arXiv subjects

Eshan Chattopadhyay

Publications and source records attributed to Eshan Chattopadhyay.

At least 19 recordsLinked to original sources

arXiv2026

Two-Sided Product Expanding Codes via Rademacher Matrices

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

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arXiv2026

Frustration Free Stoquastic Local Hamiltonian with Sub-Constant Gap is in NP

We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the $β$-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when $β= 1/\mathrm{poly}(n)$. Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap $β= Ω(1)$. We present an improved algorithm and analysis, establishing membership in NP even when $β= Ω(1/(\log\log n))$. We complement our result with an explicit example demonstrating why the analysis does not extend directly to $β= o(1/\log\log n)$.

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arXiv2026

Exponential Correlation Bounds for Polynomials

We prove that the XOR of $k$ majorities on disjoint blocks of $\ell$ bits has correlation at most $(2d/\sqrt{\ell})^k$ with every degree-$d$ polynomial over $\mathbb F_2$. By known techniques, this implies pseudorandom generators with polylogarithmic seed length for low-degree polynomials over $\mathbb F_2$ and for alternating circuits with parity gates.

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arXiv2026

A Resolution of Friedgut's Conjecture on Influential Coalitions

We prove that, for every constant $\varepsilon>0$ and every function $f:Σ^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.

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arXiv2026

Improved Bounds for Coin Flipping, Leader Election, and Random Selection

Random selection, leader election, and collective coin flipping are fundamental tasks in fault-tolerant distributed computing. We study these problems in the full-information model where despite decades of study, key gaps remain in our understanding of the trade-offs between round complexity, communication per player in each round, and adversarial resilience. We make progress by proving improved bounds for these problems. We first show that any $k$-round coin flipping protocol over $\ell$ players, each player sending one bit per round, can be biased by $O(\ell/\log^{(k)}(\ell))$ bad players. We obtain a similar lower bound for leader election. This strengthens prior best bounds [RSZ, SICOMP 2002] of $O(\ell/\log^{(2k-1)}(\ell))$ for coin flipping protocols and $O(\ell/\log^{(2k+1)}(\ell))$ for leader election protocols. Our result implies that any (1-bit per player) protocol tolerating linear fraction of bad players requires at least $\log^* \ell$ rounds, showing existing protocols [RZ, JCSS 2001; F, FOCS 1999] are near-optimal. We next initiate the study of one-round, (1-bit per player) random selection. For all $m\ge (\log(\ell))^2$, we obtain an optimal protocol (a first in the full information model for any task): We construct a protocol resilient to $O(\ell / m)$ bad players that outputs $m$ uniform random bits. And, we show that any protocol that outputs $m$ uniform random bits can be corrupted using $O(\ell / m)$ bad players. This also implies a one-round leader election protocol resilient to $\ell / (\log \ell)^2$ bad players, improving the prior best protocol [RZ, JCSS 2001] which was resilient to $\ell / (\log \ell)^3$ bad players. Our resilience matches that of the best one-round coin flipping protocol by Ajtai & Linial. To obtain our lower bound, we introduce multi-output influence, an extension of influence of boolean functions to the multi-output setting.

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arXiv2025

Condensing and Extracting Against Online Adversaries

We study the tasks of deterministically condensing and extracting from Online Non-Oblivious Symbol Fixing (oNOSF) sources, a natural model of defective randomness where extraction is impossible in many parameter regimes [AORSV, EUROCRYPT'20]. A $(g,\ell)$-oNOSF source is a sequence of $\ell$ blocks where at least $g$ blocks are good (independent, with min-entropy) and the remaining bad blocks are controlled by an online adversary and can be arbitrarily correlated with prior blocks. Previously, [CGR, FOCS'24] proved impossibility of condensing beyond rate $1/2$ when $g\le 0.5 \ell$ and showed existence of condensers for when $g \ge 0.51\ell$ and $n$ is exponential in $\ell$. In this work, not only do we construct the first explicit condensers matching the existential results of [CGR, FOCS'24], but we make a doubly exponential improvement by handling the case when $g\ge 0.51\ell$ and $n$ is only polylogarithmic in $\ell$. We also obtain a much improved explicit construction for transforming low-entropy oNOSF sources into uniform oNOSF sources. Next, we essentially resolve the question of the existence of condensers for oNOSF sources by showing the existence of condensers even when $n$ is a large enough constant and $\ell$ is growing (provided $g \ge 0.51\ell$). We apply our condensers to collective coin flipping and collective sampling, widely studied problems in fault-tolerant distributed computing, and provide very simple protocols for them. Finally, we study the possibility of extraction from oNOSF sources. For lower bounds, we introduce the notion of online influence - extending the notion of influence of boolean functions - and establish tight bounds that imply extraction lower bounds. We also construct explicit extractors via leader election protocols that beat standard resilient functions [AL, Combinatorica'93].

