We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) with$B$Byzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In [1], where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In [2], Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when symmetric privacy is required. To that end, we define Byzantine servers, inspired by [1], as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.
We consider the problem of finding the asymptotic capacity of symmetric private information retrieval (SPIR) withBByzantine servers. Prior to finding the capacity, a definition for the Byzantine servers is needed since in the literature there are two different definitions. In [1], where it was first defined, the Byzantine servers can send any symbol from the storage, their received queries and some independent random symbols. In [2], Byzantine servers send any random symbol independently of their storage and queries. It is clear that these definitions are not identical, especially when symmetric privacy is required. To that end, we define Byzantine servers, inspired by [1], as the servers that can share everything, before and after the scheme initiation. In this setting, we find an upper bound, for an infinite number of messages case, that should be satisfied for all schemes that protect against this setting and develop a scheme that achieves this upper bound. Hence, we identify the capacity of the problem.
Byzantine-Eavesdropper Alliance: How to Achieve Symmetric Privacy in Quantum X-Secure B-Byzantine E-Eavesdropped U-Unresponsive T-Colluding PIR?
We consider the quantum \emph{symmetric} private information retrieval (QSPIR) problem in a system with N databases and K messages, with U unresponsive servers, T-colluding servers, and X-security parameter, under several fundamental threat models. In the first model, there are E1 eavesdropped links in the uplink direction (the direction from the user to the N servers), E2 eavesdropped links in the downlink direction (the direction from the servers to the user), where ∣E1∣,∣E2∣≤E; we coin this eavesdropper setting as \emph{dynamic} eavesdroppers. We show that super-dense coding gain can be achieved for some regimes. In the second model, we consider the case with Byzantine servers, i.e., servers that can coordinate to devise a plan to harm the privacy and security of the system together with static eavesdroppers, by listening to the same links in both uplink and downlink directions. It is important to note the considerable difference between the two threat models, since the eavesdroppers can take huge advantage of the presence of the Byzantine servers. Unlike the previous works in SPIR with Byzantine servers, that assume that the Byzantine servers can send only random symbols independent of the stored messages, we follow the definition of Byzantine servers in \cite{byzantine_tpir}, where the Byzantine servers can send symbols that can be functions of the storage, queries, as well as the random symbols in a way that can produce worse harm to the system. In the third and the most novel threat model, we consider the presence of Byzantine servers and dynamic eavesdroppers together. We show that having dynamic eavesdroppers along with Byzantine servers in the same system model creates more threats to the system than having static eavesdroppers with Byzantine servers.
We introduce the problem of symmetric private information retrieval (SPIR) on replicated databases modeled by a simple graph. In this model, each vertex corresponds to a server, and a message is replicated on two servers if and only if there is an edge between them. We consider the setting where the server-side common randomness necessary to accomplish SPIR is also replicated at the servers according to the graph, and we call this as message-specific common randomness. In this setting, we establish a lower bound on the SPIR capacity, i.e., the maximum download rate, for general graphs, by proposing an achievable SPIR scheme. Next, we prove that, for any SPIR scheme to be feasible, the minimum size of message-specific randomness should be equal to the size of a message. Finally, by providing matching upper bounds, we derive the exact SPIR capacity for the class of path and regular graphs.
In symmetric private information retrieval (SPIR), a user communicates with multiple servers to retrieve from them a message in a database, while not revealing the message index to any individual server (user privacy), and learning no additional information about the database (database privacy). We study the problem of SPIR on graph-replicated database systems, where each node of the graph represents a server and each link represents a message. Each message is replicated at exactly two servers; those at which the link representing the message is incident. To ensure database privacy, the servers share a set of common randomness, independent of the database and the user's desired message index. We study two cases of common randomness distribution to the servers: i) graph-replicated common randomness, and ii) fully-replicated common randomness. Given a graph-replicated database system, in i), we assign one randomness variable independently to every pair of servers sharing a message, while in ii), we assign an identical set of randomness variable to all servers, irrespective of the underlying graph. In both settings, our goal is to characterize the SPIR capacity, i.e., the maximum number of desired message symbols retrieved per downloaded symbol, and quantify the minimum amount of common randomness required to achieve the capacity. To this goal, in setting i), we derive a general lower bound on the SPIR capacity, and show it to be tight for path and regular graphs through a matching converse. Moreover, we establish that the minimum size of common randomness required for SPIR is equal to the message size. In setting ii), the SPIR capacity improves over the first, more restrictive setting. We show this through capacity lower bounds for a class of graphs, by constructing SPIR schemes from PIR schemes.
We study the symmetric private information retrieval (SPIR) problem under arbitrary collusion and eavesdropping patterns for replicated databases. We find its capacity, which is the same as the capacity of the original SPIR problem with the number of serversNreplaced by a numberF*. The numberF* is the optimal solution to a linear programming problem, and it is a function of the joint pattern, which is the union of the collusion and eavesdropping pattern. This is the first result that shows how two arbitrary patterns collectively affect the capacity of the SPIR problem. We draw the conclusion that for SPIR problems, the collusion and eavesdropping constraints are interchangeable in terms of capacity, i.e., the two patterns play the same role in the SPIR problem and the capacity remains unchanged if we exchange the colluding and eavesdropping patterns. As corollaries of our result, the capacity of the SPIR problem under arbitrary collusion patterns, and the capacity of the PIR problem where each colluding set is included in some eavesdropping set, are also found. Some extensions with restrictions to finite message lengths are provided, and in this case, upper and lower bounds on the capacity are given. The lower bound is described with a solution to an integer linear programming problem.
Private information retrieval (PIR) is the problem of retrieving, as efficiently as possible, one out of K messages from N non-communicating replicated databases (each holds all K messages) while keeping the identity of the desired message index a secret from each individual database. Symmetric PIR (SPIR) is a generalization of PIR to include the requirement that beyond the desired message, the user learns nothing about the other K - 1 messages. The information theoretic capacity of SPIR (equivalently, the reciprocal of minimum download cost) is the maximum number of bits of desired information that can be privately retrieved per bit of downloaded information. We show that the capacity of SPIR is 1-1/N regardless of the number of messages K, if the databases have access to common randomness (not available to the user) that is independent of the messages, in the amount that is at least 1/(N -1) bits per desired message bit. Otherwise, if the amount of common randomness is less than 1/(N -1) bits per message bit, then the capacity of SPIR is zero. Extensions to the capacity region of SPIR and the capacity of finite length SPIR are provided.
We study the equivalence between non-perfect secret sharing (NSS) and symmetric private information retrieval (SPIR) with arbitrary response and collusion patterns. NSS and SPIR are defined with an access structure, which corresponds to the authorized/forbidden sets for NSS and the response/collusion patterns for SPIR. We prove the equivalence between NSS and SPIR in the following two senses. 1) Given any SPIR protocol with an access structure, an NSS protocol is constructed with the same access structure and the same rate. 2) Given any linear NSS protocol with an access structure, a linear SPIR protocol is constructed with the same access structure and the same rate. We prove the first relation even if the SPIR protocol has imperfect correctness and secrecy. From the first relation, we derive an upper bound of the SPIR capacity for arbitrary response and collusion patterns. For the special case of n-server SPIR with r responsive and t colluding servers, this upper bound proves that the SPIR capacity is (r−t)/n. From the second relation, we prove that a SPIR protocol exists for any response and collusion patterns.