3,414 research outputs found
Resource Bounded Immunity and Simplicity
Revisiting the thirty years-old notions of resource-bounded immunity and
simplicity, we investigate the structural characteristics of various immunity
notions: strong immunity, almost immunity, and hyperimmunity as well as their
corresponding simplicity notions. We also study limited immunity and
simplicity, called k-immunity and feasible k-immunity, and their simplicity
notions. Finally, we propose the k-immune hypothesis as a working hypothesis
that guarantees the existence of simple sets in NP.Comment: This is a complete version of the conference paper that appeared in
the Proceedings of the 3rd IFIP International Conference on Theoretical
Computer Science, Kluwer Academic Publishers, pp.81-95, Toulouse, France,
August 23-26, 200
Limitations of semidefinite programs for separable states and entangled games
Semidefinite programs (SDPs) are a framework for exact or approximate
optimization that have widespread application in quantum information theory. We
introduce a new method for using reductions to construct integrality gaps for
SDPs. These are based on new limitations on the sum-of-squares (SoS) hierarchy
in approximating two particularly important sets in quantum information theory,
where previously no -round integrality gaps were known: the set of
separable (i.e. unentangled) states, or equivalently, the
norm of a matrix, and the set of quantum correlations; i.e. conditional
probability distributions achievable with local measurements on a shared
entangled state. In both cases no-go theorems were previously known based on
computational assumptions such as the Exponential Time Hypothesis (ETH) which
asserts that 3-SAT requires exponential time to solve. Our unconditional
results achieve the same parameters as all of these previous results (for
separable states) or as some of the previous results (for quantum
correlations). In some cases we can make use of the framework of
Lee-Raghavendra-Steurer (LRS) to establish integrality gaps for any SDP, not
only the SoS hierarchy. Our hardness result on separable states also yields a
dimension lower bound of approximate disentanglers, answering a question of
Watrous and Aaronson et al. These results can be viewed as limitations on the
monogamy principle, the PPT test, the ability of Tsirelson-type bounds to
restrict quantum correlations, as well as the SDP hierarchies of
Doherty-Parrilo-Spedalieri, Navascues-Pironio-Acin and Berta-Fawzi-Scholz.Comment: 47 pages. v2. small changes, fixes and clarifications. published
versio
Nondeterministic functions and the existence of optimal proof systems
We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class . An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system
Strong ETH Breaks With Merlin and Arthur: Short Non-Interactive Proofs of Batch Evaluation
We present an efficient proof system for Multipoint Arithmetic Circuit
Evaluation: for every arithmetic circuit of size and
degree over a field , and any inputs ,
the Prover sends the Verifier the values and a proof of length, and
the Verifier tosses coins and can check the proof in about time, with probability of error less than .
For small degree , this "Merlin-Arthur" proof system (a.k.a. MA-proof
system) runs in nearly-linear time, and has many applications. For example, we
obtain MA-proof systems that run in time (for various ) for the
Permanent, Circuit-SAT for all sublinear-depth circuits, counting
Hamiltonian cycles, and infeasibility of - linear programs. In general,
the value of any polynomial in Valiant's class can be certified
faster than "exhaustive summation" over all possible assignments. These results
strongly refute a Merlin-Arthur Strong ETH and Arthur-Merlin Strong ETH posed
by Russell Impagliazzo and others.
We also give a three-round (AMA) proof system for quantified Boolean formulas
running in time, nearly-linear time MA-proof systems for
counting orthogonal vectors in a collection and finding Closest Pairs in the
Hamming metric, and a MA-proof system running in -time for
counting -cliques in graphs.
We point to some potential future directions for refuting the
Nondeterministic Strong ETH.Comment: 17 page
Keyword-Based Delegable Proofs of Storage
Cloud users (clients) with limited storage capacity at their end can
outsource bulk data to the cloud storage server. A client can later access her
data by downloading the required data files. However, a large fraction of the
data files the client outsources to the server is often archival in nature that
the client uses for backup purposes and accesses less frequently. An untrusted
server can thus delete some of these archival data files in order to save some
space (and allocate the same to other clients) without being detected by the
client (data owner). Proofs of storage enable the client to audit her data
files uploaded to the server in order to ensure the integrity of those files.
In this work, we introduce one type of (selective) proofs of storage that we
call keyword-based delegable proofs of storage, where the client wants to audit
all her data files containing a specific keyword (e.g., "important"). Moreover,
it satisfies the notion of public verifiability where the client can delegate
the auditing task to a third-party auditor who audits the set of files
corresponding to the keyword on behalf of the client. We formally define the
security of a keyword-based delegable proof-of-storage protocol. We construct
such a protocol based on an existing proof-of-storage scheme and analyze the
security of our protocol. We argue that the techniques we use can be applied
atop any existing publicly verifiable proof-of-storage scheme for static data.