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arXiv2025

Leakage-Resilient Extractors against Number-on-Forehead Protocols

Given a sequence of $N$ independent sources $\mathbf{X}_1,\mathbf{X}_2,\dots,\mathbf{X}_N\sim\{0,1\}^n$, how many of them must be good (i.e., contain some min-entropy) in order to extract a uniformly random string? This question was first raised by Chattopadhyay, Goodman, Goyal and Li (STOC '20), motivated by applications in cryptography, distributed computing, and the unreliable nature of real-world sources of randomness. In their paper, they showed how to construct explicit low-error extractors for just $K \geq N^{1/2}$ good sources of polylogarithmic min-entropy. In a follow-up, Chattopadhyay and Goodman improved the number of good sources required to just $K \geq N^{0.01}$ (FOCS '21). In this paper, we finally achieve $K=3$. Our key ingredient is a near-optimal explicit construction of a new pseudorandom primitive, called a leakage-resilient extractor (LRE) against number-on-forehead (NOF) protocols. Our LRE can be viewed as a significantly more robust version of Li's low-error three-source extractor (FOCS '15), and resolves an open question put forth by Kumar, Meka, and Sahai (FOCS '19) and Chattopadhyay, Goodman, Goyal, Kumar, Li, Meka, and Zuckerman (FOCS '20). Our LRE construction is based on a simple new connection we discover between multiparty communication complexity and non-malleable extractors, which shows that such extractors exhibit strong average-case lower bounds against NOF protocols.

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arXiv2025

Two-Sided Lossless Expanders in the Unbalanced Setting

We present the first explicit construction of two-sided lossless expanders in the unbalanced setting (bipartite graphs that have polynomially many more nodes on the left than on the right). Prior to our work, all known explicit constructions in the unbalanced setting achieved only one-sided lossless expansion. Specifically, we show that the one-sided lossless expanders constructed by Kalev and Ta-Shma (RANDOM'22) -- that are based on multiplicity codes introduced by Kopparty, Saraf, and Yekhanin (STOC'11) -- are, in fact, two-sided lossless expanders. Moreover, we show that our result is tight, thus completely characterizing the graph of Kalev and Ta-Shma. Using our unbalanced bipartite expander, we easily obtain lossless (non-bipartite) expander graphs on $N$ vertices with polynomial degree $\ll N$ and expanding sets of size $N^{0.49}$.

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arXiv2024

On the Existence of Seedless Condensers: Exploring the Terrain

We prove several new results for seedless condensers in the context of three related classes of sources: Non-Oblivious Symbol Fixing (NOSF) sources, online NOSF (oNOSF) sources [AORSV, EUROCRYPT'20], and adversarial Chor-Goldreich (aCG) source [DMOZ, STOC'23]. We think of these sources as a sequence of random variables $\mathbf{X}=\mathbf{X}_1,\dots,\mathbf{X}_\ell$ on $\ell$ symbols where at least $g$ out of these $\ell$ symbols are "good" (i.e., have some min-entropy requirement), denoted as a $(g,\ell)$-source, and the remaining "bad" $\ell-g$ symbols may adversarially depend on these $g$ good blocks. The difference between each of these sources is realized by restrictions on the power of the adversary. Prior to our work, the only known seedless condenser upper or lower bound in these settings is due to [DMOZ, STOC'23], where they explicitly construct a seedless condenser for a restricted subset of $(g,\ell)$-aCG sources. We show: 1) oNOSF sources a) When $g\leq\ell/2$, we prove that condensing with error 0.99 above rate $\frac{1}{\lfloor \ell/g \rfloor}$ is impossible. In fact, we show that this is tight. b) For $g> \ell/2$, we show the existence of excellent condensers for uniform oNOSF sources. In addition, we show the existence of similar condensers for oNOSF sources with only logarithmic min-entropy. 2) aCG sources a) We observe that uniform aCG sources are equivalent to uniform oNOSF sources and consequently inherit the same results. b) We show that one cannot condense beyond the min-entropy gap of each block or condense low min-entropy CG sources above rate $1/2$. 3) NOSF sources a) We show that condensing with constant error above rate $\frac{g}{\ell}$ is impossible for uniform NOSF sources for any $g$ and $\ell$, thus ruling out the possibility of any non-trivial condensing. This shows a distinction between NOSF sources and oNOSF sources.

