All reports by Author Gil Cohen:

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TR18-066
| 8th April 2018
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Avraham Ben-Aroya, Gil Cohen, Dean Doron, Amnon Ta-Shma#### Two-Source Condensers with Low Error and Small Entropy Gap via Entropy-Resilient Functions

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TR18-032
| 14th February 2018
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Gil Cohen, Bernhard Haeupler, Leonard Schulman#### Explicit Binary Tree Codes with Polylogarithmic Size Alphabet

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TR17-161
| 30th October 2017
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Mark Braverman, Gil Cohen, Sumegha Garg#### Hitting Sets with Near-Optimal Error for Read-Once Branching Programs

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TR16-114
| 30th July 2016
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Gil Cohen#### Two-Source Extractors for Quasi-Logarithmic Min-Entropy and Improved Privacy Amplification Protocols

Revisions: 1

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TR16-052
| 7th April 2016
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Gil Cohen#### Making the Most of Advice: New Correlation Breakers and Their Applications

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TR16-030
| 7th March 2016
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Gil Cohen#### Non-Malleable Extractors with Logarithmic Seeds

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TR16-014
| 3rd February 2016
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Gil Cohen, Leonard Schulman#### Extractors for Near Logarithmic Min-Entropy

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TR15-183
| 16th November 2015
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Gil Cohen#### Non-Malleable Extractors - New Tools and Improved Constructions

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TR15-095
| 14th June 2015
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Gil Cohen#### Two-Source Dispersers for Polylogarithmic Entropy and Improved Ramsey Graphs

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TR15-038
| 11th March 2015
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Gil Cohen#### Local Correlation Breakers and Applications to Three-Source Extractors and Mergers

Revisions: 1

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TR14-160
| 27th November 2014
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Gil Cohen, Igor Shinkar#### Zero-Fixing Extractors for Sub-Logarithmic Entropy

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TR14-099
| 7th August 2014
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Gil Cohen, Igor Shinkar#### The Complexity of DNF of Parities

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TR14-023
| 19th February 2014
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Gil Cohen, Anat Ganor, Ran Raz#### Two Sides of the Coin Problem

Revisions: 1

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TR13-155
| 10th November 2013
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Gil Cohen, Amnon Ta-Shma#### Pseudorandom Generators for Low Degree Polynomials from Algebraic Geometry Codes

Revisions: 2

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TR13-145
| 20th October 2013
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Gil Cohen, Avishay Tal#### Two Structural Results for Low Degree Polynomials and Applications

Revisions: 1

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TR13-138
| 5th October 2013
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Itai Benjamini, Gil Cohen, Igor Shinkar#### Bi-Lipschitz Bijection between the Boolean Cube and the Hamming Ball

Revisions: 1

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TR13-107
| 7th August 2013
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Gil Cohen, Ivan Bjerre Damgard, Yuval Ishai, Jonas Kolker, Peter Bro Miltersen, Ran Raz, Ron Rothblum#### Efficient Multiparty Protocols via Log-Depth Threshold Formulae

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TR12-133
| 21st October 2012
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Noga Alon, Gil Cohen#### On Rigid Matrices and Subspace Polynomials

Revisions: 1

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TR12-050
| 25th April 2012
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Avraham Ben-Aroya, Gil Cohen#### Gradual Small-Bias Sample Spaces

Revisions: 3

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TR11-096
| 2nd July 2011
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Gil Cohen, Ran Raz, Gil Segev#### Non-Malleable Extractors with Short Seeds and Applications to Privacy Amplification

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TR11-002
| 9th January 2011
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Gil Cohen, Amir Shpilka, Avishay Tal#### On the Degree of Univariate Polynomials Over the Integers

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TR10-039
| 10th March 2010
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Gil Cohen, Amir Shpilka#### On the degree of symmetric functions on the Boolean cube

Comments: 1

Avraham Ben-Aroya, Gil Cohen, Dean Doron, Amnon Ta-Shma

In their seminal work, Chattopadhyay and Zuckerman (STOC'16) constructed a two-source extractor with error $\varepsilon$ for $n$-bit sources having min-entropy $poly\log(n/\varepsilon)$. Unfortunately, the construction running-time is $poly(n/\varepsilon)$, which means that with polynomial-time constructions, only polynomially-large errors are possible. Our main result is a $poly(n,\log(1/\varepsilon))$-time computable two-source condenser. For any $k ... more >>>

Gil Cohen, Bernhard Haeupler, Leonard Schulman

This paper makes progress on the problem of explicitly constructing a binary tree code with constant distance and constant alphabet size.

