The Capacity of Private Information Retrieval From Coded Databases

• Published in: IEEE transactions on information theory Vol. 64; no. 3; pp. 1945 - 1956
• Main Authors: Banawan, Karim, Ulukus, Sennur
• Format: Journal Article
• Language: English
• Published: New York The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 01-03-2018
• Subjects: Codes; Information retrieval; Information theory; Messages; Storage
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2018.2791994

We consider the problem of private information retrieval (PIR) over a distributed storage system. The storage system consists of N non-colluding databases, each storing an MDS-coded version of M messages. In the PIR problem, the user wishes to retrieve one of the available messages without revealing the message identity to any individual database. We derive the information-theoretic capacity of this problem, which is defined as the maximum number of bits of the desired message that can be privately retrieved per one bit of downloaded information. We show that the PIR capacity in this case is C = (1 + K/N + K2/N2 + ⋯ + KM-1/NM-1)-1 = (1 + Rc + Rc2 + ⋯ + RcM-1)-1 = (1 - Rc)/(1 - RcM), where Rc is the rate of the (N, K) MDS code used. The capacity is a function of the code rate and the number of messages only regardless of the explicit structure of the storage code. The result implies a fundamental tradeoff between the optimal retrieval cost and the storage cost when the storage code is restricted to the class of MDS codes. The result generalizes the achievability and converse results for the classical PIR with replicated databases to the case of MDS-coded databases.