6 research outputs found
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
Extractors for Polynomial Sources over
We explicitly construct the first nontrivial extractors for degree
polynomial sources over . Our extractor requires min-entropy
. Previously, no
constructions were known, even for min-entropy . A key ingredient in
our construction is an input reduction lemma, which allows us to assume that
any polynomial source with min-entropy can be generated by 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
. In more detail, we show that sumset extractors cannot even
disperse from degree polynomial sources with min-entropy . 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
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
How to Extract Useful Randomness from Unreliable Sources
For more than 30 years, cryptographers have been looking for public sources of uniform randomness in order to use them as a set-up to run appealing cryptographic protocols without relying on trusted third parties. Unfortunately, nowadays it is fair to assess that assuming the existence of physical phenomena producing public uniform randomness is far from reality.
It is known that uniform randomness cannot be extracted from a single weak source. A well-studied way to overcome this is to consider several independent weak sources. However, this means we must trust the various sampling processes of weak randomness from physical processes.
Motivated by the above state of affairs, this work considers a set-up where players can access multiple potential sources of weak randomness, several of which may be jointly corrupted by a computationally unbounded adversary. We introduce SHELA (Somewhere Honest Entropic Look Ahead) sources to model this situation.
We show that there is no hope of extracting uniform randomness from a SHELA source. Instead, we focus on the task of Somewhere-Extraction (i.e., outputting several candidate strings, some of which are uniformly distributed -- yet we do not know which). We give explicit constructions of Somewhere-Extractors for SHELA sources with good parameters.
Then, we present applications of the above somewhere-extractor where the public uniform randomness can be replaced by the output of such extraction from corruptible sources, greatly outperforming trivial solutions. The output of somewhere-extraction is also useful in other settings, such as a suitable source of random coins for
many randomized algorithms.
In another front, we comprehensively study the problem of Somewhere-Extraction from a weak source, resulting in a series of bounds. Our bounds highlight the fact that, in most regimes of parameters (including those relevant for applications), SHELA sources significantly outperform weak sources of comparable parameters both when it comes to the process of Somewhere-Extraction, or in the task of amplification of success probability in randomized algorithms. Moreover, the low quality of somewhere-extraction from weak sources excludes its use in various efficient applications
Applications of Derandomization Theory in Coding
Randomized techniques play a fundamental role in theoretical computer science
and discrete mathematics, in particular for the design of efficient algorithms
and construction of combinatorial objects. The basic goal in derandomization
theory is to eliminate or reduce the need for randomness in such randomized
constructions. In this thesis, we explore some applications of the fundamental
notions in derandomization theory to problems outside the core of theoretical
computer science, and in particular, certain problems related to coding theory.
First, we consider the wiretap channel problem which involves a communication
system in which an intruder can eavesdrop a limited portion of the
transmissions, and construct efficient and information-theoretically optimal
communication protocols for this model. Then we consider the combinatorial
group testing problem. In this classical problem, one aims to determine a set
of defective items within a large population by asking a number of queries,
where each query reveals whether a defective item is present within a specified
group of items. We use randomness condensers to explicitly construct optimal,
or nearly optimal, group testing schemes for a setting where the query outcomes
can be highly unreliable, as well as the threshold model where a query returns
positive if the number of defectives pass a certain threshold. Finally, we
design ensembles of error-correcting codes that achieve the
information-theoretic capacity of a large class of communication channels, and
then use the obtained ensembles for construction of explicit capacity achieving
codes.
[This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi