3 research outputs found
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
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 with Eavesdroppers
We consider the problem of private information retrieval (PIR) with colluding
servers and eavesdroppers (abbreviated as ETPIR). The ETPIR problem is
comprised of messages, servers where each server stores all
messages, a user who wants to retrieve one of the messages without
revealing the desired message index to any set of colluding servers, and an
eavesdropper who can listen to the queries and answers of any servers but
is prevented from learning any information about the messages. The information
theoretic capacity of ETPIR is defined to be the maximum number of desired
message symbols retrieved privately per information symbol downloaded. We show
that the capacity of ETPIR is
when , and when . To
achieve the capacity, the servers need to share a common random variable
(independent of the messages), and its size must be at least symbols per message symbol. Otherwise, with less amount of shared
common randomness, ETPIR is not feasible and the capacity reduces to zero.
An interesting observation is that the ETPIR capacity expression takes
different forms in two regimes. When , the capacity equals the inverse
of a sum of a geometric series with terms and decreases with ; this form
is typical for capacity expressions of PIR. When , the capacity does
not depend on , a typical form for capacity expressions of SPIR (symmetric
PIR, which further requires data-privacy, {\it i.e.,} the user learns no
information about other undesired messages); the capacity does not depend on
either. In addition, the ETPIR capacity result includes multiple previous
PIR and SPIR capacity results as special cases