143,732 research outputs found
The Capacity of Private Computation
We introduce the problem of private computation, comprised of distributed
and non-colluding servers, independent datasets, and a user who wants to
compute a function of the datasets privately, i.e., without revealing which
function he wants to compute, to any individual server. This private
computation problem is a strict generalization of the private information
retrieval (PIR) problem, obtained by expanding the PIR message set (which
consists of only independent messages) to also include functions of those
messages. The capacity of private computation, , is defined as the maximum
number of bits of the desired function that can be retrieved per bit of total
download from all servers. We characterize the capacity of private computation,
for servers and independent datasets that are replicated at each
server, when the functions to be computed are arbitrary linear combinations of
the datasets. Surprisingly, the capacity,
, matches the capacity of PIR with
servers and messages. Thus, allowing arbitrary linear computations does
not reduce the communication rate compared to pure dataset retrieval. The same
insight is shown to hold even for arbitrary non-linear computations when the
number of datasets
On the Capacity of Private Nonlinear Computation for Replicated Databases
We consider the problem of private computation (PC) in a distributed storage
system. In such a setting a user wishes to compute a function of messages
replicated across noncolluding databases, while revealing no information
about the desired function to the databases. We provide an
information-theoretically accurate achievable PC rate, which is the ratio of
the smallest desired amount of information and the total amount of downloaded
information, for the scenario of nonlinear computation. For a large message
size the rate equals the PC capacity, i.e., the maximum achievable PC rate,
when the candidate functions are the independent messages and one arbitrary
nonlinear function of these. When the number of messages grows, the PC rate
approaches an outer bound on the PC capacity. As a special case, we consider
private monomial computation (PMC) and numerically compare the achievable PMC
rate to the outer bound for a finite number of messages.Comment: 5 pages, 1 figure, 1 table. Presented at the 2019 IEEE Information
Theory Workshop (ITW). Figure 1 is updated as it contained incorrect
data-points for and . arXiv admin note: text overlap with
arXiv:2003.1000
Private Computation of Systematically Encoded Data with Colluding Servers
Private Computation (PC), recently introduced by Sun and Jafar, is a
generalization of Private Information Retrieval (PIR) in which a user wishes to
privately compute an arbitrary function of data stored across several servers.
We construct a PC scheme which accounts for server collusion, coded data, and
non-linear functions. For data replicated over several possibly colluding
servers, our scheme computes arbitrary functions of the data with rate equal to
the asymptotic capacity of PIR for this setup. For systematically encoded data
stored over colluding servers, we privately compute arbitrary functions of the
columns of the data matrix and calculate the rate explicitly for polynomial
functions. The scheme is a generalization of previously studied star-product
PIR schemes.Comment: Submitted to IEEE International Symposium on Information Theory 2018.
Version 2 fixes some typos and adds some clarifying remark
A Shannon Approach to Secure Multi-party Computations
In secure multi-party computations (SMC), parties wish to compute a function
on their private data without revealing more information about their data than
what the function reveals. In this paper, we investigate two Shannon-type
questions on this problem. We first consider the traditional one-shot model for
SMC which does not assume a probabilistic prior on the data. In this model,
private communication and randomness are the key enablers to secure computing,
and we investigate a notion of randomness cost and capacity. We then move to a
probabilistic model for the data, and propose a Shannon model for discrete
memoryless SMC. In this model, correlations among data are the key enablers for
secure computing, and we investigate a notion of dependency which permits the
secure computation of a function. While the models and questions are general,
this paper focuses on summation functions, and relies on polar code
constructions
Energy-constrained two-way assisted private and quantum capacities of quantum channels
With the rapid growth of quantum technologies, knowing the fundamental
characteristics of quantum systems and protocols is essential for their
effective implementation. A particular communication setting that has received
increased focus is related to quantum key distribution and distributed quantum
computation. In this setting, a quantum channel connects a sender to a
receiver, and their goal is to distill either a secret key or entanglement,
along with the help of arbitrary local operations and classical communication
(LOCC). In this work, we establish a general theory of energy-constrained,
LOCC-assisted private and quantum capacities of quantum channels, which are the
maximum rates at which an LOCC-assisted quantum channel can reliably establish
secret key or entanglement, respectively, subject to an energy constraint on
the channel input states. We prove that the energy-constrained squashed
entanglement of a channel is an upper bound on these capacities. We also
explicitly prove that a thermal state maximizes a relaxation of the squashed
entanglement of all phase-insensitive, single-mode input bosonic Gaussian
channels, generalizing results from prior work. After doing so, we prove that a
variation of the method introduced in [Goodenough et al., New J. Phys. 18,
063005 (2016)] leads to improved upper bounds on the energy-constrained
secret-key-agreement capacity of a bosonic thermal channel. We then consider a
multipartite setting and prove that two known multipartite generalizations of
the squashed entanglement are in fact equal. We finally show that the
energy-constrained, multipartite squashed entanglement plays a role in bounding
the energy-constrained LOCC-assisted private and quantum capacity regions of
quantum broadcast channels.Comment: 31 pages, 6 figure
The Asymptotic Capacity of -Secure -Private Linear Computation with Graph Based Replicated Storage
The problem of -secure -private linear computation with graph based
replicated storage (GXSTPLC) is to enable the user to retrieve a linear
combination of messages privately from a set of distributed servers where
every message is only allowed to store among a subset of servers subject to an
-security constraint, i.e., any groups of up to colluding servers must
reveal nothing about the messages. Besides, any groups of up to servers
cannot learn anything about the coefficients of the linear combination
retrieved by the user. In this work, we completely characterize the asymptotic
capacity of GXSTPLC, i.e., the supremum of average number of desired symbols
retrieved per downloaded symbol, in the limit as the number of messages
approaches infinity. Specifically, it is shown that a prior linear programming
based upper bound on the asymptotic capacity of GXSTPLC due to Jia and Jafar is
tight by constructing achievability schemes. Notably, our achievability scheme
also settles the exact capacity (i.e., for finite ) of -secure linear
combination with graph based replicated storage (GXSLC). Our achievability
proof builds upon an achievability scheme for a closely related problem named
asymmetric -secure -private linear computation with
graph based replicated storage (Asymm-GXSTPLC) that guarantees non-uniform
security and privacy levels across messages and coefficients. In particular, by
carefully designing Asymm-GXSTPLC settings for GXSTPLC problems, the
corresponding Asymm-GXSTPLC schemes can be reduced to asymptotic capacity
achieving schemes for GXSTPLC. In regard to the achievability scheme for
Asymm-GXSTPLC, interesting aspects of our construction include a novel query
and answer design which makes use of a Vandermonde decomposition of Cauchy
matrices, and a trade-off among message replication, security and privacy
thresholds.Comment: 39 pages, 2 figure
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