Three-Source Extractors for Polylogarithmic Min-Entropy

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 min-entropy as small as $\log n+O(1)$. Previously, the best known result in the two-source case is an extractor by Bourgain \cite{Bourgain05}, which works for min-entropy $0.49n$; and the best known result in the general case is an earlier work of the author \cite{Li13b}, which gives an extractor for a constant number of independent sources with min-entropy $\mathsf{polylog(n)}$. However, the constant in the construction of \cite{Li13b} depends on the hidden constant in the best known seeded extractor, and can be large; moreover the error in that construction is only $1/\mathsf{poly(n)}$. In this paper, we make two important improvements over the result in \cite{Li13b}. First, we construct an explicit extractor for \emph{three} independent sources on $n$ bits with min-entropy $k \geq \mathsf{polylog(n)}$. In fact, our extractor works for one independent source with poly-logarithmic min-entropy 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 $1/\mathsf{poly(n)}$ to $2^{-k^{\Omega(1)}}$, which is almost optimal and crucial for cryptographic applications. Some of the techniques developed here may be of independent interests

Distributed computing and cryptography with general weak random sources

The use of randomness in computer science is ubiquitous. Randomized protocols have turned out to be much more efficient than their deterministic counterparts. In addition, many problems in distributed computing and cryptography are impossible to solve without randomness. However, these applications typically require uniform random bits, while in practice almost all natural random phenomena are biased. Moreover, even originally uniform random bits can be damaged if an adversary learns some partial information about these bits. In this thesis, we study how to run randomized protocols in distributed computing and cryptography with imperfect randomness. We use the most general model for imperfect randomness where the weak random source is only required to have a certain amount of min-entropy. One important tool here is the randomness extractor. A randomness extractor is a function that takes as input one or more weak random sources, and outputs a distribution that is close to uniform in statistical distance. Randomness extractors are interesting in their own right and are closely related to many other problems in computer science. Giving efficient constructions of randomness extractors with optimal parameters is one of the major open problems in the area of pseudorandomness. We construct network extractor protocols that extract private random bits for parties in a communication network, assuming that they each start with an independent weak random source, and some parties are corrupted by an adversary who sees all communications in the network. These protocols imply fault-tolerant distributed computing protocols and secure multi-party computation protocols where only imperfect randomness is available. The probabilistic method shows that there exists an extractor for two independent sources with logarithmic min-entropy, while known constructions are far from achieving these parameters. In this thesis we construct extractors for two independent sources with any linear min-entropy, based on a computational assumption. We also construct the best known extractors for three independent sources and affine sources. Finally we study the problem of privacy amplification. In this model, two parties share a private weak random source and they wish to agree on a private uniform random string through communications in a channel controlled by an adversary, who has unlimited computational power and can change the messages in arbitrary ways. All previous results assume that the two parties have local uniform random bits. We show that this problem can be solved even if the two parties only have local weak random sources. We also improve previous results in various aspects by constructing the first explicit non-malleable extractor and giving protocols based on this extractor.Computer Science

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

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 $k \geq n-n^{\Omega(1)}$, with output size $n^{\Omega(1)}$ and error $2^{-n^{\Omega(1)}}$. (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 $n^{\Omega(1)}$, which works for min-entropy $k \geq n-n^{\Omega(1)}$, with output size $n^{\Omega(1)}$ and error $2^{-n^{\Omega(1)}}$. 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 $n^{\Omega(1)}$, relative rate $n^{\Omega(1)}/n$, and error $2^{-n^{\Omega(1)}}$.Comment: 50 pages; see paper for full abstrac