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arXiv2024

Extractors for Polynomial Sources over $\mathbb{F}_2$

We explicitly construct the first nontrivial extractors for degree $d \ge 2$ polynomial sources over $\mathbb{F}_2^n$. Our extractor requires min-entropy $k\geq n - \tildeΩ(\sqrt{\log n})$. Previously, no constructions were known, even for min-entropy $k\geq n-1$. A key ingredient in our construction is an input reduction lemma, which allows us to assume that any polynomial source with min-entropy $k$ can be generated by $O(k)$ uniformly random bits. We also provide strong formal evidence that polynomial sources are unusually challenging to extract from, by showing that even our most powerful general purpose extractors cannot handle polynomial sources with min-entropy below $k\geq n-o(n)$. In more detail, we show that sumset extractors cannot even disperse from degree $2$ polynomial sources with min-entropy $k\geq n-O(n/\log\log n)$. In fact, this impossibility result even holds for a more specialized family of sources that we introduce, called polynomial non-oblivious bit-fixing (NOBF) sources. Polynomial NOBF sources are a natural new family of algebraic sources that lie at the intersection of polynomial and variety sources, and thus our impossibility result applies to both of these classical settings. This is especially surprising, since we do have variety extractors that slightly beat this barrier - implying that sumset extractors are not a panacea in the world of seedless extraction.

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arXiv2023

Recursive Error Reduction for Regular Branching Programs

In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length $n$ and width $w$, both of which were proved in the work of Chen et al. using their error reduction: $\bullet$ There is a WPRG with error $\varepsilon$ that has seed length $\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)).$ $\bullet$ There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error $\pm\varepsilon$ with space complexity $\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon)).$ (This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler.) Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al.

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arXiv2022

Low-Degree Polynomials Extract from Local Sources

We continue a line of work on extracting random bits from weak sources that are generated by simple processes. We focus on the model of locally samplable sources, where each bit in the source depends on a small number of (hidden) uniformly random input bits. Also known as local sources, this model was introduced by De and Watson (TOCT 2012) and Viola (SICOMP 2014), and is closely related to sources generated by $\mathsf{AC}^0$ circuits and bounded-width branching programs. In particular, extractors for local sources also work for sources generated by these classical computational models. Despite being introduced a decade ago, little progress has been made on improving the entropy requirement for extracting from local sources. The current best explicit extractors require entropy $n^{1/2}$, and follow via a reduction to affine extractors. To start, we prove a barrier showing that one cannot hope to improve this entropy requirement via a black-box reduction of this form. In particular, new techniques are needed. In our main result, we seek to answer whether low-degree polynomials (over $\mathbb{F}_2$) hold potential for breaking this barrier. We answer this question in the positive, and fully characterize the power of low-degree polynomials as extractors for local sources. More precisely, we show that a random degree $r$ polynomial is a low-error extractor for $n$-bit local sources with min-entropy $Ω(r(n\log n)^{1/r})$, and we show that this is tight. Our result leverages several new ingredients, which may be of independent interest. Our existential result relies on a new reduction from local sources to a more structured family, known as local non-oblivious bit-fixing sources. To show its tightness, we prove a "local version" of a structural result by Cohen and Tal (RANDOM 2015), which relies on a new "low-weight" Chevalley-Warning theorem.

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arXiv2021

Extractors for Sum of Two Sources

We consider the problem of extracting randomness from \textit{sumset sources}, a general class of weak sources introduced by Chattopadhyay and Li (STOC, 2016). An $(n,k,C)$-sumset source $\mathbf{X}$ is a distribution on $\{0,1\}^n$ of the form $\mathbf{X}_1 + \mathbf{X}_2 + \ldots + \mathbf{X}_C$, where $\mathbf{X}_i$'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 $0.51 n$. As our main result, we construct an explicit extractor for sumset sources in the setting of $C=2$ for min-entropy $\mathrm{poly}(\log n)$ and polynomially small error. We can further improve the min-entropy requirement to $(\log n) \cdot (\log \log n)^{1 + o(1)}$ 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 $\mathbf{X_1} + \mathbf{X_2}$ is not much larger than $2^k$).

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arXiv2021

Improved Extractors for Small-Space Sources

We study the problem of extracting random bits from weak sources that are sampled by algorithms with limited memory. This model of small-space sources was introduced by Kamp, Rao, Vadhan and Zuckerman (STOC'06), and falls into a line of research initiated by Trevisan and Vadhan (FOCS'00) on extracting randomness from weak sources that are sampled by computationally bounded algorithms. Our main results are the following. 1. We obtain near-optimal extractors for small-space sources in the polynomial error regime. For space $s$ sources over $n$ bits, our extractors require just $k\geq s\cdot$polylog$(n)$ entropy. This is an exponential improvement over the previous best result, which required $k\geq s^{1.1}\cdot2^{\log^{0.51} n}$ (Chattopadhyay and Li, STOC'16). 2. We obtain improved extractors for small-space sources in the negligible error regime. For space $s$ sources over $n$ bits, our extractors require entropy $k\geq n^{1/2+δ}\cdot s^{1/2-δ}$, whereas the previous best result required $k\geq n^{2/3+δ}\cdot s^{1/3-δ}$ (Chattopadhyay, Goodman, Goyal and Li, STOC'20). To obtain our first result, the key ingredient is a new reduction from small-space sources to affine sources, allowing us to simply apply a good affine extractor. To obtain our second result, we must develop some new machinery, since we do not have low-error affine extractors that work for low entropy. Our main tool is a significantly improved extractor for adversarial sources, which is built via a simple framework that makes novel use of a certain kind of leakage-resilient extractors (known as cylinder intersection extractors), by combining them with a general type of extremal designs. Our key ingredient is the first derandomization of these designs, which we obtain using new connections to coding theory and additive combinatorics.

