28 research outputs found
Short seed extractors against quantum storage
Some, but not all, extractors resist adversaries with limited quantum
storage. In this paper we show that Trevisan's extractor has this property,
thereby showing an extractor against quantum storage with logarithmic seed
length
Better short-seed quantum-proof extractors
We construct a strong extractor against quantum storage that works for every
min-entropy , has logarithmic seed length, and outputs bits,
provided that the quantum adversary has at most qubits of memory, for
any \beta < \half. The construction works by first condensing the source
(with minimal entropy-loss) and then applying an extractor that works well
against quantum adversaries when the source is close to uniform.
We also obtain an improved construction of a strong quantum-proof extractor
in the high min-entropy regime. Specifically, we construct an extractor that
uses a logarithmic seed length and extracts bits from any source
over \B^n, provided that the min-entropy of the source conditioned on the
quantum adversary's state is at least , for any \beta < \half.Comment: 14 page
Extensions to the Method of Multiplicities, with applications to Kakeya Sets and Mergers
We extend the "method of multiplicities" to get the following results, of
interest in combinatorics and randomness extraction. (A) We show that every
Kakeya set (a set of points that contains a line in every direction) in
\F_q^n must be of size at least . This bound is tight to within a factor for every as , compared to previous bounds
that were off by exponential factors in . (B) We give improved randomness
extractors and "randomness mergers". Mergers are seeded functions that take as
input (possibly correlated) random variables in and a
short random seed and output a single random variable in that is
statistically close to having entropy when one of the
input variables is distributed uniformly. The seed we require is only
-bits long, which significantly improves upon
previous construction of mergers. (C) Using our new mergers, we show how to
construct randomness extractors that use logarithmic length seeds while
extracting fraction of the min-entropy of the source.
The "method of multiplicities", as used in prior work, analyzed subsets of
vector spaces over finite fields by constructing somewhat low degree
interpolating polynomials that vanish on every point in the subset {\em with
high multiplicity}. The typical use of this method involved showing that the
interpolating polynomial also vanished on some points outside the subset, and
then used simple bounds on the number of zeroes to complete the analysis. Our
augmentation to this technique is that we prove, under appropriate conditions,
that the interpolating polynomial vanishes {\em with high multiplicity} outside
the set. This novelty leads to significantly tighter analyses.Comment: 26 pages, now includes extractors with sublinear entropy los
Extractor Lower Bounds, Revisited
We revisit the fundamental problem of determining seed length lower bounds for strong extractors and natural variants thereof. These variants stem from a "change in quantifiers" over the seeds of the extractor: While a strong extractor requires that the average output bias (over all seeds) is small for all input sources with sufficient min-entropy, a somewhere extractor only requires that there exists a seed whose output bias is small. More generally, we study what we call probable extractors, which on input a source with sufficient min-entropy guarantee that a large enough fraction of seeds have small enough associated output bias. Such extractors have played a key role in many constructions of pseudorandom objects, though they are often defined implicitly and have not been studied extensively.
Prior known techniques fail to yield good seed length lower bounds when applied to the variants above. Our novel approach yields significantly improved lower bounds for somewhere and probable extractors. To complement this, we construct a somewhere extractor that implies our lower bound for such functions is tight in the high min-entropy regime. Surprisingly, this means that a random function is far from an optimal somewhere extractor in this regime. The techniques that we develop also yield an alternative, simpler proof of the celebrated optimal lower bound for strong extractors originally due to Radhakrishnan and Ta-Shma (SIAM J. Discrete Math., 2000)