All reports by Author Eshan Chattopadhyay:

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TR22-082
| 27th May 2022
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Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, João Ribeiro#### Low-Degree Polynomials Extract from Local Sources

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TR21-147
| 22nd October 2021
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Eshan Chattopadhyay, Jyun-Jie Liao#### Extractors for Sum of Two Sources

Revisions: 1

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TR21-106
| 22nd July 2021
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Eshan Chattopadhyay, Jesse Goodman, David Zuckerman#### The Space Complexity of Sampling

Revisions: 1

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TR21-075
| 4th June 2021
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Eshan Chattopadhyay, Jesse Goodman, Jyun-Jie Liao#### Affine Extractors for Almost Logarithmic Entropy

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TR20-121
| 3rd August 2020
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Eshan Chattopadhyay, Jason Gaitonde, Abhishek Shetty#### Fractional Pseudorandom Generators from the $k$th Fourier Level

Revisions: 2

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TR20-106
| 15th July 2020
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Eshan Chattopadhyay, Jesse Goodman#### Explicit Extremal Designs and Applications to Extractors

Revisions: 5

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TR20-069
| 4th May 2020
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Eshan Chattopadhyay, Jyun-Jie Liao#### Optimal Error Pseudodistributions for Read-Once Branching Programs

Revisions: 1

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TR20-060
| 23rd April 2020
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Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li#### Leakage-Resilient Extractors and Secret-Sharing against Bounded Collusion Protocols

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TR20-023
| 10th February 2020
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Marshall Ball, Eshan Chattopadhyay, Jyun-Jie Liao, Tal Malkin, Li-Yang Tan#### Non-Malleability against Polynomial Tampering

Revisions: 1

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TR19-184
| 13th December 2019
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Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li#### Extractors for Adversarial Sources via Extremal Hypergraphs

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TR19-145
| 31st October 2019
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Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, David Zuckerman#### XOR Lemmas for Resilient Functions Against Polynomials

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TR18-100
| 18th May 2018
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Eshan Chattopadhyay, Anindya De, Rocco Servedio#### Simple and efficient pseudorandom generators from Gaussian processes

Revisions: 1

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TR18-070
| 13th April 2018
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Eshan Chattopadhyay, Xin Li#### Non-Malleable Extractors and Codes in the Interleaved Split-State Model and More

Revisions: 3

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TR18-015
| 25th January 2018
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Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett#### Pseudorandom Generators from Polarizing Random Walks

Revisions: 1
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Comments: 1

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TR17-171
| 6th November 2017
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Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal#### Improved Pseudorandomness for Unordered Branching Programs through Local Monotonicity

Revisions: 1

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TR17-027
| 16th February 2017
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Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, Amnon Ta-Shma#### A reduction from efficient non-malleable extractors to low-error two-source extractors with arbitrary constant rate

Revisions: 1

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TR16-180
| 15th November 2016
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Eshan Chattopadhyay, Xin Li#### Non-Malleable Codes and Extractors for Small-Depth Circuits, and Affine Functions

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TR16-036
| 13th March 2016
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Eshan Chattopadhyay, Xin Li#### Explicit Non-Malleable Extractors, Multi-Source Extractors and Almost Optimal Privacy Amplification Protocols

Revisions: 3

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TR15-178
| 10th November 2015
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Eshan Chattopadhyay, Xin Li#### Extractors for Sumset Sources

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TR15-151
| 14th September 2015
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Eshan Chattopadhyay, David Zuckerman#### New Extractors for Interleaved Sources

Revisions: 1

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TR15-119
| 23rd July 2015
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Eshan Chattopadhyay, David Zuckerman#### Explicit Two-Source Extractors and Resilient Functions

Revisions: 5

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TR15-075
| 29th April 2015
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Eshan Chattopadhyay, Vipul Goyal, Xin Li#### Non-Malleable Extractors and Codes, with their Many Tampered Extensions

Revisions: 1

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TR14-102
| 4th August 2014
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Eshan Chattopadhyay, David Zuckerman#### Non-Malleable Codes Against Constant Split-State Tampering