For every constant $\delta < 1$ we give an explicit binary tree code with distance $\delta$ and alphabet size $(\log{n})^{O(1)}$, where $n$ is the depth of the tree. This ... more >>>

Mark Braverman, Gil Cohen, Sumegha Garg

Nisan (Combinatorica'92) constructed a pseudorandom generator for length $n$, width $n$ read-once branching programs (ROBPs) with error $\varepsilon$ and seed length $O(\log^2{n} + \log{n} \cdot \log(1/\varepsilon))$. A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal $O(\log{n}+\log(1/\varepsilon))$, or to construct improved hitting sets, as ... more >>>

Gil Cohen

This paper offers the following contributions:

* We construct a two-source extractor for quasi-logarithmic min-entropy. That is, an extractor for two independent $n$-bit sources with min-entropy $\widetilde{O}(\log{n})$. Our construction is optimal up to $\mathrm{poly}(\log\log{n})$ factors and improves upon a recent result by Ben-Aroya, Doron, and Ta-Shma (ECCC'16) that can handle ... more >>>

Gil Cohen

A typical obstacle one faces when constructing pseudorandom objects is undesired correlations between random variables. Identifying this obstacle and constructing certain types of "correlation breakers" was central for recent exciting advances in the construction of multi-source and non-malleable extractors. One instantiation of correlation breakers is correlation breakers with advice. These ... more >>>

Gil Cohen

We construct non-malleable extractors with seed length $d = O(\log{n}+\log^{3}(1/\epsilon))$ for $n$-bit sources with min-entropy $k = \Omega(d)$, where $\epsilon$ is the error guarantee. In particular, the seed length is logarithmic in $n$ for $\epsilon> 2^{-(\log{n})^{1/3}}$. This improves upon existing constructions that either require super-logarithmic seed length even for constant ... more >>>

Gil Cohen, Leonard Schulman

The main contribution of this work is an explicit construction of extractors for near logarithmic min-entropy. For any $\delta > 0$ we construct an extractor for $O(1/\delta)$ $n$-bit sources with min-entropy $(\log{n})^{1+\delta}$. This is most interesting when $\delta$ is set to a small constant, though the result also yields an ... more >>>

Gil Cohen

A non-malleable extractor is a seeded extractor with a very strong guarantee - the output of a non-malleable extractor obtained using a typical seed is close to uniform even conditioned on the output obtained using any other seed. The first contribution of this paper consists of two new and improved ... more >>>

Gil Cohen

In his 1947 paper that inaugurated the probabilistic method, Erdös proved the existence of $2\log{n}$-Ramsey graphs on $n$ vertices. Matching Erdös' result with a constructive proof is a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in ... more >>>

Gil Cohen

We introduce and construct a pseudorandom object which we call a local correlation breaker (LCB). Informally speaking, an LCB is a function that gets as input a sequence of $r$ (arbitrarily correlated) random variables and an independent weak-source. The output of the LCB is a sequence of $r$ random variables ... more >>>

Gil Cohen, Igor Shinkar

An $(n,k)$-bit-fixing source is a distribution on $n$ bit strings, that is fixed on $n-k$ of the coordinates, and jointly uniform on the remaining $k$ bits. Explicit constructions of bit-fixing extractors by Gabizon, Raz and Shaltiel [SICOMP 2006] and Rao [CCC 2009], extract $(1-o(1)) \cdot k$ bits for $k = ... more >>>

