52 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
A composition theorem for parity kill number
In this work, we study the parity complexity measures
and .
is the \emph{parity kill number} of , the
fewest number of parities on the input variables one has to fix in order to
"kill" , i.e. to make it constant. is the depth
of the shortest \emph{parity decision tree} which computes . These
complexity measures have in recent years become increasingly important in the
fields of communication complexity \cite{ZS09, MO09, ZS10, TWXZ13} and
pseudorandomness \cite{BK12, Sha11, CT13}.
Our main result is a composition theorem for .
The -th power of , denoted , is the function which results
from composing with itself times. We prove that if is not a parity
function, then In other words, the parity kill number of
is essentially supermultiplicative in the \emph{normal} kill number of
(also known as the minimum certificate complexity).
As an application of our composition theorem, we show lower bounds on the
parity complexity measures of and . Here is the sort function due to Ambainis \cite{Amb06},
and is Kushilevitz's hemi-icosahedron function \cite{NW95}. In
doing so, we disprove a conjecture of Montanaro and Osborne \cite{MO09} which
had applications to communication complexity and computational learning theory.
In addition, we give new lower bounds for conjectures of \cite{MO09,ZS10} and
\cite{TWXZ13}
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
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
Constructive Relationships Between Algebraic Thickness and Normality
We study the relationship between two measures of Boolean functions;
\emph{algebraic thickness} and \emph{normality}. For a function , the
algebraic thickness is a variant of the \emph{sparsity}, the number of nonzero
coefficients in the unique GF(2) polynomial representing , and the normality
is the largest dimension of an affine subspace on which is constant. We
show that for , any function with algebraic thickness
is constant on some affine subspace of dimension
. Furthermore, we give an algorithm
for finding such a subspace. We show that this is at most a factor of
from the best guaranteed, and when restricted to the
technique used, is at most a factor of from the best
guaranteed. We also show that a concrete function, majority, has algebraic
thickness .Comment: Final version published in FCT'201
Subspace Polynomials and Cyclic Subspace Codes
Subspace codes have received an increasing interest recently due to their
application in error-correction for random network coding. In particular,
cyclic subspace codes are possible candidates for large codes with efficient
encoding and decoding algorithms. In this paper we consider such cyclic codes
and provide constructions of optimal codes for which their codewords do not
have full orbits. We further introduce a new way to represent subspace codes by
a class of polynomials called subspace polynomials. We present some
constructions of such codes which are cyclic and analyze their parameters
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
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