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.
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.
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 ofn-server SPIR withrresponsive andtcolluding 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.
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, and zero otherwise.
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.