropy and seed length. Using the connection between non-malleable-extractors and privacy amplification (established first in the quantum setting by Cohen and Vidick [2017]), we get a 222-round privacy amplification protocol that is secure against active quantum adversaries with communication poly(log⁡(nε))\textsf{poly}\left(\log \left( \frac{n}ε \right)\right)poly(log(εn​)), exponentially improving upon the linear communication required by the protocol due to [2019]. 2) We construct an explicit quantum secure 222-source non-malleable-extractor for min-entropy k≥n−nΩ(1)k \geq n- n^{Ω(1)}k≥n−nΩ(1), with an output of size nΩ(1)n^{Ω(1)}nΩ(1) and error 2−nΩ(1)2^{- n^{Ω(1)}}2−nΩ(1). 3) We also study their natural extensions when the tampering of the inputs is performed ttt-times. We construct explicit quantum secure ttt-non-malleable-extractors for both seeded (t=dΩ(1)t=d^{Ω(1)}t=dΩ(1)) as well as 222-source case (t=nΩ(1)t=n^{Ω(1)}t=nΩ(1)).
The known constructions of negligible error (non-malleable) two-source extractors can be broadly classified in three categories: (1) Constructions where one source has min-entropy rate about 1/21/21/2, the other source can have small min-entropy rate, but the extractor doesn't guarantee non-malleability. (2) Constructions where one source is uniform, and the other can have small min-entropy rate, and the extractor guarantees non-malleability when the uniform source is tampered. (3) Constructions where both sources have entropy rate very close to 111 and the extractor guarantees non-malleability against the tampering of both sources. We introduce a new notion of collision resistant extractors and in using it we obtain a strong two source non-malleable extractor where we require the first source to have 0.80.80.8 entropy rate and the other source can have min-entropy polylogarithmic in the length of the source. We show how the above extractor can be applied to obtain a non-malleable extractor with output rate 12\frac 1 221​, which is optimal. We also show how, by using our extractor and extending the known protocol, one can obtain a privacy amplification secure against memory tampering where the size of the secret output is almost optimal.
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.
In [1], Dodis and Wichs introduced the notion of a non-malleable extractor. A non-malleable extractor is a much stronger version of a seeded extractor. Dodis and Wichs showed that such an object can be used to give optimal privacy amplification protocols with an active adversary. Previously, there are only two known constructions of nonmalleable extractors [2], [3]. Both constructions only work for (n, k)-sources with k >; n/2. Interestingly, both constructions are also two-source extractors. In this paper, we present a strong connection between nonmalleable extractors and two-source extractors. The first part of the connection shows that non-malleable extractors can be used to construct two-source extractors. This partially explains why previous constructions of non-malleable extractors only work for entropy rate >; 1/2, and why explicit non-malleable extractors for small min-entropy may be hard to get. The second part of the connection shows that certain two-source extractors can be used to construct non-malleable extractors. Using this connection, we obtain the first construction of non-malleable extractors for k; 0. This dramatically improves previous results and answers an open problem in [2].
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.