Gil Cohen, Igor Shinkar

We study depth 3 circuits of the form $\mathrm{OR} \circ \mathrm{AND} \circ \mathrm{XOR}$, or equivalently -- DNF of parities. This model was first explicitly studied by Jukna (CPC'06) who obtained a $2^{\Omega(n)}$ lower bound for explicit functions. Several related models have gained attention in the last few years, such as ... more >>>

Gil Cohen, Anat Ganor, Ran Raz

In the Coin Problem, one is given n independent flips of a coin that has bias $\beta > 0$ towards either Head or Tail. The goal is to decide which side the coin is biased towards, with high confidence. An optimal strategy for solving the coin problem is to apply ... more >>>

Gil Cohen, Amnon Ta-Shma

Constructing pseudorandom generators for low degree polynomials has received a considerable attention in the past decade. Viola [CC 2009], following an exciting line of research, constructed a pseudorandom generator for degree d polynomials in n variables, over any prime field. The seed length used is $O(d \log{n} + d 2^d)$, ... more >>>

Gil Cohen, Avishay Tal

In this paper, two structural results concerning low degree polynomials over the field $\mathbb{F}_2$ are given. The first states that for any degree d polynomial f in n variables, there exists a subspace of $\mathbb{F}_2^n$ with dimension $\Omega(n^{1/(d-1)})$ on which f is constant. This result is shown to be tight. ... more >>>

Itai Benjamini, Gil Cohen, Igor Shinkar

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even $n \in {\mathbb N}$ there exists an explicit bijection $\psi \colon \{0,1\}^n \to \left\{ x \in \{0,1\}^{n+1} \colon |x| > n/2 \right\}$ such that for every ... more >>>

Gil Cohen, Ivan Bjerre Damgard, Yuval Ishai, Jonas Kolker, Peter Bro Miltersen, Ran Raz, Ron Rothblum

We put forward a new approach for the design of efficient multiparty protocols:

1. Design a protocol for a small number of parties (say, 3 or 4) which achieves

security against a single corrupted party. Such protocols are typically easy

to construct as they may employ techniques that do not ...
more >>>

Noga Alon, Gil Cohen

We introduce a class of polynomials, which we call \emph{subspace polynomials} and show that the problem of explicitly constructing a rigid matrix can be reduced to the problem of explicitly constructing a small hitting set for this class. We prove that small-bias sets are hitting sets for the class of ... more >>>

Avraham Ben-Aroya, Gil Cohen

A $(k,\epsilon)$-biased sample space is a distribution over $\{0,1\}^n$ that $\epsilon$-fools every nonempty linear test of size at most $k$. Since they were introduced by Naor and Naor [SIAM J. Computing, 1993], these sample spaces have become a central notion in theoretical computer science with a variety of applications.

When ... more >>>

Gil Cohen, Ran Raz, Gil Segev

Motivated by the classical problem of privacy amplification, Dodis and Wichs (STOC '09) introduced the notion of a non-malleable extractor, significantly strengthening the notion of a strong extractor. A non-malleable extractor is a function $nmExt : \{0,1\}^n \times \{0,1\}^d \rightarrow \{0,1\}^m$ that takes two inputs: a weak source $W$ and ... more >>>

Gil Cohen, Amir Shpilka, Avishay Tal

We study the following problem raised by von zur Gathen and Roche:

What is the minimal degree of a nonconstant polynomial $f:\{0,\ldots,n\}\to\{0,\ldots,m\}$?

Clearly, when $m=n$ the function $f(x)=x$ has degree $1$. We prove that when $m=n-1$ (i.e. the point $\{n\}$ is not in the range), it must be the case ... more >>>

Gil Cohen, Amir Shpilka

In this paper we study the degree of non-constant symmetric functions $f:\{0,1\}^n \to \{0,1,\ldots,c\}$, where $c\in

\mathbb{N}$, when represented as polynomials over the real numbers. We show that as long as $c < n$ it holds that deg$(f)=\Omega(n)$. As we can have deg$(f)=1$ when $c=n$, our

result shows a surprising ...
more >>>