456 research outputs found
Affine extractors over large fields with exponential error
We describe a construction of explicit affine extractors over large finite
fields with exponentially small error and linear output length. Our
construction relies on a deep theorem of Deligne giving tight estimates for
exponential sums over smooth varieties in high dimensions.Comment: To appear in Comput. Comple
Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties
We consider the problem of uniform sampling of points on an algebraic
variety. Specifically, we develop a randomized algorithm that, given a small
set of multivariate polynomials over a sufficiently large finite field,
produces a common zero of the polynomials almost uniformly at random. The
statistical distance between the output distribution of the algorithm and the
uniform distribution on the set of common zeros is polynomially small in the
field size, and the running time of the algorithm is polynomial in the
description of the polynomials and their degrees provided that the number of
the polynomials is a constant
Linear-algebraic list decoding of folded Reed-Solomon codes
Folded Reed-Solomon codes are an explicit family of codes that achieve the
optimal trade-off between rate and error-correction capability: specifically,
for any \eps > 0, the author and Rudra (2006,08) presented an n^{O(1/\eps)}
time algorithm to list decode appropriate folded RS codes of rate from a
fraction 1-R-\eps of errors. The algorithm is based on multivariate
polynomial interpolation and root-finding over extension fields. It was noted
by Vadhan that interpolating a linear polynomial suffices if one settles for a
smaller decoding radius (but still enough for a statement of the above form).
Here we give a simple linear-algebra based analysis of this variant that
eliminates the need for the computationally expensive root-finding step over
extension fields (and indeed any mention of extension fields). The entire list
decoding algorithm is linear-algebraic, solving one linear system for the
interpolation step, and another linear system to find a small subspace of
candidate solutions. Except for the step of pruning this subspace, the
algorithm can be implemented to run in {\em quadratic} time. The theoretical
drawback of folded RS codes are that both the decoding complexity and proven
worst-case list-size bound are n^{\Omega(1/\eps)}. By combining the above
idea with a pseudorandom subset of all polynomials as messages, we get a Monte
Carlo construction achieving a list size bound of O(1/\eps^2) which is quite
close to the existential O(1/\eps) bound (however, the decoding complexity
remains n^{\Omega(1/\eps)}). Our work highlights that constructing an
explicit {\em subspace-evasive} subset that has small intersection with
low-dimensional subspaces could lead to explicit codes with better
list-decoding guarantees.Comment: 16 pages. Extended abstract in Proc. of IEEE Conference on
Computational Complexity (CCC), 201
Subspace Evasive Sets
In this work we describe an explicit, simple, construction of large subsets
of F^n, where F is a finite field, that have small intersection with every
k-dimensional affine subspace. Interest in the explicit construction of such
sets, termed subspace-evasive sets, started in the work of Pudlak and Rodl
(2004) who showed how such constructions over the binary field can be used to
construct explicit Ramsey graphs. More recently, Guruswami (2011) showed that,
over large finite fields (of size polynomial in n), subspace evasive sets can
be used to obtain explicit list-decodable codes with optimal rate and constant
list-size. In this work we construct subspace evasive sets over large fields
and use them to reduce the list size of folded Reed-Solomon codes form poly(n)
to a constant.Comment: 16 page
Two Structural Results for Low Degree Polynomials and Applications
In this paper, two structural results concerning low degree polynomials over
finite fields are given. The first states that over any finite field
, for any polynomial on variables with degree , there exists a subspace of with dimension on which is constant. This result is shown to be tight.
Stated differently, a degree polynomial cannot compute an affine disperser
for dimension smaller than . Using a recursive
argument, we obtain our second structural result, showing that any degree
polynomial induces a partition of to affine subspaces of dimension
, such that is constant on each part.
We extend both structural results to more than one polynomial. We further
prove an analog of the first structural result to sparse polynomials (with no
restriction on the degree) and to functions that are close to low degree
polynomials. We also consider the algorithmic aspect of the two structural
results.
Our structural results have various applications, two of which are:
* Dvir [CC 2012] introduced the notion of extractors for varieties, and gave
explicit constructions of such extractors over large fields. We show that over
any finite field, any affine extractor is also an extractor for varieties with
related parameters. Our reduction also holds for dispersers, and we conclude
that Shaltiel's affine disperser [FOCS 2011] is a disperser for varieties over
.
* Ben-Sasson and Kopparty [SIAM J. C 2012] proved that any degree 3 affine
disperser over a prime field is also an affine extractor with related
parameters. Using our structural results, and based on the work of Kaufman and
Lovett [FOCS 2008] and Haramaty and Shpilka [STOC 2010], we generalize this
result to any constant degree
Deterministic Extractors for Additive Sources
We propose a new model of a weakly random source that admits randomness
extraction. Our model of additive sources includes such natural sources as
uniform distributions on arithmetic progressions (APs), generalized arithmetic
progressions (GAPs), and Bohr sets, each of which generalizes affine sources.
We give an explicit extractor for additive sources with linear min-entropy over
both and , for large prime , although our
results over require that the source further satisfy a
list-decodability condition. As a corollary, we obtain explicit extractors for
APs, GAPs, and Bohr sources with linear min-entropy, although again our results
over require the list-decodability condition. We further
explore special cases of additive sources. We improve previous constructions of
line sources (affine sources of dimension 1), requiring a field of size linear
in , rather than by Gabizon and Raz. This beats the
non-explicit bound of obtained by the probabilistic method.
We then generalize this result to APs and GAPs
Two Source Extractors for Asymptotically Optimal Entropy, and (Many) More
A long line of work in the past two decades or so established close
connections between several different pseudorandom objects and applications.
These connections essentially show that an asymptotically optimal construction
of one central object will lead to asymptotically optimal solutions to all the
others. However, despite considerable effort, previous works can get close but
still lack one final step to achieve truly asymptotically optimal
constructions.
In this paper we provide the last missing link, thus simultaneously achieving
explicit, asymptotically optimal constructions and solutions for various well
studied extractors and applications, that have been the subjects of long lines
of research. Our results include:
Asymptotically optimal seeded non-malleable extractors, which in turn give
two source extractors for asymptotically optimal min-entropy of ,
explicit constructions of -Ramsey graphs on vertices with , and truly optimal privacy amplification protocols with an active adversary.
Two source non-malleable extractors and affine non-malleable extractors for
some linear min-entropy with exponentially small error, which in turn give the
first explicit construction of non-malleable codes against -split state
tampering and affine tampering with constant rate and \emph{exponentially}
small error.
Explicit extractors for affine sources, sumset sources, interleaved sources,
and small space sources that achieve asymptotically optimal min-entropy of
or (for space sources).
An explicit function that requires strongly linear read once branching
programs of size , which is optimal up to the constant in
. Previously, even for standard read once branching programs, the
best known size lower bound for an explicit function is .Comment: Fixed some minor error
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