Private information retrieval and function computation for noncolluding coded databases

The rapid development of information and communication technologies has motivated many data-centric paradigms such as big data and cloud computing. The resulting paradigmatic shift to cloud/network-centric applications and the accessibility of information over public networking platforms has brought information privacy to the focal point of current research challenges. Motivated by the emerging privacy concerns, the problem of private information retrieval (PIR), a standard problem of information privacy that originated in theoretical computer science, has recently attracted much attention in the information theory and coding communities. The goal of PIR is to allow a user to download a message from a dataset stored on multiple (public) databases without revealing the identity of the message to the databases and with the minimum communication cost. Thus, the primary performance metric for a PIR scheme is the PIR rate, which is defined as the ratio between the size of the desired message and the total amount of downloaded information. The first part of this dissertation focuses on a generalization of the PIR problem known as private computation (PC) from distributed storage system (DSS). In PC, a user wishes to compute a function of f variables (or messages) stored in n noncolluding coded databases, i.e., databases storing data encoded with an [n, k] linear storage code, while revealing no information about the desired function to the databases. Here, colluding databases refers to databases that communicate with each other in order to deduce the identity of the computed function. First, the problem of private linear computation (PLC) for linearly encoded DSS is considered. In PLC, a user wishes to privately compute a linear combination over the f messages. For the PLC problem, the PLC capacity, i.e., the maximum achievable PLC rate, is characterized. Next, the problem of private polynomial computation (PPC) for linearly encoded DSS is considered. In PPC, a user wishes to privately compute a multivariate polynomial of degree at most g over f messages. For the PPC problem an outer bound on the PPC rate is derived, and two novel PPC schemes are constructed. The first scheme considers Reed-Solomon coded databases with Lagrange encoding and leverages ideas from recently proposed star-product PIR and Lagrange coded computation. The second scheme considers databases coded with systematic Lagrange encoding. Both schemes yield improved rates compared to known PPC schemes. Finally, the general problem of PC for arbitrary nonlinear functions from a replicated DSS is considered. For this problem, upper and lower bounds on the achievable PC rate are derived and compared. In the second part of this dissertation, a new variant of the PIR problem, denoted as pliable private information retrieval (PPIR) is formulated. In PPIR, the user is pliable, i.e., interested in any message from a desired subset of the available dataset. In the considered setup, f messages are replicated in n noncolluding databases and classified into F classes. The user wishes to retrieve any one or more messages from multiple desired classes, while revealing no information about the identity of the desired classes to the databases. This problem is termed as multi-message PPIR (M-PPIR), and the single-message PPIR (PPIR) problem is introduced as an elementary special case of M-PPIR. In PPIR, the user wishes to retrieve any one message from one desired class. For the two considered scenarios, outer bounds on the M-PPIR rate are derived for arbitrary number of databases. Next, achievable schemes are designed for n replicated databases and arbitrary n. Interestingly, the capacity of PPIR, i.e., the maximum achievable PPIR rate, is shown to match the capacity of PIR from n replicated databases storing F messages. A similar insight is shown to hold for the general case of M-PPIR