2,059 research outputs found
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
Convex Hull Formation for Programmable Matter
We envision programmable matter as a system of nano-scale agents (called
particles) with very limited computational capabilities that move and compute
collectively to achieve a desired goal. We use the geometric amoebot model as
our computational framework, which assumes particles move on the triangular
lattice. Motivated by the problem of sealing an object using minimal resources,
we show how a particle system can self-organize to form an object's convex
hull. We give a distributed, local algorithm for convex hull formation and
prove that it runs in asynchronous rounds, where is the
length of the object's boundary. Within the same asymptotic runtime, this
algorithm can be extended to also form the object's (weak) -hull,
which uses the same number of particles but minimizes the area enclosed by the
hull. Our algorithms are the first to compute convex hulls with distributed
entities that have strictly local sensing, constant-size memory, and no shared
sense of orientation or coordinates. Ours is also the first distributed
approach to computing restricted-orientation convex hulls. This approach
involves coordinating particles as distributed memory; thus, as a supporting
but independent result, we present and analyze an algorithm for organizing
particles with constant-size memory as distributed binary counters that
efficiently support increments, decrements, and zero-tests --- even as the
particles move
3-Message Zero Knowledge Against Human Ignorance
The notion of Zero Knowledge has driven the field of cryptography since its conception over thirty years ago. It is well established that two-message zero-knowledge protocols for NP do not exist, and that four-message zero-knowledge arguments exist under the minimal assumption of one-way functions. Resolving the precise round complexity of zero-knowledge has been an outstanding open problem for far too long.
In this work, we present a three-message zero-knowledge argument system with soundness against uniform polynomial-time cheating provers. The main component in our construction is the recent delegation protocol for RAM computations (Kalai and Paneth, TCC 2016B and Brakerski, Holmgren and Kalai, ePrint 2016). Concretely, we rely on a three-message variant of their protocol based on a key-less collision-resistant hash functions secure against uniform adversaries as well as other standard primitives.
More generally, beyond uniform provers, our protocol provides a natural and meaningful security guarantee against real-world adversaries, which we formalize following Rogaway’s “human-ignorance” approach (VIETCRYPT 2006): in a nutshell, we give an explicit uniform reduction from any adversary breaking the soundness of our protocol to finding collisions in the underlying hash function.National Science Foundation (U.S.) (Award CNS-1350619)National Science Foundation (U.S.) (Award CNS-1413964
Keeping Authorities "Honest or Bust" with Decentralized Witness Cosigning
The secret keys of critical network authorities - such as time, name,
certificate, and software update services - represent high-value targets for
hackers, criminals, and spy agencies wishing to use these keys secretly to
compromise other hosts. To protect authorities and their clients proactively
from undetected exploits and misuse, we introduce CoSi, a scalable witness
cosigning protocol ensuring that every authoritative statement is validated and
publicly logged by a diverse group of witnesses before any client will accept
it. A statement S collectively signed by W witnesses assures clients that S has
been seen, and not immediately found erroneous, by those W observers. Even if S
is compromised in a fashion not readily detectable by the witnesses, CoSi still
guarantees S's exposure to public scrutiny, forcing secrecy-minded attackers to
risk that the compromise will soon be detected by one of the W witnesses.
Because clients can verify collective signatures efficiently without
communication, CoSi protects clients' privacy, and offers the first
transparency mechanism effective against persistent man-in-the-middle attackers
who control a victim's Internet access, the authority's secret key, and several
witnesses' secret keys. CoSi builds on existing cryptographic multisignature
methods, scaling them to support thousands of witnesses via signature
aggregation over efficient communication trees. A working prototype
demonstrates CoSi in the context of timestamping and logging authorities,
enabling groups of over 8,000 distributed witnesses to cosign authoritative
statements in under two seconds.Comment: 20 pages, 7 figure
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