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TR12-127
| 3rd October 2012
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Eshan Chattopadhyay, Adam Klivans, Pravesh Kothari#### An Explicit VC-Theorem for Low-Degree Polynomials

Omar Alrabiah, Eshan Chattopadhyay, Jesse Goodman, Xin Li, João Ribeiro

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, ... more >>>

Eshan Chattopadhyay, Jyun-Jie Liao

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 ... more >>>

Eshan Chattopadhyay, Jesse Goodman, David Zuckerman

Recently, there has been exciting progress in understanding the complexity of distributions. Here, the goal is to quantify the resources required to generate (or sample) a distribution. Proving lower bounds in this new setting is more challenging than in the classical setting, and has yielded interesting new techniques and surprising ... more >>>

Eshan Chattopadhyay, Jesse Goodman, Jyun-Jie Liao

We give an explicit construction of an affine extractor (over $\mathbb{F}_2$) that works for affine sources on $n$ bits with min-entropy $k \ge~ \log n \cdot (\log \log n)^{1 + o(1)}$. This improves prior work of Li (FOCS'16) that requires min-entropy at least $\mathrm{poly}(\log n)$.

Our construction is ...
more >>>

Eshan Chattopadhyay, Jason Gaitonde, Abhishek Shetty

In recent work by Chattopadhyay et al.[CHHL19,CHLT19], the authors exhibit a simple and flexible construction of pseudorandom generators for classes of Boolean functions that satisfy $L_1$ Fourier bounds. [CHHL19] show that if a class satisfies such tail bounds at all levels, this implies a PRG whose seed length depends on ... more >>>

Eshan Chattopadhyay, Jesse Goodman

An $(n,r,s)$-design, or $(n,r,s)$-partial Steiner system, is an $r$-uniform hypergraph over $n$ vertices with pairwise hyperedge intersections of size $0$, we extract from $(N,K,n,k)$-adversarial sources of locality $0$, where $K\geq N^\delta$ and $k\geq\text{polylog }n$. The previous best result (Chattopadhyay et al., STOC 2020) required $K\geq N^{1/2+o(1)}$. As a result, we ... more >>>

Eshan Chattopadhyay, Jyun-Jie Liao

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 ... more >>>

Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li

In a recent work, Kumar, Meka, and Sahai (FOCS 2019) introduced the notion of bounded collusion protocols (BCPs), in which $N$ parties wish to compute some joint function $f:(\{0,1\}^n)^N\to\{0,1\}$ using a public blackboard, but such that only $p$ parties may collude at a time. This generalizes well studied models in ... more >>>

Marshall Ball, Eshan Chattopadhyay, Jyun-Jie Liao, Tal Malkin, Li-Yang Tan

We present the first explicit construction of a non-malleable code that can handle tampering functions that are bounded-degree polynomials.

Prior to our work, this was only known for degree-1 polynomials (affine tampering functions), due to Chattopadhyay and Li (STOC 2017). As a direct corollary, we obtain an explicit non-malleable ... more >>>

Eshan Chattopadhyay, Jesse Goodman, Vipul Goyal, Xin Li

Randomness extraction is a fundamental problem that has been studied for over three decades. A well-studied setting assumes that one has access to multiple independent weak random sources, each with some entropy. However, this assumption is often unrealistic in practice. In real life, natural sources of randomness can produce samples ... more >>>

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett, David Zuckerman

A major challenge in complexity theory is to explicitly construct functions that have small correlation with low-degree polynomials over $F_2$. We introduce a new technique to prove such correlation bounds with $F_2$ polynomials. Using this technique, we bound the correlation of an XOR of Majorities with constant degree polynomials. In ... more >>>

Eshan Chattopadhyay, Anindya De, Rocco Servedio

We show that a very simple pseudorandom generator fools intersections of $k$ linear threshold functions (LTFs) and arbitrary functions of $k$ LTFs over $n$-dimensional Gaussian space.

