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Reports tagged with extractors:
TR95-058 | 20th November 1995
Amnon Ta-Shma

On Extracting Randomness From Weak Random Sources

We deal with the problem of extracting as much randomness as possible
from a defective random source.
We devise a new tool, a ``merger'', which is a function that accepts
d strings, one of which is uniformly distributed,
and outputs a single string that is guaranteed ... more >>>

TR98-017 | 29th March 1998
Oded Goldreich, Madhu Sudan

Computational Indistinguishability: A Sample Hierarchy.

We consider the existence of pairs of probability ensembles which
may be efficiently distinguished given $k$ samples
but cannot be efficiently distinguished given $k'<k$ samples.
It is well known that in any such pair of ensembles it cannot be that
both are efficiently computable
(and that such phenomena ... more >>>

TR98-047 | 21st August 1998
Salil Vadhan

Extracting All the Randomness from a Weakly Random Source

Revisions: 1 , Comments: 1

In this paper, we give explicit constructions of extractors which work for
a source of any min-entropy on strings of length $n$. The first
construction extracts any constant fraction of the min-entropy using
O(log^2 n) additional random bits. The second extracts all the
min-entropy using O(log^3 n) additional random ... more >>>

TR98-055 | 4th September 1998
Luca Trevisan

Constructions of Near-Optimal Extractors Using Pseudo-Random Generators

Comments: 1

We introduce a new approach to construct extractors -- combinatorial
objects akin to expander graphs that have several applications.
Our approach is based on error correcting codes and on the Nisan-Wigderson
pseudorandom generator. An application of our approach yields a
construction that is simple to ... more >>>

TR99-046 | 17th November 1999
Ran Raz, Omer Reingold, Salil Vadhan

Extracting All the Randomness and Reducing the Error in Trevisan's Extractors

We give explicit constructions of extractors which work for a source of
any min-entropy on strings of length n. These extractors can extract any
constant fraction of the min-entropy using O(log^2 n) additional random
bits, and can extract all the min-entropy using O(log^3 n) additional
random bits. Both of these ... more >>>

TR00-009 | 21st February 2000
Russell Impagliazzo, Ronen Shaltiel, Avi Wigderson

Extractors and pseudo-random generators with optimal seed length

We give the first construction of a pseudo-random generator with
optimal seed length that uses (essentially) arbitrary hardness.
It builds on the novel recursive use of the NW-generator in
a previous paper by the same authors, which produced many optimal
generators one of which was pseudo-random. This is achieved ... more >>>

TR00-044 | 26th June 2000
Tzvika Hartman, Ran Raz

On the Distribution of the Number of Roots of Polynomials and Explicit Logspace Extractors

Weak designs were defined by Raz, Reingold and Vadhan (1999) and are
used in constructions of extractors. Roughly speaking, a weak design
is a collection of subsets satisfying some near-disjointness
properties. Constructions of weak designs with certain parameters are
given in [RRV99]. These constructions are explicit in the sense that
more >>>

TR00-059 | 11th August 2000
Omer Reingold, Ronen Shaltiel, Avi Wigderson

Extracting Randomness via Repeated Condensing

On an input probability distribution with some (min-)entropy
an {\em extractor} outputs a distribution with a (near) maximum
entropy rate (namely the uniform distribution).
A natural weakening of this concept is a condenser, whose
output distribution has a higher entropy rate than the
input distribution (without losing
much of ... more >>>

TR01-018 | 23rd February 2001
Omer Reingold, Salil Vadhan, Avi Wigderson

Entropy Waves, the Zig-Zag Graph Product, and New Constant-Degree Expanders and Extractors

The main contribution of this work is a new type of graph product, which we call the zig-zag
product. Taking a product of a large graph with a small graph, the resulting graph inherits
(roughly) its size from the large one, its degree from the small one, and ... more >>>

TR01-036 | 2nd May 2001
Amnon Ta-Shma, David Zuckerman, Shmuel Safra

Extractors from Reed-Muller Codes

Finding explicit extractors is an important derandomization goal that has received a lot of attention in the past decade. This research has focused on two approaches, one related to hashing and the other to pseudorandom generators. A third view, regarding extractors as good error correcting codes, was noticed before. Yet, ... more >>>

TR05-025 | 20th February 2005
Zeev Dvir, Ran Raz

Analyzing Linear Mergers

Mergers are functions that transform k (possibly dependent)
random sources into a single random source, in a way that ensures
that if one of the input sources has min-entropy rate $\delta$
then the output has min-entropy rate close to $\delta$. Mergers
have proven to be a very useful tool in ... more >>>

TR05-061 | 15th June 2005
Ronen Gradwohl, Guy Kindler, Omer Reingold, Amnon Ta-Shma

