435 research outputs found
Bloom Filters in Adversarial Environments
Many efficient data structures use randomness, allowing them to improve upon
deterministic ones. Usually, their efficiency and correctness are analyzed
using probabilistic tools under the assumption that the inputs and queries are
independent of the internal randomness of the data structure. In this work, we
consider data structures in a more robust model, which we call the adversarial
model. Roughly speaking, this model allows an adversary to choose inputs and
queries adaptively according to previous responses. Specifically, we consider a
data structure known as "Bloom filter" and prove a tight connection between
Bloom filters in this model and cryptography.
A Bloom filter represents a set of elements approximately, by using fewer
bits than a precise representation. The price for succinctness is allowing some
errors: for any it should always answer `Yes', and for any it should answer `Yes' only with small probability.
In the adversarial model, we consider both efficient adversaries (that run in
polynomial time) and computationally unbounded adversaries that are only
bounded in the number of queries they can make. For computationally bounded
adversaries, we show that non-trivial (memory-wise) Bloom filters exist if and
only if one-way functions exist. For unbounded adversaries we show that there
exists a Bloom filter for sets of size and error , that is
secure against queries and uses only
bits of memory. In comparison, is the best
possible under a non-adaptive adversary
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Probabilistic Black-Box Checking via Active MDP Learning
We introduce a novel methodology for testing stochastic black-box systems,
frequently encountered in embedded systems. Our approach enhances the
established black-box checking (BBC) technique to address stochastic behavior.
Traditional BBC primarily involves iteratively identifying an input that
breaches the system's specifications by executing the following three phases:
the learning phase to construct an automaton approximating the black box's
behavior, the synthesis phase to identify a candidate counterexample from the
learned automaton, and the validation phase to validate the obtained candidate
counterexample and the learned automaton against the original black-box system.
Our method, ProbBBC, refines the conventional BBC approach by (1) employing an
active Markov Decision Process (MDP) learning method during the learning phase,
(2) incorporating probabilistic model checking in the synthesis phase, and (3)
applying statistical hypothesis testing in the validation phase. ProbBBC
uniquely integrates these techniques rather than merely substituting each
method in the traditional BBC; for instance, the statistical hypothesis testing
and the MDP learning procedure exchange information regarding the black-box
system's observation with one another. The experiment results suggest that
ProbBBC outperforms an existing method, especially for systems with limited
observation.Comment: Accepted to EMSOFT 202
The White-Box Adversarial Data Stream Model
We study streaming algorithms in the white-box adversarial model, where the
stream is chosen adaptively by an adversary who observes the entire internal
state of the algorithm at each time step. We show that nontrivial algorithms
are still possible. We first give a randomized algorithm for the -heavy
hitters problem that outperforms the optimal deterministic Misra-Gries
algorithm on long streams. If the white-box adversary is computationally
bounded, we use cryptographic techniques to reduce the memory of our
-heavy hitters algorithm even further and to design a number of additional
algorithms for graph, string, and linear algebra problems. The existence of
such algorithms is surprising, as the streaming algorithm does not even have a
secret key in this model, i.e., its state is entirely known to the adversary.
One algorithm we design is for estimating the number of distinct elements in a
stream with insertions and deletions achieving a multiplicative approximation
and sublinear space; such an algorithm is impossible for deterministic
algorithms.
We also give a general technique that translates any two-player deterministic
communication lower bound to a lower bound for {\it randomized} algorithms
robust to a white-box adversary. In particular, our results show that for all
, there exists a constant such that any -approximation
algorithm for moment estimation in insertion-only streams with a
white-box adversary requires space for a universe of size .
Similarly, there is a constant such that any -approximation algorithm
in an insertion-only stream for matrix rank requires space with a
white-box adversary. Our algorithmic results based on cryptography thus show a
separation between computationally bounded and unbounded adversaries.
(Abstract shortened to meet arXiv limits.)Comment: PODS 202
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