Leakage-resilient coin tossing

Proceedings 25th International Symposium, DISC 2011, Rome, Italy, September 20-22, 2011.The ability to collectively toss a common coin among n parties in the presence of faults is an important primitive in the arsenal of randomized distributed protocols. In the case of dishonest majority, it was shown to be impossible to achieve less than 1 r bias in O(r) rounds (Cleve STOC ’86). In the case of honest majority, in contrast, unconditionally secure O(1)-round protocols for generating common unbiased coins follow from general completeness theorems on multi-party secure protocols in the secure channels model (e.g., BGW, CCD STOC ’88). However, in the O(1)-round protocols with honest majority, parties generate and hold secret values which are assumed to be perfectly hidden from malicious parties: an assumption which is crucial to proving the resulting common coin is unbiased. This assumption unfortunately does not seem to hold in practice, as attackers can launch side-channel attacks on the local state of honest parties and leak information on their secrets. In this work, we present an O(1)-round protocol for collectively generating an unbiased common coin, in the presence of leakage on the local state of the honest parties. We tolerate t ≤ ( 1 3 − )n computationallyunbounded Byzantine faults and in addition a Ω(1)-fraction leakage on each (honest) party’s secret state. Our results hold in the memory leakage model (of Akavia, Goldwasser, Vaikuntanathan ’08) adapted to the distributed setting. Additional contributions of our work are the tools we introduce to achieve the collective coin toss: a procedure for disjoint committee election, and leakage-resilient verifiable secret sharing.National Defense Science and Engineering Graduate FellowshipNational Science Foundation (U.S.) (CCF-1018064

Multi-Source Non-Malleable Extractors and Applications

We introduce a natural generalization of two-source non-malleable extractors (Cheragachi and Guruswami, TCC 2014) called as \textit{multi-source non-malleable extractors}. Multi-source non-malleable extractors are special independent source extractors which satisfy an additional non-malleability property. This property requires that the output of the extractor remains close to uniform even conditioned on its output generated by tampering {\it several sources together}. We formally define this primitive, give a construction that is secure against a wide class of tampering functions, and provide applications. More specifically, we obtain the following results: \begin{itemize} \item For any $s \geq 2$, we give an explicit construction of a $s$-source non-malleable extractor for min-entropy $\Omega(n)$ and error $2^{-n^{\Omega(1)}}$ in the {\it overlapping joint tampering model}. This means that each tampered source could depend on any strict subset of all the sources and the sets corresponding to each tampered source could be overlapping in a way that we define. Prior to our work, there were no known explicit constructions that were secure even against disjoint tampering (where the sets are required to be disjoint without any overlap). %Our extractor is pre-image sampleable and hence, gives rise to non-malleable codes against the same tampering family. % \item We show how to efficiently preimage sample given the output of (a variant of) our extractor and this immediately gives rise to a $s$-state non-malleable code secure in the overlapping joint tampering model (via a generalization of the result by Cheragachi and Guruswami). \item We adapt the techniques used in the above construction to give a $t$-out-of-$n$ non-malleable secret sharing scheme (Goyal and Kumar, STOC 2018) for any $t \leq n$ in the \emph{disjoint tampering model}. This is the first general construction of a threshold non-malleable secret sharing (NMSS) scheme in the disjoint tampering model. All prior constructions had a restriction that the size of the tampered subsets could not be equal. \item We further adapt the techniques used in the above construction to give a $t$-out-of-$n$ non-malleable secret sharing scheme (Goyal and Kumar, STOC 2018) for any $t \leq n$ in the \emph{overlapping joint tampering model}. This is the first construction of a threshold NMSS in the overlapping joint tampering model. \item We show that a stronger notion of $s$-source non-malleable extractor that is multi-tamperable against disjoint tampering functions gives a single round network extractor protocol (Kalai et al., FOCS 2008) with attractive features. Plugging in with a new construction of multi-tamperable, 2-source non-malleable extractors provided in our work, we get a network extractor protocol for min-entropy $\Omega(n)$ that tolerates an {\it optimum} number ($t = p-2$) of faulty processors and extracts random bits for {\it every} honest processor. The prior network extractor protocols could only tolerate $t = \Omega(p)$ faulty processors and failed to extract uniform random bits for a fraction of the honest processors. \end{itemize

