10,842 research outputs found
Multiparty Computation for Modulo Reduction without Bit-Decomposition and A Generalization to Bit-Decomposition
Bit-decomposition, which is proposed by Damgård \emph{et al.}, is a powerful tool for multi-party computation (MPC). Given a sharing of secret , it allows the parties to compute the sharings of the bits of in constant rounds. With the help of bit-decomposition, constant-rounds protocols for various MPC problems can be constructed. However, bit-decomposition is relatively expensive, so constructing protocols for MPC problems without relying on bit-decomposition is a meaningful work. In multi-party computation, it remains an open problem whether the \emph{modulo reduction problem} can be solved in constant rounds without bit-decomposition.
In this paper, we propose a protocol for (public) modulo reduction without relying on bit-decomposition. This protocol achieves constant round complexity and linear communication complexity. Moreover, we show a generalized bit-decomposition protocol which can, in constant rounds, convert the sharing of secret into the sharings of the digits of , along with the sharings of the bits of every digit. The digits can be base-\emph{m} for any . Obviously, when \emph{m} is a power of 2, this generalized protocol is just the original bit-decomposition protocol
Fast Computation of Small Cuts via Cycle Space Sampling
We describe a new sampling-based method to determine cuts in an undirected
graph. For a graph (V, E), its cycle space is the family of all subsets of E
that have even degree at each vertex. We prove that with high probability,
sampling the cycle space identifies the cuts of a graph. This leads to simple
new linear-time sequential algorithms for finding all cut edges and cut pairs
(a set of 2 edges that form a cut) of a graph.
In the model of distributed computing in a graph G=(V, E) with O(log V)-bit
messages, our approach yields faster algorithms for several problems. The
diameter of G is denoted by Diam, and the maximum degree by Delta. We obtain
simple O(Diam)-time distributed algorithms to find all cut edges,
2-edge-connected components, and cut pairs, matching or improving upon previous
time bounds. Under natural conditions these new algorithms are universally
optimal --- i.e. a Omega(Diam)-time lower bound holds on every graph. We obtain
a O(Diam+Delta/log V)-time distributed algorithm for finding cut vertices; this
is faster than the best previous algorithm when Delta, Diam = O(sqrt(V)). A
simple extension of our work yields the first distributed algorithm with
sub-linear time for 3-edge-connected components. The basic distributed
algorithms are Monte Carlo, but they can be made Las Vegas without increasing
the asymptotic complexity.
In the model of parallel computing on the EREW PRAM our approach yields a
simple algorithm with optimal time complexity O(log V) for finding cut pairs
and 3-edge-connected components.Comment: Previous version appeared in Proc. 35th ICALP, pages 145--160, 200
On the Complexity of Local Distributed Graph Problems
This paper is centered on the complexity of graph problems in the
well-studied LOCAL model of distributed computing, introduced by Linial [FOCS
'87]. It is widely known that for many of the classic distributed graph
problems (including maximal independent set (MIS) and -vertex
coloring), the randomized complexity is at most polylogarithmic in the size
of the network, while the best deterministic complexity is typically
. Understanding and narrowing down this exponential gap
is considered to be one of the central long-standing open questions in the area
of distributed graph algorithms. We investigate the problem by introducing a
complexity-theoretic framework that allows us to shed some light on the role of
randomness in the LOCAL model. We define the SLOCAL model as a sequential
version of the LOCAL model. Our framework allows us to prove completeness
results with respect to the class of problems which can be solved efficiently
in the SLOCAL model, implying that if any of the complete problems can be
solved deterministically in rounds in the LOCAL model, we can
deterministically solve all efficient SLOCAL-problems (including MIS and
-coloring) in rounds in the LOCAL model. We show
that a rather rudimentary looking graph coloring problem is complete in the
above sense: Color the nodes of a graph with colors red and blue such that each
node of sufficiently large polylogarithmic degree has at least one neighbor of
each color. The problem admits a trivial zero-round randomized solution. The
result can be viewed as showing that the only obstacle to getting efficient
determinstic algorithms in the LOCAL model is an efficient algorithm to
approximately round fractional values into integer values
Sublinear-Time Distributed Algorithms for Detecting Small Cliques and Even Cycles
In this paper we give sublinear-time distributed algorithms in the CONGEST model for subgraph detection for two classes of graphs: cliques and even-length cycles. We show for the first time that all copies of 4-cliques and 5-cliques in the network graph can be listed in sublinear time, O(n^{5/6+o(1)}) rounds and O(n^{21/22+o(1)}) rounds, respectively. Prior to our work, it was not known whether it was possible to even check if the network contains a 4-clique or a 5-clique in sublinear time.