The two analyses of our PRG (for intersections versus arbitrary functions of LTFs) are quite different from each other and from previous analyses of ... more >>>

Eshan Chattopadhyay, Xin Li

We present explicit constructions of non-malleable codes with respect to the following tampering classes. (i) Linear functions composed with split-state adversaries: In this model, the codeword is first tampered by a split-state adversary, and then the whole tampered codeword is further tampered by a linear function. (ii) Interleaved split-state adversary: ... more >>>

Eshan Chattopadhyay, Pooya Hatami, Kaave Hosseini, Shachar Lovett

We propose a new framework for constructing pseudorandom generators for $n$-variate Boolean functions. It is based on two new notions. First, we introduce fractional pseudorandom generators, which are pseudorandom distributions taking values in $[-1,1]^n$. Next, we use a fractional pseudorandom generator as steps of a random walk in $[-1,1]^n$ that ... more >>>

Eshan Chattopadhyay, Pooya Hatami, Omer Reingold, Avishay Tal

We present an explicit pseudorandom generator with seed length $\tilde{O}((\log n)^{w+1})$ for read-once, oblivious, width $w$ branching programs that can read their input bits in any order. This improves upon the work of Impaggliazzo, Meka and Zuckerman (FOCS'12) where they required seed length $n^{1/2+o(1)}$.

A central ingredient in our work ... more >>>

Avraham Ben-Aroya, Eshan Chattopadhyay, Dean Doron, Xin Li, Amnon Ta-Shma

We show a reduction from the existence of explicit t-non-malleable

extractors with a small seed length, to the construction of explicit

two-source extractors with small error for sources with arbitrarily

small constant rate. Previously, such a reduction was known either

when one source had entropy rate above half [Raz05] or ...
more >>>

Eshan Chattopadhyay, Xin Li

Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs as an elegant relaxation of error correcting codes, where the motivation is to handle more general forms of tampering while still providing meaningful guarantees. This has led to many elegant constructions and applications in cryptography. However, most works so far only ... more >>>

Eshan Chattopadhyay, Xin Li

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 ... more >>>

Eshan Chattopadhyay, Xin Li

We propose a new model of weak random sources which we call sumset sources. A sumset source $\mathbf{X}$ is the sum of $C$ independent sources $\mathbf{X}_1,\ldots,\mathbf{X}_C$, where each $\mathbf{X}_i$ is an $n$-bit source with min-entropy $k$. We show that extractors for this class of sources can be used to give ... more >>>

Eshan Chattopadhyay, David Zuckerman

We study how to extract randomness from a $C$-interleaved source, that is, a source comprised of $C$ independent sources whose bits or symbols are interleaved. We describe a simple approach for constructing such extractors that yields:

(1) For some $\delta>0, c > 0$,

explicit extractors for $2$-interleaved sources on $\{ ...
more >>>

Eshan Chattopadhyay, David Zuckerman

We explicitly construct an extractor for two independent sources on $n$ bits, each with min-entropy at least $\log^C n$ for a large enough constant $C$. Our extractor outputs one bit and has error $n^{-\Omega(1)}$. The best previous extractor, by Bourgain [B2], required each source to have min-entropy $.499n$.

A key ... more >>>

Eshan Chattopadhyay, Vipul Goyal, Xin Li

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 \emph{seeded non-malleable extractors}, introduced by Dodis and Wichs \cite{DW09}; \emph{seedless non-malleable extractors}, introduced by Cheraghchi and Guruswami ... more >>>

Eshan Chattopadhyay, David Zuckerman

Non-malleable codes were introduced by Dziembowski, Pietrzak and Wichs \cite{DPW10} as an elegant generalization of the classical notions of error detection, where the corruption of a codeword is viewed as a tampering function acting on it. Informally, a non-malleable code with respect to a family of tampering functions $\mathcal{F}$ consists ... more >>>

Eshan Chattopadhyay, Adam Klivans, Pravesh Kothari

Let $X \subseteq \mathbb{R}^{n}$ and let ${\mathcal C}$ be a class of functions mapping $\mathbb{R}^{n} \rightarrow \{-1,1\}.$ The famous VC-Theorem states that a random subset $S$ of $X$ of size $O(\frac{d}{\epsilon^{2}} \log \frac{d}{\epsilon})$, where $d$ is the VC-Dimension of ${\mathcal C}$, is (with constant probability) an $\epsilon$-approximation for ${\mathcal C}$ ... more >>>