Improved Computational Extractors and their Applications

Recent exciting breakthroughs, starting with the work of Chattopadhyay and Zuckerman (STOC 2016) have achieved the first two-source extractors that operate in the low min-entropy regime. Unfortunately, these constructions suffer from non-negligible error, and reducing the error to negligible remains an important open problem. In recent work, Garg, Kalai, and Khurana (GKK, Eurocrypt 2020) investigated a meaningful relaxation of this problem to the computational setting, in the presence of a common random string (CRS). In this relaxed model, their work built explicit two-source extractors for a restricted class of unbalanced sources with min-entropy $n^{\gamma}$ (for some constant $\gamma$) and negligible error, under the sub-exponential DDH assumption. In this work, we investigate whether computational extractors in the CRS model be applied to more challenging environments. Specifically, we study network extractor protocols (Kalai et al., FOCS 2008) and extractors for adversarial sources (Chattopadhyay et al., STOC 2020) in the CRS model. We observe that these settings require extractors that work well for balanced sources, making the GKK results inapplicable. We remedy this situation by obtaining the following results, all of which are in the CRS model and assume the sub-exponential hardness of DDH. - We obtain ``optimal\u27\u27 computational two-source and non-malleable extractors for balanced sources: requiring both sources to have only poly-logarithmic min-entropy, and achieving negligible error. To obtain this result, we perform a tighter and arguably simpler analysis of the GKK extractor. - We obtain a single-round network extractor protocol for poly-logarithmic min-entropy sources that tolerates an optimal number of adversarial corruptions. Prior work in the information-theoretic setting required sources with high min-entropy rates, and in the computational setting had round complexity that grew with the number of parties, required sources with linear min-entropy, and relied on exponential hardness (albeit without a CRS). - We obtain an ``optimal\u27\u27 {\em adversarial source extractor} for poly-logarithmic min-entropy sources, where the number of honest sources is only 2 and each corrupted source can depend on either one of the honest sources. Prior work in the information-theoretic setting had to assume a large number of honest sources

Quantum secure non-malleable-extractors

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 $k \geq \textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$ ($n$ is the length of the source and $\epsilon$ 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 $n$) 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 $2$-round privacy amplification protocol that is secure against active quantum adversaries with communication $\textsf{poly}\left(\log \left( \frac{n}{\epsilon} \right)\right)$, exponentially improving upon the linear communication required by the protocol due to [2019]. 2) We construct an explicit quantum secure $2$-source non-malleable-extractor for min-entropy $k \geq n- n^{\Omega(1)}$, with an output of size $n^{\Omega(1)}$ and error $2^{- n^{\Omega(1)}}$. 3) We also study their natural extensions when the tampering of the inputs is performed $t$-times. We construct explicit quantum secure $t$-non-malleable-extractors for both seeded ($t=d^{\Omega(1)}$) as well as $2$-source case ($t=n^{\Omega(1)}$)

Secrecy without Perfect Randomness: Cryptography with (Bounded) Weak Sources

Cryptographic protocols are commonly designed and their security proven under the assumption that the protocol parties have access to perfect (uniform) randomness. Physical randomness sources deployed in practical implementations of these protocols often fall short in meeting this assumption, but instead provide only a steady stream of bits with certain high entropy. Trying to ground cryptographic protocols on such imperfect, weaker sources of randomness has thus far mostly given rise to a multitude of impossibility results, including the impossibility to construct provably secure encryption, commitments, secret sharing, and zero-knowledge proofs based solely on a weak source. More generally, indistinguishability-based properties break down for such weak sources. In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirna (SV) sources. The quantification relies on a novel relaxation of indistinguishability by a quantitative parameter. We call the resulting notion differential indistinguishability in order to reflect its structural similarity to differential privacy. More concretely, we prove that indistinguishability with uniform randomness implies differential indistinguishability with weak randomness. We show that if the amount of weak randomness is limited (e.g., by using it only to seed a PRG), all cryptographic primitives and protocols still achieve differential indistinguishability