For even-length cycles, C_{2k}, we give an improved sublinear-time algorithm, which exploits a new connection to extremal combinatorics. For example, for 6-cycles we improve the running time from O~(n^{5/6}) to O~(n^{3/4}) rounds. We also show two obstacles on proving lower bounds for C_{2k}-freeness: First, we use the new connection to extremal combinatorics to show that the current lower bound of Omega~(sqrt{n}) rounds for 6-cycle freeness cannot be improved using partition-based reductions from 2-party communication complexity, the technique by which all known lower bounds on subgraph detection have been proven to date. Second, we show that there is some fixed constant delta in (0,1/2) such that for any k, a Omega(n^{1/2+delta}) lower bound on C_{2k}-freeness implies new lower bounds in circuit complexity.
For general subgraphs, it was shown in [Orr Fischer et al., 2018] that for any fixed k, there exists a subgraph H of size k such that H-freeness requires Omega~(n^{2-Theta(1/k)}) rounds. It was left as an open problem whether this is tight, or whether some constant-sized subgraph requires truly quadratic time to detect. We show that in fact, for any subgraph H of constant size k, the H-freeness problem can be solved in O(n^{2 - Theta(1/k)}) rounds, nearly matching the lower bound of [Orr Fischer et al., 2018]
Nondeterministic quantum communication complexity: the cyclic equality game and iterated matrix multiplication
We study nondeterministic multiparty quantum communication with a quantum
generalization of broadcasts. We show that, with number-in-hand classical
inputs, the communication complexity of a Boolean function in this
communication model equals the logarithm of the support rank of the
corresponding tensor, whereas the approximation complexity in this model equals
the logarithm of the border support rank. This characterisation allows us to
prove a log-rank conjecture posed by Villagra et al. for nondeterministic
multiparty quantum communication with message-passing.
The support rank characterization of the communication model connects quantum
communication complexity intimately to the theory of asymptotic entanglement
transformation and algebraic complexity theory. In this context, we introduce
the graphwise equality problem. For a cycle graph, the complexity of this
communication problem is closely related to the complexity of the computational
problem of multiplying matrices, or more precisely, it equals the logarithm of
the asymptotic support rank of the iterated matrix multiplication tensor. We
employ Strassen's laser method to show that asymptotically there exist
nontrivial protocols for every odd-player cyclic equality problem. We exhibit
an efficient protocol for the 5-player problem for small inputs, and we show
how Young flattenings yield nontrivial complexity lower bounds
Applying Grover's algorithm to AES: quantum resource estimates
We present quantum circuits to implement an exhaustive key search for the
Advanced Encryption Standard (AES) and analyze the quantum resources required
to carry out such an attack. We consider the overall circuit size, the number
of qubits, and the circuit depth as measures for the cost of the presented
quantum algorithms. Throughout, we focus on Clifford gates as the
underlying fault-tolerant logical quantum gate set. In particular, for all
three variants of AES (key size 128, 192, and 256 bit) that are standardized in
FIPS-PUB 197, we establish precise bounds for the number of qubits and the
number of elementary logical quantum gates that are needed to implement
Grover's quantum algorithm to extract the key from a small number of AES
plaintext-ciphertext pairs.Comment: 13 pages, 3 figures, 5 tables; to appear in: Proceedings of the 7th
International Conference on Post-Quantum Cryptography (PQCrypto 2016
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