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arXiv2020

Fractional Pseudorandom Generators from Any Fourier Level

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay {et al.} [CHHL19,CHLT19] that exploit $L_1$ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the $k$-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with $k$. This interpolates previous works, which either require Fourier bounds on all levels [CHHL19], or have polynomial dependence on the error parameter in the seed length [CHLT10], and thus answers an open question in [CHLT19]. As an example, we show that for polynomial error, Fourier bounds on the first $O(\log n)$ levels is sufficient to recover the seed length in [CHHL19], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor's theorem and bounding the degree-$k$ Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the \emph{level-k unsigned Fourier sum}, which is potentially a much smaller quantity than the $L_1$ notion in previous works. By generalizing a connection established in [CHH+20], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for $\mathbb{F}_2$ polynomials with seed length close to the state-of-the-art construction due to Viola [Vio09], which was not known to be possible using this framework.

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arXiv2020

Optimal Error Pseudodistributions for Read-Once Branching Programs

In a seminal work, Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$ and width $w$ read-once branching programs with seed length $O(\log n\cdot \log(nw)+\log n\cdot\log(1/\varepsilon))$ and error $\varepsilon$. It remains a central question to reduce the seed length to $O(\log (nw/\varepsilon))$, which would prove that $\mathbf{BPL}=\mathbf{L}$. However, there has been no improvement on Nisan's construction for the case $n=w$, which is most relevant to space-bounded derandomization. Recently, in a beautiful work, Braverman, Cohen and Garg (STOC'18) introduced the notion of a pseudorandom pseudo-distribution (PRPD) and gave an explicit construction of a PRPD with seed length $\tilde{O}(\log n\cdot \log(nw)+\log(1/\varepsilon))$. A PRPD is a relaxation of a pseudorandom generator, which suffices for derandomizing $\mathbf{BPL}$ and also implies a hitting set. Unfortunately, their construction is quite involved and complicated. Hoza and Zuckerman (FOCS'18) later constructed a much simpler hitting set generator with seed length $O(\log n\cdot \log(nw)+\log(1/\varepsilon))$, but their techniques are restricted to hitting sets. In this work, we construct a PRPD with seed length $$O(\log n\cdot \log (nw)\cdot \log\log(nw)+\log(1/\varepsilon)).$$ This improves upon the construction in [BCG18] by a $O(\log\log(1/\varepsilon))$ factor, and is optimal in the small error regime. In addition, we believe our construction and analysis to be simpler than the work of Braverman, Cohen and Garg.

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arXiv2018

Non-Malleable Extractors and Codes for Composition of Tampering, Interleaved Tampering and More

Non-malleable codes were introduced by Dziembowski, Pietrzak, and Wichs (JACM 2018) as a generalization of standard error correcting codes to handle severe forms of tampering on codewords. This notion has attracted a lot of recent research, resulting in various explicit constructions, which have found applications in tamper-resilient cryptography and connections to other pseudorandom objects in theoretical computer science. We continue the line of investigation on explicit constructions of non-malleable codes in the information theoretic setting, and give explicit constructions for several new classes of tampering functions. (1) Interleaved split-state tampering: Here the codeword is partitioned in an unknown way by an adversary, and then tampered with by a split-state tampering function. (2) Linear function composed with split-state tampering: In this model, the codeword is first tampered with by a split-state adversary, and then the whole tampered codeword is further tampered with by a linear function. In fact our results are stronger, and we can handle linear function composed with interleaved split-state tampering. (3) Bounded communication split-state tampering: In this model, the two split-state tampering adversaries are allowed to participate in a communication protocol with a bounded communication budget. Our results are the first explicit constructions of non-malleable codes in any of these tampering models. We derive all these results from explicit constructions of seedless non-malleable extractors, which we believe are of independent interest. Using our techniques, we also give an improved seedless extractor for an unknown interleaving of two independent sources.

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arXiv2016

Explicit Non-Malleable Extractors, Multi-Source Extractors and Almost Optimal Privacy Amplification Protocols

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 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 $s=Ω(\sqrt{k})$, 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 $O(s/\sqrt{k})$ 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 $\log^{1+o(1)} n$.

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