Finally, we discuss the efficiency of our construction.Comment: A preliminary version of this work has been published in
International Conference on Information Security Practice and Experience
(ISPEC 2018
MPC for MPC: Secure Computation on a Massively Parallel Computing Architecture
Massively Parallel Computation (MPC) is a model of computation widely believed to best capture realistic parallel computing architectures such as large-scale MapReduce and Hadoop clusters. Motivated by the fact that many data analytics tasks performed on these platforms involve sensitive user data, we initiate the theoretical exploration of how to leverage MPC architectures to enable efficient, privacy-preserving computation over massive data. Clearly if a computation task does not lend itself to an efficient implementation on MPC even without security, then we cannot hope to compute it efficiently on MPC with security. We show, on the other hand, that any task that can be efficiently computed on MPC can also be securely computed with comparable efficiency. Specifically, we show the following results:
- any MPC algorithm can be compiled to a communication-oblivious counterpart while asymptotically preserving its round and space complexity, where communication-obliviousness ensures that any network intermediary observing the communication patterns learn no information about the secret inputs;
- assuming the existence of Fully Homomorphic Encryption with a suitable notion of compactness and other standard cryptographic assumptions, any MPC algorithm can be compiled to a secure counterpart that defends against an adversary who controls not only intermediate network routers but additionally up to 1/3 - ? fraction of machines (for an arbitrarily small constant ?) - moreover, this compilation preserves the round complexity tightly, and preserves the space complexity upto a multiplicative security parameter related blowup.
As an initial exploration of this important direction, our work suggests new definitions and proposes novel protocols that blend algorithmic and cryptographic techniques
Quantum Proofs
Quantum information and computation provide a fascinating twist on the notion
of proofs in computational complexity theory. For instance, one may consider a
quantum computational analogue of the complexity class \class{NP}, known as
QMA, in which a quantum state plays the role of a proof (also called a
certificate or witness), and is checked by a polynomial-time quantum
computation. For some problems, the fact that a quantum proof state could be a
superposition over exponentially many classical states appears to offer
computational advantages over classical proof strings. In the interactive proof
system setting, one may consider a verifier and one or more provers that
exchange and process quantum information rather than classical information
during an interaction for a given input string, giving rise to quantum
complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum
analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit
some properties from their classical counterparts, they also possess distinct
and uniquely quantum features that lead to an interesting landscape of
complexity classes based on variants of this model.
In this survey we provide an overview of many of the known results concerning
quantum proofs, computational models based on this concept, and properties of
the complexity classes they define. In particular, we discuss non-interactive
proofs and the complexity class QMA, single-prover quantum interactive proof
systems and the complexity class QIP, statistical zero-knowledge quantum
interactive proof systems and the complexity class \class{QSZK}, and
multiprover interactive proof systems and the complexity classes QMIP, QMIP*,
and MIP*.Comment: Survey published by NOW publisher
Minimal pairs of polynomial degrees with subexponential complexity
AbstractThe goal of extending work on relative polynomial time computability from computations relative to sets of natural numbers to computations relative to arbitrary functions of natural numbers is discussed. The principal techniques used to prove that the honest subrecursive classes are a lattice are then used to construct a minimal pair of polynomial degrees with subexponential complexity; that is two sets computable by Turing machines in subexponential time but not in polynomial time are constructed such that any set computable from both in polynomial time can be computed directly in polynomial time
Algorithmic Bayesian Persuasion
Persuasion, defined as the act of exploiting an informational advantage in
order to effect the decisions of others, is ubiquitous. Indeed, persuasive
communication has been estimated to account for almost a third of all economic
activity in the US. This paper examines persuasion through a computational
lens, focusing on what is perhaps the most basic and fundamental model in this
space: the celebrated Bayesian persuasion model of Kamenica and Gentzkow. Here
there are two players, a sender and a receiver. The receiver must take one of a
number of actions with a-priori unknown payoff, and the sender has access to
additional information regarding the payoffs. The sender can commit to
revealing a noisy signal regarding the realization of the payoffs of various
actions, and would like to do so as to maximize her own payoff assuming a
perfectly rational receiver.
We examine the sender's optimization task in three of the most natural input
models for this problem, and essentially pin down its computational complexity
in each. When the payoff distributions of the different actions are i.i.d. and
given explicitly, we exhibit a polynomial-time (exact) algorithm, and a
"simple" -approximation algorithm. Our optimal scheme for the i.i.d.
setting involves an analogy to auction theory, and makes use of Border's
characterization of the space of reduced-forms for single-item auctions. When
action payoffs are independent but non-identical with marginal distributions
given explicitly, we show that it is #P-hard to compute the optimal expected
sender utility. Finally, we consider a general (possibly correlated) joint
distribution of action payoffs presented by a black box sampling oracle, and
exhibit a fully polynomial-time approximation scheme (FPTAS) with a bi-criteria
guarantee. We show that this result is the best possible in the black-box model
for information-theoretic reasons
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