On the Error Parameter of Dispersers

Optimal dispersers have better dependence on the error than
optimal extractors. In this paper we give explicit disperser
constructions that beat the best possible extractors in some
parameters. Our constructions are not strong, but we show that
having such explicit strong constructions implies a solution
to the Ramsey graph construction ... more >>>

TR05-067 | 28th June 2005
Zeev Dvir, Amir Shpilka

An Improved Analysis of Mergers

Mergers are functions that transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has min-entropy rate $\delta$ then the output has min-entropy rate close to $\delta$. Mergers have proven to be a very useful tool in ... more >>>

TR06-003 | 8th January 2006
Joshua Buresh-Oppenheim, Rahul Santhanam

Making Hard Problems Harder

We consider a general approach to the hoary problem of (im)proving circuit lower bounds. We define notions of hardness condensing and hardness extraction, in analogy to the corresponding notions from the computational theory of randomness. A hardness condenser is a procedure that takes in a Boolean function as input, as ... more >>>

TR06-013 | 24th January 2006
Luca Trevisan

Pseudorandomness and Combinatorial Constructions

In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science,
probabilistic algorithms are sometimes simpler and more efficient
than the best known ... more >>>

TR06-128 | 5th October 2006
Shankar Kalyanaraman, Chris Umans

On obtaining pseudorandomness from error-correcting codes.

A number of recent results have constructed randomness extractors
and pseudorandom generators (PRGs) directly from certain
error-correcting codes. The underlying construction in these
results amounts to picking a random index into the codeword and
outputting $m$ consecutive symbols (the codeword is obtained from
the weak random source in the case ... more >>>

TR07-056 | 10th July 2007
Zeev Dvir, Ariel Gabizon, Avi Wigderson

Extractors and Rank Extractors for Polynomial Sources

In this paper we construct explicit deterministic extractors from polynomial sources, namely from distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial ... more >>>

TR07-085 | 2nd September 2007
Ran Raz, Amir Yehudayoff

Multilinear Formulas, Maximal-Partition Discrepancy and Mixed-Sources Extractors

We study multilinear formulas, monotone arithmetic circuits, maximal-partition discrepancy, best-partition communication complexity and extractors constructions. We start by proving lower bounds for an explicit polynomial for the following three subclasses of syntactically multilinear arithmetic formulas over the field C and the set of variables {x1,...,xn}:

1. Noise-resistant. A syntactically multilinear ... more >>>

TR08-015 | 23rd January 2008
Anup Rao

Extractors for Low-Weight Affine Sources

We give polynomial time computable extractors for low-weight affine sources. A distribution is affine if it samples a random point from some unknown low dimensional subspace of F^n_2 . A distribution is low weight affine if the corresponding linear space has a basis of low-weight vectors. Low-weight ane sources are ... more >>>

TR08-042 | 6th April 2008
Zeev Dvir

Deterministic Extractors for Algebraic Sources

An algebraic source is a random variable distributed
uniformly over the set of common zeros of one or more multivariate
polynomials defined over a finite field $F$. Our main result is
the construction of an explicit deterministic extractor for
algebraic sources over exponentially large prime fields. More
precisely, we give ... more >>>

TR08-058 | 1st June 2008
Zeev Dvir, Avi Wigderson

Kakeya sets, new mergers and old extractors

A merger is a probabilistic procedure which extracts the
randomness out of any (arbitrarily correlated) set of random
variables, as long as one of them is uniform. Our main result is
an efficient, simple, optimal (to constant factors) merger, which,
for $k$ random vairables on $n$ bits each, uses a ... more >>>

TR09-063 | 29th July 2009
Matt DeVos, Ariel Gabizon

Simple Affine Extractors using Dimension Expansion

Revisions: 2

Let $\F$ be the field of $q$ elements. An \emph{\afsext{n}{k}} is a mapping $D:\F^n\ar\B$
such that for any $k$-dimensional affine subspace $X\subseteq \F^n$, $D(x)$ is an almost unbiased
bit when $x$ is chosen uniformly from $X$.
Loosely speaking, the problem of explicitly constructing affine extractors gets harder as $q$ gets ... more >>>

TR09-077 | 16th September 2009
Zeev Dvir

From Randomness Extraction to Rotating Needles

The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this problem ... more >>>

TR10-037 | 8th March 2010
Boaz Barak, Guy Kindler, Ronen Shaltiel, Benny Sudakov, Avi Wigderson

Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors

We present new explicit constructions of *deterministic* randomness extractors, dispersers and related objects. We say that a
distribution $X$ on binary strings of length $n$ is a
$\delta$-source if $X$ assigns probability at most $2^{-\delta n}$
to any string of length $n$. For every $\delta>0$ we construct the
following poly($n$)-time ... more >>>

