28 research outputs found
Cross Subspace Alignment and the Asymptotic Capacity of -Secure -Private Information Retrieval
-secure and -private information retrieval (XSTPIR) is a form of
private information retrieval where data security is guaranteed against
collusion among up to servers and the user's privacy is guaranteed against
collusion among up to servers. The capacity of XSTPIR is characterized for
arbitrary number of servers , and arbitrary security and privacy thresholds
and , in the limit as the number of messages .
Capacity is also characterized for any number of messages if either or if . Insights are drawn from these results, about aligning
versus decoding noise, dependence of PIR rate on field size, and robustness to
symmetric security constraints. In particular, the idea of cross subspace
alignment, i.e., introducing a subspace dependence between Reed-Solomon code
parameters, emerges as the optimal way to align undesired terms while keeping
desired terms resolvable
The Capacity of Private Information Retrieval from Decentralized Uncoded Caching Databases
We consider the private information retrieval (PIR) problem from
decentralized uncoded caching databases. There are two phases in our problem
setting, a caching phase, and a retrieval phase. In the caching phase, a data
center containing all the files, where each file is of size bits, and
several databases with storage size constraint bits exist in the
system. Each database independently chooses bits out of the total
bits from the data center to cache through the same probability
distribution in a decentralized manner. In the retrieval phase, a user
(retriever) accesses databases in addition to the data center, and wishes
to retrieve a desired file privately. We characterize the optimal normalized
download cost to be . We
show that uniform and random caching scheme which is originally proposed for
decentralized coded caching by Maddah-Ali and Niesen, along with Sun and Jafar
retrieval scheme which is originally proposed for PIR from replicated databases
surprisingly result in the lowest normalized download cost. This is the
decentralized counterpart of the recent result of Attia, Kumar and Tandon for
the centralized case. The converse proof contains several ingredients such as
interference lower bound, induction lemma, replacing queries and answering
string random variables with the content of distributed databases, the nature
of decentralized uncoded caching databases, and bit marginalization of joint
caching distributions.Comment: Submitted for publication, November 201
On the Asymptotic Capacity of -Secure -Private Information Retrieval with Graph Based Replicated Storage
The problem of private information retrieval with graph-based replicated
storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity
remains open in almost all cases. In this work the asymptotic (large number of
messages) capacity of this problem is studied along with its generalizations to
include arbitrary -privacy and -security constraints, where the privacy
of the user must be protected against any set of up to colluding servers
and the security of the stored data must be protected against any set of up to
colluding servers. A general achievable scheme for arbitrary storage
patterns is presented that achieves the rate , where
is the total number of servers, and each message is replicated at least
times. Notably, the scheme makes use of a special structure
inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is
also presented. The two bounds are shown to match for many settings, including
symmetric storage patterns. Finally, the asymptotic capacity is fully
characterized for the case without security constraints for arbitrary
storage patterns provided that each message is replicated no more than
times. As an example of this result, consider PIR with arbitrary graph based
storage () where every message is replicated at exactly servers.
For this -replicated storage setting, the asymptotic capacity is equal to
where is the maximum size of a -matching in a
storage graph . In this undirected graph, the vertices correspond
to the set of servers, and there is an edge between vertices
only if a subset of messages is replicated at both servers and
Private Polynomial Computation from Lagrange Encoding
Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers that store the dataset. In this paper it is shown that Lagrange encoding, a recently suggested powerful technique for encoding Reed-Solomon codes, enables private computation in many cases of interest. In particular, we present a scheme that enables private computation of polynomials of any degree on Lagrange encoded data, while being robust to Byzantine and straggling servers, and to servers that collude in attempt to deduce the identities of the functions to be evaluated. Moreover, incorporating ideas from the well-known Shamir secret sharing scheme allows the data itself to be concealed from the servers as well. Our results extend private computation to non-linear polynomials and to data-privacy, and reveal a tight connection between private computation and coded computation