TR10-144 | 20th September 2010
Eli Ben-Sasson, Noga Ron-Zewi

From Affine to Two-Source Extractors via Approximate Duality

Revisions: 1

Two-source and affine extractors and dispersers are fundamental objects studied in the context of derandomization. This paper shows how to construct two-source extractors and dispersers for arbitrarily small min-entropy rate in a black-box manner from affine extractors with sufficiently good parameters. Our analysis relies on the study of approximate duality, ... more >>>

TR11-129 | 22nd September 2011
Eli Ben-Sasson, Ariel Gabizon

Extractors for Polynomials Sources over Constant-Size Fields of Small Characteristic

Let $F$ be the field of $q$ elements, where $q=p^{\ell}$ for prime $p$. Informally speaking, a polynomial source is a distribution over $F^n$ sampled by low degree multivariate polynomials. In this paper, we construct extractors for polynomial sources over fields of constant size $q$ assuming $p \ll q$.

More generally, ... more >>>

TR15-017 | 20th January 2015
Bruno Bauwens, Marius Zimand

Linear list-approximation for short programs (or the power of a few random bits)

A $c$-short program for a string $x$ is a description of $x$ of length at most $C(x) + c$, where $C(x)$ is the Kolmogorov complexity of $x$. We show that there exists a randomized algorithm that constructs a list of $n$ elements that contains a $O(\log n)$-short program for $x$. ... more >>>

TR15-038 | 11th March 2015
Gil Cohen

Local Correlation Breakers and Applications to Three-Source Extractors and Mergers

Revisions: 1

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 >>>

TR15-051 | 5th April 2015
Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, Guang Yang

Incompressible Functions, Relative-Error Extractors, and the Power of Nondeterminsitic Reductions

Revisions: 2

A circuit $C$ \emph{compresses} a function $f:\{0,1\}^n\rightarrow \{0,1\}^m$ if given an input $x\in \{0,1\}^n$ the circuit $C$ can shrink $x$ to a shorter $\ell$-bit string $x'$ such that later, a computationally-unbounded solver $D$ will be able to compute $f(x)$ based on $x'$. In this paper we study the existence of ... more >>>

TR15-144 | 1st September 2015
Raghu Meka

Explicit resilient functions matching Ajtai-Linial

Revisions: 1

A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n-q variables; in other words, no coalition of at most q variables has significant influence on the ... more >>>

TR15-151 | 14th September 2015
Eshan Chattopadhyay, David Zuckerman

New Extractors for Interleaved Sources

Revisions: 1

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 >>>

TR15-183 | 16th November 2015
Gil Cohen

Non-Malleable Extractors - New Tools and Improved Constructions

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 >>>

TR16-014 | 3rd February 2016
Gil Cohen, Leonard Schulman

Extractors for Near Logarithmic Min-Entropy

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 >>>

TR16-030 | 7th March 2016
Gil Cohen

Non-Malleable Extractors with Logarithmic Seeds

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 >>>

TR16-052 | 7th April 2016
Gil Cohen

Making the Most of Advice: New Correlation Breakers and Their Applications

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 >>>

TR16-088 | 1st June 2016
Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

Explicit two-source extractors for near-logarithmic min-entropy

We explicitly construct extractors for two independent $n$-bit sources of $(\log n)^{1+o(1)}$ min-entropy. Previous constructions required either $\mathrm{polylog}(n)$ min-entropy \cite{CZ15,Meka15} or five sources \cite{Cohen16}.

Our result extends the breakthrough result of Chattopadhyay and Zuckerman \cite{CZ15} and uses the non-malleable extractor of Cohen \cite{Cohen16}. The main new ingredient in our construction ... more >>>

TR16-106 | 15th July 2016
Avraham Ben-Aroya, Dean Doron, Amnon Ta-Shma

Low-error two-source extractors for polynomial min-entropy

Revisions: 1

We construct explicit two-source extractors for $n$ bit sources,
requiring $n^\alpha$ min-entropy and having error $2^{-n^\beta}$,
for some constants $0 < \alpha,\beta < 1$. Previously, constructions
for exponentially small error required either min-entropy
$0.49n$ \cite{Bou05} or three sources \cite{Li15}. The construction
combines somewhere-random condensers based on the Incidence
Theorem \cite{Zuc06,Li11}, ... more >>>

TR17-136 | 10th September 2017
Salman Beigi, Andrej Bogdanov, Omid Etesami, Siyao Guo

Complete Classi fication of Generalized Santha-Vazirani Sources

Let $\mathcal{F}$ be a finite alphabet and $\mathcal{D}$ be a finite set of distributions over $\mathcal{F}$. A Generalized Santha-Vazirani (GSV) source of type $(\mathcal{F}, \mathcal{D})$, introduced by Beigi, Etesami and Gohari (ICALP 2015, SICOMP 2017), is a random sequence $(F_1, \dots, F_n)$ in $\mathcal{F}^n$, where $F_i$ is a sample from ... more >>>

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